THE UNIVERSITY OF MtICHFIGAN' INDUSTRY PROGRAM OF THE COLIE E OF E.NGINEERING FINITE DIFFERE\!'CE (COMP'TA7TIO N OF NATURA L CONVECTIOIN HAT TRA'TSFER Jesse Davf21 i'rtIl.;i.st A dissertat:ion. submitted in. par.tm... i.f':i fi.mer-t of the requirernents.for?the iegre.e otf Dpctor of Pbhilopohy.n the ni'versi ty o.f' Mic:biga.n l 960 August, 960 IP- 46.

Doctoral Committee: Professor Stuart W. Churchill, Chairman Professor John A. Clark Assistant Professor Bernard A. Galler Professor Joseph J. Martin Professor Edwin H. Young

ACKNOWLEDGMENTS The alithor wishes to express his appreciation for the assistance and guidance rp ovided by the members of the Doctoral Committee during this work; especia ly that of Professor Stuart Wo Churchill, the committee chairman. The aid and cooperation of the staff of the University of Michigan Computing Center is also greatly appreciated. The author extends his gratitude to the Bendix Aviation Corporation for financial support during the academic year 1959-1960 Finally, the work in preparing the manuscript by the Industry Program of the College of Engineering is appreciated~ ii

FINITE DIFFERENCE CONMPUTATION OF NATURAL CONVECTION HEAT TRANSFER Jesse David Hellums ABSTRACT The use of finite difference methods for the solution of the partial differential equations describing the conservatio eervation o mass, energy and momentum in natural convection was investigated. The great advantage of the finite difference approach is that the idealizations required to obtain analytical solutions are not necessary. The main problems associated. with the method are the stability and convergence of the difference equations and the amount of computation required. These problems have retarded the widespread use of difference methods in convection problems which would seem to be warranted by the great advances in computer technology of the last few years. Explicit difference equations were devised that are stable and that require only moderate amounts of computer time and storage by mrodern. standards. The equations are written in time dependent form and treated as an initial value problem. Starting from a motionless, isothermal initial condition, the velocity and temperature distributions are compat;,d. as functions of space and time. The complete transient solution, including the steady state as a limiting condition, is obtained. This time-dependent approach is indicated to be preferable to methods in which steady state is assumed at the outset even if the steady state solution is of primary interest. iii

The infiniteL isot herall flait plate- aji. t:e region insideg a:i infinite, horizontal c'ylinde l r wil the:etirlal halyes of thte -wall z ai tained at diffe rent -znifor tm;peiraurlx.s ware> chosen for illui tratire calculations on an. IM7Oc4'ecaause. of thte a:.1aila-flity of eperinental data and analytical solutions as ar and.: test- for converge!>ncg-.o The flat plate solution rwas btained for a' Prandtl nmber of 0.733. Trie solu:tion, is companed.sith tet. short tine solu~tion for con0:>-c tion alone, and with Ostrach s soluti.ron for the steady state. The -result:s are in good agreement in both case:s In the intermediate time rZage the problem has not been sol ed before so the re-.lts resuIts represnt ntI information..The cylinder probl<im'was solved for a Prandtl number of 0.' and three different values of the Grashof ntnmbr.r An additional solution rwas obtained for a Prandtl nmber of 10. The results ar.e brought,og.tr.; dimensional analysis so that the- fo0r soltiot.s pe.pit predictio o<f,- -ao transfer rates in the cylinder oxar;^xAin rages$ of botIh- parame-te rs ", results are shown to be in good agra-n-iexrert with the -;p.erimental mas-: of Martini and. Chichillo A discussion is give o. th-i application of finite diffe-r?::cmethods to other problems,:e:.-thod. l^sda. in this work applies pzrac+i:-.all, without cbange to any problem Ln flid.d motion in which the pressur=; dl.di'::.. < tion is specified or can be calculate.d from perfect fluid the or, E-',atI the most difficult probles in fluid, mechaniTcs and heat traa-fe w:ill al-to;^ certainly be solved by the finit, adif.fer-.mce approach. This work con.ts:;i.,...-. a significant step in. that d.irJcto*.,i iv

TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii ABSTRACTi LIST OF TABLESri LIST OF FIGURES vii NOMENCLATURE x I, INTRODUCTION II. REVIEW OF PRIOR WORK AND THEORETICAL BACKGROUND A. The Mathematical Model 4 B. The Flat Plate 7 1. A General Model 8 2. The Schmidt-Beckmann Model 10 3. A Highly Simplified Model 14 4. Comparison of Models15 C. Confined Fluids - The Horizontal Cylinder 16 1. Previous Experimental Measurements7 2. Analytical Solutions 18 3. A General Model 21 4. A Simplified Model 23 5. A Highly Simplified Model 2.5 III. FINITE DIFFERENCE METHODS 29 A. Methods of Attack 31 1. The Steady State Approach 3 2. The Unsteady State Approach 3B. Stability and Convergence 33 C. Stability Analysis 3 1. Positive Type Difference Equations 3 2. The von Neumann Method of Stability Analysis 38 IV. THE FLAT PLATE 43 A. The Differential Problem44 B. The Difference Problem k4 1. The Space Grid4 2. The Difference Equations 47 3. The Stability Criterion 4' 4. The Calculations4 C. Results51 1. The Leading Edge 2. Principal Results 4 v

TABLE OF CONTENTS (cont.) Page V. THE HORIZONTAL CYLINDER 62 A. The Differential Problem 63 B. The Difference Problem 64 1. The Space Grid 65 2. The Difference Equations 70 3. The Stability Criterion 72 4. The Calculations 75 C. Results 79 1. The Transient Solution 80 2. Comparison of Two Models 84 3. Effect of Subdivision of the Grid 86 4. Direct Comparison with Experiment 94 5. Variation of Parameters and Additional Comparison with Experiment 99 VI. DISCUSSION OF APPLICATIONS TO RELATED PROBLEMS 118 VII. SUMMARY OF RESULTS 123 REFERENCES 125 APPENDIX A - ANALYSIS FOR THE CYLINDER 128 APPENDIX B - RESULTS FOR THE FLAT PLATE 131 APPENDIX C - RESULTS FOR THE CYLINDER 135 APPENDIX D - COMPUTER PROGRAM 153 vi

LIST OF TABLES Table Page Summary of Calculations for the Cylinder 77 II Summary of Solutions for the Cylinder 99 III Fluctuations in the Heat Transfer Results of Solution 4 108 IV Parameters from the Results of Martini and Churchill 112 V Comparison with the Results of Martini and Churchill 114 VI Transient Results for the Flat Plate 132 VII The Steady State Flat Plate Solution after Extension of the Y Coordinate 133 VIII Transient Heat Transfer Group for the Flat Plate 134 IX Transient Nusselt Numbers for the Cylinder 136 X Results for the Cylinder 138 XI Steady State Heat Transfer Results for the Cylinder 149 XII Velocities from Martini and Churchill 151 XIII Temperatures from Martini and Churchill 152 vii

LIST OF FIGURES Figure Page 1 The Horizontal Cylinder 17 2 The Space Grid 46 3 Effect of Leading Edge Error on Heat Transfer Group 53 4 Effect of Leading Edge Error on Velocity Profile 53 5 Transient Velocity Profiles 55 6 Transient Temperature Profiles 56 7 Transient Velocity at Various Positions 57 8 Transient Temperature and Heat Transfer Group 58 9 The Cylindrical Space Grid 66 10 Subdivision of Outer Grid 67 11! Approximation of Boundary Temperature 69 12 Transient Velocities in the Cylinder 81 13 Transient Nusselt Numbers in the Cylinder 83 14 Velocity and Temperature Profiles from First Grid 85 15 Effect of Subdivision on Velocity and Temperature Profiles for Solution 1 88 16 Effect of Subdivision on Nusselt Number for Solution 1 90 17 Effect of Subdivision or. Velocity and Temperature Profiles for Solution 3 92 18 Effect of Subdivision on Nusselt Number for Solution 3 93 19 Comparison of Velocities with Measurements 95 20 Comparison of Temperatures with Measurements 97 21 Comparison of Nusselt Numbers with Measurements 98 22 Collected Profiles Near the Discontinuity 101 23 Collected Profiles on the Hot Side of the Cylinder 102 viii

LIST OF FIGURES (conto) Figure Page 24 Collected Heat Transfer Results 104 25 Profiles from Solution 4 Showing Fluctuations 106 26 Velocity Fluctuations in Solution 4 107 27 Comparison of Solution with Experiments for Several Grashof Numbers 115 28 Comparison of Overall Heat Transfer Results 117 ix

NOMEN CLCATURE A Rayleigh Number, (Gr Pr) a, b constants appearing in Equations 40 and 41 B yZ/(AY)2 as used in Equation 54. C Z R/( AR)2 as used in Equation 63 C heat capacity3 Btu/lb-~F D diameter of the cylinder. ft D AI/ (R A)2 as used in Equation 63 93 total differential operator div divergence operator g acceleration due to gravity2 ft/sec Gr Grashof Number~ g3\Tx3/y2 for the flat plate g ATr03/y2 for the cylinder h heat transfer coefficient heat flux divided by overall temperature drop j integer denoting grid position in the direction parallel to the -boundary k thermal conductivity2 Btu/lb-ft-~F kklk2 integers in a term in a Fourier Series integer denoting grid position normal to the boundary n integer denoting time increment Nu Nusselt Numbehr hx/k for the plate ~ hD/k for the cylinder Pr the Prandtl number, 7/ //K p pressure, Lbs/ft-sec2 p1 excess in pressure above the initial condition, lbs/ft-sec2 P' dimensionless excess pressure as defined in Equation 8 or 25 r radial position, ft

ro radius of the boundary, ft R dimensionless radial position = r/rO. R the gas law constant in Equation 71 T temperature, OR t time, sec u velocity parallel to the boundary, ft/sec U dimensionless velocity parallel to the boundary as defined in Equations 52 or Equations 26. v velocity normal to the boundary, ft/sec V dimensionless velocity normal to the boundary as defined in Equations 52 or Equations 26. V the velocity vector, ft/sec x distance, parallel to the plate, ft X dimensionless distance as defined in Equations 52 y distance normal to the plate, ft Y dimensionless distance as defined in Equations 52 Greek Letters ic thermal diffusivity, ft2/sec coefficient in Equation 2, oR-l 3 either |u/Zt/(RAO) or /U/At/AX when used in discussions of stability' either JV/AZT/AR or |V/At//AY when used in discussions of stability AT overall temperature drop = Tw-Ti for the plate and TH-TC for the cylinder, OF 7, amplification factors in a Fourier Series /eC viscosity, lb(mass)/ft-sec ~y kinematic viscosity, ft2/sec (3 angle as indicated in Figure 1 <6 dimensionless temperature = (T-Ti)/AT xi

dimensionless time as defined in Equation 26 or Equation 52 7- an eigenvalue A density, lbs(mass)/ft3 Subscripts C denotes the cold side of the cylinder H denotes the hot side of the cylinder i denotes the initial condition j,X denotes position in the space grid m denotes a mean value o denotes a reference quantity w denotes the condition at the flat plate Superscript (n) denotes the time increment xii

I, INTRODUCTION Numerical finite difference methods for solving partial differential equations are of increasing interest and importance since the advent of high speed digital computers. The methods have been employed extensively in heat conduction (diffusion) problems. However, heat convection, which involves conduction plus fluid motion, has received very little attention. The principal attraction of the finite difference approach to partial differential equations lies in the fact that the methods should yield solutions to the great class of problems which resist ordinary methods of analysis. In actual practice there are some difficulties associated with convergence and stability of the difference equations and with the relatively large amount of computation required. These difficulties have retarded the widespread application of the methods which would seem to be warranted by the great advances in computer technology of the last few years. However, the promise is there, and it seems almost certain that the methods will eventually be used for solving the most difficult problems in fluid mechanics and heat convection, The purpose of this work is to investigate the solution of natural convection problems by difference methods. Methods and problems associated with the methods are studied, and two conditions are selected for illustrative calculations: the isothermal, vertical flat plate, and the region confined by an infinite, horizontal cylinder with the vertical halves of the walls at different uniform temperatures. These conditions 1

-2are chosen because of the availability of analytical solutions and experimental measurements to compare with the results. In problems of the type considered in this thesis, there are two basic and difficult questions. First, does the system of equations under consideration adequately describe the actual physical phenomenon over the ranges of interest of the variables? Secondly, does the finite difference method of computation give the solution to the system of equations or an adequate approximation of the solution? The first question theoretically can be answered without ever solving the system of equations if the required derivatives are measured or obtained from measured quantities. Unfortunately, the precision of the measurements required to test the equations in this way is usually prohibitively high so that it is usually necessary to solve the equations and compare the solution with measurements. The second question in many cases can be answered by analysis. However, the mathematical theory is often inadequate and it is necessary to compare the approximate solution either with an exact solution or with measurements. The question of stability in initial value problems is a crucial part of the second basic question in that no meaningful results can be expected unless stable difference equations are used. The flat plate problem was chosen for calculation because the results of the calculations could be compared with a solution which may be regarded as exact. Such a solution exists for very small time where conduction alone prevails and for very large time where the steady state is reached. The cylinder problem was selected as a more difficult problem for which an analytical solution is not available, but for which experimental data exist, The calculations would be of little value if it were

-3always necessary to have solutions or measurements by which to verify the calculations. Such verification is essential in exploratory investigations such as the present work, but once the validity of the result is established, the methods can be used to provide new information and to solve new problems. This thesis is divided into seve p.parts: 1. Introduction. 2. Review of prior work and theoretical background. 3. A discussion of finite difference methods. 4. Results for the flat plate. 5. Results for the cylinder. 6. Discussion of related problems. 7. Summary of results.

II. REVIEW OF PRIOR WORK AND THEORETICAL BACKGROUND A. The Mathematical Model Conservation of mass, energy, and momentum are described by the equations given below. The thermal conductivity, viscosity and heat capacity are assumed to be constant, and viscous dissipation and work of compression are neglected, _' St =l /V V (la) *T = (a'VT (lb) -B rn j t ~ v~ E cLl VJ (lc) o v =_ LvV / dfvrJ ( Ld ) 0 A-1 J7- -f -TT^ ^Y-^ v 3 (1 The force of gravity is taken to be in the negative x direction. For three-dimensional or turbulent motion a third momentum balance should be included. However, for the purpose of dimensional analysis, it can be omitted since it is of the same form as Equation ld, Associated with Equations 1 are boundary and initial conditions. In this work the initial condition is that the velocity is zero and the temperature is some constant, Tio On the boundary the velocity is zero and the temperature is prescribed as either a constant or a function of position. 4

-5If density variations due to pressure are negligible, the followv ing equation of state is a good approximation for most gases and liquids. Ae = C ( /X)^ (2) For ideal gases the coefficient,e, equals 1/Ti. The initial components of the pressure gradient are: b/ -a 6 (3) and Sk' = 0 (4) (4) In natural convection it may be expected that the pressure gradient will depart very little from the initial, static pressure gradient. So it is desirable to divide the pressure into two parts: pi, the initial pressure and p' the increase in pressure due to motion and variations in density. Later on it will be assumed that in some cases p' 0. y - _L i- i - ^ Ad P axax-TX a W (5a), = P' V - t = o t zt (5b) The expressions for the components of the pressure gradient given by Equations 5 may be substituted into equations Ic and Id to give:

-6^-s = — +) ^ f^ LL#VI/gJ (6a) 0 c f,y t ByC rVL lr t~b V (6b) Now the density may be eliminated from all the conservation equations by use of the equation of state to give at: L,'t + P-(T-T T (7b) ^ -- z (r-T- 5' ) ah = e f (T-7T) )- t0^ -L J,+7 (7c) 0- V — - o~iV =,- -t t j V f to L/t4(F-) g07ff + i jv (7d) Equations 7 may be considered as a general model for natural convection. The only assumption made to this point which seems subject to serious challenge is the assumption of constant viscosity which will not be a good approximation for viscous fluids under large temperature difference. However, Sparrow and Gregg (39) have investigated the effect of variable viscosity and their results tend to support the present practice among engineers of using a constant viscosity evaluated at a mean temperature. It should be mentioned that there is some confusion in the literature regarding Equations 7. Many books and papers introduce the

-7term gl(9i/4 - l) by reference to "Archimedes' law of buoyancy" without noting that p' differs from p. Hermann (12) discusses this point and emphasizes that Archimedes' law applies only for a particle of a given density immersed in a fluid of uniform density, which is certainly not the case in natural convection. It is also a common practice to start with a system of equations like Equations 7 except that the terms 1 + (;(T - Ti) are omitted along with the terms div v. These simplifications will be used later in this work, but it should be noted that such simplifications imply that (T - Ti) -1/3. That is to say, the maximum overall temperature difference must be small relative to L/a (Ti for gases). B. The Flat Plate Consider an isothermal, vertical plate of infinite width, extending from — = 0 to /= o immersed in a Newtonian fluid of infinite extent, initially at a uniform temperature, Ti, and at rest. At some time, t = 0, the plate instantaneously takes on a temperature Tw, different from the initial fluid temperhture, Ti. Let u - u(x,y,t), then the boundary and initial conditions may be written u(x,0,t) a u(x,Ct) = u(x,y,0) = 0 (X t) =(xgO,t) = v(x,y,0) = 0 T(x,O,t) = Tw, T(xet) = T(x,y,O) = Ti Notice that only two parameters, Tw and Ti, appear in these equations. If the variable T is replaced with a new variable,, such that./= (T-Ti)/A T where AT = Tw - Ti, the two parameters are eliminated and the conditions on a are: i(-,~O)-=/, ~f(,Xc>y )z= </(,o - oj=0

-8Now if any of the variables u, v, x, y, or t (not f) is replaced by a new variable differing from the old by only a constant multiplier, the boundary and initial conditions using the new variable will be the same as before. This fact gives one the freedom to choose new, dimensionless: variables without introducing parameters into the boundary and initial conditions. 1. A General Model A systematic technique for choosing dimensionless variables so that the number of parameters and independent variables in the problem is reduced to a minimum has been given by Hellums and Churchill (10), and the technique is discussed and illustrated later on a simple system of equations. For Equations 7 the dimensionless variables may be shown by this technique to be: _ L73 i -a 7) /3I / h13 3 _3'p"- (-_~~T'/V- (? ~ ( T)'/; _____ ~cx f ( Y Y (7 ) x - Lti /' # (OF' T) /f*nT p ( ~ +iL d'(8a) __ z - #(it rnrL)(-' * 4 L/ + A/ LI (8c) voS _ jFiXT~y)( 1 + 41-l (8d) TTZ -r- 3Equations 8 with the boundary and initial conditions contain only two parameters: (YT and 7k, the Prandtl number. That is to say U,

-9V, and. 2 depend on these two parameters in addition to X, Y and'-. The seven parameters which appeared in the original problem, Tw, Ti./L, (ti, Ad g, and Ox have been combined or included in the new variables such that only two parameters remain. Such a reduction in the number of parameters is of great value if the equations are to be solved numerically or if experimental data are to be correlated. From these equations it is apparent that (T - Ti)/ZlT, u/(vygeT)1/3, v/(/gpiT)1/3 and p'/i (Yg T)2/3 are functions only of x( gp T/y/2)1/3, y(g T/-y2)1/3. t(g(fz T)2/3/1/3, T., and7//. The local heat transfer coefficient can be introduced and evaluated as follows: hdT - -j/( /,:,:=) - (Jr- (9) - j-phence /3 hence k-!-^1 -~ 7/ // r ( -/'6 r3 (0o) Equation (10) is a very general result. No assumption was made either explicitly or implicitly relative to the type of flow, and this functional relationship is presumed to hold for both laminar and turbulent flow. For a steady state, the time group may be dropped. For turbulent conditions the heat transfer coefficient then becomes the time mean value over a sufficient interval of time to dampen out the turbulent fluctuations. The parameter (/ T is observed to occur only in the form 1 + f'ATT) in Equations 8. Since 0 L y4 i O1.0, /T may be dropped out of

-10the functional relationships if dT(<< 1/. The simplified result for steady state is X i Ap- (11) The Prandtl numberY//, is the only parameter remaining in Equation 11. Therefore a single experiment or numerical calculation in which h is determined as a function of x should be sufficient to define the problem for a given Prandtl number if dT<< 1/?. Equations 8 have been useful in the past only for dimensional analysis. The equations have never been solved in such a general form. 2. The Schmidt-Beckmann Mode Schmidt and Beckmann aided by a mathematician, Pohlhausen (31), made a number of simplifications of Equations 8 and obtained an analytical solution to the simplified equations. The simplifications, which are of the type now often referred to as of the boundary layer type, consist of dropping ju2/Jx2, ~2T/3x2, 6p/ x and the entire momentum balance in the y direction, and again assuming &-AT to be negligible with respect to unity. Ostrach (26) gives a detailed discussion of the implications of the assumptions. The resulting equations are given below except that here the time derivatives are included. Schmidt and Beckmann worked only with the steady-state case. F + t =- 0 (12a) -g o< (12b) 5 +9 7/ (412c)

-11" The boundary and initial conditions for this system of equations have some implications which should be discussedo First, the velocity normal to the plats, v, cannot be restrained to zero for large y. At y = 0 v is zero but v approaches some non-zero function of x as y increases. Secondly, along the leading edge of the plate (x o) u - 0 and v 9., The behavior of v does not correspond to that of a real fluid except at positions which in some sense are near the plate and far from the leading edge. The solution to the system of equations cannot be expected to be valid near the leading edge. Dimensional analysis of this system of equations will be carried out in detail as an illustration of the technique mentioned before, A new set of variables will be adopted: U m u/uO, V v/vo, X - x/xo, Y Y/YO and ^- t/tO where u09 Vo0 x09 and YO are constants or parameters. The variable/' will. be left unaltered since any multiplier would introduce a parameter into the boundary conditions as was mentioned above. It is important to note that the quantities Uo0, VO0, X0, YO and to are entirely arbitrary and can be chosen in such a way that the problem is simplified. That is to say the objective of the analysis is to eliminate as many aS possible of the parameters and independent variables fromr the problem by appropriate choice of values for the arbitrary quantities~ The equations in terms of the new variables are given -belowo A^i _- =^J i =Y:)- (L3a) i7 \/) C) j. 0 (13b) Z/T-L' /Z i Yot j\ dl = r7/{yTh (LOc) PC)~~~~~~~~~~~~~~~2.14V

-12Now it is desired to eliminate as many as possible of the dimensionless groups in the equations. Each group can be equated to unity which gives a system of eight equations (not all independent) in the five arbitrary quantities UO0 VO, x0, y0O and t0o These equations can be solved to give the arbitrary quantities in terms of the parameters of the original problemnC %y, g,, and aTo If all of the eight equations can be satisfied, all of the parameters can be eliminated from the partial differential equations. In the case at hand, the two equations OtO/y02 1 and V/to/Y2 I cannot both be satisfied so one parameter will remain. The solution to the algebraic problem for the arbitrary quantities is,Co - (o&- T) i j ( C1T/A o - o - (~o/^ATy i o (Jv /(fT)' k (14);)6 - o~.hlt,.a7 0 = or= teay All the parameters in the problem except one can be eliminated without specifying xo, and the equations are not affected by the choice of a value for xO. Substitution of the quantities of Equations 14 into Equations 13 gives; -, = V (15a) c- - ~ (-b (A9+ = > $ ~~~~~~~ (15b n = / + rant- (15c) in which /O, the Prandtl number, is the only parameter. So it is concluded that u/xogyAT)l/2, v(xOYf2g/AT)l/4 and(T Ti}w - Ti) depend on

-13x/xo, y(gz T/T2Xo)l/4, t(g T/xo)l/2 and</<. The system of differential equations with the boundary and initial conditions is independent of the choice of x0. So it must be concluded that the solution is also independent of the choice of xO. A new set of variables which does not contain xO can be formed by multiplying each old variable by x/x0 raised to an appropriate power. Birkhoff (2) has given a formal justification to such a procedure by much the same line of reasoning as given here. The revised conclusion is that u/(xgl T)K/2, v(x/YV2gnT)i/4 and(T - T)/Tw - T depend on y(g/L\Tt/2x)l/4, t(g(4T/x)l/2 and vY The validity of the new choice of variables can always be checked by computing the required derivatives and substituting them into the original system of equations. The local heat transfer coefficient can be introduced in the same way as before to give X8~ Ti'fisT io(V. (16) If attention is restricted to steady state, the terms involving time may be dropped; a single independent variable remains, and the partial differential equations can be reduced to ordinary differential equations. The system of ordinary differential equations was first developed and solved for a Prandtl number of 0.733 by Schmidt, Beckmann and Pohlhausen (31). The solution was found to agree well with the experimental data of Schmidt and Beckmann. The original solution was by series. Since that time others have solved the equations for a variety of conditions using numerical methods. References 7, 26, 30, 31, 33, 35, 36, 37, 38, 39 and 40 all pertain to solutions of this system of equations.

-l4Transient free convection has been studied by Illingworth (13), Sugawara and Michiyoshi (41) and, most recently, Siegel (34)0 None of these workers obtained a solution which is valid over the whole range of time out to the steady state condition. Siegel used the Karman-Pohlhausen approximation method in attacking the problem. He did not compute heat transfer coefficientsbeyond the short time when conduction alone prevails. However, he did produce an estimate of the time required to reach steady state. 3. A Highly Simplified Model As a final case the further simplification of steady state and very slow motion such that the inertial terms in the momentum equations can be dropped will be examined, Morgan and Warner (23) indicate that this latter idealization is justifiable for fluids with large Prandtl numbers, The resulting equations are O (17a) U T + V1 = 0 (L7b) g (l- t/) + -v = 0 (l7c) Proceeding as before reveals that (T - T)/ T, u(Y/xog 4T)l/2 and v(xV/^?3gp4T)1/4 are functions only of y(g?4 T/Yx)l/4 and hence that h~ ( f i = C (18) where C is an unknown constanto

-15This result is practically as useful as a complete solution since a single experimental or computed value for the heat transfer coefficient defines the coefficient for all other positions and conditions within the range of the applicability of Equations 17. Equation 18 can be rearranged in the more familiar but less explicit form X = C (O if/f (19) This relationship, including the a value for C was first derived by Lorenz (19). However, Lorenz made more simplifications than were necessary in the analysis presented above. Morgan and Warner apparently were the first to derive Equation 19 from Equations 17. 4. Comparison of Models It is interesting to compare the different functional representations obtained for the local, steady state, heat transfer coefficient for the successive idealizations. Equation 19 should be regarded as a first order approximation and would be expected to become a poorer representation as the Prandtl number is decreased. Equations 16, 11, and 10 should be successively better approximations. The dimensionless. heat transfer groups on the left side of Equations 16, 11, and 10 can be changed to the same form as the heat transfer group on the left side of Equation 19 by multiplying through by the dimensionless groups on the right side to the appropriate powers, thus obtaining L 321f = F7 / A(22)'/- (^

and Mt, a /) = FD7/o (23) respectively. Thus the dimensionless heat transfer group h(Ylx/graT)l/4/k can be regarded as invariant as a first approximation, as a function of 7/c/ as a better approximation, and as a function of the dimensionless position x(gzAT/]y2)l/3 and zAT in the more general case. This suggests plotting data in the form of h(yVx/g(3AT)l/4/k versus al/7or if necessary as h4c/x/g z dT)l/4/k versus x(g CHT)/V2)l/3 with e(/7yand gAT as parameters. It should be remembered that the variation of/a, k and Cp with temperature and of e with pressure were neglected in all of the analyses. Analytical expressions for these variations would introduce additional dimensionless groups. C. Confined Fluids - The Horizontal Cylinder A large part of the prior work on natural convection has been on the flat plate problem discussed above. The flat plate is the simplest condition of practical importance from both the theoretical and experimental standpoints. A larger and more important class of conditions are those associated with confined fluids. Specific examples include air spaces within the walls and rooms of buildings, and within many refrigerators and heaters of both household and industrial use. Practically all problems in heating, cooling, boiling and insulating involve natural convection of confined fluids to some degree. Analytical or numerical solutions to convection problems almost invariably are approximate solutions to simplified models so that it is highly desirable to compare the solution with experimental data. If the

-17solution agrees with experimental data, the inference is that the solution will provide information on conditions for which no experimental data exist. Comparison of heat transfer rates is necessary but such a comparison does not provide a critical test of the method. A critical test is provided by comparison of local velocity and temperature distributions. 1. Previous Experimental Measurements The only experimental measurements of local velocities and temperatures for natural convection in an enclosed region seem to be those of Martini and Churchill (21). In the great majority of investigations only the overall rate of heat transfer or only the temperature distribution is measured. Martini and Churchill studied natural convection of air inside a horizontal cylinder 36 inches long by 4.3 inches diameter. The cylinder was divided longitudinally at the vertical diameter and a small layer of insulation was inserted between the two halves so that the two sides of the cylinder could be maintained at different temperatures (see Figure 1). The length of the cylinder was sufficient so that near the center the motion of the air was considered to be two dimensional. Local air temperatures were measured directly by thermocouple L traverses. Velocity data were obtained T-Tc ~ ~T= T by taking multiple exposure pictures of small dust particles suspended in the air. This method of determining velocities gives both the flow lines and the magnitude of the velocity. The results Figure 1. The Horizontal Cylinder

-18will be discussed in comparison with the calculations of the present work. It should be mentioned that accurate measurement of local velocities is extremely difficult —which undoubtedly accounts for the dearth of such measurements. The problems of measurement are more difficult for confined fluids than for unconfined fluids, and the comparatively low velocities associated with natural convection pose more problems than cases of high velocity forced convection. Martini and Churchill considered their work to be exploratory in that they were seeking an effective method to make such measurements. The technique was only partly successful and they do not claim a high accuracy of measurement. Unfortunately, the difficulties of measurement by any technique are multiplied near the boundary where the results are of most interest. Ostroumov (25) studied natural convection in a horizontal cylinder by an optical method. His discussion is limited for the most part to a description of the method. No data are reported except photographs showing lines of constant components of the temperature gradiento Natural convection in rectangular regions has been investigated by several workers. Jakob (14), Globe and Dropkin (8), Schmidt and Silveston (32), and de Graff and van der Held (3) give discussions and correlations. Natural convection in vertical tubes closed at one end has been studied by Hartnett and Welsh (9), Eckert and Diagula (5), Foster (6), and Martin (20), As was mentioned above, none of these workers measured local velocities. 2. Analytical Solutions All efforts toward solving the confined fluid problem have been restricted to the case of steady, two dimensional, laminar flowo

Batchelor (1) considers a rectangular region such as an air space in the wall of a house with the two vertical walls at different temperatures. A single solution to the system of equations is not given as - such, but three solutions each of which is expected to be a valid approximation for a limited range of the parameters. Working with the parameters' Rayleigh number, A (product of Grashof and Prandtl numbers), and aspect ratio, L/D (ratio of height to thickness of the air layer), solutions are developed for three limiting cases: (a) very small A, L/D not restricted; (b) very large A, L/D not restricted; and (c) L/D very large, A not restricted. The three solutions give a fairly complete overall qualitative view of the phenomenon, although the heat transfer coefficients so predicted are about 50 to 100L higher than those measured by Mull and Reiher (24). Zhukhovitski (42) considers an infinitely long cylindrical cavity in a solid medium having a horizontal temperature gradient perpendicular to the axis of the cylinder. A solution is sought in terms of powers series in a modified Rayleigh number. The coefficients are determined by a method of successive approximation. The results of the calculations are compared with the measurements of Ostroumov and are found to be in qualitative agreement for a Rayleigh number of 500. Zhukhovitski indicates that there is no certainty that the method will apply to large Rayleigh numbers. The series may not converge for Rayleigh numbers greater than unity although it is asserted that calculations indicate convergence at least up to a Rayleigh number of 1,000o Batchelor (1) uses the power series approach in his solution for small Rayleigh numbers and gives an argument that 1,000 is the highest value of Rayleigh number for which the power series is useful. Unfortunately Rayleigh numbers of 1,000 or less

-20are of no practical interest in Batchelor's problem since most of the heat transfer is by conduction in this range. It can be concluded that Zhukhovitski s solution shows some promise, but more work is needed to determine the value of the method over wide ranges of the parameters and on conditions other than those of the original investigation. Poots (27) gives a solution to the same problem considered by Batchelor —the rectangular regiono The temperature and the stream function are expressed as double series of orthogonal functions of the space variables. By means of Fourier transforms the equations for the coefficients are reduced to two infinite sets of coupled simultaneous algebraic equations. An iterative method for solving the algebraic equations is outlined and numerical values of the coefficients are given for a Prandtl number of 0.73, aspect ratio of unity (a square region), and several Rayleigh numbers between 500 and 10,000. The calculations become more difficult with increasing Rayleigh number, and the number of coefficients required in the series also increases. It is indicated that the determination of the coefficients is impractically laborious for Rayleigh numbers greater than 10,000 or aspect ratios greater than 4. There are no data on square regions to compare with Poots' solution. However, for a Rayleigh number of 10,000, Poots' solution agrees with Jakob's (14) empirical formula for the overall heat transfer coefficient. Poots' method of solution seems to hold promise. However, more work is needed to determine if it can be adapted to larger values of the parameters Rayleigh number and aspect ratio. Lighthill (18) has analyzed the case of natural convection in heated vertical tubes closed at the lower end and opening into a reservoir of cool fluid at the top. The Karman-Pohlhausen integral approximation method is used on a system of equations in which it is assumed that the

-21Prandtl number is large so that the non-linear terms can be omitted from the momentum balance. In this method the shape of the velocity and temperature profiled must be assumed or deduced in advance by physical considerations. It can be concluded that the methods of attack used to date on the equations governing natural convection in enclosed regions have had only limited success and that each method appears to be severely limited in its range of applicability. 3. A General Model Equations 1 with appropriate boundary conditions apply to confined fluids as well as unconfined fluids. The calculations to be described later on were performed for a cylindrical region so it is desirable to consider the equations in cylindrical coordinates. The angle e is measured clockwise from the vertical such that the force of gravity is in the radial direction where 9 = 0. The motion is assumed to be two dimensional so that all gradients in the z or axial direction are zero. The equations are given below where u is the velocity component in the 8 direction and v is the velocity component in the radial direction. t ir -u — ar t -+ (24a) Kd27 + viY / - ^~ - L,fZ) 936 3 (24b) e L Tr~- /rX /r 6rz r —Z A z Ao- + 3 I V | (

-22-. -T + LT - cT <rlZT + } / z l (24e) b/, 4 d/' L- /fe - If,2- ~ (24 H + X a (4e- + ( = o (24d) The boundary and initial conditions for the problem are as indicated below and in Figure 1 where u - u(r, ~, t). u(r09e, t) u(r,, 0) = v(rO, 6, t) v(r,0, O) = 0 T(r0, e, t) TH if 0 < (< 7i T(ro, & t)= TC if?(t@.V T(r,,8 O) Ti In the cases for which calculations were performed, the initial temperature Ti was taken to be (TH + TC)/2. Equations 24 can be simplified somewhat if attention is restricted to cases of small temperature differences (see the discussion just after Equations 7 in Part II)o Making this assumption, dividing pressure into two components in the same way as was done in obtaining Equations 7, and putting the equations into dimensionless form gives - t u ~ L +V + Vl VI]r - C _ + / + L _ - = 6i - f+ R2 1 - + - L(25a) ^ + -I- _LY ^-f -,a- (25b) ^ ^ -^-T- rPb ^

-23at Z;/ a 0( + d cf 18 if 7 R J (25c) v -'a - (25d) where the dimensionless variables are as indicated below T - Ti ur0 vr H:i2 r and P P 9 r se&) 02:0 The two parameters are Gr/ rO3gZT/y2, a Grashof number based on the radius.: of the cylinder, andy/o( the Prandtl number. In the case of the unconfined fluid the corresponding set of equations contain only a single parameter. Here, the boundary conditions are slightly more complex and two parameters are required. For a still more complicated boundary such as a rectangle, a third parameter involving the ratio of the two dimensions of the rectangle is requiredo 40 A Simplified Model The variable P' in Enuations 25 represents the deviation of the pressure from the initial hydrostatic pressure distribution and the gradient of P' should be in some sense smallo Equation 25a is a momentum balance in the e direction, the direction of principal motion, and U and the derivatives of U with respect to R should be large compared to V and the derivatives of V which appear in Equation 25bo If it is assumed that }P'/Ae is negligible compared to the largest terms of Equation 25a, Equations 25a, 25c, and 25d

-24are sufficient for the determination of U, V, and.o Equation 25b could then be used to determine P' if desired. As a second simplification the Coriolis force term, UV/R, will be neglected in Equation 25a along with the term 2/R2)(-V/ ) to give the system of equations which were used in the computation part of this worko ^ b fz L + bVR 6& 4Er f A (26a) L~ + Va i (26b) }( I I 0 (26c) The boundary conditions are now that at R ~ 1: U and V are zero, / = 1/2 for O0 9 ~E,4 and?= - 1/2 for Th G < 27. Initially U, V, and are zero. The implications of the idealizations given above deserve some discussion. Dropping the terms UV/R and(2/R2)( V//O) can be justified very simply by estimating the magnitude of the terms in relation to other terms in the equation by use of Martini's data and by use of solutions of Equations 26. Neglecting ~P'/& in Equation 25a leads to anomalies which are analogous to those of the flat plate, Eouations 12. The radial velocity, V, is that velocity required to satisfy the continuity equation without regard to momentum changes in the radial direction. Outside the boundary layer where U and its derivatives are small, the quantity RV becomes a non-zero function of 0, and at the center V is infinite. This behavior of V away from the boundary, as in the case of the flat plate, need not prevent the model from being useful. It is postulated that bU/~R,

-252U/)R2, )9/~R and b20/dR2 are large near the boundary and U approaches zero with increasing distance from the boundary after going through a maximum. Beyond the point where U is small, values of V predicted by Equations 26 are meaningless, but they are of no interest. Values of U, V, and 9 from the equations should be valid in the region near the boundary where U and differ appreciably from zero. In the central region values of U and should be valid (approximately zero) and values of V are meaningless. The idealization required to simplify Equation 26 to the form of Equations 25 are discussed in more detail in Appendix A. It is shown there that the validity of the idealizations depends on the Grashof number. The model should approximate the actual behavior of fluids more closely as the Grashof number increases. For very small Grashof numbers the model should be expected to be inadequate. 5. A Highly Simplified Model It is instructive to consider a highly simplified model from which a first approximation of the effect of the parameters is obtained. Suppose that the motion is in some sense slow such that the inertial terms in Equations 26 may be neglected and that U and < are different from zero only in a narrow region near the boundary so that rdS 5 r0d~ where ro is the radius of the boundary. As additional simplifications, suppose that l/r2 2u/e& and(l/r2)( 2~/8 2)are negligible compared to d2u/jr2 and ~2 /r2. The resulting simplified equations are given below for steady state. NT~ ie'' -, Y O (27a) V 4 + IL- (27b) v5 f hi EC = 0 (27o) A2 ~ Or

-26If the variable r is replaced by y = ro - r, the distance from the boundary, the equations are unchanged except that some terms change sign. By the postulation of narrow boundary layer it is possible to neglect the curvature of the wall when thinking of the boundary conditions associated with yo For example one condition where U = O0 is at y = 0 and the other condition where U = 0 may be taken to be at y scO o Multiplying y by any constant will not alter the two conditions so a new dimensionless variable, y/yO, may be chosen in which yo is completely arbitrary. Using this freedom and carrying out dimensional analysis in the manner illustrated before leads to the result that u (t/rOggATc)2, v (roV//gAgTo)4W and ( depend on Q and (1 - r/ro) 1 1 (gATrO3/lo/)4; and h/k (Yc/rO/g AT:)4 depends only on'-), This result can be rearranged in terms of the Grashof, Nusselt and Prandtl numbers and in terms of the variables of Equations 26 as indicated below ^~ - (ti / IV(6,,fa - i} )l(W (28) P3)/r (I - (29) ___ = ((l{g) (31) Equations 28 through 31 are extremely useful in predicting the qualitative effect of the Grashof and Prandtl parameters. As will be shown in the discussion of results the simple model agrees well with the computations based on a more complex modelo The equations can even serve

-27in a limited way to establish the conditions under which the simplifying assumptions by which the equations were devised may be expected to be valid. Such a procedure admittedly involves circuitous reasoning. However, the procedure seems to give the right answers to it will be outlined. For simplicity in the notation, let Z be the second argument of fl in Equation 28 and let U = ur/y, V = vr0/y, and R = r/r0 as in Equations 26. As a first example, consider the assumption of a thin boundary layer. This assumption tends to fail as the Rayleigh number decreases since for Z to have a fixed value 1 - r/ro must increase as the Rayleigh number decreases. The boundary layer thickness is proportional to (GrPr)-l/4. As a second example consider the assumption that (l/R2) 2U/02)may be neglected compared to ~U/~R2. From Equation 28 which shows that the maximum radial derivative increases with the Grashof number whereas the azimuthal derivative does not. The assumption then tends to fail as the Grashof number decreases. By a similar argument it can be shown that other of the idealizations tend to fail for small Prandtl numbers. These conclusions must be tempered by the knowledge that if the Grashof number is very large, turbulent flow occurs and the model also fails. Herman (11,12) Merk and Prins (22), and Morgan and Warner (23) give other discussions of the idealizations used here. In the analysis given here the effects of both the Grashof and Prandtl numbers are taken into account. By Equation 31 the Nusselt number is proportional to the product of the two numbers to the one-quarter power.

-28In Appendix A the analysis is given in more detail. The analysis given there shows: I. The dependence of the solution on the Grashof number can be established by assuming only a large Grashof number —it is not necessary to neglect the inertial terms. In other words the asymPtoti c solution for large Grashof numbers is that Nuo<Gri. 2. The inertial terms in the momentum balance become less and less important relative to the other terms as the Prandtl number increases. In other words the analysis given above in which Nu oc(GrPr) is the asymptotic solution for both large Gr and large Pro

III. FINITE DIFFERENCE METHODS It has been mentioned that finite difference methods hold promise for solution of problems which are too difficult for ordinary methods of analysis. In this section the methods will be discussed in some detail. It should be mentioned at the outset that basic questions of convergence and stability can not be resolved with mathematical rigor because the theory is generally inadequate. Nevertheless it is possible to cope with the major difficulties and produce an apparently satisfactory solution for the system of equations under consideration. The basic idea of finite difference approximation comes directly from the definition of the derivative. Suppose u - f(x,y,t) and its first derivatives are continuous; then the definition of the partial derivative with respect to x may be written in three different ways: i} A->o 7 6(o /7 /,, f(l#h1,A0 o) - f(ra-h, I0 fd) in which the alternate definitions are identical. If the limit process is not carried out, h is finite and the result is called a divided difference approximation to the derivative: 29

-302/7 6/76 ~-~ " to to —in which the alternate forms may not be expected to be the same. The three differences in the numerator of the equation are called forward, backward and central differences, respectivelyo From Taylor's formula the central difference would be expected to be the best of the three from the standpoint of accuracy. However, there are other factors to consider in selecting the form of the difference as will be discussed later. The error in the approximation clearly depends on the size of the increment, h, as well as on the behavior of the function. An approximation to the second derivative can be obtained by repeating the process whereby the first derivative was approximated. A different method which illustrates the use of Taylor's formula is outlined below f fx +br -e t A ) ( /9 At hf' t A I ) #ff(-,/to,)=k, 0)-/(o, t&) =A 4 _ - _:. ( t/'iy ) where xo < x9 < xo + h and xo-h < x < xo (the values x+ and x- are chosen to satisfy the equality). Adding the two formulas and rearranging gives:

-31+'- - _ (- (32) The first term on the right hand side of Equation 32 is called a central difference approximation of the second derivative, and the other term represents the truncation error incurred by replacing the derivative by the divided difference. The truncation error vanishes as h-> O and is zero for any h if the third derivative is identically zero. The basic concept of the finite difference approach can be stated in very simple terms. The derivatives in a system of equations are replaced by divided differences giving a system of algebraic equations which presumably can be solved by some method. The solution to the difference equations is expected to approximate the solution to the differential equations. In actual practice there are some difficulties which are to be discussed below. It is clear that if the increment size is in some sense large the solution to the difference equations might be a very poor approximation to the solution to the original problem. A. Methods of Attack In the problems under consideration there are two space variables, x and y; a time variable, t, and three dependent variables u, v, and T. If it is supposed that at large times a steady state is reached such that u, v, and T no longer depend on time and that this steady state solution is of primary interest, then there are two alternate methods of attacking the problem: the steady state approach and the unsteady state approach.

-32L, The Steady State Approach In the steady state approach the derivatives with respect to time in the equations are dropped reducing the number of independent variables from three to twos If the x and y dimensions of the region of interest ame divided into M - 1 and N - 1 increments, respectively, there will be MN "grid pointso" At each grid point there are the three independent variables so that there are 3MN algebraic equations to be solved. As an indication of the enormity of the task, the number 3MN was as high as 6,000 in this work. The algebraic equations are not linear. Methods of solving non-linear algebraic equations, in contrast to those for linear equations, are not highly developedo There is a stability problem associated with unsteady state coai. culations which is not present in the steady state approacho It might therefore be expected that the steady state approach is preferable if only the steady state solution is desired~ However, the system of algebraic equations- will almost certainly have to be solved by some iterative procedure, and the problems associated with finding a method of iteration which converges with some rapidity are considerable. Douglas and Peaceman (4) indicate that, even for conduction problems which involve only linear equations, the unsteady state approach is preferable to the steady state approach. In view of the difficulties involved in solving non-linear algebraic equations and for the reasons given below, the unsteady state approach was used throughout this work, 2, The Unsteady State Approach In the unsteady state approach, the equations are written as an initial value problem in which the velocities and the temperature are computed as functions of space and time starting from some initial condition. There are several advantages to this method of attack:

-331. Both the transient and steady state solutions are obtained; the steady state solution being the limiting value of the transient solution. 2. The unsteady state calculations may be thought of as an iterative method of solving the steady state problem in which the intermediate values of the dependent variables have physical significance. If the transient solution is not desired, the initial condition can be replaced by an estimate of the steady state solution thereby reducing the amount of computation required. The steady state solution is independent of the choice of initial conditions. 3. No direct assumption of laminar flow is required. The steady state approach is clearly limited to laminar flow. In the unsteady state approach the time dependent form of the equations is preserved along with the intriguing possibility of actually computing the fluctuations which characterize turbulent flow. The direct calculation of turbulent flow is very likely more difficult than laminar flow by orders of magnitude and even may be essentially impossible. However, it is a matter of such interest and importance that it is desirable to learn as much as possible about the behavior of the difference equations with respect to time in the hope that the work may constitute a step in the direction of computing turbulence. The main difficulty in the unsteady state method is that the difference equations may be unstable unless care is taken in selecting the form of the differences and the size of the time step. B. Stability and Convergence The remainder of this section is based for the most part on Richtmyer's book on difference methods (28) in which work by Lax, von

-34Neumann and others is presented along with a number of examples, The reader is referred to this book for an excellent discussion of the theory and practice, Richtmyer gives the development of the theory for a class of linear equations with constant coefficients, but points out that the theory is inadequate for complicated problems of the most interest. He then shows that the von Neumann methodof stability analysis can be applied successfully to problems for which a rigorous stability analysis is unknown. The stability criterion so predicted is shown to be an excellent approximation cf the necessary condition for stability in several cases by actually performing ex"* perimental calculations and observing the behavior of the solution. Stability is a necessary condition for the solution of the differ-.ence problem to converge to solution of the differential problem as the size of the increments, ax, dy and At tend to zero. Convergence is essential ~for the results to be meaningful in that the fundamental idea of an approxination is that the error can be made as small as one wishes. In practice:an unstable scheme of calculation usually yields meaningless numbers which overflov the accumulator of the computer after a relatively few time steps. The essense of stability is that there should be a limit to the extent to which any part of the' initial data can be amplified in the numerical proce. dure. Suppose the x and y dimensions of the region of interest are divided into increments of size Ax and Ay respectively such that x m JAx.and'y.- -/y where J and /are integers., Let n. an.integer, tdeote the number of time steps starting from the initial condition such that t s n At. Let ut1) be an independent variable associated with the nth time step (n here is a superscript not an exponent) and the position denoted by the subscripts. Now in the scheme of calculations a new set of the variables,

-35(nl) perhaps using values u.4, is somehow calculated from the old, u(n), perhaps using values of other dependent variables and perhaps using values of the variables at other time levels (for example, u(nl)). However the calculations are carried out the scheme will be said to be stable if the following inequality holds for any choice of initial data:, I / i</),) i 4[ 1W /U (/W-/ytA -b; t +/1iA7 (33) for some M 0, M2 0. The numerical procedure should be thought of as one of a sequence in which zx, By, and At are made smaller and smaller with the expectation of convergence. For stability it is required that the values of u at some time, t, be bounded independent of the increment sizes. By repeated application of the inequality, u(n) can be bounded in terms of the initial data u(O) and the..time, t as indicated below:;2j 4 if] I/ i ilrltt /f/af t /Yfa f J4 (p, }fj ^^'^ +19/i')*4/ ht t * - t(i^,/f~\ i,~,.L L.:I J. =:-. Using the fact that (1, MlAt)n B enln(l+M1Zdt) and that ln(1 $ M!At) z= ML LAt gives'1 en'/ A et/4/tpsli, / (1 MAdz n

-36which is the desired result. For a fixed t as At, 0 the number of time steps, n, becomes infinite, but the solution is bounded independent of t. In problems for which the theory is well developed stability implies convergence under fairly general circumstances if the truncation error incurred by replacing derivatives by divided differences vanishes as 4x, Ay, and At 4 0 It should be mentioned that stability is ^defined in different ways by some workers. The definition given above is essentially the same as that originated by Fritz John (16) ana used by Richtmyer (28). C. Stability Analysis The importance of the concept of stability has been indicated above. In this section methods of stability analysis will be discussed in association with some examples. It will be shown that the choice of the form of the differences used to approximate the derivatives is of crucial importance. Some choices lead to schemes which are unconditionally unstable. As a general rule a stability criterion involves a restriction on At in terms of Ax, Ay and the parameters of the system of equations. However, there are schemes which are unconditionally stable or unconditionally unstabiLe. In the case of non-linear problems the stability criterion may also involve the dependent variables. As a very simple example of a non-linear problem, consider the equation given below in which only the leading terms of a momentum balance are included. t _ - M (35) b ^

-37In association with Equation 35 it is supposed that u is specified on a boundary and initially although it is not necessary to think of the equation as representing a physical situation. Consider three different approximations to Equation 35 in which Mu/l t is replaced by a forward divided difference and Qu/<x is replaced by a forward, central or backward divided difference. >7+/) (ir.) (i / 1r) fi - "' - -a, ~ - (36)' At - 1' ZA^' ^u?(,: (:,l,r,?.(g) ) 9- " -B -( ~':, -M _/ (38) n A-^ --- - (i; (8^82!..The three schemes are called explicit since the values of u at the n. Lth time level can be solved for directly from those at the nth time level. It will be shown that Equation 37 is unconditionally unstable; that Equation 36 can be stable only if u O 0, and that Equation 38 can be stable only if u = O. In the stable cases it is required that u(n) At/Ax il. It is interesting to notice that Equation 37 which would intuitively be expected to be preferred is completely useless because of instability. 1i Positive Type Difference Equations A useful method of stability analysis is based on the use of difference equations in which all the coefficients are positive. In such cases a sufficient condition for stability can often be established by inspection. By way of illustration Equation 36 can be rearranged to give

-38(/ (/() at (4my ((- )ie Ai from which no conclusions are obvious unless u(n) K 0. Considering u(n) < 0 the equation can be rewritten J t/ )) /;,m\ (a/UA LA) Now the sum of coefficients of jn) and J.) is unity and each coefficient is positive or zero providing aZt/ax |ug)j i o Then u(n+l) always falls ~Ii J between u(n) and u(n) Hence Equation 36 is stable if un L 0 and J+l At/Ax /ti)/ v 1. Notice that the stability criterion depends on the solution so that it is not generally possible to select a time step in advance which will insure stability. The computer must test the criterion and alter the time step as necessary to maintain stability. The simple way of looking at the stability problem outlined above is useful for only a small class of problems. There is a much more general method of stability analysis due to von Neumann. 2. The von Neumann Method of Stability Analysis The von Neumann method of stability analysis employing Fourier series can be applied to a great variety of problems. Theoretically the method only applies to a small class of linear equations with constant coefficients. In practice it has been found to give a good approximation to the stability criterion even for non-linear problems. The method can be applied to explicit or implicit schemes inrvoling any number of time levels and any number of variables. By way of introduction the method will be used to show that Equation 37 is unstable as previously asserted.

-392 - i)= ( 37) The equation could be made linear by writing u t u in place u, where u is a small quantity of the first order, and dropping quantities of the second and higher order. For Equation 37 this only amounts to thinking of the coefficient of the derivative as a constant, so the notation of the equation will be retained except the subscript and superscript on the coefficient will be dropped. By rearrangement (hU) ~)(I (in) (,1) /Y+ = USA - ~!af^' - 2v'1') (38) where - uAt/2Ax. Now it is assumed that the solution of the equation can be written in the form (I, e(k) By the assumption of separation of variables each term in the series grows or decays independently and a general term of the series can be considered. Substitution of $neikjJx into equation gives tf/ 12e > m in4Y M -k ( k (ze ) which can be solved for 5 o I= / - -(e& e44) / - (39) The absolute value of 5 determines the rate of growth of u with time: /5/ < 1 corresponds to no growth and / - 1 + M t (M not dependent on k) corresponds to exponential growth which is permissible by the

.40o definition of stability, However, by Equation 39, / ).> 1 without regard to At for all values of k except those where sin k4x I 0. In other words at a fixed t, as At. 0 and n e with g fixed, some terms of the series are amplified without bound, Therefore, the scheme is unstabLe. The technique is easily adapted to systems of equations in seVeral space variablesa Consider a simple example to illustrate the method. The example is too simple to represent a physical situation, but contains terms of the type of interesto g- -_^ + t w4 )d _- -_u + 6 C41. In this example u is taken to be positive and the difference equat.ibs'l given below in which u is treated asa constant where it is a coefficient in the manner of the first example.^.... - It vill be Supposed that the general terms of the series for u and ~aare of the form egeiki JIAx ik2 e Ay and {neiklja4oy ik29t AY respectively, Substitution and rearrangement gives ~^+-/ _ 5^r'- t p7aa (Ii2)

-41^: f//-(</- e-Ik/d~)LI Jde'-z r e.(43) where s= uzt/Ax and. b At/(Ay)2. In a more general problem it would not be possible to solve the equations directly. The system of equations might be of the form A z = BfV or v, ABV, where V1 is the vector with components ~n+l and f~nl and V2 is the vector with components gn and r. Let G = A-B; G is called the amplification matrix and it plays the part of f in the first example. From Equation 42 and 43 for this example i 1g~~i~dxa,, ~ k a 2-Ik2 where CL 1 - (l - eiklAX) and C2 = [ik2A - 2 e 2X[cos k2,Y - i2 The von Neumann necessary condition for stability is that /-/ 1 + M t for some M X 0O where X is the largest eigenvalue of G. The eigenvalues of G in the example are -ik,4dI 7= c, = /-8f/-e J If =+ 2y A 1 both the eigenvalues lie within or on the unit circle ( - O, b O). The stability requirement then is that u dt/^x * 2bAt/ ( dy)2 L i, where u' O. If u < 0 the equations are unconditionally unstable,

-42It should be mentioned that known sufficient conditions for stability may be more stringent than necessary conditions. However, Richtmyer asserts that the von Neumann condition has always been found to be both necessary and sufficient in those cases where the gap between the two has been narrowed. Almost every step in the procedure of stability analysis given above is unsatisfactory from the mathematical standpoint. Nevertheless the procedure has been found to give the correct answers to the stability question in many cases. Richtmyer (28) gives a number of examples wherein the stability criterion is determined by experiment and found to agree with that predicted by the procedure given above. Round off error has not been mentioned at all in this section despite the fact that many workers define stability in terms of round off error. It can be shown that the alternate definitions of stability are essentially the same for linear equations as well as non-linear equations in those cases for which the theory is well developed. Richtmyer states that in his opinion round off error is generally not of much importance.

IV. THE FLAT PLATE The flat plate problem was selected for finite difference calculations partially because a solution to the problem exists both for small time where practically all the heat transfer is by conduction and for large time where steady state is reached. The results of the calculations will be compared with the existing solutions at both ends of the time scale. In the intermediate time range the problem has never been solved before so the results represent new information. The existing solutions will be reviewed very briefly before discussion of the finite difference solution. Initially in the flat plate problem there is no fluid motion, and after the motion starts for some time the motion is essentially one dimensional. During this initial interval the heat transfer is almost entirely by conduction for which the classical solution is l -' / - ^ (9 i (50) where ~ is the dimensionless temperature as before. By differentiating Equation 50 the heat transfer coefficient is found to be X = 1/'$ (51) which will be used later in a comparison of results. At very large times the steady state solution of Schmidt, Beckmann and Pohlhausen which was discussed earlier applies. The original solution was by series in which the coefficients were determined numerically by iteration. More recent solutions have been by nurrmerical integration. 43

Numerical methods of solving ordinary differential equations yield accurate results with very little computation so such a solution will be regarded here as exact. Ostrach's solution (26) will be used in the comparison with results, Ostrach gives a comparison of the solution with the measurements of Schmidt and Beckmann. Excluding points near the leading edge and those where turbulent flow may be starting, the agreement is good. The velocities and temperatures agree remarkably well near the plate where they are of the most interests The deviation in the velocity increases with dis. tance from the plate. The velocity of the solution seems to approach zero more quickly with distance than the measured velocity. A, The Differential Problem The equations of the Schmidt-Beckmann model used in the finite difference calculations are given below in dimensionless form. I L L f \ = / f Wz (52a) XY? V.- f } (52a) v a - 0 (52c) The boundary and initial conditions are: X- o:' == o Y=0; / =V- v = / Y a_ u =vpo= o u='- -0

and the dimensionless variables are: A= (vg"TJ/ j /= I= > 7 1 / y.. ~'v —~,'- ( "". i T3 AT) B, The Difference Prob lem It is possible to approximate Equations 52 by a system of explicit difference equations. The difference equations will be stable for sufficiently small time steps only if certain types of differences are used for the non-linear terms. The terms U bU/aX, V bU/y, U i /)X, and V b $/bY must be approximated using either a forward or backward difference depending on the sign of U or V, whichever appears as a coefficient of the derivative. A forward difference is used where the coefficient velocity is negative and a backward difference is used where the coefficient velocity is positive. This method of dealing with terms of this type is due to Lelevier according to Richtmyer (28). In general the velocities may be expected to change sign in the space-time region of interest so that four different sets of equations are required and the machine must determine the signs of U and V at each point and select the equations to be used. However, the flat plate problem is somewhat simpler than the most general problem in that U is always positive or zero and V is always negative or zeroo As a result a single system of difference equations can be uaed, 1, The Space Grid The space grid used for the problem is shown in Figure 2. The point j sl Qs 1I corresponds to the origin, X a (j - l)AX and Y -

-46( l)AYo The line Y a 0 corresponds to the plate with X E 0 being the leading edge. The plate -- - is of infinite extent in the X t-7 direction and the fluid is of in- -=6 finite extent in the Y direction. t5.. It is also assumed here that the' -- plate is of infinite extent in the -=3 direction normal to the X - Y plane, k=2 In the numerical procedure it is 3 -f: 1-t necessary to work with a finite region, In this work the X dimension of the region was 100 which corresponds to a Figure 2. The Space Grid. Grashof number of 106 The choice of the X dimension is somewhat arbitrary since the solution can be rearranged into a form valid for all X. The integers j ranged up to 40, That is j 40 corresponds to X - 100 so that A X = 100/390 In the Y direction there are conditions that U and ( are zero at Y = < so that some finite Y dimension which can be regarded as infinite must be used. An infinite distance may be thought of as some distance so large that it no longer matters how large it is, The problem was first solved forcing U and ~ to be zero at Y - 25 corresponding to /0 40 so that AY Y 25/39~ To determine the effect of the choice of the Y dimension the problem was reworked forcing U and < to be zero at Y = (25)(49)/39 which corresponds to,- 50 using the same 4Y. It was found that the velocities and temperatures near the plate were the same in the two cases

to at least four significant figures. It is concluded that either case is a satisfactory solution with respect to the infinite condition. 2. The Difference Equations The difference equations corresponding to Equations 52 are given below, uie^/;,! _ f / -''/ +? S -t/ g - A 7L/*L ->t - -zzr; (53a) i,^ n _ a( _ 4/tl -z2.o 42-/ (53b)'-eKe +' al = o (53c) The primed variables are at the time level' TACt and the unprimed variables are at the time 7. The procedure of calculation using the equations is very simple. Starting from the initial condition} values of U' and I are computed using Equations 53a and 53b, respectively for the whole grid excluding the boundary. Then the corresponding values of V are computed from Equation 53c, working from the boundary. where V = 0, outward. The procedure is repeated over and over giving the velocity and temperature distribution for increasing values of time. 3. The Stability Criterion The criterion for stability of Equations 53 can be obtained by either of the methods described earlier. The simple approach using a positive

-48type of difference equation applies here and gives the same results as the von Neumann method. First consider the energy balance, Equation 53b, which can be rearranged to give: y = da,2(i - - ^ f - ^Q f -iat B f e',<, ( i9 +) (54) where I, /I I - B Y The absolute value of V is used where V is a coefficient so that y ~ and B all exceed or equal zero. The coefficients of Equation 54 add up to unity and the coefficients are all positive or zero if + t 2B'</y i 1, in which case p is always between the extreme of,and three neighboring temperatures at the previous timeO Initially, all the temperatures are zero except those on the boundary which are unityo Then f at any point at any time cannot exceed unity nor be less than zero, and stability is assured. The stability requirement for the energy equation is then SUo + IV/^ * fI v/L (55 / The momentum balance can be treated in exactly the same waye It is found that Ul cannot exceed AtjR, plus the largest of Ujs and neighboring values of U providing 1-H i t A///^ -^ f - 2_ / (56) "^Td) ^r (F5?

.49If inequalities 55 and 56 are both satisfied 0 i1 and 0 4 U w 2which assures stability. Inequality 55 is more restrictive for Prandtl numbers of less than unity, whereas inequality 56 is more restrictive for Prandtl numbers greater than unity, 4. The Calculations Equations 53 were solved on an IBM 704 computer for a Prandtl number of 0.733. This Prandtl number was selected since it was used by Schmidt and Beckmann (31) as well as by Ostrach (26), and one purpose of the flat plate work was to provide a comparison of a finite difference solution with exact solutions. The maximum values of U and V to be expected were estimated and on this basis a time step of ZT = 0.1 was used. It will be shown in the discussion of results that steady state was reached after about 400 time steps although the calculations were carried out for 680 time steps to be sure that no further change occurred. For the first ten time steps all the independent variables which differed from zero were printed by the machine; then selected values were printed after 20, 40, 80, 120... time steps. A total of about six million values of the indepen. dent variables were calculated during the work so it was not practical to print them all. The "Fortran" language was used for all the programs in this work. The procedure of calculation for the flat plate is very simple relative to that for the cylinder problem. The basic procedure was given in the discussion of the difference equations. It should be mentioned that in the calculations numbers smaller than 10-38 were encountered since U and p tend to zero at large Y. Such numbers are below the capacity of the machine used so that an error is incurred unless some corrective action is taken. All numbers less than 10-10 were set identically equal to zero. By this

=50procedure the generation in the calculation of numbers between 10-38 and zero was avoided. The first solution using the 40 by 40 space grid was terminated after 680 time steps, Then the Y dimension corresponding to Y s X was extended by 25 per cent as indicated in the discussion of the space grid. The calculationsaweze then continued for 240 time steps using the extended (40 by 50) space grid. The calculations on the extended grid were for the purpose of determining if the choice of the Y dimension influenced the solution near the plate. As indicated earlier there was no significant difference in the resultso Tabular results are given in the appendix for both cases. It is highly significant that by modern standards very little calculation was required to obtain an excellent solution to the system of equations The solution was essentially complete after less than two hours of computing time. Much more than two hours computer time was actually used in investigating the effect of the conditions at Y a O and in continuing the calculations past the time where steady state was reached. By making use of the knowledge gained in this work the steady state solution of a problem of similar difficulty could probably be obtained in about one hour of calculation on the IBM 704 or some similar machine. The amount of calculation could be reduced by using a more course grid at large distances from the boundary where derivatives are small. Also, the size of the time step could be increased by about 50% if the derivatives U bU/^X Ub5//x, V U//Y and V 0/bY were approximated using implicit differences. This procedure will be discussed in a later section of this thesis where it will be shown that using implicit differences for these terms does not complicate the calculations~ The larger time step was not used in the calculations since it would presumably reduce the accuracy of the transient

-51solution. The transient solution was considered to be of particular importance in this work since such a solution has not been obtained before, C. Results The direct results of the calculations are values of U, V, and as functions of X, Y, and Z, the dimensionless variables of Equations 52. The results can be placed in a more compact form by use of the analysis previously given in Part II. In terms of the dimensionless variables of Equation 52 for a fixed Prandtl number U/X1/2, VX1/4, and $ depend on y/Xl/4 and t/Xl/2 or, in terms of the variables of the original problem u/(xg(aT)l/2, V (x/;2?fAT)l/4 and T-TiTw-Ti depend on y(gAT//2x)l/4 and t(gAT/x)l/21 and h/k (y2x/grAT)l/4 depends on t (geaT/x)l/2. The heat transfer coefficient, h, was defined as follows rh( 7A)! 2 - Y(57) The derivative in Equation 57 was evaluated by simply taking a straight line through the point corresponding to the plate and the point nearest the plate. Inspection of the computed temperatures subsequently confirms that the temperature does vary almost linearly with distance near the plate. A parabola passed through three points instead of the straight line through two points, yields values of h differing only about 0.2%, providing the points under consideration are not near the leading edge. The effect of the leading edge is discussed below. 1. The Leading Edge It has been mentioned before that Equations 52 cannot be expected to describe the actual behavior of fluids near the leading edge of the plate. Nevertheless there is a solution to Equations 52 near the leading

-52edge whether it is physically meaningful or not. There is some difficulty in.approximating this solution at the leading edge by finite difference methods. The solution to the difference problem will depart from the solution to the differential problem more and more as the leading edge of the plate is approached from values of x above the leading edge. The reason for this departure is that v and h are infinite at the leading edge in the true solution. This behavior of v and h causes no difficulty in a solution where one works with quantities of the form vxl/4 and hxl/4 which are bounded at x = O. In the numerical procedure where v and h are computed directly, the variables cannot be obtained at all at the leading edge of the plate, and the variables have some large but finite value at the level of the first x increment above the leading edgeo The departure of the solution of difference problem from the solution of the difference problem near the leading edge is not of consequence providing the final solution is taken at some distance from the leading edge. In terms of the variables used in the calculations the space-time grid is three dimensional. However, according to the analysis given earlier, the number of independent variables can be reduced from three to two. Therefore, the solution along any line X - constant must be the same as that along any other line X = constant providing the solutions are expressed in terms of the composite variables Deviations from the analysis can only be due to error, If there were no error the values of the dependent variables taken along any line of constant X would constitute a complete solu-t tion valid for all Xo Figure 3 shows steady state values of h/k (y2x/g~T)l/4 versus position. The group h/k (Y2x/gf4AT)l/4 would be a constant at steady state if there were wno error and its deviation at small j represents the departure mentioned above. Figure 4 shows steady state velocity profiles at

-530.55 + 0.50 * +-+-+ *-+-++ -— + - + —-+-+ -- - 0.35 0.30 I 2 10 20 30 40 j WHERE x (j-I)(Ax) Figure 3. Effect of Leading Edge Error on Heat Transfer Group. 0.6 + j=40 0.5 - jO0 A j =5 0.4A j- 3 s ^"/^^^ 0 j: 2 0.3 0.2 0.I 0. 0 I 2 3 4 5 6 7 8 9 (vx3 Figure 4. Effect of Leading Edge Error on Velocity Profile.

various positior:,so The vrfi wo;.d a'l be the same if there were no error and the deviat:tio-s at smal- l valUs of also repnresent the departure:irbertoned above. I a e oeen i' b7 rt fig.re thiat the error due to the leadi:.g edge is confined to sI a luCes of to Tkae- resu ts to boe presented henceforth are all adt Ilarge, ^K.ier;-e t;hae re'sult;s are independent of' j I should Abe me. tio;3:ed, hat the leadi.xg edge error is easily avoided onily because t4he ntbe:.. dee:lndern; variables can'be reduced from three;to two i. this particular pro-Lle- o te Otherwise it would be necessary to suabdiv.ide the grid near the leadil.g edge to obtain a better approximation to the soe:tio9 However, the f's'. o e ction v could znever be approximated at the leading edge in a problem s -cLh as the one at LLanod sin.ce th e function is not definred there In, problems of mo:re nearly complete physi cal significance te. velocity would be everywhere fir.nteo 2 Oa" ts - -,,. _ =, *I =1 The_ rel;-ts oj f t':e calcuatiotios on the late ae given in Flgure s'5 6f I an:ld 80 Figure 5 s1hows t,;h.e velocity perofile at various times StartingL from te, inittal. oltiiuless co::,ditio*, e vtelocity at a given point. fninreases uinformy vlt ime unti,. L a mLaxiimjm is reached, and then decreases sl$ightl;y -to it ste&ady F state val'*.e The trian gula points were takeni from Ostracch^s steady i2tate soitio'1 (26>~ It can be seen that the agree.ienit betweee. the calcuLatio.s and. OstraachIs work is excellent. The largest differ,.ee beetweern the two is about 2%0 Figure 6 sbhows t;he teqmerature profile at various times. Starting from an. initial value of ser:o the tem perat.re at a given point increases with time, goes through a maxizmum and then. decreases slightly to a steady state value. The triangular poirts again?:i are from OstrachVs solution for steady state an..d the agreem-nt ca:.. be seen to be excellent.

-550.5 0 2.0 A t ( 2. + rl 6.8 (STEADY STATE) 0.4 1.6 A OSTRACH (26) -J 0.3 1.2 z 0.2 0.2 0 I 2 3 4 7 Figure 5. Transient Veloeity Profiles.

-561.0 + 6.8 (STEADY STATE) lA POINTS FROM OSTRACH (26) 0.9 0.8 0.7 Io.6 0.5 CO0.4 0.3 0.2 -09\VALUES OF t (xT)' 1.2 0.1 0.8 0.09 01357 0 1 2 3 4 5 6 /gai T \'/4 Figure 6. Tri:anent T',perature Proflles.

-57~e6 -g i,L j ~0.6 / ~ g3AT \-I/4 Y ) 1.4190 *I 0.5810 Fa //t.^^-O^^^rt 0.6081 0o.4 -- o.. 0.4054 >0.3 g 02-. 2J 0.2027 O. i+^^ 4.6623 013 0 0 I 2 3 4 DIMENSIONLESS TIMEt ons Figure 7. Transeient Velocity at Various Positions.

1.0 \B.C'~~~~~~~ Q~~~~~~~0.2027 o~sL \ + + + + + +. + + + + - -I- +-'1''"'+ —------- 0.9.++ - + /\0.~~~~ ++-^^8~ ^ ~+-~_ _~O_ _ 0.4054 0-8 O"\ + -. 0.6081 +-+ ^4.-^^^^ ---- ----— + ++_ 0.8108 i 0.7+ + /+ TEMPERATURES +1 f +/ + 1.0136 +I/ +lI I;/ ---- ^( w — + ~~ ++ 0.6 + I O+ 1/4 + + )2X' DIMENSIONLES TIME, t ( ) c: —.-.-._._+ _+ + w + 0.4 0 I U.l 0.3 / OSTRACH'S SOLUTION (26) +z + k -- () CONDUCTION ALONE w 0.2 0.1 D1MENSI0NLECS TETIME, t (6 ) 1giure 8. Tranrient Temperatures and Heat Transfer Group.

-59Figures 7 and 8 show more clearly the variation of velocity and temperature with time. In these figures the points given at intervals of time represent one fortieth of the points actually calculated. The time scale on the figures terminates where the time group equals 4.0 since there are essentially no changes in the variables beyond this point. The calculations were actually carried out to a value of the time group of 6.8. Each curve may be thought of as showing the variation of velocity or temperature at a fixed point with time although in this and the other figures the composite variables are used for compact presentation of the results. Figure 8 also gives the most important partof the hresults from an engineering standpoint: the variation of the heat transfer group with time. The heat transfer group is infinite initially since the wall temperature changes discontinuously at time zero. The group then decreases with time, goes through a minimum, and finally increases to a steady state value. Ostrach's steady state value is denoted in Figure 8 by a triangle. The difference between this work and Ostrach's is about 2% which again constitutes excellent agreement. The cause of the 2% difference is not clear. It might be due to the influence of the leading edge error discussed earlier or it might be due to the size of the increments used in the finite difference method of calculations. There is of course no assurance that Ostrach's solution is more accurate than the solution presented here. However, it seems likely that finite difference method is the less accurate. The heat transfer coefficient for the initial interval was previously given from the analytical solution to be which can be put into the form of the variables of this work to give:

.60o +(- t="17 y= L+~/LiXJ)2.o(58) Equation 58 which is expected to hold for small times is represented it Figure 8 as a dashed curve~ The dashed curve departs from the full curve representing the numerical solution for values of the time group greater than 2.4. For times below this value of 2o4 the two cur-ves are indistinguishable. It is remarkable that the solution for conduction alone holds for times up to fairly near the time at which steady state is reached, Siegel (34) studied the transient convection problem using the KaFrman-Pohlhausen approximation method, and developed expressions for the time at which steady state is reached and for the time at which the initial interval endso Equation 59 is Siegel's formula for the end of the initial interval and Equation 60 is his formula for the time to reach steady stateo _(<_/ = ___<n/v5 ~f" i (/)(O.C f) (59) 4ext/ ~=I (32. 2p fr F.55a377~+ (60) According to Equation 59 the initial interval should end when the time group equals 2.7 as cdnpared to a value of about 2 4 from inspection of Figure 8. According to Equation 60 steady state should be reached when the time group equals 7.1, as compared to a value of about 3.5 or 410 from inspection of Figure 8. Siegelts estimate of the time at which the initial interval ends must be considered as good~ His estimate of the time to reach steady state is high but even here the estimate is of the right order

-61and could be considered to be good depending on how one wishes to define the time at which steady state is reached. Siegel did not obtain a direct solution for the transient problem. He estimated the times at which the two existing solutions should be valid as outlined above, and then simply interpolated between the two limiting solutions. The confirmation of the existence of a minimum with respect to time in the heat transfer group and corresponding maximums in the temperature and velocity are of particular interest. The only experimental work on transient natural convection seems to be the limited work of Klei (17) in which such a minimum in the heat transfer coefficient was found. Klei's measurements were for a plate with constant energy input rather than constant temperature so his results are not comparable on a quantitative basis with those of this work. The existence of the minimum was also predicted by Siegel in his analysis.

V TEE HORIZONTAL CYLINDER The horizontal cylinder problem was selected for finite differs ence calculations as a problem for which no solution is available, but for which experimental data exist. The measurements of Martini and Churchill (21) which were discussed earlier will be compared with the results of the calculations, In the case of the flat plate the finite difference solution was compared with a more exact solution arnd the validity of the finite difference calculations was immediately evident. Comparing a finite difference solution with experimental data is less conclusive verification of the method of calculation because there are several additional uncertainties: 1. The results of the numerical procedure will always deviate from the exact solution of the difference problem because of rounding error, and because the calculatdioIs may not be continued until steady state is reached. 2. The exact solution to the difference problem will differ from the solution to the differential problem because of truncation error. 3. The exact solution of the differential problem may not adequately describe the physical situation. This remark applies to the conservation equations even in their most general form since they have never been tested critically; extensive simplifications of the equations have always been required to obtain a solution.v 4. The mathematical model selected for the calculations may differ from the physical situation~ For example, in the cylinder model 62

-63the wall temperature change discontinuously at two points, whereas the wall temperature changes continuously in the physical situation. The discontinuous change can only be approximated in the physical situation. 5. There is always an unknown experimental error. In this section the method of calculation and the results for the horizontal cylinder will be discussed and compared with measurements. The results will be shown to be in general good agreement with the measurements. There are deviations for some ranges of the Grashof number which seem to be due mostly to the model rather than to the finite difference method of calculation or to experimental errors. A. The Differential Problem The differential equations used in the calculations are Equations 26 which were discussed in Part IIC. The equations are repeated below in dimensionless form. / t 6 -- _i -- -t- (26a) a+ t a v^ V) =... V. - (26b ) 0v) -' = ~ (26c) The boundary and initial conditions are: R —1, o e< T -. 0 - z' O.. — 0;: J= V L- =O

and the dimensionless variables and parameters are: UL= 9Ar/, v/ g/r y = /v/M - /~/f j 6 r - gm - y = v'< is the Prandtl number. The angle, e, is measured from the vertical. Equations 26 are symmetrical in a certain sense and this symmetry can be used to reduce the amount of calculation required to solve the equations. The functions U and V are periodic with period, and the function sat a given angle is the negative of at some angle different by A7: U(R, e, t) a U(R, & +, C) (61a) V(Re, I) = V(R, 6-+/t, t) (6lb) I(R, a, t) = -4 (R, e,/7-, A) (6lc) Equations 61 are satisfied initially by the choice of initial conditions, and they can be shown to hold for subsequent times by formally integrating Equations 26 with respect to time. The important result or Equations 61 is that only half of the cylinder need be considered in the calculations. In a more general problem in which physical properties of the fluid vary, the equations would not be symmetrical. B. The Difference Problem. Equations 26 can be approximated using explicit difference equations providing care is used in selecting the form of the difference. However, it does not seem to be possible to use a single system of explicit

-65equations over the whole space-time region of interest. Four separate systems of equations were used in this work corresponding to the combinations of signs of the velocity components: U? 0O V > O; U> 0, V ( O; U 0, V > 0; and U < O0 V < O. The use of several alternate sets of equations causes no particular difficulty although the calculations are complicated very slightly by the fact that the machine must determine which equations are to be used at every stage in the process. The equations and their application will be discussed after the space grid is described. 1. The Space Grid The space grid used in the calculations is shown in Figures 9 and 10. As indicated in Figure 9 the integers, denote radial position with I= 1 being the origin and,= 26 being the boundary such that AR is 1/25. The integers j denote angle with j = 2 corresponding to e 0 and j = 10 corresponding to & = P such that -> inr/8. Near the boundary the radial increments are smaller than the angular increments: (R) (d) is about ten times AR. The increments were so selected in expectation of radial gradients being much larger than aximuthal gradients in which case the finer divisions are required to maintain accuracy. The half cylinder j = 2 through j - 10 was used in the calculations. From the symmetry U, V, and ( at a given radius on the ray j = 1 are always equal to U, V, and /, respectively, at the same radius on the ray j - 9. Values of the variables along the ray j = 11 are related to those along j = 3 in the same way. The rays j = 1 and 11 serve the same purpose as a boundary in providing exterior points where the values of the independent variables are always known. The use of these rays in the calculations will be explained later,

.66j 10 26j=0 Fige 9. The Cylindica Spare G j =.= 2o~~~~ =.~

-6w._ _67_oj7 j=7 ~I?! =-6 -4 ofOu- G= Figure 0 S in of Oute' Grid.! Figure 10. Subdivision of Outer Grid.

The solution to the system of equations using the space grid of Figure 9 was considered to be a first approximation and the outer part of the grid was subdivided in most cases to obtain a better approximation. Subdividion of the grid in this way not only gives an improved solution but gives an indication of the truncation error. Figure'O is a fragment of the grid showing how the subdivision was carried out. The full lines are those from the first grid and the broken lines are those added in subdivision. The increment sizes for the outer third of the cylinder were cut in half. The unsubdivided solution to the temperature and velocity distributions in the inner region was accepted as valid when the grid was suibdivided. That is to say U and,' were held constant at the inner boundary of the subdivided region using the values from the first approximation. This procedure is acceptable since U and f and their derivatives are small in the inner region and a first approximation to the values here should suffice. In the differential problem the temperature on the boundary changes discontinuously at (= 0 from the value - 1/2 to the value + 1/2 and the temperature at 0- 0 is not defined. Similarly there is a discontinuity at ^=1'. In the difference problem the temperature is specified. only at discrete points and, since any number of functions could be passed through these points, the function being described by the discrete values of the variable is somewhat arbitrary. The method by which the grid is subdivided determines which wall temperature distribution is being approximated. By way of illustration, Figure 11 gives the values of the wall temperature near 9 = 0 for the two grids in comparison with the wall temperature distribution of the differential problem. If one looks at the grids individually, a function with two discontinuities (the full line), or a continuous function (the broken line) are among the many which could be thought

-69+V2 AFIRST GRID / 0 + / / / / -1/2 -+ + +1/2 — +- + — + +SUBDIVIDED GRID I + 1/2 - /! -1/2- /+ -+ — + — J + 1/2 DIFFERENTIAL PROBLEMS -1/2 --------------- - r/4 - /8 0/o +r/8 +t/4 ANGLE, 8 Figure 11. Approximation of Boundary Temperature.

-70of as the function being approximated. However, in the problem at hand the grids should be thought of as two of a sequence of smaller and smaller increment sizes such that the single discontinuity of the differential problem is approached. As would be expected from inspection of Figure 11 the subdivision of the grid makes a difference in the solution near the discontinuity since the boundary conditions are actually different after the grid is subdivided. 2. The Difference Equations The difference equations corresponding to Equations 26 are given in this section for the case where U = 0 and V! 0. In this case the terms where U and V appear as coefficients are approximated using backward differences, The equations used in the other cases are obtained by simply replacing the backward difference by a forward difference whenever the coefficient velocity is less than zero. Three of the terms in the momentum balance, Equation 26a, can be combined into a single term as indicated below. ^ _ /_- - _ _ >(zi^) >7 7 /,- X,4 [ J Then the single term can be approximated: 7_ - 7 7,7 - ai'/2 j)(L)2 Two of the terms in the energy balance, Equation 26b, can be combined into a single term as indicated below. e- - ALL "

-71Then the single term can be approximated V-Jr^ f A ^ (R^M/2)(4!+/ - of) — (R-a ^Y(! - ^S-/) (/,'IW -1 L an + iJ|I + t LI J The complete difference equations are given below where A denotes the approximation of?/]R[1/R'(UB)/bR] and B denotes the approximation of (1/R) {/RE R /R BI] }. = #Ae.A^ ^-^ # + - ~ +,^c* f (62a): t uP ~;- X$580 6 4le Wi44-/ Th-' V eX 4i4,2 P-'- (62b) {KtAK)(V& (A')54,) t 4 /-'' jf/ t4Lw'S-4fj- i ~ (62c) The primed variables are at the time level 72 +4't as before and the unprimed variables are at the time'Z:I The procedure of calculation using the equations is much the same as for the flat plate problem: the energy equation is used to determine values of ~ at a new time level over the whole grid.; the momentum balance is then used to similarly advance the values of UT Finally, the corresponding values of V are computed using

-72Equation 62c, working from the boundary in toward the center. The procedure is repeated over and over again giving values of U, V} and i for increasing values of time. The use of the equations will be given in more detail after the stability requirement is discussedo 3. The Stability Criterion Sufficient conditions for stability of Equations 62 can be established by either of the methods described earlier. In the case of the energy equation, the two methods give the same condition. However, in the case of the momentum equation, the results are slightly different. As would be expected, the von Neumann method of analysis permits a slightly larger time step. The von Neumann condition is supposed here to be necessary and sufficient for stability* whereas the condition from the simple analysis using a positive type difference equation is only sufficient for stability. The stability analysis using a positive type difference equation will be omitted since such an analysis involves only rearranging the equation and inspecting the coefficients as was illustrated in the flat plate problem. From the analysis of the energy equation, it can be shown that the temperature will always fall between the extremes of the boundary temperature (- 1/2 & 1/2) providing /dA' IV I 2 o(, t a f 1 (63) _e r t (AV)2'A 4R 6It will be shown that the stability requirement for the momentum equation is the same as inequality 63 except that V/V is replaced by unity. Equations 62a and 62b can be rearranged to give -The condition actually is nown to be sufficient only for a small class of problems as indicated in Part III.

-73-. = a,, S t -f - tc 3. = ck( -!- i / (64) The superscripts in the equations denote time level to be consistent with Z =, ('.3 = N) ~,- /- -~-2C 6'- - c-~-ZC j ^-7J 44=^ ~~~, - I( -- 2 C 4 -fbf's a_ j' 4s - CD, D where = e_/n Y /V 7 -, KdO R A / A I LT ( Now the amplification matrix can be formed in exactly the same way as in the example of Part III and the eigenvalues are, by inspection - l e /tl ^ / iA\ x (67)

-74The coefficients are all real and they are all positive except al and bl which may be either positive or negative. Therefore the largest absolute values of -1 and /2 will occur when all the terms in Equations 66 and 67 are real. That is to say when kl AX = k2LAY 27W or when klZlX = k2ZY =T Consider the maximum real value of i/2: ik),, = Dh/ + b *+ b 3 f^ go b5 Substituting the definitions of the coefficients and simplifying gives tat, = / C A-/ - R -f / i Since R is never less than AR (the equations are notused at the origin) it is concluded that the largest real value of 7q2 cannot exceed unity. If there is to be a stability restriction, it will be to prevent /'2 from being less than -1. To examine this possibility, consider:the minimum real value of-A2. -~" - _. < 2 (rnP) " A ~; Substituting for the coefficients and simplifying gives X,,~: / Xy w- - _t -.h which will be less than -1 unless...-i. I C! -'" /" } _4 (68) Inequality 68 is the condition for stability, By applying the same technique to -Al the condition is found to be Equation 63 as was asserted previouslyo In summary, the results of the stability analysis are that the following inequality must be satisfied if the PrandtL number. (7/cx) is less than unity.

&4T5/z t I.-; 2- -i - - - " */C -^ ^/(69) If the Prandtl number exceeds unity, 2/y in the inequality should be replaced by unity. The analysis given above was for the case of U I O0 V 0O. The stability criterion so obtained holds irrespective of the sign of U or V providing the absolute values of U and V are used in the inequalities as indicated, and providing the difference equations are changed according to the sign of U or V as prescribed in Section 2, above. 4. The Calculations Equations 62 were first solved for a Prandtl number of 0.70 and a Grashof number (based on radius) of 6.15 x 105. These conditions were selected to correspond to the conditions of the experiment which Martini and Churchill presented in most detail so that a direct comparison with the measurements could be made. The equations were then solved for the same Prandtl number and two additional values of the Grashof number. Finallyr the equations were solved for a Prandtl number of 10 and the original value of the Grashof number. Transient results were obtained in the first solution. For subsequent solutions the velocity and temperature distributions were estimated by use of the highly simplified model discussed earlier so that the amount of computation was reduced. In all cases except that of the lowest Grashof number, the grid was subdivided to improve on the solution obtained by the first grid. The incentive for subdividing the grid decreases with decreasing Grashof number as will be seen later.

=76 A summary of the calculations is given in Table I. Solutions 1, 2, 3, 4 are the solutions mentioned above. Solution 1A is the same as solution 1 except that ce-rtain termns were omitted, from the equations as will be discussed later on. Thee tmach ie te requireda for most solutions was between one and two hours which is surprisingly smiall considering the complexity of the problem. A mac.hie time requirement of 3,5 hours is shown in Table I for Solution 4. This larger amount of machine time was used in Solution 4 because the behavior of the solution in this case was somewhat different from the others as will be discussed later. The procedure for computing U, V, and i at time t+ A Ir from the variables at time t is given below. (1) The values of U, V, ard. along the rays j 1 and j w 11 are established from those along j 9 and j 3/respectively in the way indicated in the discussion on the symmetry of the equations. The variables are specified on the boundary and U ard ( are equal to zero at the origin,by symmetry. Hence values of U and. d at points exterior to the part of the grid to be advanced (j 2 through 10 and., 2 through 25, i.ncl-irve) are now fixed. (2) The quantity!UjtA-/RhA + |Iva!/zR + 2c./(R FA3)2 + 20agt/y(RA')2 is calculated for the interior points, the largest value o the quantity is fo un.d, and tahe ti.:ie ~icremect is altered as -ecessary to assur.e stability. This st(-,ep wovul..d.. c _-ot be neeeded i+f a very good estimate of U and V could be made in adiTvanrce. However, the extra work.involved in the step is well spent and actI;ally sa ves com-vuter time in most cases. A conservatively small value of _- w-ou.ld hae o'e used. if the criterion were not testedo

-77TABLE I SUMMARY OF CALCULATIONS FOR THE CYLINDER Solution Solution Solution Solution Solution 1 1A 2 3 4 Grashof Number 6.15 x 105 6.15 x 105 4.5 x 104 107 6.15 x l05 Prandtl Number 0.7 0.7 0.7 0.7 10.0 First Grid Number of time steps 960 400 800 480 1280 Time increment at end of calculations 3.4 x 10-5 2.8 x 10-5 7.4 x 10-5 2.7 x 10-5 4.1 x 10-5 Elapsed time, tro/V~f 0.047 0.0121 0.030 0.0125 0.070 Machine time, hours 1.5 0.5 1.0 0.5 1.5 Subdivided Grid Number of time steps 480 -- - 480 1920 Time increment, tr02/7 6.0 x 10-5 --- -- 3.0 x 10-5 5.0 x 10-5 Elapsed time, tro2/t 0.029 0.0144 0.096 Machine time, hours 0.5 -- - 0.5 2.0 Total Machine time, hours 2.0 0,5 1.0 1.0 3.5

(3) The new values of ) and U are computed at each interior point using Equations 62a and 62b o The values of t are computed first since U appears in the stability criterion and an increase in U would make the energy equation unstable. The calculations -roceed one ray at a time and the values calculated are not suhbstituted diectly into the ray where they belong until after the next ray in the sequence is completed. This "holding out" of the values is required so that all of the values used in computing a given difference are from the same time levelo Before advsancing any given point the machine must determine the sign of U and V and select the equations to be used accordingly. (4) The new values of V are compu;.ted from Equation 62c. V is specified as zero on the boundary and starting here the new values of V are computed working in toward the origLi. (5) The procedure starting with item 1 above is repeated. The calculations were continued as long as az.y of the Lindependent variables changed appreciably with time. It will be recalled Prom the discusion of the stab ili.ty require ment that the time step depends on. both. U and V. In the central region of the cylinder U and its deriva tves are small and. RV is a constant at any fixed angleo At the origi' V would be infinie e but since TI and w were specified as equal to zero at the ori.gie te equations are not applied here and the value of V at the or-ign is not a eeded in he calth alcuations, However, the val ues of V near the origi. are la-ge and these -valaues contribute heavily to the restriction on the time step imposed by the stability criterior. For example ir, the first so ut,;io. at'E 0.06 the largest value of jVJ \'t/A~R is about 0O83 whicoh indicates tha+t the allowable time step is almost directly proportion.al to V There is no point ic. carrying out

-79" the calculations very near the or-gin because no information is gained and the time step is restrictedo In this work the smallest value of R for which the calculations were carried out was 0O08 in every solution except solution 3, the' case of the largest Grashof numbero In the case of solution 3 the smallest value of R was selected as 0,16 based on the analysis given before which indicated that the boundary layer thickness is propor1 1 1 tional to Gr4o The ratio of Gr4 in this case to Gr4 of solution 1 is about twoo Use of a larger minimum radius for the larger Grashof number compensates for the fact that the velocities increase with Grashof number which reduces the allowable time step, After the solution using the first grid was completed the solution was used as input for the calculations using the subdivided grido The calcutlations using the subdivided grid were carried out in the same way as those siing the first grid with two exceptionso the minimum radius was much larger as indicated before, and the time step was held constant since the first solution gave excellent estimates of the maximum magnitudes of U and V for use in determin.ing the allowable time stepo Co Results The direct results of the calculations are values of U, V. and. as functions of R, G0 and _, the dimensionless variables of Equations 62o The heat transfer coefficient is calculated from the temperature distribution in the same way as for the flat plate. In the. firs parts of this section the results of the calculations will be presented in terms of U, V, and P o In the final part of this section in which the. effects of the Grashof and Prandtl parameters are discussed, different variables will be used: (uro/7/)(Pr/Gr)

(vrdO/)Pr3/Gr and n are expressed as function of 6 and (1 - r/ro) (GrPrJ.) These variables from the analysis of a highly simplified model of Part, IC are expected to take the parameters into account to a first approximation so that the results can be presented. in compact form, 1, The Transient Solution The transient solution for the cylinder problem was obtained for a Grashof number, (gk,,0Tr3/), of 6.15 x 105 and a Prandtl number of 0.7. The initial condition was that of a motionless isothermal cylinder in which one half of the wall suddenly assumes a high temperature (= + 1/2) and the other half assumes a low temperature (= - 1/2)o After the change in temperature of the boundary the fluid near the hotter part of the boundary tends to rise and that near the colder part tends to fallo Eventually a steady state is reached where, according to the observations of Martini and Churchill (21), as well as the results to be presented here, a narrow layer of fluid near the boundary circulates while the inner part of the fluid is practically isothermal and motionless, The steady state solution is of primary interest and. is independent of'the choice of initial conditions, The transient solution obviously depends on the choice of inial conditions4 The initial conditions used in the cylinder problem were chosen primarily to facilitate obtaining the steady state solution rather than to correst pond to any condition of actual engineering importance, For this reason only the most important results of the transient solution will be given here, Figure 12 shows the velocity as a function of time as various positions, The points shown are for every fortieth time step as in the case of the flat plateo The calculations were carried out using the space grid shown in Figure 9, The first and most imraportant conclusion from

400 e = D-i- = 0.92 (MAXIMUM VELOCITY) L~, 300 f~ + ft~-)-3 -+ -hC,'4. 81 4 ro.0.96' o / 9++++6+'+' -300 -" /'+_ —+-+.+"-+-'"+ +-+-+-t-+-+-+-+-+- ~ /r ^ *i-' 096 Iz o 200 r r 07 /f^ JL^^^-^-^-^K^- 0 =0.96 ^/ i^^ ^'-r^8096 Z~t~~~~~t Fur12.StsltTeoy t Velocities in the Cylind0. 0.02 0.03 04 0.05

-82inspection of Figure 12 is that steady state was reached at a value of the time group of about 0.04, The somewhat erratic behavior with time of the velocity at various points as shown in Figure 12 is interestingo The system seems to almost reach a steady state early in the calculations -while the velocity at Q= 0 (and & =7) is practically zero. Then the velocity at = 0 (and ~,=' ) increases and adjusts to its equilibrium value causing adjustments in the velocities elsewhere. This behavior can be explained by the fact that the fluid on the rays 0 and 1 A?' near the boundary has no incentive to move. That is to say the "buoyant force" on these rays acts perpendicular to the boundary. Eventually the fluid motion elsewhere is carried across the rays -= 0 and =:,'by momentum. Figure 13 gives the most important results of the transient solution: the variation of the Nusselt number at various positions with time. Notice that the Nusselt number appears to reach steady state somewhat earlier than the velocity as given in Figure 12. This behavior is characteristic of all the calculations: the Nusselt number depends on the velocity distribution, but is insensitive to small variations in the velocity distribution. For this reason the approach to steady state is best judged by observing the variation of velocity with time. When the velocil.y becomes steady, the temperature distribution and the Nusselt n-umber certainly also will be steady. The Nusselt numbers vary less with time than the velocities except at small times. The boundary temperature changes discontinuously when 7i= 0 everywhere except at 0 and' -t at these two points the

17 G=lr/8 16 - ^9- 0 J..'4 15 14 13 0 01 0.2 03 04 0r/ o II I0a 8= 3/8 0i IMENSIONLESS TIME, -i TA 13. Aim K1 t a t ar./ Z 9 94-v/8 DIMENSIONLESS TIME, + 64 ~~~~+ ~Fiu 1986v=iuNsItN r t6 Cyl/8i Flgux. 13. tranlent Nusslt Nubr e te 2heC1~

-84Nusselt number is zero at 7 = O. Elsewhere the Nusselt number is infinite at A= 0. 2. Comparison of Two Models It will be recalled that in the development of Equations 26 it was postulated that radial gradients are large relative to azimuthal gradients so it might be expected that the terms 1/R2 >'2 //2 and 1/R2/'2U/ 2 as well as U/R2 are negligible. This expectation was investigated by solving the equations with and without these terms. There is no difficulty associated with the terms in finite difference calculations —as opposed to other methods of attack in which retaining the terms is often not possible if a solution is to be obtained. Finite difference methods are thus well suited for testing the validity of the idealizations which are often used in fluid mechanics. Figure 14 gives a comparison of velocity and temperature profiles at various positions obtained by solving Equations 62 with and without the terms mentioned above. The solution of the simpler equations is denoted by broken curves and the solution of the more general equations by full curves. Near the boundary the two solutions are virtually indistinguishable and the deviations in the central region are slight. Hence, the expectation is confirmed. The comparison of Figure 14 is based on solutions using the first (unsubdivided) space grid. At some positions, especially near:- 0, there are relatively few points near the boundary where the temperature changes rapidly with position so the need for subdivision of the space is evident. However, it will be shown in the next section that-the results of Figure l4 constitute a surprisingly good solution despite the apparently coarse space grid used. The solution of Figure 14 is for the same values of the Grashof and Prandtl numbers as the transient solution.

-850.5 I 400 r0.4- -0 e =-/8 C~-Is — +'- MORE GENERAL EQUATIONS - - MORE GENERAL EQUATIONS i — "0- SIMPLER EQUATIONS -— 0 —-SIMPLER EQUATIONS.300 -0.3 200 - 0. 2 -0..1 - + I 0.2 j.I-%~~-I r/4 8 l. /8 + -- MORE GENRAL BOUATIONS + - MOE GENERAL E~JATION$ rw00-'.y3 - - 0.2- r _r o a o o 1 / I-0. 1 -30 0. 0.40.5 0.6 0.7 G8 0.9 0.4 0. 0.6 G? 0.8 09 1.0 Q Ir/r r /ro

-86 The temperature profile at various positions is interesting. Consider the fluid moving near the boundary in the direction of increasing. Just before the fluid gets to the bottom of the cylinder ( - -= 0) the fluid has passed the entire cold side of the cylinder and the temperature profile is "well developed;" it is monotone decreasing from the center (!v = O) out to the boundary ( - - 1/2). The profile at = - /8 is the reflection across the line 0 = 0 of the profile at — = 71t8. At 0= the boundary temperature abruptly changes from /) - 1/2 to, = 0, then at - =;/8 the boundary changes temperature abruptly again to = 1/2. T he fluid heats as it passes up the hot side of the cylinder unrtil, near the top, the temperature is monotone increasing with radius. At -?= the same sequence of changes starts only in the opposite direction. 3. Effect of Subdivision of the Grid It has been mentioned before that subdivision of the space grid is expected to give an improved solution, and also to give an indication of the error involved in the approximation. This later expectation deserves some discussion. Suppose some function, f, satisfies a differential problem. This function in general will not satisfy the difference equations used to approximate the differential problems. By use of Taylor's formula as was illustrated before, it is possible to compute differences in f, substitute them into the difference equations and find an expression for the trunrcation error. For any consistent approximation the truncation error must vanish as the increment sizes approach zero. In the problems of thiswork the truncation error varies linearly with increment size from which it is inferred that the change in the solution on subdividing the grid by half is of the same order as the error in the resulting solution. This approach gives some indication of the error in the finite difference approximation

-87despite its obvious weaknesses. There is no other general method of estimating the error for complicated problems of the most interest. There is some difficulty in considering the temperature profile near the boundary at a = O. It will be recalled that here the temperature varies discontinuously with both R ande at the boundary in the differential problem. There is no reason to try to approximate a function where the function is not defined and it must be concluded that the heat transfer coefficients computed at = 0 are meaningless. It is instructive to consider the behavior of the solution near the discontinuity, however, since the approximation is intended to be useful everywhere except at 0 = 0 (and ~ = ). The results of Figure 14 for e= O, Tr/8, and 1-/4 are shown again in Figure 15 in comparison with the solution to the same problem after subdivision of the grid. The fluid is generally colder after the grid is subdivided than before. This difference comes about because the boundary conditions on the difference problem were altered on subdivision as indicated earlier. The fraction of the boundary surface at the mean temperature ( = 0) is 1/16 as opposed to 1/8 before subdivision. Consequently the fluid passes by more cold surface in traversing the cold side and more hot surface in traversing the hot side. The result is that the minimum and maximum temperatures in the fluid are both increased in magnitude. The effect of subdivision on the solution is seen to be small even along the ray e = t/8 which is the ray nearest the discontinuity which is common to both grids. The effect of the subdivision is less for rays farther removed from the discontinuity. For values of 9 greater than Tr/4 the solutions are practically indistinguishable. The subdivision has very little effect on the velocity profile. Only two points from the

0.5 0,o~~r C~ ~8 = 0 8 8Mr/8 8 7= r/4 ^ 400 0.4 O Iw o~~ F + FIRST GRID + FIRST GRID + FIRST GRID 300 0- SUBDIVIDED 0 SUBDIVIDED GRID 0 SUBDIVIDED GRID 300 0.3 GRID cn Cn~~Q | 200 w 0.2 \ C 100 ~ 0.1 u z w U U w a.+ \ + +.-+-' 2 m +~~~~~~~~~~~~~~~ a d ++~+6 + +.+ + + 0 oi~~C, CoCDr \ 110.1 4,Bj+ -i J z z -0.2- + ____I_______1___1____ I ______________________ 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 r/r R/ro r/ro Figure 15. Effect of Subdivision on Velocity and Temperature Profiles for Solution 1 (Gr = 6.15 x 105).

-89subdivided solution are given along the ray /= i/4 because the points are coincident to the points from the first grid. The effect of subdivision would have been much less if the boundary conditions on temperature had been treated differently. An alternate way of treating the conditions would involve avoiding placing a grid point at the discontinuity. Suppose the first grid contained points to the left and right of the discontinuity but not at the discontinuity. Then it could be supposed that the temperature changes discontinuously half way between two points. The angular increments should not be subdivided in half in this case because this would entail specification of the temperature at the discontinuity. However, the angular increments could be subdivided by one third and the discontinuity would again fall half way between two points. This procedure is preferable to the one used in this work. The procedure used in this work is sound and the difference between the two procedures dimininishes with diminishing increment size. However, the alternate procedure outlined above is superior for large increment sizes. The effect of the subdivision on the heat transfer coefficient is shown in Figure 16. In drawing Figure 16 the Nusselt number at & = 0 and 6>= 7. was taken to be unbounded as in the differential problem despite the fact that finite heat transfer coefficients are actually calculated in the difference problem as indicated on the figure. It was mentioned earlier that the interpretation of the results near discontinuities is somewhat arbitrary. From inspection of Figure 16 it is concluded that the subdivision of the grid has practically no influence on the solution which in turn indicates that there is little error in the approximate solution.

20 |20.~ | + FIRST GRID 15-' 0 SUBDIVIDED GRID 9 0 POINTS COINCIDE 10 IL 0 0 5 c 10 I + 0 - 5 -20 - 5 -." 9 I"' - I - /4 0 7r/4 r/2 3'r/4 5 5ni/4 3r/2 ANGLE, 8 Figure 16. Effect of Subdiividing the Grid on Nusselt Number for Solution 1 (Gr = 6.15 x 105).

-91The necessity for subdividing the grid depends on the Grashof and Prandtl parameters. An increase in either of these parameters results in a decrease in the boundary layer thickness so that a finer mesh size is desired near the boundary to maintain accuracy. For this reason the subdivided grid was used in the calculations for the cases to be described in which either of the parameters exceeded the values in the solution of Figure 16 (Gr = 6.15 x 105, Pr = 0.7). The subdivision of the space grid should be most important for solutions of the largest Grashof number. Not only does the boundary layer thickness decrease, but the maximum velocity increases-with increasing Grashof number. The effect of subdivision on Solution 3 (Gr = 107) is shown in Figures 17 and 18. The velocity and temperature profiles given in Figure 17 are very similar in appearance to those discussed earlier from Solution 1 (Gr = 6.15 x 105). This similarity is not due to coincidence: the ratio of the Grashof numbers for the two cases is about 16 to 1 from which it is to be expected that the boundary layer would be one half as thick and the maximum velocity would be four times as great. The distance scale in Figure 17 is twice that of Figure 15 and the velocity scale is four times as great. As a result the two figures look much alike. The effect of the Grashof number will be discussed in more detail later on. The comments given earlier on the subdivision of Solution 1 apply to Solution 3 except that the subdivision alters the large Grashof solution slightly more as was expected. The difference between the first grid solution and the subdivided solution is given in Figure 18 in terms of the Nusselt number. Here, in contrast to Solution I, there is a discernable difference in the Nusselt number near the discontinuity. At

0.5 3 i 5 a8 0=o 8 = /8 8 =./4' 1600 0.4..... = |- + FIRST GRID 1 1200 Q0.3 0 SUBDIVIDED 1200-. 0.3 - G +G'-G GRID o w U w l f 3 r 10 > 800 - 0.2 U I| 400- O.I z' o w,+z 0 cn 0 w 0.7 0.8 0.9 0.7 0.8 0.9 0.7 0.8 0.9 1.0 _ -o.2 0.7 0.8 0.9 0.7 0.8 0.9 0.7 0.8 0.9 - 1.0 r/ro r/ro r/ro Figure 17. Effect of Suibdivision on Velocity and Temperature Profiles for Solution 5 (Gr = 107).

40 4\0 -— + —- FIRST GRID 30 --- SUBDIVIDED GRID 10 1 C2 0 F 20 -e o S+lui 30 -140 — /4 0 7r/4 r/2 37 —/4 5r/4 3/2 ANGLE 8 Figize 18. Effect of Subdivision on Nlusselt IhTumber for Solution 5 (Gr = 107).

-94-'-= /8, the point nearest the discontinuity and common to both grids, the first grid solution gives a Nusselt number 3 per cent lower than that from the subdivided solution. The difference between the two solutions is less and less farther from the discontinuityo At the discontinuity there is a larger difference between the two solutions; but, as indicated earlier, the results here are meaningless or open to interpretation. It is concluded that even in the large Grashof number solution the difference equations and the space grid used give a good approximation to the solution to the differential problem. 4 Direct Comparison with Experiment In this section the results of the calculations for the case of Grashof number of 6.15 x l05 and Prandtl number of 0.7 will be compared with the measurements of Martini and Churchill (21). The values of the parameters were selected to correspond to Martini and Churchill's experiment number 4 for which experimental results were presented in most detail. Some additional comparisons with the experiments will be given in the next section. Figure 19 shows calculated velocity profiles in comparison with the measurements. The points are calculated and the curves without points are from Martini and Churchill. The broken parts of the experimental curves are estimated and are uncertain. The overall agreement is good considering the great difficulty encountered in measuring velocities in natural convection in enclosed spaces. The largest deviation near the maximum velocity is about 30o. Martini and Churchill's results are the most certain on the ray O- 3</2: in the middle of the cold side. It is satisfying to notice that the measured velocity profile is in the best agreement with the calculations on this ray.

DIMENSIONLESS VELOCITY) / O E VE LCITy ur0/u DIMENSIONLESS VELOCITY, u0/ C)~0 o cuV L C T,u0 02 0 0Y ~~ ~~~~O cnn ~ ~>~+ +I;I +,, 0 0" |V0I+C ps 01 +s ( ~ Lz I;o 4 +*

_96A comparison of temperature distributions is given in Figure 20. Martini and Churchill measured temperatures along horizontal and vertical lines rather than along radial lines. The calculations were carried out in cylindrical coordinates so it was necessary to interpolate between calculated values to give temperature distributions along horizontal lines as shown in the figure. From Figure 20 it is seen that the agreement is excellent near the boundary where the results are of most interest. The agreement in the central region is not as good as it is near the boundary although the overall agreement is good. The calculationst show a more nearly isothermal central region than was measured. The difference is probably due in part to a deficiency in the model which has been mentioned before: the radial velocity is that velocity required to satisfy the continuity equation without regard to radial momentum changes. The radial fluid motion in the central region in the calculations is large which tends to make the central region isothermal. Figure 21 gives a comparison of the calculated and measured Nusselt numbers. Here the agreement is remarkably goodo The only deviations which are not within the accuracy of the measurements occur near the discontinuities in the boundary temperature, and, at these points, the mathematical model was different from the physical situation. In the physical situation the boundary temperature changes continuously at 93 O0 and the Nusselt number is a smooth function of -. Martini and ChurchillUs measurements for both sides of the cylinder are given on Figure 21 to show that the physical situation is symmetrical in the same way as the mathematical model. As indicated before9 only one half of the cylinder need be considered, although more than one half is given in the figure for clarity.

-970.5 - + 0.4 0.3 I" ABOVE CENTER (y/ro =0.465) 1 0.2 0.1 0 ft- " +-.+- -++ -0. I -0.2 -- + CALCULATED -0.3 - MEASURED (21) 1 -0.4 - + -0.5 0.4 0.3- AT CENTER (y/r0 =O) 0.2 + c -0.4 -0.3 ~ 0,C 3 - tx) I" BELOW CENTER (y/r = -0.465) | 0.1 0o, ++- + —++-+ —+ _. -0.2 -0.4 _ -0.5 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 HORIZONTAL DISTANCE FROM CENTER OF CYLINDER, x/ro Figure 20. Comparison of Temperature with Measurements.

20 - \ - THIS WORK 0 MEASURED - COLD SIDE (21) 15 1 \ + MEASURED- HOT SIDE (21) + + I0 Jc 5 0 D -5 I —' 0 -10 0 3 c,) ^~~~~~~~~~~~~~~~~ +4 z -15 -20 -7 /4 0 7r/4 7r/2 37-/4 7" 5- /4 37r/2 ANGLE, 8 Figcre 21. Comparison of IT'l-sselt ITeaers Wiswith asuireents.

-99It is concluded that the calculations and the measurements of Martini and Churchill's experiment number 4 are in good agreement. In the next section it will be shown that the calculations do not agree with all the measurements because the idealizations of the mathematical model are not valid under all conditions. 5. Variation of Parameters and Additional Comparisons with Experiment The cylinder problem was solved for two additional values of the Grashof number and one additional value of the Prandtl number. The results are sufficient to allow prediction of heat transfer rates in the clinder over wide ranges of both parameters. In this section most of the results will be presented in terms of. the variables from the simple analysis as mentioned before. A summary of the cases solved is given in Table II along with the principal results. TABLE II SUMMARY OF SOLUTIONS Solution 1 Solution 2 Solution 3 Solution 4 Grashof Number, g qanTro3/V2 6.15 x 105 4.5 x 104 o710 6.15 x 105 Prandtl Number, v//< 0.7 0.7 0.7 10.0 Rayleigh Number, (g ATr03)/(yo.) 4.3050 x 105 3.1500 x 104 7.0000 x 10 6.1500 x 106 (-g Tr04 25.615 13.322 51.437 49.799 hmD/k 9.434 4.562 18.338 16.291 hm o(roV) 0. 1841 0.1712 0.1782 0.1629 k lgg^T/

-100The results of most interest were considered to be those for gases so a wide range of the Grashof group was covered in three solutions for a fixed Prandtl number of 0.7. Then a solution for a Prandtl number of 10 was developed to give an indication of how well the simple analysis predixits the effect of the Prandtl number. Figures 22 and 23 give velocity and temperature profiles for the four solutions at various positions. Where the points are coincident only the point for Solution 1 is given. It is seen that the profiles for the three solutions of Pr = 0.7 are brought together remarkably well by use of the composite variables. Solutions 1 and 3 are very nearly indistinguishable and Solution 2 deviates from the others by no more than 15 per cent near the maximum velocity. The deviation is much less near the boundary. The deviation is undoubtedly due in large part to the fact that Solution 2 was not s ubdivided whereas all other solutions were. The solution for Pr = 10, Solution 4, deviates considerably from the other solutions especially in the velocity profile. All the velocity profiles show that the fluid accelerates while traversing the hot so:- cold' side of the cylinder. Then on passing the top (or' bottocm) of the cylide-9 the fluid decelerates before once again picking up speedo The large Prandtl number fluid is much more prone to accelerate and decelerate t.i. the other fluid. On starting up the hot side the fluid practically:o;syet by the time the fluid has completed a traverse of a side of fr.e cyl'daerO it has accelerated relatively more than the fluid of Pr 0.oo Vh-.es. the fluid first starts up the hot side (or down the cold side ) of t9e c'y.i-.der - the "'buoyant force" as well as drag opposes the motion o T'-E opipo- lio:, must be overcome by the inertia of the fluid if the motion is ta co:nt,:;i....e o Therefore, the behavior of the large Prandtl mnumber fluid ie iri ao:rda':~,:e

0.7 ------- 0.6 9o + SOLUTION I, 3ra6.15 x 10s5 8ir/8 A SOLUTION 2, Gr a 4.50 x 04 0.5 - 0 SOLUTION 3. Gra 107 13 SOLUTION 4, Pr3 10 0.4 0.3 z 4 0.2 o H f \Q'^ ^'"Q' N. 0.1 a.o~ 0 -0.1I -0.2 ^. O _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I I I I I 16 14 12 10 8 6 4 2 14 12 10 8 6 4 2 0 (I-R) (GrPr)4 (I-R)(Gr Pr) Figure 22. Collected Profiles Near the Discontinuity,

0.7T + SOLUTION I (Gr = 6.15 x 10) 0 SOLUTION 3 (Gr= 107) o0.6 A SOLUTION 2 (Gr 4.50 x 104) 0 SOLUTION 4 (Pr= 10)) 0.5 0.4 7/4 8 =/4 1 1?0r/2 0.3 0.2 "' 0.1. - 16 14 12 10 8 6 4 2 14 12 10 8 6 4 ( -R)(Gr Pr)14 (I - R)(Gr Pr)1/4 Figure 25. Collected Profiles on the Eot Side of the Cylinder.

-103with the analysis given earlier (also see Appendix A) which showed that the relative importance of the inertial forces is inversely proportional to the Prandtl number. The large Prandtl number fluid has relatively little inertia or relatively little resistance to acceleration and deceleration in comparison to the viscous and buoyant force terms. The temperature profiles for the fluids of different Prandtl number are in better agreement than the velocity profiles. Along the ray 9 - 7r/8 the profiles almost coincide. However, there are appreciable differences close to the bottom of the cylinder. The heat transfer results for all four solutions are given in Figure 24. Notice that the heat transfer group from the simple analysis serves to bring all the results together. The three solutions for Pr = 0.7 are brought together remarkably well, and the solution for a Prandtl number of 10 deviates from the others only near the discontinuity. At 0'T/16 the large Prandtl number solution is about 32 per cent below the other solutions. The results of Solution 4 have an interesting property tha'$, ha not been mentioned: in contrast to the other solutions, fluctua.ions oc. curred in a part of the space grid after the grid was subdividedO rb.e re. sults presented heretofore did not show the fluctuations beca9.se the a tude is practically too small to be distinguished on the graphs excep. in c-"eo small part of the space grido The calculations were continued much longer than in the other solutions to study the behavior of several cylces of tflutuation. It was established that the fluctuations were decaying a:n.d that they were restricted in area and amplitude. The fluctuations were not decaying very rapidly, however, so the calculations were discontirnued before steady state was reached.

0. 4 0.3 + ALL POINTS COINCIDE 0.2 0- 0.1 op-~~~~~~~~~~~~~~~~~3 0 0^ ^ -0' -— +'+. 0. I -0.2- + SOLUTION I (Gr =6.15x 105) A SOLUTION 2 (Gr= 4.50 x 104) + -0.3 0 SOLUTION 3 (Gr = 107) 0 SOLUTION 4 (Pr = 10) -0.4 - 7/4 0 ir/4 7/2 3v/4 I 5i/4 37/2 ANGLE 8 Figure 24. Collected Heat Transfer Results.

-105Figure 25 gives velocity and temperature profiles near = 0 for Solution 4. The crosses represent the profiles at dimensionless time of o0.0o68, and the circular points represent the profiles at a dimensions less time of 0.084 where the time is measured starting from the subdivision of the grid. The difference between the profiles represents the maximum amplitude of the last cycle of fluctuations in the calculations. Where only crosses are given on the figure, such as at e = -/616, the fluctuations are too small to show on the graph. Notice that the fluctuation in the velocity along the ray -=T/16 is pronounced, but it does not carry over to adjacent rays to any appreciable degree. The fluctuation in the temperature profile is much less than that of the velocity profile. The velocity at the point of maximum fluctuation is shown versus time in Figure 26 for the three rays where the fluctuations are the largest. The scale of Figure 26 is enlarged and the maximum and minimum values of the velocity on the ray - 7T/16 are indicated on the figure. The maximum amplitudes of the first three cycles are 87o7, 62o6, and'596, respectively. Hence it is clear that the fluctuation is decaying such that steady state woul. be reached if the calculations were continued. It is equally clear that steady state is being approached only very slowly. The points of Figure 26 represent 1920 time steps which took two hours of computer tima. Only one out of forty of the points are shown oia the graph The influence of the fluctuations on the heat transfer coefficient is indicated in Table III, The heat transfer coefficienit fluet.c.ae by as much as ~ 2.4 per cent at one point, but the mean fluctuiatlo i. on ly ~ 0.8 per cent.

TEMPERATURE, 4,(T-Ti)/(TH-Tc) -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 8=0 I8^.St^. V=-r/ 16 $ e.ir/8 L? 0.8+ 0.9 ~'r 1 0.4 + U + + u i+ O~~f t) + = 7/16 U.?0.87 8= + + 0+ " 0 = == 8 8 8= -/16 -h 8=0 + +1 S10 0.8 +t' 0.9 + min max * + o +~~ 1.0 0 50 100 150 0 50 100 150 -50 0 50 -50 0 50 VELOCITY uro/v Figure 25. Profiles from Solution 4 (Pr 10) Showing Fluctuations.

125 + L —- - ++ + + + + + + + + + + + +'- + + + + + + + + + ++ + ++.+-++++++++++++++++++++-4-++ +++++++++++ 100 = 0, r/r 0.82 75,62. 1 47.0 v/16, r/ro 0.74 h..ht -'~~ ~~'N +" / + ^'.N- N +\ 50 ~ ~ ~ ~~~~~ i i4 ~~~~~~~~~5.0 25 + + 50 2 ~ \ /~$ \ /v + / ++// s ~- \ ~~ ~~~/ \ 1 -25 - = 2r/8,r/ro = 0.82 -15.6 14.6 - 501 - I I I I rI j 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.0Z 0.1 TIME, tyv/r2 AFTER SUBDIVISION OF SPACE GRID Figure 26. Velocity Fluctuattions in Solution 4 (?r = 10).

-1o8TABLE III FLUCTUATIONS IN THE HEAT TRANSFER RESULTS OF SOLUTION 4 b,/k Angle, maximum minimuim deviation from,,__,,_ ______ the mean, % 0 15.728 15.508 0.7 T716 28.040 26.736 2.4 2 -/16 22.263 21.555 1.6 3 77/16 20.498 20.019 1.2 4 1m/16 19.578 19.229 0.9 5 1/16 18.907 18.643 0.7 6?/L6 18.317 18.113 0o6 7 /16 17.741 17.581 0.5 8 /16 17.143 17.017 0.4 9,/1 6 16.497 16.398 0.3 10o /16 15.777 15.700 0.2 11?/1l6 14.933 14,874 0.2 12 P/16 13.925 13.880 0.2 13 /16 12.692 12.662 0o 14 T716 11.092 11.073 0.1 15 /-6 8.716 8 694 0.1 16 T/16 -15.727 ~15 o-08 0.7 Overall 16.423 16.158 0.8

- 09It is interesting to consider the cause of the fluctuation. No direct assumption of laminar flow has been made in this work and the time dependent form of the basic differential equations has been preserved. However the basic equations were simplified to such a degree that it seems highly unlikely that the simplified equations are capable of describing turbulent flow. The fluctuations almost certainly do not correspond to the actual behavior of the fluid, but were induced by the particular method by which the grid was subdivided. This expectation is supported by the fact that no fluctuations occurred prior to subdivision of the grid. The large Prandtl number fluid practically stops near the bottom (and top) of the cylinder as indicated before. In doing so the fluid which was moving relatively rapidly near the boundary is forced out into the central region causing the boundary layer to thicken as shown earlier in Figures 22, 23, and 25. On subdivision of the grid it was assumed that the boundary layer was thin such that the result from the first grid served as an adequate approximation to the solution in the central region. This asisumptio:-n is less satisfactory for the large Prandtl number solution than for the other solutions. The main difficulty is associated with the fact that the rays added on subdivision do not have a direct connection with the centrral region of the first grid. The connecting points of the added rays were establisLed by linear interpolation using the values on adjacent rays. This proced'i.ure was adequate in all solutions except Solution 4 because the aximr.uthal gradients were relatively small. In Solution 4 the azimuthal gradients are much larger near 0 = O as indicated before and linear interpolationT is less accurate. Refer back to Figure 25, The two rays 0= _ I/l6 were added on subdivision. The points are those from the subdivided grid and the dotted lines in the central region were found by linear inrterpola-ton

-110Notice that the velocity profile at = fP/16 seems to tend to collapse much like the profile at 9 =t/8. However, the velocity at the connection to the central region is artificially prevented from decreasing and is held constant at the value found by linear interpolation. This artificial restraint on the velocity at e= r/l6 probably induced the fluctuation, The error due to this restraint is difficult to assess. It seems likely that the velocity profile at - T//16 would be considerably different were it not restrained. However, the fluctuation has only a small influence on the temperature profile, and the fluctuation is confined to a small part of the grid. Therefore it seems likely that there is little error in the solution considered as a whole. It should be mentioned that the fluctuation does notseem to be caused by instability of the difference equations as previously defined. The magnitudes of the independent variables are bounded and they seem to be approaching values independent of time. It has been shown that the results of all the calculations for a Prandtl number of 0.7 are brought together by a choice of variLable~s from analysis of a simplified model, As a final part of thLs sectio::;r. e generalized results will be compared with Martini and Churchill~s I 21) measurements. Martini and Churchill treated the two sides of tihe cyli...e-r separately. That is, they calculated Nusselt, Grashof anrd Prandtl+ groupsF based on the overall temperature drop and the fluaid properties ev'i.:.a.ed at the temperature of the side of the cylinder in question There is a substantial difference in the Grashof numbers for the two ides of the cylinder in the experiments at large temperature differenceso Foo exa'mpLe0 in Martini and Churehill' s experiment 15 the Grashof number on the ccld

-11lside is 19 times as large as that on the hot side. The model for this work is based on an assumption of a constant Grashof number as well as a small overall temperature drop so that it is expected that the agreement with experiment will decrease with increasing temperature difference. It has been pointed out before that the model is theoretically restricted to large Grashof numbers so agreement with experiment cannot be expected for very small temperature differences. It will be shown that these expectations are confirmed by comparison with the experiments insofar as the velocity profiles are concerned. However, the heat transfer results agree with the experiments more closely than might be expected. Martini and Churchill's results were recalculated to conform to the model used in this work. The viscosity and the Prandtl number were taken as the average of the values at the hot and cold sides of the cylindero The heat transfer coefficient was taken to be that reported by Martini and Churchill as the mean value for the hot sideo Martini and Churchill found the mean heat transfer coefficient on the cold side to be consisteantly lower than that from the hot side by two to twenty per cento This difference must be due to heat losses since from the definition the two mean -v;a-es must be equal. The recalculated results are given in Table IV It is interesting to notice that increasing the temperatire difference for air does not necessarily produce an increase in the Grashof nrumber_ as calculated here. For example, compare experiments 6 an-d 15, The overall tmpeeat''re drop is increased from 200C to 2000~C yet the Grasn.of rnum ber is pracstFcally unchanged because the viscosity of the air and the mean temperature bothL increase. A comparison of the calculations with the four experiments for which Martini and Churchill measured velocit-y profiles with t-he most

-112= TABLE IV PARAMETERS FROM THE RESULTS OF MARTINI AND CHURCHILL ~^ VX104^ "' l~ hDk Experi- AT'Ti ft2 Gr P Experi /T ti (vo/se XK -4 Pr (GrPr)4 (mean-) h l ( ).,................OK........ k.g.;.' (avg.) XLO (hot side) 3 24,0 326.7 2.03 33l. 0.701 22.0 7047 0.1700 4 38.9 317.2 1.93 61.5 0.702 25.6 8.74 0.1705 5 25.7 302.6 1.78 49.7 0.707 24.3 6.83 0.1405 6 20.7 303.1 1.78 40.2 0.706 23.1 6.13 0.1325 7 Wr).5'72 293.2 1.67 12.89 0.705 17. 34 3 95 o.1140 8:1.93 290.4 1.66 4.50 0.710 13.38 6.85 0.2550 9 51.9 319.7 1.97 78.0 0.699 27.2 7o06 0.1300 10 90.5 344.6 2.37 86.7 0.697 27.9 8.13 o0.4 5 11 75.2 335.8 2.15 90.0 0o698 28.i 8.41 0.1h49 12 110.5 365.3 2,81 71l1 0.693 26.5 7o04 0o.330 13 111.2 364.3 2.80 72.3 0.695 26.6 7 33 0.1375 14 174.1 406.4 3.98 50.3 0.716 24.5 7.26 0.1482 15 203.7 425.5 4.55 43.1 0.20 23.6 7.32 0,15,2

-113confidence is given in Figure 27 and Table V The comparison is along the ray e= 3172 (in the middle of the cold side). On this ray the illumination of Martini and Churchillts dust particles was the most intense so the measurements should be the most accurate, Martini and Churchill made one experiment, experiment 8, at a very small Grashof number. The agreement between this work and that experiment as shown in Figure 27 is poor as expected. It should be mentioned that the measurement of the velocity profile by Martini and Churchill's method should be more accurate for small Grashof numbers than for large Grashof numbers because the boundary layer is thicker and the most important measurements are made at a greater distance from the wall. Glare from reflection of light at the boundary caused difficulty in Martini and Churchill's worko Since this difficulty should be the least pronounced for small Grashof numbers it seems unlikely that the deviation between experiment 8 and this work is due to experimental error. The deviation is due to inadequacy of the model as indicated earlier. The deviationsi between experiments 4, 9, and 10 (which are at larger Grashof numbers) and this work are much less, and the trend of increasing deviation with increasing temperature difference is followed as expectedo The temperature profiles of Figure 27 are all in good agreement with this work. It should be pointed o-ut, however, that the agreement in the case of experiment 8 on the ray e= 3m-/2 is not typical of the othe-r rays. In experiment 8 the overall temperature drop was less than 2~C axrd Martini and Churchill obtained their heat transfer coefficients by measuring gradients in temperature near the boundaary. Since they were measuring such small differences in the case of experiment 8, the data scatter more than in the other experiments, and. the results are presumably somewhat less accurate.

-114TABLE V COMPARISON WITH THE RESULTS OF MARTINI AND CHURCHILL Experiment Experiment Experiment Experiment 4 8 9 10 Grashof Number g CATro3 6.15 x 105 4.50 x 104 7.80 x 105 8.67 x 105 Prandt1 Number 0.702 0.710 0,699 0.697 Rayleigh Number 4.32 x 105 3.20 x 104 5.45 x 105 6.04 x 105 T, ~C 38.9 1.93 51.85 90.50 AT/Ti 0.123 oo066 o.162 0.262 Agreement with velocity of this Excellent Very Poor Fair Poor work The h Group, h1 0.1705 0.255 0.1300 0.1455 hm /cxyro4 k g^ATJ Deviation in the heat transfer -3% 44% -26% -8%o group from 0.178

-1150 0.1 0 F -0.2 _ +THIS WORK tF 0/ I -0.3 -0.4 -. -6 / 0.3-.. 0.1 0.5 / + 4 0.123:\ # * 8 0.066 0.6 - / 9 0.162 0 10 0.262 0.7 0 1 3 4 6 7 8 9 10 (I -r/ro)(Gr Pr)114 Figure 27. Comparison of Solution with Experiments for Several Grashof Numbers,

_ll6The final results of this work in terms of the mean heat transfer coefficient are presented in Figure 28 in comparison with all of Martini and ChurchiLl's results. The mean heat transfer coefficients were calculated using Simpson's ruleo Two lines are shown on the figure: the full line representing the results of thiswork for a Prandtl number of 0.7, and the dotted line representing Martini and Churchill's results. The results of this work for Pr = 0o7 can be expressed in a very concise form: ^D - 4b ^T /ro3 VM } or, the equivalent form: Tn ) =oQ/^_\ o 0./78 For a Prandtl number of 10 the constant 0.178 should be replaced by 0.163, a change in the constant of only 8.5 per cent. Martini and Churchill's results seem to be in good agreement with this work although there are insufficient data at low Grashof numbers-to establish the trend of their results there with certainty. The dotted line through Martini and Churchill's results falls 16 per cent lower than the results of this worko The physical situation is expected to give lower heat transfer results than the mathematical model, In the physical situation the wall temperature varies continuously between the two extremes. That is to say the discontinuous change in wall temperature of the mathematical model is only approximately realized in the physical situation. As a rem suit the difference between the wall temperatures in the physical situation tends to be somewhat less than that of the mathematical model near the top and bottom of the cylinderO The results are given in tabular form in Appendix Co

-11719 18 17 1'6 15 - + 14-, 12 IIo -/ 7 ++ MEASURED (2) 0.25 + 0.1-. / + I / 0. -/ -+I.. 4 -^ ^ + 0 THIS WORK Pr =0.7 3 — A THIS WORK Pr =10: + MEASURED (21) 0 15 20 25 30 35 40 45 50 1 -5 —---— 3-7( —-GrPr)1 -Tr3 c)1/4 (G oPr) o -O- u V2 k... Figur7e 2.8, Comparison of Overall Heat Transfer Result s.

VI. DISCUSSION OF APPLICATIONS TO RELATED PROBLEMS The main difficulties in problems of the type considered in this work are associated with stability of the difference equations. It has been shown that difference equations which intuitively appear to be the best choice and which give the lowest truncation error are often useless because of instabilityo There are many possible choices of difference approximations to a given differential problem and finding a stable choice may be difficult. In seeking a useful system of difference equations the following comments may be helpful. o1 Stability criteria may be additive in a certain sense. In first attacking a complicated problem it is advisable to work with one term (other than the time derivative) at a time. If a stable scheme is found using a single term then other terms can be considered and the stability criteria may be additive in a certain sense. For example, three equations and their stability criteria are given below for differences of the types used in this works.._ _ _g > j -' / it ~ y 7 y (^ =~ - / 118

-119There is no assurance in general that the criteria will add as indicated above but this was found to be true in all cases of this work. An analysis of the complete equations should always be made but the simple approach is useful for eliminating differences which lead to unconditional instability. 2. It may be necessary to use different formulas in the approximation according to the sign of the coefficient of the derivative in the non-linear terms. This point was discussed in connection with the approximations used in this work. 3. Implicit schemes are ordinarily more strongly stable than explicit schemes. A partly implicit scheme is not necessarily difficult to solve at each time step. As an example consider the leading terms In the flat plate momentum balance using implicit differences: - _ L - /L; _ _ -_,_) a vs (If' - 2) + -.... (70) In the explicit scheme of calculation the stability depended on velocity as indicated by the terms IU/zl-//AX and Iv/AAt/AY in the criterion. However, if implicit differences such as those of Equation 70 are used, the terms involving velocity no longer appear in the stability criteri.orio Equation 70 can be solved directlyo At X = 0 and Y - there is a "corner" of the boundary (Y oo0 is actually approximated by Y = constant in the calculations by finite differences in accordance with the discussion given earlier). U is always knowrn on the boundary so that Uj for the interior point nearest the corner can be computed directly using UtL, and U1 which are on the boundary. The other values of U then can be calculated in a sequence of deceasing d ireasi asing j since the IP values needed will be kniown0 The same remarks apply to the energy equations.

-120= 4. The conduction terms of the energy equation probably should be approximated using implicit differences for very low Prandtl number fluids. The conduction terms in the energy equation as treated in the cylinder problem contributed the terms 24t/(lPr,)(RAO)2 and 2AC/(Pr)( AR)2 to the stability criteriono For very small Prandtl numbers a correspondingly small time step would be required to assure stability. If the conduction terms are approximated using implicit differences the time step is independent of the Prandtl number. Use of implicit differences here need not cause undue difficulty because methods of direct solution of the equations are highly developed. For example, Douglas and Peaceman (4) give a direct, partly implicit scheme in which the time step limitation is avoided. 5. In applying finite difference methods to the most general problems in fluid mechanics, there may be stability problems associated with the pressure or with the way the momentum and continuity equations are coupled. In this work the pressure distribution was known and assumed to be constant, and only the momentum balance in the direction of principal motion was taken into accounto Such idealizations will not be admissible in many problems, and in such problems the momentum balances are coupled through the pressure or the continuity equations Consider as an example a fragment of a simple isothermal, onedimensional problem in which the pressure distribution is variable. A similar example is given by Richtmyer (28.) Pressure is replaced by'RT in the equations where R is a gas law constant. fc = RT. f (7) _ -_ C at *e, (72)

-121L Equations 71 and 72 could be approximated as indicated below ~. - T - ( K At ~ = i. (73) "< -:, i - %.j ( U;-f, - CA/ (74) and the equations will be stable for a sufficiently small time step. This scheme is actually explicit in the procedure of calculation despite the fact that an implicit difference is used in the continuity equation. Equation 73 is used first to get the values of u? used in Equation 74. It does not seem to be possible to construct a stable, wholly explicit system of equations using simple differences involving only two time levels. Equations 73 and 74 are stable if the following inequality is satisfied: A-dr = z/ / (75) An indication of the relattude f time step can be obtained by considering dimensionless variables of the type used previously: b tV/L2, U - uL/,/, X - x/L where L is some length such as the radius in the cylinder problem. The maximum time step in terms of the same dimensionless time as was used in the cylinder problem is wc- foZr t dLe of which for the conditions and increment size of the cylinder problem is of the order of l0'8 In the cylinder problem a time step of about 10-5 was used. It is concluded that coupling of the continuity and momentum equations through the ressure or density laces a severe restriction on the time step if the equations are approximated in the way shown above.

-122Some additional work on the stability problem is needed before finite difference methods can be applied to the most general problems. It seems likely that to avoid the time step limitation either an implicit scheme or an explicit scheme involving three or more time levels will need be used. There is a different approach to the more general problems which may be preferred to that discussed heretofore. The pressure can be eliminated between the momentum balance equations giving equations involving the components of the curl of the velocity (the vorticity transport equations). It appears that the stability problems associated with the vorticity equations would be much less difficult than those associated with the basic Navier-Stokes equations. For two-dimensional motion the vorticity has only a single non-zero component.

VIIo SUMMARY OF RESULTS The principal results and conclusions are given belowv 1. The system of partial differential equations governing fluid motion and heat transfer were solved by finite difference methods for two physical situations: the classical case of the isothermal vertical flat plate, and the case of the fluid confined by an infinite horizontal cylinder with the vertical halves of the walls at different uniform temperatures. The flat plate problem was solved for a Prandtl number of 0733 to correspond to prior analytical solutions. The cylinder problem was solved for three different values of the Grashof number with the Prandtl number held constant at 0.7. An additional solution to the cylinder problem was obtained for a Prandtl number of 10. In both problems the most general equations were simplified to correspond to laminar boundary layer flow. 2. The initial value (time dependenrt) approach to the problems was used thereby obtaining both the transient and steady state solutions in one operation. This approach seems to be preferable to methods in which steady state is assumed at the outset. 3. In problems of laminar boundary layer flow explicit difference equations can be developed which will be stable for sufficiently small time increments providing different equations are used in different parts of the space-time grid depending on the sign of the velocity components. 4. The solution of the difference problems required very little computation and very little computer storage capacity by modern stand.ardso 123

5o The solution of the flat plate problem is in good agreement with the short time analytical solution for conduction alone^ and with the steady state solution of Ostracho The steady state heat transfer results for a Prandtl number of 0733 are: h c,y f= O. 5653 The constant, 0o3653, is only 2 per cent larger than that obtained by Ostracho 6, The solutions to the cylinder problem are in good agreement with the measurements of Martini and Churchill except at very low Grashof numbers where the idealizations of the mathematical model are inadequate. The heat transfer results for a Prandtl number of 0.7 are: — t 0./78 P ATJ For a Prandtl number of 10 the constant, 0ol78, should be replaced by 0.163, a change in the constant of only 8 5 per cent. 7. Finite difference methods apply to a great variety of problems which resist ordinary methods of analysis, The methods are well suited for testing the idealizatiors often used in fluid mechanicso It seems certain that the most, difficult problems in fluid mechanics and heat transfer will eventually -be solved by finite difference methods, This work constitutes a step in that. directiono

REFERENCES 1. Batchelor, G. Ko, "Heat Transfer by Free Convection Across a Closed Cavity Between Vertical Boundaries at Different Temperatures " Quart. Appl. Math., 12, 209-33 (1954). 2. Birkhoff, G., "Hydrodynamics," Dover Publications, Inc., New York (1955). 3. de Graaf, J. G. A. and van der Held., E. F, M., "The Relation Between the Heat Transfer and the Convection Phenomena in Enclosed Plane Gas Layers," App. Sci. Res., A3, 393 (1954)o 4. Douglas J., and Peaceman, D, W., "'Numerical Solution of Two Dimensional Heat Flow Problems," Am. Inst. Chem. Engrso J., L1 505 (L955). 5.- Eckert, E. R. Go and Diaguila, A5 J, "Experimental Investigation of Free Convection Heat Transfer in Vertical Tube at Large Grashof Numbers," Natl, Advisory Commo Aeronaut. Report 1211 (1955). 6. Foster, C. V., "Heat Transfer by Free Convection to Fluids Contained in Vertical Tubes," PhoD. Thesis, University of Delaware (1953). 7. Fujii, T,^ "Mathematical Analysis of Transfer from Vertical Flat Plate by Free Convection," Trans. Jap. Soco Mech. Engrs, 24, 957-72 (1958). 8. Globe S., and Dropkin, D., "Natural Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and Heated from Below,"' J' Heat Transfer, 81, 24 (1959)o 9. Hartnett, J, P. and Welsh, Wo E., "Experimental Studies of Free Convection Heat Transfer in a Vertical Tube with Uniform Wall Heat Flux," Am. Soc. Mech. Engrso Paper 56-A-113 (1956). 10. Hellums, J. Do and Churchill, S, Wo, "Dimensional Analysis and Natural Convection," Presented at the Joint Am. Inst. Chem. Engrs.-Am. Soc. Mech. Engrs. Heat Trans Conf. August, 1960o 11. Herman, R., "Heat Transfer by Free Convection from Horizontal Cylinders in Diatomic Gases," V. Do I. Forchungsheft, No. 379 (1936). Translation from the German: Natl. Advisory Comm. Aeronaut. Techo Memo, 1366 (1954). 12. Herman, R., "Warmeubergang bei fr-eier Konvektion," Physik Zeitshr.., 3^ 425 (1932). 13. Illingworth, C. RR, "Unsteady Laminar Flow of Gas Near an Infinite Flat Plate," Proc. Cambridge Phil. Soc. 46, 603 (1950)o 14. Jakob, Mo, "Heat Transfer," Volume 1, John Wiley and Sons, New York, pp. 443-450 (1949). 125

-12615, Jodlbauer, K^'"The Temperature and Velocity Fields in the Vicinity of a Tube Under Free Convection Conditions, Forschug auf dem Gebiete des Ingenieurwesens 4_ 157 (1933). 16. John, Fritza "On the Integration of Parabolic Equations by Difference Methods, Cormm Pure and Appiat. 5 155 (1952) 17. Klei, H, Ea, "A Study of Unsteady State Natural Convection for a Ve.rical Plate" B. S, Thesis, MIT, Jize, 1957. 18o Lighthill, M, Ji, "Theoretical Considerations on Free Convection in Tubes," Quart. J, Mechanics Appl. Math., 6, 398-439 (1953). 19. Lorenz, L,, "The Conductivity of Metals for Heat and Electricity," Annalen d_ Physik u. Chemie, 13 582 (1881). 20. Martin, B. W., "Free Convection in an Open Thermosyphon, with Special Reference to Turbulent Flow," Proc. Royal Soo (London) A231 502 (1955), 21, Martini, W, R. and S. W. Churchill, "Natural Convection inside a Horizontal Cylinder " Am. Inst. Chem. Engrs o JT 6, 251 (1960) 22. Merk, H. J. and Prins, Jo A,, "Thermal Convection in Laminar Boundary Layers," Appl. Sci, Research, A4, 1l124 195-224, 207-221 (1954). 23. Morgan, G. E, and Warner, W, Ho., iOn Heat Transfer in Laminar Boundary Layers at High Prandtl Numbers, J. Aero. Sciences 23 937 (1956)o 24 Mull, W. and Reiher, Jo. as reported in Reference 14, pp4 535-39. 25. Ostroumov, GO A,, "Free Convect'ion UJder the Conditions of the Internal Problem, V S4ate Publishing House, Technico -Theoretical Literature Moscow-Leningrad (1952)o Translation from the Russian~ Natl, Advisory Comm.r Aeronaut, Techl Memio 407i (198)0 26, Ostrach, S T "kA Analysis of Laminar Free-o Fwvection Flow ard Heat Transfer About a Flat Plate Parallel to the Direction of the Generation Body Force," Natl. Advisory Comm. Aerona;uto Techo Report 1111 (1953o 270 Poots, Go, "'Laminar Free Convection in Enclosed Plane Gas Layers, /' Quart. Jo Mech, and ApplI Matho II, 257 (1958)o 28, Richtmyer, R. D., "Difference Methods for Tnitial Value Problems, Interscience Publishers, Inc.. New York (1957). 29, Schlichtings Ho, "Boundary Layer Theory.' Permagon Press, New York, 278-279 (1955)o 30. Schuh, H., "Temperaturgrenzschichten' Gottinger Monographien B 6 (1946)o (See Reference 29), 31. Schmidt, E and Beckmann^ Wo, "Das Temperat-ure-und Geswindigkeitsfeld vor einer Warme abgeben senkrechter Platte bei naturlicher Konvection,' Techno Mecho Uo Thermodynamik, 341, 391 (1930).

_12732. Schmidt, I. E. and Silveston, p, iL, "Natural Convection in Horizontal Liquid Layers," Preprint 24 of paper presented at Am, Inst. Chem, Engrs.Am. Soc4 Mech. Engrs. Heat Transfer Conference (1958), 33. Siegel, R., "Analysis of Laminar and Turbulent Free Convection from a Smooth Vertical Plate with Uniform Heat Dissipation per Unit Surface Area," G. E. Report R546L89, 1954o 34. Siegel, Robert, "Transient Free Convection from a Vertical Flat Plate," Trans. Am. Soc. Mech. Eng., 80 347 (1958)0 35. Sparrow,, E. M., "Free Convection with Variable Properties and Variable Wall Temperature," PhoD. Thesis, 1956, Harvard University, Cambridge, Mass. 36. Sparrow, E, M. and Gregg, Jo Lo, "Details of Exact Low Prandtl Number Boundary-Layer Solutions for Forced and for Free Convection, Nat'l. Advisory Comm. Aeronaut. Memo 2-27-59E (1959). 37. Sparrow, E, Mo and Gregg, J. L., "Laminar Free Convection from a Vertical Plate with Uniform Surface Heat Flux," Trans. Am, Soc. Mech. Engrs-, 78, 435-440 (1956)o 38. Sparrow, E. M. and Gregg, J. Lo, "Similar Solutions for Free Convection from a Nonisothermal Vertical Plate," Trans. Am. Soc. Mech. Engrs., 80, 379 (1958). 39. Sparrow, E. M. and Gregg, J. L., "The Variable Fluid-Property Problem in Free Convection," Trans. Am. Soco Mecho Engrs., 80, 879 (1958). 40. Sugawara, So and Michiyoshi, I,,'Effects of Prandtl Number of Heat Transfer by Natural Convection," Proc. Third Japan Nat. Cong. Applo Mech., 315 (1953). 41. Sugawara, S, and Michiyoshi, T1 "The Heat Transfer by Natural Convection in the Unsteady State on a Vertical Flat Wall," Proc. Third JTapan Nat. Cong. Appl. Iech., 501 (1951). 42. Zhukhovitskii, E. M.,'Free Stationary Convection in an Infinite Horizontal Tube," Zhur. Techo Fiz.o, 22 pp. 832-835 (1952). (See Reference 25).

APPEND IX A ANALYSIS FOR THE CYLINDER Equations 25a, 25c^ arnd 25d as given previously are R U dL/R VR =~ (3/ _ 9 i_ b / t-^ tVf, zr - - + _f z r^(25ab ^^M ^-fr^" U t/+o] (25b) g!) t = ~0 (25d) A new set of. variables can be introduced as indicated below U = U(Pu,/6r^ j V- v(PS/aS) f r= vI-R)(jG^P j' = vt( /- 9V In terms of these variables the equations become~ Jfiz;/;t^' (W U) - V'L t v' r1 = 6' e LI a2 /,I / LI/ t / / — T(f(I ri'/' - _ L2 128

-129^ ^ (^,'4 u A _ /l - =yz z)'t-rjr'6i )/i-rj j (c )r PI 6 b From inspection of these equations it is evident that the idealizations made in developing the system of equations used in the calculations are valid only for large Grashof numbers. Suppose that Pr is bounded away from zero which is valid from physical considerations. Then in the asymptotic case as Gr-oeo, GrPr —oo, and the equations can be simplified by dropping the terms corresponding to )P'/e), VU/R, U/R2, (l/R))U/IR, (2/R2 )('V/~e),(l/R )(32 U/f!'. (j/RR) /) and (l/R2)(2 /. Other simplifications follow from the fact that 0 4 Y 1 so that Y may be neglected beside (GrPr)4. It should be mentioned that in this procedure of dropping terms it is assumed that as the coefficient of a term in the differential equation tends to zero, the effect of the term on the solution also tends to zero. This assumption cannot be justified from the mathematical standpoint although it has been used successfully in boundary layer theory for many years. Birkhoff (2) gives a good discussion of this difficulty, The simplified equations are given below,

,130[t + --./.. = ~~o + _ ~ i )U1K y I-L-Z U 1 LoEZ The variables in these equations are the same as those used in the analysis of a highly simplified model given in Part IIC, The functional dependence is given below in terms of the original variables of the problem,? f - 9, ( 7 33 / The Grashof number has been taken into account in the analysis. However, the Prandtl number still appears as a parameter in the prdblemo Notice that the importance of the inertial terms relative to the viscous terms in the momentum balance decreases with increasing Prandtl number, In the limit as Pr->~o the inertial terms may be dropped and the problem is reduced to the form of that of the analysis of a highly simplified model given in Part IICo

APPENDIX B RESULTS FOR THE FLAT PLATE The transient results for the flat plate are given in Table VI in terms of the variables used in the calculations. These results are for the line X = 100 for the 40 by 40 grid. To put the results in terms of the composite variables the velocity parallel to the plate as well as the time step number need only be divided by 10. The velocity normal to the plate should be multiplied by,/.. The composite distance variables corresponding to the values of are given in Table VII. Where there is a blank space in Table VI (such as all the V values for the first 120 time steps) the variable is less than 10~10L Table VII gives the steady state solution after the Y coordinate (corresponding to Y - oO) was extended and 240 additional time steps were carried out. Comparison of the steady state values in Table VI with those of Table VII shows the influence of the Y coordinate. Table VIII gives transient values of the heat transfer group along with Ostrach's result (26). The symbol E is used in the tables to denote an exponent of 10, For example 22.5E-02 is the same as 0,225. 131

TABLE VI. TRANSIENT RESULTS FOR THE FLAT PLATE -n.L..- 6 72.. 8 _ 12 16 2 1 2o Values of u/(gPtT-,V) 1 33.200E-03 2 61.396E-03 19.102E-03 87.92 E-03 43.176E-03 83.083E-04 11.234E-02 70.173E-03 22.120E-03 32.369E-04 13.516E-02 98.274E-03 40.256E-03 98.925E-04 11.911E-04 6 15.659E-02 12.677E-02 bl. 44E-03 19.919E-03 40.999E-0 42.380E-0] 7 17.687E-02 15.525E-02 84.8 6E-03 932E-03 90.41bE-04 16.192E-Ol 14.760E-05 8 19.614E-02 18.349E-02 10.983E-02 8.527E-03 16.094E-03 38.773E-OJ 61.885E-o0 5o.682E-o6 9 21.45 E-02 21.13bE-02 13.595E-02 6.313E-03 25.219E-03 73.900E-04 15.974E-04 23.106E-05 20 36.724E-02 46.494E-02 41. 53E-02 3.890E-02 20.080E-02 11.631E-02 60.513E-03 28.349E-03 15.24E-04 5 0. 78E-07 29. 789E-1~o 58.925E-02 86.934E-02 93. 44E-02 b.714E-02 73.079E-02 5. 226E.o2 42,19E.-o2 27. 27E-o2 7.2.-o3 59.0 978E- 22.077-o5 38.16E-07 2.9-09 ^fc?^ ^: |:^ I K 1^:%^ ^*j|^ ^~~~~~~5.22E-0 42119-0 27g7:2g21-0 59:Og0 2:674-0 38.156-07 24.599E-09 20 1.7E-021 14.7b1-779 E217.3187E-01 18.75E-01.27 E01 1.4.8E-01 12.31E-01 778E-02 1. E-2 37.223E-03 52.706E-04 53. 125E-0 5 38.026E-0 6 19. 060E-07 2^0 17.0L2E-1 30.376E-1 h0.4033-1 47.522E-1 ^2.123E- 16-841E-01 14.854E-01 1263 lE01 7787E-0 018 8E% *n " -0 169E0 030E029.1E0 120 1.75E0-01 172E-01 24.5637E-01 27.019E-01 2.787E-01 15.710E-01 69.047 E-02 62-.91-014E03 13.485E-03 23.209E-0 4 1.181.E-0 16o 13.6 23:626E-01 30-555E-01 -E-01 9.52E-0612.E ~oo ~ ~o~ ~ ~o~ ~ ~E~~~~~~~~~~~47078E- 01 4Z0E-O~ 0E-0l ~6Z2EO01:.02E-O1 26:g -01 14 016.?5E0 ~1E-03 13048E-03 23 9-04 10. E0 3200 17.27E-01 237.26E-01 350.70E-01 7.088E-01 15.281E-01 0.16E-01 6. 08E-01.2*oE-01 26.3E -0101iO 1423.740E-02 73.308E-03 19.3 1E-03 0 240 17.042E-01 30.376E-01 40.403E-01 7.522E-01 52.123E-01 54.590 01 55.288E-01 1. 3 7 E-01 231 -01 18. 18 E-01 550-02 2. 805E-02 18.352E-02 19.317E-03 o7E-01 31.341E-01.5958-01 4 2 53.157-01 55.311.-01 55.616E-o1 s.477-01 45.810o-01 3oo.7-o1 18.ooo-oE-01.ooE-o 417-o 7 8.6E-02 47.87-0 3 320 17.273E-01 30.726E-01 40.705E-01 7.604E-01 %1.847E-01 53.867E-01 517.78-024.331E72.902E-01 3 17.169E-01 30.538E-01 0 3462E-01 93.683015E-02 18.372E-023 ~9E-01 7-337E 1 51-562E-01 53.626E-01 53.885E-01 52.752E-01 435.-012975E-012771E0 6u0007228-0-01 29315E01 51 32E- 529.0E7 5-01.713E-E 01 431 0 2 0 77E 96.112E-02 47.136E-02 19.7 9E-02 55.522E-03 00 17.15-01 30.09E-01 422E. 7- 5.2E 53575E-01 5386E-01 52. 707E-01.292E-01 29.770E-01 17.790E-02 8.178-02 20. OE-02 57.882E-03 460 - 317.13E0 30 L7b-01 0.377E-01 723 -01 51. 07E-01 53 -01 53. 71E-01 52.629E-01.218E-01 29.721E-01 17.733E-01 97.511E-02 48.6148E-02 20.758E-02 59.034E-03 60 17.130-01 30.463E-01 0.3588-01 71.210E-01 51.040E-01 53.473E-01 53.727E-01 52.593E-01.174E-01 29.6718-01 17:36E-01o7 5 o. o 20 6E2:7'30 ~~~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~ ~~1774E-13-457E-02039E-01 7-1E-01 17.436E-01 97.158E-02 48.406E-02 20.65E-02 58.784-0 68o 17.1278-01 30.457E-01 40.3498-01 47.198E-01 51.4258-01 53.456E-01 53.708E-01 52.571E-01 4 144E-01 29.629E-01 17.888E-01 9b.716E-02 48.091E-02 20.5001-02 58.286E-03 Values of v/(gOATy) 160 160 w ~ 69. 138-07 -18.0158~-06.0 -38.81 -1 2 780 5 19.732 0 -2.7K -74.505:g,-10s: — 93.132E-10 -10.244E-09 -11.641E-09 -11.699E-09 -11.772E-09 11. 780 0 200 -18.477E-07 -6913E-0 18 015E-6 -38817E-06 -73567E-0 -12.578E-05 -19732E-05 -2878E-05 -6047E-05 - 11.309E-04 -14.825E-04 -16.791E-04 -17. 710E-04 -18. 080E-O -18.215- 04 -18.2E2;~0 -71.6118-05 -23. 27E-0 -51.829E-04 -95.988E-04 -15.868E-03 -24.1818-03 -34.-03 -03 -92.204E-03 -15. 2E-02 -.938E-02 -25.657E-02 -26.470E-02 -26.821-02 -26.886E-02 2) -71.bll01 - 339.54E-03 -0.15813-03 -3.4510E-02 -139.1 E-02 -521.46E-02 -315.90E-02 -39.8678-02 -44.920E-02 -47.7518-02 -49.1378-02 -49.657E-02 -49.719E-02 36o -32.667E-O0 -97.330E-04 -19.254E-03 -31.602E-03 -46. 269E-03 -63.477E-03 -82.192E-03 -10.217E-02 -lb.516E-02 -~4.2948-02 -30.231E-02 -34.148E-02 -3b.418E-02 -37.514/E-02 -37.970E-02 -38.024E-02 F 40-32.3608-04 -960S1E-04 -19.1108-03 -31.4028-03 -46. 2278-03 -63.2158-03 -81.9428-03 -10. 1958-02 -16. 5198-02 -24.356E-02 -30. 3698-02 -34.3678-02 -36. 713E-02 -37. 901E-02 -38.3o0E-02 -38.4198-02!\-~ 48 -32.118E-o0 -95.785E-04 -18.969E-03 -31.173E-03 -45.894E-03 -62. 767E-03 -81. 369E-03 -10.125-02 -16.410-02 -24. 209E-02 -30.207-02 -3.21oE-02 -36.591E-02 -37.807E-02 -38.20E-02 -38.3 1-02 560 -31.998 E-0 -195.931E-0o -18.901 -3 63E-03 - 45.736E-03 -62.554E-03 -81.099-0 3 - 10.091E-02 -16.357E-02 -24.19E-02 -30.102E-02 -34.09E-02 -3-6.4589E-02 -37.067E-02 -38.1426-02 -38.202L-02 o 68O -31.932E-04 -95.238E-04 -18.863E-03 -31.002E-03 -45.647E-03 -62.435E-03 -80.944E-03 -10.072E-02 -16.325E-02 -24.074E-02 -30.022E-02 -33.9 E-02 -36.332E-02 -37.531E-02 -37.998 -02 -38.058:-0 2 Values of (T-Ti)/(Tw-Ti) 133. 200E-02 2 4.355E-02 11.022E-02 51.763E-02 18.129E-02 36.596E-03 56.711E-02 2J.592E-02 73 483E-03 12.150E-03 60.419E-02 29.531E-02 11.037E-02 28.479E-03 40.339E-04 63.305E-02 33.646E-02 14.458E-02 47.552E-03 10.810E-03 13.392E-04 7 65.641E-02 37.122E-02 17.607E-02 67.569E-03 19.864E-03 19 864E-03 40.391E-04 44.464E-05 8 67.580E-02 40.111E-02 20.484E-02 87.755E-03 30.448E-03 80.999E-04 14.904E-04 14.762E-05 9 69.22hE-02 421715E-02 23.113E-02 10.760E-02 42.054E-03 13.325E-03 32.389E-04 5.44~E-05 9 8.696E-02 58.859E-02 41.666E-02 27.807E-02 17.426E-02 10.215E-02 55.776E-03 28.23E-03 22.014E-04 16.193E-06 58.889E-10 40 4.731E-02 70.012E-02 56.333E-02 44.080E-02 33.502E-02 24.703E-02 17.652E-02 12.211E-02 32.935E-03 34.224E-04 18.382E-05 40.261E-07 47.100E-09 10.837E-11 60 89.128E-02 78.457E-02 68.175E-02 58.450E-02 49.422E-02 41.195E-02 33.837E-02 27.378E-02 13.206E-02 39.704E-03 90.392E-04 15.338E-04 19.096E-05 17.148E-06 10.808E-07 120 91.103E-02 82.316E-02 73.745E-02 65.488E-02 57.631E-02 50.249E-02 43.397E-02 37.118E-02 21.867E-02 93.252E-03 33.339E-03 99.139E-04 24.348E-04 49.027E-05 76.978-06 16o 92.286E-02 84.644E-02 77.143E-02 69.850E-02 62.825E-02 56.120E-02 49.781E-02 43.848E-02 28.660E-02 1l.6o4E-02 65.148E-03 25.655E-03 87.555E-04 25 311E-O 5q.84O.-05 200 93.095E-02 86.242E-02 79.491E-02 72.889E-02 66.482E-02 60.311E-02 54.410E-02 48.8128-02 34.038E-02 19.346E-02 99.4i223-03 46.007E-03 19.090E-03 70. 1286-0o4 1q.9386-04 24 93.571E-02 87.156E-02 80.755E-02 74.498E-02 68.343E-02 62.372E-02 56.632E-02 51.167E-02 36.693E-02 22.107E-02 12.338E-02 63.312'-03 29.6471-03 12 361E-03 37.671-o0 20 93.005E-02 85.991E-02 78.990E-02 72.068E-02 65.309E-02 58.802E-02 52.627E-02 46.8)L6E-02 32.198E-02 18.726E-02 10.450E-02 55.133E-03 26.784!-03 ll 282-03 32.,05~-0o 320 92.632E-02 85.283E-02 78.001E-02 70.8*4E-02 63.920E-02 57.278E-02 50.998E-02 45.137E-02 30.373E-02 17.075E-02 92.908E-03 48.751E-03 23.9338-03 10 26)v-03 30.000-04 360 92.625E-02 85.272E-02 77.989E-02 70.841E-02 63.904E-02 57.253E-02 50.957E-02 4 E.0738-02 30 203E-02 16.772E-02 89.726E-03 46.525E-03 22.856E-03 99.41-04 29 7>4L-0) 400 92.616E-02 85.254E-02 77.959E-02 70.799E-02 63.848E-02 57.181E-02 50.869E-02 4698E-02 30.047E-02 6.567E-02 87.699E-03 44.9848-03 21.9513-03 95.323E-0) 28.1 ~i-0o 480 92.600E-02 85.222E-02 77.912E-02 70.737E-02 63.770E-02 57.089E-02 50.761E-02 44.843E-02 29.872E-02 16.343E-02 85.426E-03 43.129E-03 20.726E-03 986E-o 26.,L0!-04 56o 92.595E-02 85.212E-02 77.898E-02 70.718E-02 63.747E-02 57.060E-02 50.726E-02 44.803E-02 29.812E-02 16.259E-02 84.490E-03 42.297E-03 20.132E-03 85 7078-,o 25.377b-0l 68o 92.594E-02 85.209E-02 77.894E-02 70.712E-02 63.739E-02 57.051E-02 50.715E-02 44.789E-02 29.790E-02 16.224E-02 84.060E-03 41.885E-03 19.821E-03 83 929- 04 24.770E-OL

-133TABLE VII THE STEADY STATE FIAT PLATE SOLUTION AFTER EXTENSION OF THE Y COORDINATE ~ Y4 ~) (xg4)/2 - V x 14 (T -Ti/TW-Ti) 2 20.271E-02 17.127E-02 -10.094E-03 92 594E-02 3 40.5 42E-02 30.457E-02 -30 107E-03 85.210E-02 4 60.813E-02 40.349E-02 -59.633E-03 77.895E-02 5 81. 084E-02 47.199E-02 -98.009E-03 70.714Em-02 6 10.135E-01 51.427E-02 -14.431E-02 63.742E-02 7 12.162E-01 53.460E-02 -19.739E-02 57.054E-02 8 14.189E-O1 535.713E-02 -25.591E-02 50.719E-02 9 16.216E-01 52.579E-02 -31.847E-02 44.794E-02 10 18.245E-01 50.412E-02 -38.361E-02 39.319E-02 11 20.271E-01 47.521E-02 -44.995E-02 34.319E-02 12 22.298E-01 44.166E-02 -51.624E-02 29.799E-02 13 24.325E-01 40.559E-02 -58. 137E-02 25.754E-02 14 26.5352E-01 36.867E-02 -64.442E-02 22.166E-02 15 28.5379E-01 33.55214E-02 -70.468E-02 19.007E-02 16 30.406E-01 29.692E-02 -76.160E-02 16.245E-02 17 32433E-01 26.5363E-02 -81. 481E-02 135.844E-02 18 34.460E-01 235.264E-02 -86.410E-02 11. 768E-02 19 36.487E-01 20.418E-02 -90.938E-02 99.818E-03 20 38.514E-01 17.830E-02 -95.064E-02 84.495E-03 21 40.542E-01 15.500E-02 -98.800E-02 71 397E-03 22 42.569E-01 13.416E-02 -10.216E-01 60.231E-03 23 44.596E-01 11.566E-02 -10.-516E-D1 50.737E-03 24 46.623E-01 99.338E-03 -10.783E-01 42.679E-Q3 25 48.650E-01 85.000E-03 -11 019E-01 35.852E-03 26 50.677E-01 72471E-03 -11.226E-01 30.077E-03 27 52.704E-01 61. 571E-03 -11.408E-01 25.198E-03 28 54.731E-01 52.126E-03 -11. 566E-01 21. 081E-03 29 56.758E-01 435.976E-03 -11.703E-01 17.611E-03 30 58.785E-01 36.967E-03 -11.822E-01 14.689E-0 31 60.813E-01 30.962E-03 -11.923E-01 12.232E-03 32 62,840E-O1 25.836E-03 -12.010E-O1 10l167E-03 33 64.867E-01 21. 474E-03 -12.084E-01 84.5331E-04 34 66,894E-01 17.775E-03 -12.146E-01 69.78 E-04 35 68.921E-01 14.649E-03 -12.199E-01 57.60oE-04 36 70.948E-01 12. 015E-03 -12. 243E-01 47.400E-o4 37 72.975E-01 98.052E-04 -12.279E-01 38.868E-0o4 38 75.002E-01 79- 550E-04 -12.310E-01 31.740E-04 59 77.029E-01 64 117E-04 -12. 54E-01 25.790E-04 40 79.056E-01 51 281E-04 -12.555E-01 20.826E-04

-134TABLE VIII TRANSIENT HEAT TRANSFER GROUP FOR THE FLAT PLATE Time 1 (2 \ 0.02 2,7450 0.1 1.4483 0.2 1o0509 0.4 0.7532 0.8 0.5362 1.2 0.4389 1.6 0.3805 2.0 0 3405 2.4 0.3171 2.8 0.3450 3.2 0.3634 3.6 0.3637 4.0 0.3642 4.4 0.3647 4.8 0.3650 5.2 0.3652 5.6 0.3652 6.o 0 o3653 6.4 0.3653 6~8 0.3653 Ostrach's result (26) 0.359

APPENDIX C RESULTS FOR THE CYLINDER The results of the calculations on the cylinder problem are given in Tables IX, X, XI, XII, and XIII. Table IX gives the transient Nusselt numbers in the cylinder for Solution 1 using the first grid. The integers j in the table denote angular position in increments of t-/8. The angle E is (j-2) (/8'). Tables X and XI give the steady state results for the cylinder. In these tables the integers j denote angular position in increments of 17'/16. The angle e is ( j-2)(/16 ). The variable Y is (l-r/ro)(GrPr)4. The odd numbered rays in the central region of the cylinder are blank in Table X because the rays added on subdivision do not extend into the central regiono The results in Table X denoted as Solutions 4a and 4b are actually the same solution, that previously called Solution 41 but are at different times to show the maximum amplitude of the velocity fluctuation~ Solution 4a is at a dimensionless time of Oo-048 and Solution 4b is at a dimensionless time of 0072o Similarly, Solutions 4c and 4d of Table XI are at different times to show the maximum amplitude of the fluctuations in the heat transfer results. Solution 4c is at a dimensionless time of 0.072 and Solution 4d is at a dimensionless time of 0.096. The fluctuations in the heat transfer rate are out of phase with the velocity fluctuations. The dimensionless time mentioned here is measured from the start of the subdivision of the grid as before. 135

-136Tables XII and XIII give the part of Martini's velocity and temperature data used in the figures and. the dimensionless variables calculated from the data.

-137TABLE IX rpj~Time TTRANSIENT NUSSELT NUMBERS FOR TEE CYLINDER t~ 7xi2 3 j=2 3 4 5 6 7 8 9 0.4o 0 15.699 15.699 15.699 15.699 15.699 15.699 15.699 0.80 0.2460 13.181 135.181 1.181 135.181 135.181 135.181 135.181 1.72 O. 180 10.859 10.105 9.967 9.837 9.707 9.593 9.583 2.55 0.172 10.299 8.984 8.680 8.412 8.109 7.821 7.668 35.50 0.170 10.197 8.429 7.909 7.505 6.996 6.438 6.oo003 4.63 o0.185 10.5328 8.272 7.565 7.036 6.367 5.498 4o574 6.07 0.252 10.525 8.542 7.531 6.943 6.238 5.189 3.678 6.97 0.5 347 10.617 8.411 7.580o 6.989 6.5312 5.277 35.594 7.71 0.483 10o.687 8.463 7.623 7.035 6. 84 5.541 35644 8.41 o. 685 10.755 8.504 7.659 7.074 6.442 5.492 35.741 9. 11 0.990 10.838 8.543 7.690 7.105 6.487 5.572 35.835 9.83 1.442 10.956 8.585 7.717 7.150 6.519 5,630 3.924 1058 2.100 11.14o0 8.640 7.743 7.152 6.543 5.673 4.009 11.42 35.027 11.451 8.729 7.778 7.171 6.561 5.705 4.1oo 12.41 4.250 12.011 8..904 7.839 7.198 6.578 5.752 4,216 135.40 5.386 12.786 9.193 7.947 7.240 6.597 5.75 4.527 14.42 6.293 135.722 9.631 8.135 7.320 6.631 5.781 4.439 15o 61 6. 984 14.765 10.279 8.478 7.490 6.711 5.828 4.555 17.01 7.400 15.639 11.024 8.988 7.800 6.887 5.927 4.675 18.54 7.574 16.138 11.603 9.512 8.201 7.166 6.108 4.819 20.51 7.579 16.331 11.911 9.890 8.572 7.486 6.361 5.o14 22.17 7.465 16.330 11.973 10.029 8.763 7.695 6.565 5.196 25.86 7.331 16,255 11.934 10.026 8.796 7.757 6.647 5.287 25.28 7.232 17-171 11.876 9.984- 8.777 7.749 6.654 5.306 26.55 7*165 16.080 11.819 9.934 8.730 7.718 6.633 5.293 27.74 7.122 12.028 11.767 9.884 8.684 7.679 6.602 5.265 28.91 7*099 15.973 11.722 9.834 8.635 7.637 6.565 5.2352 30.07 7.089 15.930 11.685 9.788 8.588 7.594 6.527 5.197 31.26 7.056 15.899 11.654 9.748 8.544 7.553 6.490 5.164 34.09 7.122 15.870 11.617 9.693 8.478 7.484 6 424 5.101 36.58 7*160 15.882 11.617 9.684 8.463 7.464 6.401 5.076 39.12 7 190 15.908 11.631 9.695 8.470 7.468 6.402 5.075 41.72 7.205 15.95533 11.647 9.713 8.488 7.482 6.413 5 086 44.536 7.208 15.948 11.660 9. 750 8.505 7.498 6.428 5. 099 47.05 7.204 15.954 11.666 9.739 8.516 7*509 6.438 5.108

TABLE 1. RESULTS FOR THE CYLINDER nmr *J=2 3..L.j 6. 7. 8 9.j 10 11 12 13 4 1 16 Values of U for solution 1 0. 0. -. 0. 0. 0. 0. - 0 40.OOOE-03 40.2'liIE 19 92.25E-01 -191 725E 00 -20.749E-02 0. 86.873E-09 52.970E-09 21.01OE 07 4.9E0 80.000E-03 9 5,E -"0 21117E 00 -41.459E 00 -24.165E-02 0. 13.326E-08 80.353E-09 27.203E 07 ~ 5.7E0 12.OOOE-02 1' Li E -'O4 _ 2_ 512E 00 S4.9E 00 -37. 047E-02 30.574E-10 71. 784E-08 45. 157E-08 5 9E01.1E0 I6.OOOE-02 I 0 1' E -1 - 23.9iE 00 4431E 00 -81.42GE-02 40.765E-09 52. 568E-07 35.557E-07 41 81ES u00.18E0 20.OOOE-02 48.00GE0- H _____ 6"83E Al -4495E 00 -19.903E-01 40.082E-03 32. 088E-OG 23.262E-06 2:91 O5E-0524.OOOE-02 I" 8- 4 73E -'OH'.7.912E 00 -'4i. 9 0 0E 00 -42.807E-01 32.331E-07 16. 250E-05 12.611lE-05 1 3.b52E "4 1.1E0 28.OOOE-02 99. ID iE -HJ2' ii0 iE U" -44.07SE-00 -75.911E-01 19.943E-06 70. 678E-05 58.546E-05 5'6 751E 04 S31E0 32.OOOE-02 19 5 57:'2E "Fj1 34.41E 00 42 2E 00 -11.280E 00 10.526E-05 27.134E-04 23.L'57E-04 21 A078E-"035SE0 36.OOOE-0235 34'!7 193 "'a5E 00i3 72E 00 -14.653E 00 49.570E-05 93. 836E-04 88.207E-104 7' 35.S4EE03 40.OOO'E -0 72)315E- 4 4 "fl7 E 00 - 691 E 00 -17.409F 00 21.214E-04 29. 667E-03 29.67(E 03 E-01 E-0 7 44.0lOE-0 3 4E0 51.liE'2"-7(S.G E rai -19.644E 00 83.357E-04 86. 725E-03 M231E- I __________ E 0 4"00H 22'I.' 50E00' 1 E" 00'7.61E 10 21.456E 00 30.295E-03 23. 648E-02 2-u 793E-'r2 *62E Hi 9 0 9:2. H'10E -1 l-,3.41.EO,7I.2 1 E: O' 957.74E-IDl -22'94fF 00 10.242E-02 GO. 569E-02 773 050E-0'19 -1 2 7-E Hi' 00 -Sis r OE0'.9 EC ""'13.134 H4 0'j L458E-'ii23G72E Ou 32.352E-02 14.654E-01 18.79 7E-01 1'-. 2OClE -0 1 0 60.J HEH0 GS.C 4 1 8 ""5E'0' 244E'il'0 -2.70E 00 95.749E-02 33.647E-01 45.75E- Fri EI'11 L14. 00E02 941.31iE O' I I. G7 E " 1 155E0 2.75E 00 26.58GE-01 73.630E-01 1u 0 5'2E fly 7 EF --' "'91''0E0 1"~.53 1 3 11E17 I 1'If j, — OI99'r'0 "+ 1'~EO021.80E'ru19 16544E 00-48. li1SE-01 69.1i9E-01 11. 163E 00 15.407E 00 19. 1009E'' "' r 1',9A 21 6O E0 7n rA'17-'-,1: —.4E,11'f. 7EL 5. G," " E 1 1 2'.'S4 E 0 1';95 3 477 E 1 "0 2 47' E 0 -159 03E 00-39.544E-01 87.670E-01 14. 319E 00 19. 731E 0 0 24.910E 00 3a 178E 01 49' I E OrO'""1'a" D 1" 090 "'"'10 0E-0 7 - 1"2 "" I I "2'7-E " 1 1 5 2 E O 10.0779 0 1 29 1579 00J 121 10'G 00-15.998E-01 11.929E 00 19.064E 00 25.970E 0 0 3.3.OO1E Arj 40.407E 005rr 9 lE " 0' 4""'E 0" ii 11 0 5 F0 74. O~ A H C i7FIE -1' 7OI1 755E - -' 1 OlE 0 1 1i 1.-5 O I -.081"'10-68.079E-Al 29.997E-01 17.117E 00 26.1illE 00 34.895E 00 44.185E 00 54. 3-1IE 11 — e 7 7' 5'1'1r'1- 4 3 EO 13r a 0 - H..CC E-O2I'''''E 0 1127EC 1.'22O - - 17 423EO11'2.0 EHII 4G.017E — a 34.6E-01 10.845E 00 25.302E 00 36.405E 00 47.481E 0 0 5194599 0'J 72.?33E O " 199 Il" l 5F 1" 707" "E0 7'-.O010F-F+ - -CE01 2 - 1 97-1, 01 1" II' 1i E C 1 i2''"'~ E C11 6~ CC" G2E 10 Ci18'F0E 00 23. 133F 00 37. 714E 00 51. 121E 0'] 64.59.3E 00 79.951E 00 96- f-"4E 0r3 1 LI f''1O 1 5 1 2 EO I' 1"'914. 1 "'E 0 30.00 -,E -I- -7. E01112 C__ 2.729E0 3 - ____ J1f0 "'1 77. 53 F1' 3'935E 00 41.096E 00 55.757E 0 0 7 1.'fOOOE 00 S8S.390E 0 0 10. 681E ul1'S0al 1 -79 1 1 725EE l E 0 E -02 7- 0 1 -175-4E 2' 4 0- E'1 2017.7i C F'1ID1 1 "S'"3'9'11 I" 95.2 E 10 97.509E 00 11.369E 0 1 13.46.3E 0 1 15.768E 0 1 1S.251E 0 1'0- 939'E 0i'-E'12 7 O " E 0 uH O2 C 2EI077-) E 12 4. "1 l E Hi'ID. 3,0E'11 1'7 "-l E''I 1 4 31. —E UI la 1 97F6 01 13.594E 0 1 15.430E 01 17.783E 0 1 20.350E 01 23. 04.3E 0 1125. 1 "E C,"f'11 0Si'W"E 01 liE H~ " E "i 1' 147 ul IL6.712F 01 17.914E 01 20.0'65E 01 22.653E 01 25.382E 01 28.121E 01 30'47E "i'+ "'' 0'IFI 112. 2'. 1OE0" 11 "''"' OE ID1 1'r453E O' I 4 —j6 E'J "H1 t 47357E 01 22.303E 0 1 24.793E 0 1 27.535E 0 1 30.26,3E 0 1 32.807E 0 1 34.'40E Oi E 3'P 2. F, F-Ii0iE -iC'''E 7i "i 0' E3011 ESU E 01 26-11.E 01521 E35E 0 1 31.457E 0 1 33.944E 0 1 36.004E 01 "1 37 0E'I "'E'1 ""'E5l J4. EI:E 1 2!' iI 7 E -I I,::,.H74fE ID r 1'1Z I7 E Hi'' 22'E ul'S W07E 0 1 27.799E 0 1 30.482E 0 1 32.904E 0 1 34.868E 0 1 36.1'39E "11 36. 41lE O 1" "1 E''i 1 "" Hrf0 0H'E"C 1" ".,74E OHI 1'1 -7"FE'.G-3E11 1"'0 - i E O I SHi.G E Hi 1 23.28 E 611 25.7S4E 0 1 259.01OE 0 1 29.801E 0 1 31.008E 0 1 31.441E Al1 30. " I"E CF "' i':[-E 1i " 5'F'I 0''E 0 "'HE ~ F4 72, - E-''0 A'E 1'C'II 12"'E''1I 1 2' O'"-'E 0i 14 al ID I 19 3E 1 741 0 6E 0 19.G65.E 0 1 20.0S4E 0 1 19. 539F I i 19.1199'O 1 7'11''1" 0 1:1 IrrI' 01 LII Fr 1.O.0 0. 0. 0. 0. 0... Values of V for Solution 1 0. 0 0. 0. 0. 0.~ ~ ~ 0.'1 - 4I. OH EF- 1. A 0. 0.F. rio".111E03 7- rC 7' F E l 1.04E'11 I 20."Sif 0 1 42.919E 0 1 2 lE 0"" iE'11 "C, C HIE -2 14.258E 0 1 4_____ _____EO I4 4 E Hi1 ii''a72E 0 1 28.600E 0 1 -10'C7'E 0 1 2____ I'''E''1 I1'. - -' IDE -'''" 542E C- C,' - 3 HG.Ci2 E C''1'-4 "'5r-E''i 77 - i3. 13E 00 21.431E 0 1 1 4. hOE 01 "''I O'rO'"10E',-7-'' V EOf H'J 7.' 7 9 0 1'3.0 E'1 4"'78E Lb 17.1lO9E 01 11____92____ E 0 1 ___ 2 " E 0'1 I' 4a -1 OE -HO 99' 5 413E'" -22'' 10E ID1 IS315.5E''1[ 30.193E 00 14.1i9lE 0 1 990 "W7E 00 1 H'""fu3'HI -"'SE "' -I'"S1 -E Il 1 2'3E0'1 17. 84E 00 1.5E0 9SE0'-"'f'~9 or'1 9 129'1 0'1 2 07E-0 120.95GE 0 1 45.1'4E 00'HE -2, -OE-02 1. 7,7E I C, 15.1997'11-4 19G3'17" 7."4uE-01 77453 0091CAE 00 2E u 44 C'' E"' -.C~2E -1 2, E011 R E9''11 I ___' 20.1OE-01 70.11SF 00 6654_"A E 007,. 14EO0 4"'7 H"'1 ""'E Hi -ii "''S'ii -79 -7-'E-01 62.074E 00 40 3E00~~~~~~~~~~~~~~~~~~~01 9. 93E00 -4'r"Hf H' H "'Sf H''0 or Sf~~~~~~~~~~~~~~~~~~~~~~-7E Al "" "H 01.94E 0 " 64'-E Ar 49 Sr 7 7Er O'H if E' )- 5,EI -1'-1''-F il __- __ ___ SR F'' 3.7E 0713.' "a94w' H E'' CIOOE-aI2 24'1 9" li: 2SE 1161 - 934 iF', 01r IH 1'ifE61 70.154E 00 iF4.0 0'' G r ~ E" "'''HOOFC I, i " 2' ""357 E' - OI a "f H'- rE 4 —1 HH",f''-1- 0 if OH S 774fEA"17 E0 62.074E 00 3.74A0-'4973F LIH ""DEf'r' 1-'""'. 0 -1 -'-Cl2-iE'HI-7 E' O' 7. FiE D'-"045 10 4' 11' 401 1 62E~f01 31 5E 00 5.6019E 00 36.5E A 9 1"'lE''I "1 0"r 74Hr - "''"'7 I 5 0 E C-2, 1 C'C 5,E O' —I4. 3 E D 0'34.195'0 "'fAaS AE A 0 1"4E120001 4.944E 00 34 70 42.3" E! 4 5iF YH "'a"'1 — O 4f' l' "'41 H"' H"' "'4720E6 5 6 272E003237E009" ID 5 G.' ""'H 2 1"4'iE4 16Sf5'11 4 349 09.471E 00 30.1 SE 90 318if 0' 9 F"'1 i'r"f'FO1 "'''r'llf'~'+f On'~-f 1''''4 "'19 ii "''lf 1 E01r" "" "' r14 0 7.0790 E."'IO'E'''0 01""'1 H.4 - C'rifE'102 - 214 -iF'4 E 0 47HI 27 1 "'E"f'iIi"1 3'81'L-fH~'_6- i"a"S 0".1524E 0 0 3614E 0 47 +1Ef 001OE.-' — E"' 7 - 2-2G 3-5E 0 -3 -9 7 1' "'-,,E 00-4r'1: il E'~ 0lI -' SI —. H-,-, i- 11E 0i 610'7977 001 11 7.294E 00 22.974E 00 22.3 4E A00' 33 43GE H O 12"' 8rl rE r 00 f1 1 -E 1 4 "'4 FF107'1" 9~~~~~ """F Fl"'''' f rirl'f-'l S- 0 —f Ill i"' ""'"f 00 17 76C~~~~~~~~~~~~~E 00 lS.S7SE 00 8E iS34.299E1 00 171 70E1OO IrC-1 E D02" E'''' 4I "O1 lOF02 2HI2I HEf5E1'F-'3 2 "'S 5 2r f "1 - 37-4697 10C6.0.5 E 00 7 —077E 00-S17 E lE 004 1332",'E13OH65-01E-1r754 00H37-"l 5"'rrloE 2 " O' -''4 91F T C 21E 4O 02 EI,-47'1 ill - 041 0 5.1" 01"'f020 14 024E 00 34.179E 0i0 34.373E 00 102.47'E LOu D 9 3 i -.'15 F-ri J j 17 Ir~rif r:r 7~'71 19 E OI 5O -".4CE0O- -7. q 9'E" " 0 f 01C 49 7 4 5E01 I s 26.6 E 01 32.722E-0i 32.314E 01 0 41 950E ul ii- 2 Or" 01'9' Jrl "1,- 74 rarlE-02-7'7-1 E 9 - -).O44EI O" -2 31E 11 97 91F-rn' i 0059-01 60 7"1E 01 9622.46E 01 27.737E 0l 27.741E 01 11 "'lI) E UlC 22 3 r Ej 6i 1 llF i "6 "'09E 0' C'4 " 1 991D4D5,EFF 9 11 E "-1 5 l4'lfE 3!- 1 4 F,:'HE 131 4+ C9979I2 0 1 7i78E0 ~ 06 01 25.152E-0lo 25.144E 01 58203SF 0 59-.:OIF, 1 -14. S E1-"'S-i'"fi 14 OIr 0E'102 1. 2,3.O4: D- 1 9'113 1" I'E'O"F 4 7' 74 37SF Fr' -'"i 1-1'' 1'7 1 2E-0' 5i:.0-71 0" "1 2i2- 7E 0"0 175 58E 02 17.279E-020 14.29SE 02 71 187E 001 2.448E 12'A 7', F8 I' -I'1'59 - 10. rrOE 1 0 0.D —E 0. 0) II E 01. SE- I Fl Z`

TABLE X. CONTINUED r- 2 3 7.i i. 8. 10 1112 22 1 V alu esg Of U(Pr/Gr)l for Solution 1- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0...0.0 0. 0.0 24.590E 00 42.095E-0 8 98422E 04 -21.044E-03 -22.137E-05 0.92. 682E- 12 5.1E1 24~ 04 8E0 2356 0592'9SE-08 22 5298 03 -44.231E-03 -25.781E-05 0. 14.217E-11 85.726E-12 29 022E-1' 0% 22.541E 00 20-284E-67 24.017E-03 -46.194E-03 -39.525E-05 32.618E-13 1 76. 584E-1I1 48.176E-11 90.887 E-10 4 21.518E0010, 25.515E u03 -47.327E-03 -86.871E-05 43.491E-12 56. 083E-1IO 37.935E-10 53.158E-091 48 I 20.492E 00 51.1 lE-O0 27.400E 03 -47.965E-03 -21.234E-04. 42.762E-1 1 34. 234E-09 24_. 83 9 E -09 -29 984E-08t8 L 19.4678 00 19 L45E-0 29.7798 03 -47.946E-03 -45.669E-04 34.493E-10 17. 336E-08 13.454E-08 1 4. 4698 079E0 18.443E 00 b' tO21E-fl. 32.770E 03 — 47.025E-03 -80. 987E-04 21.277E-09 75.399E-08 62.461E-08,It 9458 07u 17.413E 00 i6- r-4E-04 36.530E 03 -44.917E-03 -12. 035E-03 11.230E-08.28. 949E-07 25.55SE-07 li 2E 0, 01 16. 394E 00 30.2'40-E 04 41.51E 03 -41.322E-03 -15.633E-03 52.885Eo8 1001-94.105E-07 95 51tE-U06-4E 1 15.369E 00 7.190E 04 47163E 03 -.35.944E-0.3 -18.573E-03 22.632E-0 3161-631.661E-06 2' 9 771908 0 14.3448 00 1 3-'l22E -0 3,5'51E 03 -28.481E-03 -20.957E-03 8892-792.5258-06 q'S.487E-06 -9 I''-u1 13.320E 00 2 I-9E0 63.672E 03 -18.632E-03 -22.891E-03 32.321E-06 25. 230E-05 28.584E-05 17 340E-L'+ 12 295E 00 35 -64GE-0-i 74.913E 03 -61.53E 04 -24.374E-03 10.927E-05 64. 619E805 "7' 935E0 -' 278 -04.11.2718 0 E0 38.97E003 90.Lt242E 04 -25.255E-03 345 -515. 634E 040 E 10.2468L E 00-356EO7 0T 507E 13J 26.0-74E 13 -25. 146E-03 10.215E-04 35. 897E 04 4-. 1 I1E-04 I't0E5 E- -2.214E-01 1005-2 44 "5E-It2 -23.21 1E-03 28.364E-04 78. 553E 04 1 H 2-liiE- 03.7 8191 698-ul 1 -3-4 -2- E -HO 214. U7 2-E -0 _214.71E C 10 15G,8Ei 02 614.7E-0 23.2. E-03-17.650E-03-51.337E-04 73.826E-04 11.910E803 16. 437E 03'0.377E-0 )2- 4. 3. 8 7'.'38 H1. " 3'-.i: ~E76.985o 014. 1.E 21 - 0 E'-1 C' 2I H' iL -53_E U' 1 -.3 1E-u - 291.0548 u3 21.945E-0.3-16.9G7E-03-42.188E-04 9).3.533E-04 15.277E803 21. 050E 03 -'C.576EI03 " 219GE93 E 0 1'4 G P - -? 01'2EO. 1' 0 E0 18.'798 02 195'7 308E0 lH'0.7o 22~81-03-13.769E-03-17.068E-04 12.7268-03 20.338E-03 27. 70GE-03 39 207E 03 43.~10)9E- 6-7.l4)-Q G G -5 9E -0 11 7 E 0 2-'2 1.O I0E -It'- 217. i"'E 0', 1 13 "0E 0C'.36.364E-03-72.632E-04 32.002E-0 18.261E-03 27.857E-03 37.22'E 0' 4' 140E-03 5.7 42E —,I F -i 1'-C.''2~22iS3 221 9' 5 E It' 1"8 2. "E- 0 21' 2'HO i E' - 41) i'uE-03 36.5548-04 11i.570E-03 26.994E-03 38.839E-03 50. 656E-03 63.43.5E 03'77704E-A IQ lE' -,290 Li 11 iE-0 47G17 H-2 3 4'E-2 I- 1) 2"'E0 3.- lE I'0 64. i9E-03 20.1057E-03 24.680E-03 40.2368E-03 544.540E03 69.23'448-0' 89 2' G80-03 -1" 0 *-' —t - 9~~~~~~~~~~~0. 57 021.'~5E 02 10.341E-02' 69.21 1E-03 70. 133E-03 86. 185E-03 IO.577E-0'2 12.708E 02 195.035E-027 1 7. 3- -E- 02' * —n198 q4.91q4'-01 3.17 "E - 3 "8 13E- j'' 25'. G( 4E - Ii 2 "l -7-E-0~' 17.n05E-0' 1' 67SE-0'? 10.1598 02 10.403E-02 12. 129E-02 14.363E-02 16. 822E 02 19. 47'8)-O_ 20 2 -)3E-1-22 F 3 0' 4 E44-Itf' 01 -8 L E - 0 7 EH -'9 i7 E -0'2 21. 90E02 I' 8 1''18- A71 1. 2IH 7,1E-, 13.44~E 02 14.503E-02 16.4628-02 18.972E-02 21. 71 0E 02 24.984E0 C'. ". "' 3E 1 —-l q4"- - 1 -7~E-C 1 3. i'4-07 92 E-2'4 )3E02 2'' S7E- 1' 1 ____ E-CQ - 1' 045E-U2 17.8'"08-02 19. 112E-02 21.407E-02 24. 168E-02 27. 080E 02 30. 0018E-0'-' 3- 4-H3E-!- 35.1E-j23 3G E-0 HI5E- 31'2 H - C2 7.0 1 GE- 8-U 23- 5SE 257EO?075"E 0''0 9'.''0.'99E8 U 21.8028 02 23.794E-02 26.4518-02 29.376E-02 32. 2878 0' 39.0 Q~E-0''7 78 277E2 -2 14E-C 2-3. 492E0 I'21 GE-02-2 4099E 0' 210 171E 2'1 1 08 49 0'' 23079E 02 25.1448 02 27.752E-02 30.657E-02 33.5618-02 36. 214E80 38."411E2 39_9 302 41 ID'H- E -'0'2 R 5.8i:E -1'8i 9E48 1 * L4 18.- 4iE-02 19 G02'E 0''21 5E-C'E 23.99E 0' 26.748E 02 29.658E-02 32.520E-02 35. 105E-02 37~.1 99-02'38.58 927Eb-0'2 3_8 39- 7 80-CC'HHE-H'l.~ E0 H2 102S-1Iri, 5'2E-0H 14.119E'8' 14 7098 0' IC RE 70 —1- j 9.t241E02 02 4.'41E 02 27.5098-02 29.8184E-02 31.794E-02 33. 0818 E0' 33.544E-02 -32,7 -4 6E -ICr 31I 1. H-'=-"~E -0'' 5I.)E -. —4E- 11 5"-lE-1"' 89 412'E 03 10.-05E-02 1 -2.iE a2 14. 99E-0U' 16.1148 02 18.628E-02 20.01680 0.8E2 21.43SE -U 22.2 2E-7'I''3E-02187 —-Er_.I 2 5E- 0. 0. 0. ~~~ ~~~~~0 0. 00. 0. 0..0 —----'Values of v(pr3/Gr)-, for Solution 0. 0. ~~~~~~~~~00. 0...,1.9008 00 0. 0 0 0. 0..0 2.6E00 G2 —'-3E-O'21188E IJU 26808 00 55.3418-01 11.7298 00 81.4898-01'245 9E-01 J U 0' 32. 0638 0 1 I 348 Ill ~~~~__ __17 C 178 00 31.9798-01 78.1568-0~1~54. 3268-01I 5q0 I46E-01 251E00 27-203E-01 984'98-0 -13058 00 20.1718-01 58.5668-01 40. 7458-01 44'9'5EIi -'0 4"E 00'0 0'( 0 U4 8 Ii __-2_ 28 00 13.0298-01 46.7558-01 32 568-,jI_5 2-3-01 I'9 0-Hi946E00 19 1438 01 C0 42iE-01'-3.4478-01 82.5118-02 38.7818-01 27 163E-01'5..0 Hi E0-Il38 H 18.443E 00 I1I1 5"2EO 0 495E-1Gq74E-01 4 8. 5~998E 0 2 3294801 2838-01 25 3'5E01 it 41 8' LIII 7f t 02 -41 44'8-01 59298-01 23.7838-02 28.4178-01 20. 3728-01 22142E'8 HI 16.94E00 2.95E02 35.7AE01 50737-Ol55_4_E0324.758E-01 ___ i 108E-01 __ 119I 08 Hi 1 1E- CI HI 1. 00 4' "288 —' 030.108 01 43648-01 -76.3708-0.3 21.7248-01 -1.278-01 it 7088EIi 1~'S 7-_ 14.3448 U00 93 53E 02 -26.1888 01 -37.6t0iE-01 -16.4958-02 1911-01 14.8148-01 __16.0~186Ei — 1 40-Hi -1 132E00 58.31'8 03 -22Pt 4E-11 32658-01 -21.5298-02 16.9648-01 15758-01 14.71880 IHi 4 4 _.E-iI 12 058 00-11'4'E-0 20 0908 01 428-01 -23.0008-02 15.0558-01 12. 519E-01 1.F2'2 C -.8' II - I 271E00-27.888E-02 17.6368 01'2 30008- 0121.07G8 0' 13.3758-01 1.978- 01 1 2 4"2 2E 01 C'I- -I -' E-C, It O.t24E 00-43.8958-0A41 95 474E- 01- 108 18: I EI -16.00 8E02 1.8808-01 10. 7648-U1 II - 31 0 Ii I C;1' 02.214E-01-58.684E-02 -134488 01 -14.8118E1)1 -8' 285E 03 10.5288-01 99. 6668 02 10 15 8 E-9 4'3.'4 4 -,91 908E-01-71.2>58E-0' 0'928518 0'-11.4448-01-11.1938-Ill1itt4"'8-01-5 3.66E0- 16.'438 03 47.2608 02 92.8358-02 92.1ORE0' 91.374E827 U 91 L'7E-' 7. ED3 - 25 71,.8458-01-61.8848-0 -o-8.864E-02-10.608 II-011 311'83001-19.7 E'-ti- -.607E-01-3'b796E 03 8b 768GE02 10.275E-01 99.0038 0'2'13 709E08 0'44 -240-02 9. 548 — 2'7 2 lo - 71 ~ 1 —.4 -'8 3. ii'06E 1 13 7'E-It-i0 1'E ul13.7198 03 79.590E080 96.1378-02 939'8' 8 3-' 0 60 0,i 66 RJE-01-79-.l1078-02'.05E t- 02-.0.9038 0' 04. Ct3'E 0' 10 3"tt58E-01-77.3198 0' 67.'588 03 72.8548 02 89.4228-02 88.3078 02 8 3. " 1)7E -0', 4.'108-2.5 I252. 814 4bE-01-86.581E-02-88.'77 8-U —',1 9.00 OE0'2-771308EC0'"' 03, E' 9-3 ti18E 0' 12.6708 02 66o7438 02 82.612E-02 82.325E 02 77. 368E8 02 G 8.742E-02'17. o81E-27 4-I E-: 56.'383-01-92. 156,E-0) —85.34E-02-71.173E 0'2t 0.2398 1' 94.1998 0'-31'498 02 18.8398E02 61.392E 02 75.7178-02 75 _ -8 118_- 0' 270. 7558-0 —-'1.256E-t 4-9,'2 48E —0 2: 51.2- "08E- 01 -95. 13 38E-0'2 8Lt, 7'tE -0 2- 50 99 548E 0'2 4 3.C L0 4 E1112 3'. -0 C,2 1'2 7'48 02 24.804E 02 56.782E 02 68.7368-02 68.712E-02 63' 416 E02 54 11IIE 02 4 0 S8-'RE *1 2 I - 46.1078-01-94.9358-0' 72.913E020'46.918E 02 27.674E 1' 14.07E,8 31..22E 03 29.9428 02 52.6518 02 61.6358-02 61 -01 48 E0- O2 55.336E 0 45.5SRE023.3E;2IC I —1 22 40.984E-01-91.156E-0-1-oL-6.88E-0 —33.7728E112 13.L8OE-0 It i 83"E-0.3 15.3298 0' 33.5798 02 48.9498 02 54.325E-02 52.'418-02 46. 5908-0' 36.3698 0''1-2 I.4 t'E —' 218-I -—'1 143I: C -2'- 7 35'361E-011-833.633E-02-51.'098-02-20.8698 02 59.9364E-04 12-' 6-5E-O' 23.48',E-02' 35.1058 02 43.940E-02 46.6688 02 43.0748 02 37. 3768 0' 26.9078 0' I I -S9E- "08 1 —F- --- 30.738E-01-72.533E-0 -~-3.32318 0291.9448 0'31 81858E-3 1 5 8- 21 2'_4408E812 34.0848 0' 38.3498 0' 38.5478 02 34.8808 02 28. 0438 02' 17.7'2'2E -0'2 33 C, 140 1'I'i'7 3' -, 565-01-58.4618-0' 29 )54188-0-4 19 2098E04- 14.7018 02 " 2672E 022 RE0 03' ~3 380 9'1 25297t3E-02 19.1158 0- 95.C1-i' - - 8'0.491lE-01-42.559E-0' ~13.8958-02-CO.537E8 tt1G._-778-0- 21 -2'8-3E-1 2-498E 02 24.285E 0'"'36448-02'1.253E-' it, 1'9825E02 II'78E-0 2 IIE 0-;I22 -',E - — 5 19.369E-01-26.56281~-02-592 9478-03'8.587E 03 13.810E-02 it,270E-0'2 16.'94'E-02 16.59cGE 02 15'108 01, 1 2 9918 E02 9b. 5788 03 5'. 9048 03 49.819E804-7'3..-,'~E-i- I3 1 5 "'8 1'2 1 H I 10 24CE:-01-12.699E-02 58.4948E134 98.8938 03 84.5768'3'93. 0:"2:E 0' 0' 0'EIt' 86.87S'E0. LI 42E- 61.082E 03 41.04'8 03 19. 8GOE03 L15i 11 8E03 90. 7GOE-0''9I Ctr18-! -'8 I0'7 —ISE 51 2308-02-32.9.378-0' 3 0480 039 8i'-.-' 78U 4IREt10"8 ~1 56'0 380 944 L4( 380 L t 00. 0 000 00________________0 00 0 0.0 1 i

TABLE X. CONTINUED r/r0 J=2 3 __ _6 7 810 11 12 131 16 1 Values of or 3olution 3 0. 0. 0. 0.0.0.0 40.OOOE-03 0. 0. 0. 0. 0. 0. 0.0 80.OIJOE-0J 0.0.0 0. 0. 0. 0. 0. 0 12.OOOE-02 0. 0. 0. 0.0. 0. 0. 0. 0 16.OOOE-02 17.752E-05 3.6E00-463E0 1.24E-0. 0. 0. 8.1E-17.5-0 20.OOOE-02_36.986E-05 88.996E 00 -.99 01-135.48E-08 0. 14.94E101.98E1 273-93.8E 24.TOT:0E-02 5. 400E-05 -1.095E 00 -15.999E 0 1 -63.761E-;06 0. 14. 94E09 16E —I09 91091E-118540E0 28.000E-02 22.573E-04 71.220E 00 -81E01-637461E-06 0. 18.649E-09 21.67SE-09 14.217E-07 257E0 32.OOOE-02 65.628E-04 -9.403E 00 -19.133E 0 1 -9.1E0.-0.198-116493E-09 22.508E-08 13.8629E-07 562E 36OO-02 63018E035.2E0 2.0E0 7.9E-03 -83.963E-10 12. 410OE-06 15.310E-06 68. 928E-062018-. 40.OOOE-02 63.005E-03 ~~~~~-0.395E 00 -20.966E 01 -51.877E-02 -82.567E —09 74.613E-06 93.733E-06 38.080E-05 6.0E0 44.OOOE-02 19.611E-02 -13.777E 00 -21.659E 01 -26.232E-01-7.3E0390E05027E51'6E049 1E2 48.OOOE-02 59.410E-02 ~~~~~~36.184E-01 -22.058E 01 -99.425E-01 -56.796E-07 18.878E-04 24.584E 099E04 97 950E-04 *52.OOOE-02 17.196E-01 21.786E 00 -21.943E 01 -27.434E 00 -24.376E-06 83. 975E-04 1 3E0 7TR 31'E0 56.OOOE-02 46.807E-0i 43.458E 00 -20.943E 01 -5.3410-2.0E-63182-0 i04E 03 3746795EE-0 64.000E-02 27.84E6077E 00 00-87E01-80.297E 00 79.043E-05 13.617E-02 18 685E 02 57.893E 02 E 64.OOOE-02 27.467E 00 ~~~~~~10.578E 0 1 -14.380E 01 -99.741E 00 10.221E-03 50.100E-02 69 975E 02i 20772E-01 ~ CE0 68.OOOE-02 58.657E 00 10.606E 01 15.347E 01 36.969E 00-79.534E 00-96.455E 00-11.338E 0-56.645E 00 88.040E-03 91.516E-02 17. 423E-01 21.103E-01 -4.7d3E oi 4,. 477E-01 70.17IE-01 3'2 3E0 ~ E0 70OOE026039E009.7280 02 36001.40 0 1.9390-0.1E i-121E 15934E001.22E01124E-0 2.35-0 719821.93-063E71 " 1 1' 0 LI 9E0 72.OOOE-02 67.317E 00 10.501E 01 14.8558 01 15.037E 01 62.375E 00-22.029E 01-11.070E 01-55.916E 00 16.987E-02 15.714E-01 29. 0668-01I 38.465E-01 49.210E-01 88(m1E-01 11.79GE 00 38 5E" ~ 180 74.OOOE-02 82.626E 00 13.5358 1)1 19.030E 0 1 19.732E 0 1 10.0468 01-21.983E 01-10.8888 01-55.275E 00 37.785E-02 24.287E-01 43. 077E-01 58.947E-01 78.111E-0l 1' 3.30E o1 7. -908E 00) 49 5E1) 8 0 76.OOOE-02 ii.084E 01 18.3718 01 24.697E 01 25.0218 01 14.0958 01-21.4588 01-10.663E 01-54.232E 00 94.864E-02 41.805E-01 70. 2918E-01 97.659E-01 13.196E 00 21 092E 00o2-'. 0 2538 00 11 S 0 78.0008-02 15.842E 01 25.141E 01 31.7158 01 30.978E 1)1 18.4748 01-20.4238 01 -10.3668 01-52.323E 00 24.1588-01 77.599E-01 12.343E 00 17.093E 00 2.1 208 ii) 03.789E09'i521E 00 84 3E015:4E1 80.OOOE-02 23.369E 0 1 34.0748 0 1 40.091E 0 1 37.621E 0 1 23.237E 01-18.6078 01-99.11OE 00-48.462E 00 60.041E0114.988E 00 22.623E 00 30. 8188 00 41.-17'E Co *.. - 8.:E 0: 0 7.::4E OH 13HE0:'E0 -- 02.000-02324.089E 01 45. 2868 01 49.779E 01 44.907E 01 28.442E 01-15.530E 01-90.6198 00-40.235E 00 14.387E 00 29.327E 00 42. 1688 00 56.0698 00 73 5i2E LI0'1' 9 7E 0O 1 0 2-8 E1)1 20-i- 1 4"980 84.OOOE-02 48.325E 01 58.677E 01 60.5538 01 52.694E 01 34.1528 01-10.3818 01-72.7238 00-22.575E 00 33.077E 00 57.146E 00 78.467E 00 10.144E 01 12. 969 01 S.' ( 21.84E 0)1 9:8012S E0 86.000E-02 66.015E 01 73.720E 1)1 71.378E 0 1 60.u90E 0 1 40.4448 01-18.1508 00-34.049E 00 14.5378 00 72.681E 00 10.968E 0 1 14.3968 0 1 18.055E 0 1 22437E Ld -8 17 E H I E 015.0 07 E01 7:198 0 88.0008-02 86.0508 01 ~89.14GE 01 82.7388 01 68.393E 01 47.423E 01 12.288E 01 47.372E 00 89.447E 00 15.177E 01 20-544E 01 25.781E 01 31.323E 01 3.: 7:E, HII 45' CH E 031 5"''..E01 7~ 2780: SJ 9j0-0. E02 06-0 5E 0-2 ~10. 26 7 02 91.504E 0 1 75.068E 0 1 99 248E 0 1 29.3618 0 1 19.923E 0 1 23.OO8E 0 1 29.819E 0 1 37.11OE 0 1 44.568E 0 1 52.385E ul 1 6 12E 01 I1 -38 1- 4. E 0 1 1E011:IE092.OOOE-02 12.1588 02 11.063E 02 95.839E 01 79.7:92E 01 64.163E 01 48.727E 01 42.776E 01 46.250E 01 54.090E 01 63.363E 01 73.048E 01 82.9888 01':' 509E 01 I I' E "8 2) I I _ ____38E 0.' 1 94.OOE0212004 02i0800012 9.99E2151.66E01 42 E1106.88001E1.10201 7.39E0187.E1 0198672 011097620E12048 0 1704E 119E' 0 L48. -:E 96. OOE02 0. 07E02 9. 99E01 8.1 4E 1 7. 44E 1 1iSS 01 88.57E 01 91.6.14E 01 77.777E 02 1 8.944E 02 13.6083E 02 14.904E 02 14. 497E 0213.062E 92I214- -95 E 8 190'1 080 98.OOOE-02 64.562E 01i 52.775E 0 1 54.402E 01 62.014E 0 1 72.181E 0 1 83.215E 0 1 94.357E 01 10.502E 02 11.44E 02 13.083E 02 14.710E 02 13.947E 02 15341E U 2 1 1 CE I1 7 E 0I 29 13 1 ~ i0.OOOE-01 0. 0. 0. 0. 0 ~~~ ~~~ ~~~~~~~~~~~~~~~~0. 0. 0. 0. 0. 0. 0. 0. 0 Values of V for Solution 3 0. 0. 0. 0. 0. 0. 0. 0. O - 40.OOOE-03 0. 0. r: 0. 0. 0. 0 80.OOOE-03 0. 0. 0. 0. 0. 0. 0 ___12.OOOE-02 0. 0. 0. 0. 0. 0.0.I I 6.biOE-0 —o2i5. 426E-0 I -12.775E 02 -88.9038 01 62.526E 01 54.734E 01 33.637E 01 34.324E 0-1: 3 11 E 20.OOOE02 21.47E 00-98.297E 01 -73.441E 01 46.121E 01 43.787E 01 26.909E 01 27.459E 01 25-267E -01 E0 24.OOOE-02 36.33GE 00 -78.406E 0 1 6305E 01 34.926E 0 1 36.489E 0 1 22.425E 0 1'22.883E 0 1.10981GE0 28.OOOE02 44.98E 00-64.006E 01 5539 30E 0 1 26.738E 01 31.2768 0 1 19.2218 0 1 19.614E 01 1.047ELi44"E0 32.OOOE-02 49.768E 00 -53.041E 0 1 -49.506E 01 20.432E 01 27.367E 0 1 16.818E 0 1 17.162E 01 1597718H E 00 36.OOOE-02 51.674E 00 -44.373E 01 -44.748E 0 1 15.386E 01 24.3258 0 1 14.950E 0 1 15.2'55E 01 14.0''7E 01 E0 40.OOOE-02 51.353E 00 -.37..320E 0 1 740.754E 0 1 11.233E 01 21.889E 0 1 13.455E 0 1 13.730E 01 I61 32:E Hi 1i'3 2 44.OOOE-02 49.241E 00 -31.459E'01 -37.286E 01 77.451E 00 19.881E 01 1-2.2328 01 12.481E 01 I4 8U 4 4E0 48.0008-02 45.677E 00 -26.5148 01 ~~~~~~~~~~ ~~34.16GE 01 47.804E 00 18.1578 0 1 11.2128 0 1 11.441E:21 1 0 Fi2S-E -0 1 7 52.OOOE-02 40.922E.00 -22.308E 01 l3.3E 0 1 22.579E 00 16.5788 0 1 10.3508 0 1 10.561E 01:HE04L"E'0 56.0008 02 39 0408 00 ~~~~~-18.7368 01 -28.3E 01 14.6438-01 15.0218 01 96. 102E 00 98 0608 00 1 iE 003908'0 60.0008 02 27.8988 Lb 15.7398 0 1 -25.3878 0 1 -15.4058 00 13.4478 0 1 89.6858 ULI 91.4'99 01 939 1158 00 6 0 64.0008 02 19.245E 00 -3.2888 01 -22382 01 -27.5558 00 11.8888 01 84.0458 00 85. 700E 00 798 009-2 0 68.0008 02 87.4528 01-52.3G,:E 00-11.3488 01-15.322E 01-19.2978 01-11.3638 01-34.3018 00 34.7608 00 10.3828 01 91.4038 00 78.986E Ou 7' 0948 00 90.402E 00 74. 597E'10 fi:3. 7 )E 00 4970.0008-02 27.7428 00-99.47GE-01-72.3558 00-17.860E 01-39 8808 01-23.2418 0 1 10.5898 01 13.2488 01 11.7158 01 87.0568 00 81.5008 OH 76.1908 00 71 95G2E 00C I,''H: 7 1 E C,: ii G 0~0' 74.0008 02 i16 544E8 00 20.677 800u74.1008 00-15.b23E 01-3.6:68E:1-19.6148 0 1 78.0348 00 11.0068 01 10.2908 0 1 81.9558 00 76.7648 00 71.7728 00:66'7 005 2 S -'1 E~')- 5'i' 951~ E 0 4':H 0ii 480 76.0008 02 89.3038 01-28.3028 u0-76.455E 00-14.9558E L0i1928528 01-17.5678 01 65.0948 00 99.8998 00 96.3018 00 79.4638 00 74.4418 UL' 69 5952 00 64.580E 0:1 56942:E HO1- 99 5 248 00) 4: -~ 0- 18 7S.7000E02 80 1598E01 ~37 2038 00-78.5718 00-13.4048 01 24.9168 01-15.3668 0 1 53.2318 00 90.3628 00 89.9638 00 7 6. 90638E 00 72.0458 00 b7 223E 00 ul "77E ii 1148 0C2' 0 19O8 ii 80.0008 02 12 7o08 Lb 4: 64~E 0:i 79.5948 00-12.1118 01 20.R67E 01-13.0098 0 1 42.6858 00 81.3818 00 83.7828 00 74.1368 00 69.4438 00 64.6078 030 58.9778:0 91 51.11:4 1E002::01'0 1 8270608 02'0 6948 00 ~59 57 68 00-78.6368 00-10.L29E 01-10 7368 01-10.4988 0 1 33.7978 00 72.8708 00 77.6088 00 70.9498 00 6 6.42278 E00L 61.4818 OHF- 59 317E 0" 4: 7.1,1:18'1' 91 39 85 — Oi 14980:: 48 0 84.0008 02-41.4548 00-6:2 7168 00-74.8358 00-89.289E 00-11 5938 01-78.5498 00 27.0428 00 64.7458 00 71.2348 0 0 67.0418 00 62.6928 00 57.9148 00 90.,638:)r 41.4G-8 EIL - 0.:1'3E " —i.2 136 14 98:: 86.0008-02-55.2288 00-66.615E 00-67.4658 00-70.0888 00-85.8148 00-51.473E 0 0 23.0398 00 56.9468 00 64.400E 00 62.0288 00 57.8248 0 0 52.2798 00 44.48:28 00 "- --? —YEoo-':1 1 76 48E- 0: 17 0180I9 80 88.0008-02-65.5798 00-65.8398 00-56.1568 00-49.1798 00-49.5418 00-25.2638 00 22.5378 00 49.5428 00 56.8278 00 55.4878 00 51.3418 00 45.2988 00 36.3928 0023 -7EO 41-S9:108-il —""8:1:9 980 9000-0 2-69.9068 00-59.324E 00-41.2278'00-27.8368 00-19.6338 00-26.2878-01 24.7808 00 42.6878 00 48.2918 00 47.0808 00 42.8288 00 36.2198 00 26.1528 00 I I W3E': -1 1:1.135E O 4 E'261080 92.0008-02-65.8508 00-46.8868 00-24.155E 00-84.2778-01 20.4358-01 13.1268 00 26.6858 00 35.889E 00 38.6018 00 36.7278 00 32.2378 00 25.1858 00 14.3318 0:1-11.9' 3-'11 —-':.9-48:1-54 46 0' 980 94 Q Dr-02-52. 31 9E 00-o -3.o51 E 00-81.1348-01 54.2588-01 13.5668 00 19.4428 00 24.6368 00 27.5938 00 27.5008 00 24.8208 00 20.3258 00 13.4558 00 31.9008-01-10.9498-'b)O-2:3.8GE'0 108009518' 96.0098E-02-31-.1668 00-13.0688 00 19.8988-01 98.6598-01 13.8088 00 15.7938 00 16.7918 00 16.6778 00 15.2858 00 12.7108 00 91.7138-01 39.387E801 32. 133E-0I)-II 90E 08 Oil-' C6:4:8 E0 9)- 3 7 03 680 98 ---— 02 —95.488E-01-26.401E-01 24.006E-01 46. 1998-01 55.0908-01 57.6248-01 56.6508-01 52.4528-01 44.9058-01 34.1988-01 20.7068-01 16.435E-02-23.O00E-O1-50.7968'1)1'9 86o'E-u1-1 1 4o 09 88 10.0008-01 0. 0. 0. 0. 0. 0. 0. 0.. 0. 0. 0.0

TABLE X. CONTINUED y.2 3 -2j 6 8 A. 10 __ __2 116 Values of U(Pr/Gr)* for Solution 3 O.' 0.'0. 0. 0. 0 0. 0.0.0 49.379E 00 0. 0. 0. 0. 0. 0. 0. 0. 0 47.322E 00 0. 0-. 0-. 0.- 0-. 0-. 0-. 0.0 45.264E 00 0. 0. 0. 0. 0. 0. 0. 0. 43.207E 60046.967E-09 -24.623E-03 -38.715E-03 -35.043E-12 0. 0. 0. 2.3.576E-14 46.967E-09 41.149E 00 97.856E-09 -23.546E-03 -42.331E-03 -93.874E-11 0. 39.528E-14 44.934E-14 33.794E-13 _ _ 7.856E-09 3Z9.092E 00 22.595E-08 -21.456E-03 -45.144E-03 -16.870E-09 0. 49.465E-13 57.356E-13 37.504E-12 22.595E-08 37.035E 00 59.722E-08 -18.843E-03 -47.917E-03 -23.136E-08 0. 49.339E-12 58.423E-12 35.164E-1159.728-08 34.977E 00 17.364E-07 -15.716E-03 -50.622E-03 -24.795E-07 -18.573E-14 42.681E-11 51.601E-11 28.738E-101 4E-07 32.920E 00 53.255E-07 -12.098E-03 -53.180E-0.3 -20.872E-06 -22.214E-13 32.835E-10 40.507E-10 13.237E-09 97.255E-07 30.862E 00 16.670E-06 -80.417E-04 -55.471E-03 -13.725E-05 -21.845E-12 19.741E-09 24.799E-09 10.075E-08 16.670E-06 28.805E 00 51.886E-06 -36.450E-04 -57.305E-03 -69.403E-05 -19.085E-11 10.400E-08 13.303E-08 50.431E-'38 91.886E-06 26.747E 00 5.'19E-05 95.734E-05 -58.361E-03 -26.305E-04 -15.027E-10 49.946E-03 65.042E-08 23.270E-07 - 5.19E0 24.690E 00 459496E-05 57.641E-04 -58.057E-03 -72.585E-04 -64.493E-10 22.218E-07 29.447E-07 9).997E-0744964-822.632E 00 12.334E-04 1.498E-03 -55.410E-03 -14.402E-03 -55.833E-10 92.316E-07 12.449E040. 310E-612 4E-04 20.575E 00 31.284E-04 19.659E-03-49. 144E-03 -21.245E-03 20.913E-08 36.026E-06 49.435E-06 15.317E-05 38 4 18.517E 00 72.671E-047 27.986E-03 -38.045E-03 -26.389E-03 27.042E-07 13.255E-05 18.51 4E-05 54.957E-05 14 16.460E 00 15.519E-03 28.062E-03 40.605E-03 97.811E-04-21.043E-03-25.520E-03-29.997E-03-14.987E-03 23.293E-06 24.213E-05 46.096E-05 55.833E-05 65.570E-05 1 2561E 13.565E-04 8687E0 1 51 * 5.431E 00 15.9 6 4E-03 24. 794E-03 34.033E-03 27.'85E-03 56.341E-04-54.532E-03-29.680E-03-14.908E-03 27.178E-06 30.231E-05 57.242E-05 72.489E-05 88.747E-05 16. E-04 213.156E-04 *. 14.402E 00 17.310E-03 27.783E-03 39.302E-03 39.784E-03 16.503E-03-58.284E-03-29.288E-03-14.794E-03 44.943E-06 41.576E-05 76.901E-05 10.177E-04 13.020E-04 43. 4E-0 4 31.209E-04 1'.73-' 17 13.374E 00 21.'lE-03 35.809E-03 50.348E-03 52.205E-03 26.578E-03-58.162E-03-28.807E-03-14.624E-03 99.970E-06 64.258E-05 11.397E-04 15.596E-04 20.LGGE-04 35.197E-04 45.771E-08 12.5 03 2SE 2.345E 00 29.32CE-03 48.605E-03 65.343E-03 66.199E-03 37.292E-03-56.772E-03-28.212E-03-14.349E-03 25.099E-05 11.061E-04 18.597E-04 25.838E-04 734.86E-04 55.804E-04 7 O2.209E-04 19. - 2 2 11.316E 00 41.915E-03 0 6.517E-03 83.911E-03 81.961E-03 48.878E-03-54.034E-03-27.427E-03-13.843E-03 63.917E-05 20.531E-04 32.655E-04 45.224E-04 61.171E-04 92. 4'4E -!:4 11. OF-) 4 41 1.1E-0 10.297E 00 61.828E-03 90.151E-03 10.607E-02 99.535E-03 61.480E-03-49.230E-03-26.222E-03-12.822E-03 15.885E-04 39.656E-04 59.855E-04 81.536E-04 10.905E-03 1. 52SE-03 2. 2S4E-'3 3 1 8-3 92.586E-01 90.190E-03 11.982E-02 13.170E-02 11.881E-02 75.251E-03-41.088E-03-23.975E-03-10.645E-03 38.064E-04 77.593E-04 11.156E-03 14.334E-03 19.449E E-03 26. 3 E-E-C3 3 4472 E-3 5 9 0.1 "08E82.299E-01 12.78'E-Q2 15.524E-02 16.021E-02 13.942E-02 90. 358E-03-27. 465E-03-19.241E-03-59.729E-04 87.514E-04 15.119E-03 20.760E-03 26.838E-03 34.312E-03 44. 95 E -: 57.821I 1 E - 72.0128E-01 17.466E-02' 19.505E-02 19.017E-02 16.057E-02 10.70iE-02-4' 8020E-04-90.086E-04 38.461E-04 19.229E-03 29.019E-03 38.087E-03 47.768E-03 59.364E-3 74.432E- 14.51E-' 13.20E-0211 74i.E-02 61.724E-01 22.7.7E-02 20.586E-02 21.891E-02 18.095E-02 12.547E-02 32.511E-03 12.533E-03 23.666E-03 40.155E-03 54.355E-03 68.210E-03 82.872E-03 99.S92E-03 1 2.02 -72 1H91E-02 19.22E-0' 22.717E-2 51.437E-01 23.037E-02 2Io.13E-02 24.210E-02 19.61E-02 14.617E-82 77.681E-03 52.712E-03 60.873E-03 78.893E-03 98.185E-03 11.792E-02 13.860E-02 16.172E-02 1 3.374E-22 2 2' 1 12E 2 7E2 41.149E-01 32.16-.8E-02 29.1271E-02 25.357E-02 21.111-E-02 16.97E-02 12.892E-02 11.317E-02 12.237E-02 14.311E-02 16.764E-02 19.327E-02 21.957E-02 24.740E-02 27. 6199E-0' E-02 "'.8. G.7 IDE7 30.362E-01 33.242E-02 23. 74E -02 24.584E-02 21.570E-02 19.610E-02 1".492E-02 18.827E-02 20.594E-02 23.190E-02 26.106E-02 29.040E-02 31.878E-02 34.559E-02 6'.720-E- 377. 8 -722 3; 20.575E-01 2S.358E-02 23.653E-02 21.199E-02 20.749E-02 21.631E-02 23.303E-02 25.649E-02 28.514E-02 31.601E-02 34.615E02 37.3158-02 39.5468-07 40998E0 41 0.8 2 E H 0.287E-or 17. 0382E-02 13.63E-02 14..393E-02 16.407E-02 19.097E-02 22.017E-02 24.965E-02 27.785E-02 30.305E-02 32.357E-02 33.787E-02 34.465E-02 33.-55E-02 32.18230E- 2. 2 0. 0. 0.. 0 _~0_ 0. 0. 0. 0. 0. 0. 0' Values of V(Pr3/Gr)* for Solution 3 _________________________________________________________ ____________________ 0 0. 0. 0. 0. 0. 49.379E 00 0. 0. 0g 0.. 0. 0. 47. 322E 00 L. 0. 0- 0. 0. 0. 045.264E 00 0. 0. 0. 0. 0. 0. ____________ _____: 43.207E 00-20.993E-03 -~1~7.3885E 00 -12.099E 00 85.091E-01 74.486E-0145.7768-01 4L. 7118 0-0101 41. 149E 00 29.3:67E-02 -13.377E 00 -99.945E-01 62.765E-01 59.589E-01 36.621E-01 87.369E -01 4 3 23. * 739.092E 00 49.450E-02 -10.670E 00 -85.744E-01 47.530E-01 49.658E-01 30.517E-01 31.141E-01 2.1:-2 37.035E 00 61.237E-02 -87.105E-01 -75.330E-01 36.387E-01'42.564E-01 26.158E-01 2 82 1 2 7834.977 00 67.729E-02 -72.184E-01 287.37'E-01 22.88;, S 1- ~ 27.8058-01 37.2438-12.888-314..8H 32.920E 00 70.323E-02 -60.386E-01 -60.897E-01 20.939E-01 33.104E-01 20.3458-012 7618-01 14 I I1 -.-l-E30.362E 00 69.36E-C 2 -50.789E-01 -55.462E-01 15.287E-01 29.789E-01 18.310-01 -.685E-01 17 28.805E 00 67.011E-02 __ -42.813E-01 -50.743E-'01 10.540E-01 27.056E-01 16.646E-01 16.936E-0 1. 27-8 * 1 26.747E 00 62.161E-02 -36.083E-01 -46.497E-01 65.056E-02 24.710E-01 15. 259E-01 15. 70E-91 1.3ISE-C 24.690E 00 55.690E-02 -30.359E-01 -42.502E-01 *. 30.727E-02 22.560E-01 14.085E-01 1 4. 3 E 4.3 IE01 1.202E-0173 * 22.632E 00 47.66E-02 -25.497E-01 -38.556E-01 19.928E-03 20.442E-01 13.078E-01 1 E - Ii 2 20.575E 00 37.960E-02 -21.420E-01 -34.549E-01 -20.965E-02 18.300E-01 12.205E-01 _ 12.452E-01 11.311E 3. 2 18. 517E 00 26.190E-02 -18.,084E-01 -30.460E-01 -37.499E-02 16.178E-01 11.438E-01 11.63E-Il i).3'C-.E-O 2. 7 16.460E 00 11.901E-02-71.264E-02-15.443E-01-20.852E-01-26.261E-01-15.464E-01-46.680E-02 47.304E-02 14.129E-01 12.439E-01 10.749E-01 10.845E-01 10.:42E-78 1 15. 4318E 00 37.48-12- 0(.08E02-98.467E-02-24.305E-01-54.273E-01-31.628E-01 14.410E-01 18.029E-01 15.943E-01 11.847E-01 11.091E-01 10.369E-1 97.3 8E-i 48 7 3 14.402E 00 36.866E-02-19.829E-02-98.478E-02-22.699E-01-49.482E-01-29.274E-01 12.498E-01 16.454E-01 14.947E-01 11.495E-01 10.765E-01 10.065E-01 94.270E-8 2 7 -5 —- 30 13.374E 00 22.515E-02-24. 1 40E-02-10.084E-01-21.261E-01-44.455E-01-26.692E-01 10.620E-01 14.978E-01 14.004E-01 11.153E-01 10.447E-01 97.674E-02 91.140E —I 31. E-7 1 2.345E 00 12. 153E-0238. 51 E-02-1 0.405E-01-19.808E-01-39.264E-01-23.906E-01 88.586E-02 13.595E-01 13.106E-01 10.814E-01 10.131E-01 94.663E-T, 871 847E-' —6 2. 11.316E 00-10.909E-03 50.629E-0-10.693E-01-18.241E-01-33.908E-01-20.912E-0O 72.442E-02 12.297E-01 12.243E-01 10.466E-01 98. 046E-02 91. 483-0 84.344E-i 0 70' 10.287E 00-17.40(E-02-63.483E-02-10.832E-01-16.481E-01-28.398E-01-17.704E-01 58.089E-02 11.075E-01 11.402E-01 10.089E-01 94.504E-02 87.924E-0'2 3Q22E-22 5.E- 2 62. 3'4E-02 4 21 2021 92.586E-01763.314E-80-75.-6E -02-10.702E-01-14.465E-01-22.775E-01-14.286E-01 45.994E-02 99.169E-02 10.562E-01 96.5538-0290.400802 833669-02 7.281E-1. 53 7:E 1 9 777E-2-36 71 82.299E-01-56.414E-02-85. 350E-02-10.184E-01-12.151E-01-17. 137E-01-10. 690E-01 36.801E-02 88.111E-02 96.942E-02 91.236E-02 85.317E-8 2 78.27E-02 6 8.90'2.4.18E-2 41 1 2 7404- -.-..1'E —' 72.01T2E -01o-75. 160E-02-90. 656E-02- 31.813E-02-95.382E-02- 11.678E-01-70. 049E-02 31.354E-02 77.497E-02 87.641E-02 84.413E-02 78.693E-80 71.145E-02 60.533E-2' 4 173E-0 25 5 1 7 1':E H 2 61.724E-01 -89. 246E-02-89. OE-02-76.422E-02-66.927E-2-67. 420E-02-34. 380E-02 30.671E-02 67.422E-02 77.33SE-02 75.512E-02 69.869E-172 1.646E-02 409.52-2 32 597E 5 I. 3'; 4-13- 48 - 51437E-"01 95. 134E-02 80.734E-02-56.106E-02-37.882E-02-2.718E-02-35.774E-03 33.723E-02 58.093E-02.65.719E-02 64.071E-02 58.285E-02 49.290E-02 35.590E-2' 15.3 "7E-8- 1 4 2-. 7?E-E:I2 41.149E-01-89.614E-02-63.806E-80-32.872E-02-11.469E-02 27.810E-03 17.863E-02 36.316E-02 48.841E-02 52.532E-02 49.982E-02 43.872E -02 34.274E-02 1.154E-J-1 5. - 91 7E-'i2-'4.-I -— I -148 2 30.^2E 01 71 2018U'40 891801 11.0418-02 73.8398-04 18.462E-02 26.4588-02 33.5278-02 37.5518-02 37.4258-02433.777E-02 27.661E-02 18.311E-02 4'5418-"1. 18 74E-8':2-.5 2E027 2 72 20.575E-01-4.414E-02-17.784E-H2'O7.0798-03 13.426E-02 18.792E-02 21.493E-02 22.850E-02 22.695E-02 20.801E-02 17.297E-02 12.4818 02 40 8 23 034 1 8 3-13-E-02- 3'- -43 9 -'- ELI'-42. 4 E 4E-1 2 10.287E801-12.995E-02-35.929E-03 32.669E-03 62.871E-03 74.974E-03 73.420E-03 77.095E-03 71.382E-03 61.111E-03 4b. E-03 28. 178E-03 22. 366E-04-31 -I.' —3-' -8 0E-;i3' 5 02 - 0. 0. 0.00..0.0. 0. 0. 0. 0. _ _ 17

TABLE X. CONTINUED r/r0 J=2 3 97 10 11 12 6 1i6 Values of U for Solution 4a * 0. —-0.-0. 0. 0. 0. 0. 0. 0.0C.. 40.OOOE-03 24.227E-01 -10.530E-02 -40.850E-02 -29.028E-02 46.465E-03 1o0. 0. 0. 80.000E-03 20.?99E 00 -10.053E 00 -15.545E-01 -53.316E-02 -34.453E-03 -- 20.037E 01 29.454E 01 34.524E 01 - -82.550E 01 12.OOOE-02 23.438E 00 -12.252E 00 -25.066E-01 -87.027E-02 -17.023E-02 I 13.291E 01 19.485E 01 22.153E 01 -54.384E 01 16.000E-02_26.024E 00 -13.890E 00 -33.840E-01 -12.779E-01 -35.297E-02 99.041E 00 14.464E 01 15.883E 01-40.203E 01 20.OOOE-02 28.440E 00 -15.257E 00 -42.423E-01 -17.485E-01 -57.803E-02 78.596E 00 11.427E 01 12.065E 01 -31.631E 01 247000d-02 "30. 822E 00 -16.435E 00 -51.085E-01 -22M72E-01 -84.091E-02 64.8511E 00 93.828E 00 94.746E 00-25.870E 01 28.OOOE-02 33.251E 00 -17.464E 00 -59.953E-01 -28.578E-01 -11.360E-01 54.926E 00 79.O71E 00 75.865E 00 -21.719E 01 32.OOOE-02 35.793E 00 -18.360E 00 -69.066E-01 -34.819E-01 -14.558E-01 47.381E 00 67.868E.00 61.360E 00 - -18.577E 01 36.OOOE-02 38.507E 00 -19.129E 00 -78.387E-01 -41.380E-01 -17.897E-01 41.418E 00 59.029E 00 49.755Eg 00 -16.103E 01 40.OOOE-02 41.455E 00 -19.767E 00 -87.812E-01 -48.097E-01 -21.224E-01 36.559E 00 51.842E 00 40.158EO0 -14.111E01 44.OOOE-02 44.709E 00 -20.263E 00 -97.153E-01 -54.741E-01 -24.307E-01 32.500E 00 45.845E 00 31.992E 00 -12.459E 01 48000E-02 48.349EE00 -20.596E 00 -10.611E 00 -60.979E-01 -26.785E-01 29.039E 0040.722E 0024.870E 00-11.064E 01 52.OOOE-02 52.466E 00 -20.738E 00 -11.425E 00 -66.308E-01 -28.070E-01 26.037E 00 36.239E 00 18.521E 00 -98.633E 00 56.000E-02 57.154E 00 -20.649E 00 -12.090E 00 -69.942E-01 -27.187E-01 23.388E 00 32.203E 00 12.751E 00 - -88.1Q2E 00 60.OOOE-02 62.493E 00 -20.272E 00 -12.506E 00 -70.628E-01 -22.498E-01 21.012E 00 28.442E 00 74.297E-01 -78.655E 00'i64.O00E-06268.513E 00 -19.532E 00 -12.522E 00 -66.338E-01 -11.270E-01 18.831E 00 24.789E 00 24.863E-01 -69.944E 00 68.OOOE-02 75.132E 00 28.406E 00-18.321E 00-15.115E 00-11.910E 00-86.443E-01-53.785E-01-21.380E-01 11.025E-01 18.644E 00 16.768E 00 18.927E 00 21.087E 00 95.012E-01-20.842E-01-31.863E 00-61.643E 00 -70.000 —02 86.743E 00 42.807E 00-17.520E 00-16.257E 00-11.827E 00-83.746E-01-48.510E-01-13.405E-01 22.339E-01 *18.425E 00 16.830E 00 14.718E 00 14.063E 00 13.429E 00 12.955E 00-90.493E-bl-78.333E 00 72.OOOE-02 93.608E 00 47.007E 00-16.535E 00-16.865E 00-11.471E 00-77.607E-01-39.170E-01-63.578E-03 39.197E-01 17.286E 00 15.746E 00 13.714E 00 12.536E 00 11.321E 00 10.300E 00-12.000E 00-74.761E 0 74.0OOE-02 99.107E 00 47.703E 00-15.347E 00-16.904E 00-10.739E 00-66.800E-01-24.329E-01 18.574E-01 63.435E-01 16.138E 00 14.627E 00 12.647E 00 11.082E 00 93.985E-01 76.412E-01-14.961E 00-71.127E 00 76.OOOE-02 10.413E 01 46.948E 00-13.900E 00-16.313E 00-94.982E-01-49.744E-01-21.419E-02 46.320E-01 97.364E-01 14.969E 00 13.463E 00 11.509E 00 96.594E-01 75.949E-01 49.71OE-01-17.731E 00-67.150E00 78.OOOE-I02 1.-8TE 01 45.403E 00-12.143E 00-15.003E 00-75.828E-01-24.434E-01 29.730E-01 85.235E-01 14.387E 00 13.763E 00 12.244E 00 10.293E 00 82.354E-01 58.532E-01 23.140E-01-20.148EOO-62.682EOO 80.OOOE-02 11.268E 01 43.295E 00-10.027E 00-12.850E 00-47.877E-01 11.657E-01 74.220E-01 13.859E 00 20.648E 00 12.507E 00 10.962E 00 89.983E-01 67.879E-01 41.341E-01-27.350E-02-22.054E 00-57.639E 00 S2.000E-05~1TE.557E01 O40.720E 00-74.993E-01-96.859E-01-86.275E-02 61.655E-01 13.495E 00 21.040E-00 28.949E 00.11.186E 00 96.095E-01 76.265E-01 53.085E-01 24.236E-01-27.022E-01-23.286E 00-51.979E 00 84.OOOE-Q2 11.686E 01 37.765E 00-44.881E-01-52.972E-01 44.962E-01 12.936E 00 21.634E 00 30.548E 00 39.792E 00 97.878E-01 81.859E-01 61.892E-01 38.065E-01 74.479E-02-48.479E-01-23.6S5E 00-45.709E'J 36.OOOE-02 11.587E 01 34.552E O0-87.503E-02 58.164E-02 11.677E 00 21.930E 00 32.352E 00 42.931E 00 53.734E 00 83.066E-01 66.975E-01 47.107E-01 23.158E-01-83.174E-02-65.511E-01-23.108E 00-33.339E 00 88.OOOE-02 11.184E 01 31.268E 00 35.784E-01 82.493E-01 21.123E 00 33.645E 00 46.192E 00 58.735E 00 71.280E 00 67.454E-01 51.666E-01 32.370E-01 90.525E-02-21.828E-01-76.263E-01-21.446E 00-31.c50E 00 90.000OE — TO72E 01 28.161E 00 91.853E-01 17.900E 00 33.219E 00 48.474E 00 63.514E 00 78.234E 00 92.552E 00 51.292E-01 36.431E-01 18.484E-01-30.738E-02-31.346E-01-7;3.870E-01-18.654E 00-24.21OE 00 92.OOOE-02 91.364E 00 25.482E 00 16.223E 00 29.554E 00 47.909E 00 66.172E 00 83.814E 00 10.061E 01 11.634E 01 35.215E-01 22.200E-01 67.260E-02-11.546E-01-34.897E-01-71.974E-01-14.816E 00-16.937E 00 94.000E-02 73.785E00 2 3.289E 00 24.276E 00 42.178E 00 63.479E 00 84.302E 00 10.389E 01 12.186E 01 13.788E 01 20.452E-01 10.440E-01-12.OIOE-02-14.562E-01-31.025E-01-55.623E-01-10.220E 00-10.27E00 96.OOOE-02 51.568E 00 20.901E 00 30.704E 00 51.315E 00 73.555E 00 94.673E 00 11.387E 01 13.067E 01 14.465E 01 88.191E-02 28.578E-02-38.749E-02-11.339E-01-20.175E-01-32.623E-0 -54.667E-01-43.-51E-01 *58.OOOE-02 26.209E 00 15.563E 00 27.837E 00 44.895E 00 62.049E 00 77.776E 00 91.494E 00 10.281E 01 11.141E 01 20.184E-02 16.453E-03-18.761E-02-40.735E-02-65.901E-02-99.746E-02-15.o9E-01-12.4E-01 10.OOOE-01 0. 0. 0_. 0. 0. 0. 0. 0. ____ 0. 0. 0.P0. 0. 0. 0. 0. Values of V for Solution 4a 0. 0. 0. 0. 0. 0. i-. 40.0O0E-03 0. 0. 0. 0. 0. 53.246E-02 11.241E-01 13.114E-01 4. - 80.U00E-03-82.550E 01 -79.469E 01 30.1 19E 01 24.908E 01 22.977E 01 55. 645E-02 13.470E-01 35.09.3E-0120.20E 0 12.OOOE-02-54.384E 01 -51.967E 01 19.636E 01 16.524E 01 15.264E 01 58.272E-02 16.101E-01 47.690E-01 3 6. OOOE 02-40 203E 01 -38.094E 01 14.345E 01 12.307E 01 11.395E 01 60.679E-02 18.896E-01 5. 11 E-01 20.OOOE-02-31.631E 0 1 -29.685E 011 11.143E 01 97.606E 00 90.617E 000 62. 744E-02 21.819E-01 Q7.6Q5E _____2.-0 00 — 24. 0 0E-' O 4-iOE 01 -24.009E 01 89.924E 00 80. ^97E 00 74.952E 00 64.496E-02 24.868E-01 O0. 722E01 23.000E-0221.719E 01 -19.896E 01174.462E 00 63.168E 00 63.659E 00 66.109E-02 28.054E-01 835.23fiE-01 32.O00E-02-18.577E 01 -16.757E 01 62.807E 00 58.827E 00 55.091E 00 67.957E-02 31.391E-01 94.054E-01 36.000E-02 -16. 108E 01 -14.265E 01 53.715E 00 51.478E 00 48.332E 00 70. 753E-02 34.910E-01 10. 24E 0o _____ 3. 5 0'7E 40 OOE021 4. 1 1 1 E 01 -12.224E 01 46.436E 00 45.521E 00 42.836E 00 75. 794E-02 38.681E-01 1.3422E 00 E 00 44.OOOE-02-12.459E 01 -10.507E 01 40.493E 00 40.576E 00 38.254E 00 85.428E-02 42.872E-01 _ 2 43E 003 44.'3 "4.0OOF0- J11T -0-4E 0 -1 -90.298E 00 35.564E 00 36.388E 00 34.352E 00 10.388E-01 47.843E-01 3.9 00 43.?4'?E E.: 52.OOOE-02-98.633E 00 -77.336E 00 31.426E 00 32.779E 00 30.968E 00 13.868E-01 54.361E-01 _ 1.050E 00n -56.u00E-02-88.102E 00 -65.759E 00 27.918E 00 29.620E 00 27.981E 00 20.297E-01 63.890E-01 1 9E I 90 154E 960.OOOE-02-78.655E 00 -55.253E 00 24.915E 00 26.812E 00 25.298E 00 31.905E-Q1 79.195E-01 2 4.0iC E 9. 47T000E-02-69.944E60 -45.592E 00.22.318E 00 24.275E 00 22.837E 00 52.369E-01 10.516E 00 31.494E OJ1 68.OOOE-02-61.643E 00-49.130E 00-36.617E 00-82.905E-01 20.036E 00 20.985E 00 21.933E 00 21.227E 00 20.520E 00 49.288E-01 87.550E-01 11.871E 00 14.987E 00 23.I13E 00i' 2 43.89 00 58.7E 0u 7. 13E 0 70.000OE-02-78.333E 00-97.947E 00-30.276E 00 17.788E 00 23.830E 00 22.188E 00 21.415E 00 20.448E 00 19.515E 00 63.862E-01 10.643E 00 14.453E 00 18.260E 00 30.5 O0E 0O 43.523E 00i 62.05E0 i 84.782E-01 13.249E- 0E 17.845E 00 002.74'3E 90033 72.OOOE-02-74.761E 00-87.643E 00-25.086E 00 16.913E 00 22.568E 00 21.058E 00 20.300E 00 19.352E 00 18.397E 00 84.782E-01 13.249E 00 17.825E 00 22463E 003 74.OOOE-02-71.127E 00-77.545E 00-19.985E 00 16.122E 00 21.292E 00 19.943E 00 19.192E 00 18.257E 00 17.277E 00 11.406E 00 16.787E 00 22.211E 00 27.807E 00 33.624E'0 i O 4 E 00 -'*:'. 10 7 E 00 76.OOOE-02-67.150E 00-67.714E 00-15.173E 00 15.394E 00 20.010E 00 18.829E 00 18.079E 00 17.149E 00 16.141E 00 15.422E 00 21.523E 00 27.878E 00 34.540E 00 45.218E 00 57.331E 00 ^ 13i. 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y 36 87 9 10 11 11i6id Values of U(Pr/Gr)k for Solution h4a 0. 0. 0 0. 0. 0. 0. 0. 0. 47.807E 00 97.693E-04 -42.462E-05 -16.472E-04 -11.705E-04 18.736E-05 21.471E-04 45.327E-04 73.041E-04 97.693E-Q4 45.815E 00 81.492E-03- - - -40.537E-03 -62.683E-04 -21.499E-04 -13.893E-05 22.438E-04 54.315E-04 14.151E-03 - 81.492E-03 43.823E 00 94.512E-03 -49.404E-03 -10.107E-03 -35.093E-04 -68.643E-05 23.497E-04 64.924E-04 19.231E-03 94 51'E-03 41.831E 00 I.494E-02 -56.011E-03 -13.646E-03 -51.528E-04 -14.233E-04 24.468E-04 76.197E-04 23.461E-03 10.494E-02 39.839E 00 11.46E-02 -61.522E-03 -17.107E-03 -70.507E-04 -23.308E-04 25.301E-04 87.981E-04 27.261E-03 68E-02 37.847E 00 1 2.429 E-0 -66.273E-03 -20.600E-03 -91.825E-04 -33.909E-04 26.007E-04 10.028E-03 30.852E-03 12.429E-02 35.855E 00 13.408E 02 -70.420E-03 -24.175E-03 -11.524E-03 -45.809E-04 26.658E-04 11.313E-03 34.371E-03 08E-02 33.863E 00 14.433E-02"-74.033E-03 -27.850E-03 -14.040E-03 -58.705E-04 27.403E-04 12.658E-03 37.926E-0314.433E-02 31.871E 00 15.527E-02 -77.135E-03 -31.609E-03 -16.686E-03 -72.168E-04 28.530E-04 14.077E-03 41.629E-03 15 527E p2 29. 879E 00 16.71SE-02 -79.709E-03 -35.409E-03 -19.394E-03 -85.582E-04 30.563E-04 15.598E-03 45.655E-03 16E-02 27.887E 00 18.028E-02 -81.709E-03 -39. 176E-03 -22.074E-03 -98.014E-04 34.448E-04 17.288E-3)3 50.335E-03 2S. 95E 00 19.49 6- -83.053E-03 -42.789E-03 -24.589E-03 -10.801E-03 41.889E-04 19.294E-03 5G.307E-03 14E-02 23.903E 00 21.156E-02 -83.625E-03 -46.071E-03 -26.738E-03 -11.319E-03 55.923E-04 21.920E-03 64.720E-03. 0 21.911E 00 23.047E-02 -83.264E-03 -48.751E-03 -28.203E-03 -10.963E-03 81.846E-04 25.763E-03 77.418E-03'3.047E0' 19.920E 00 25.200E-02 -81.746E-03 -50.428E-03 -28.480E-03 -90.722E-04 12.866E-03 31.934E-03 97.045E-035r.E-0 17.928E 00 2 7.27E-62 -78.762E-03 -50.493E-03 -26.750E-03 -45.444E-04 21.117E-03 42.404E-03 12.700E-02. E ii 15. 936E 00 -30.296E-02 11.454E-02-73.876E-03-60.951E-03-48.026E-03-34.857E-03-21.688E-03-86.212E-04 44.459E-04 19.875E-03 35.304E-03 47.869E-03 60.435E-03 11.573E-02 17.113E-02 23 096-02 30.296E0 14.940E 00 34.97SE-02 17.216E-0-70.649E-03-65.553E-03-47.692E-03-33.770E-03-19.561E-03-54.054E-04 90.078E-04 25.752E-03 42.918E-03 58.279E-03 73.632E-03 12.335E-02 17.550E-02 25.021E-02 34.978E-0 13.944E 00 37.746E-02 18.955E-02-66.676E-03-68.007E-03-46.257E-03-31.294E-03-15.795E-03-25.637E-05 15.806E-03 34.187E-03 53. 424E-03 71.878E-03 90.579E-03 1369E-02 1.660E -02 7.122E-0' 974-0 12.9'48E 00 39.964E-02 1. 23'SE-02-61.885E-03-68. 163E-03-43. 304E-03-26. 936E-03-98. 103E-04 74.899E-04 25.579E-03 45. 9'4E -03 67. 692E-03 8.563E-03 1 1 13E -02 15.'9 5E- 0 2.5591E-i2 292 C,. 5 1E-n 1.952E 00 41.938E-02 18.931E-02r-56.048E-03-65.780E-03-38.390E-03-20.059E-03-86.368E-05 18.678E-03 39.261E-03 62.186E-03 86.790E-03 11.241E- SE-03.928E0 1.23 4E-02 23.93 2 0 E 3 3: -1 3 10.95bE 00 4.845-2 1. 308E-02-48. 963E-03-60. 499E-C3-30. 577E-03-9. 527E-04 11.988E-03 34.370E-03 58.012E-03 84.004E-03 11'006-0' 14.171E-02 171 27E2 21 31.651E-02 43.345E-' *99.598E-01 45.439E-02 1 7.453E-024'0 433E-03-51.815E-03-1 9.36E-03 47.005E-04 29.928E-03 55.886E-03 83.262E-03 11.295E-02 14.485E-02 17.892E-02 21.513E-9D2 2i6.. i 35 E-O2 3". 723E-0 42.191E-02 4 3 89.638E-01 4i.S604E-02 1*S. 420E-0-390.240E-03-39.057E-03-34.789E-04 24.862E-03 54.419E-03 84.843E-03 11.673E-02 15. 00E-02 1".70E-02 22. 568E-02 26.644E-02 31.437E-02 7.3376-02 "4E 79''786E-L 47.12- E-0 -02-18. 098E-03-21.360E-03 18.131E-03 52.163E-03 87.235E-03 12.318E-02 16.046E-02 19.957E-02 24.OG1E-Q2 28.363E-02 32. 831E-02 7.72 E-0 2. 702E-CL 1 5 0' 1 69.718E-01 4S. 7 -02 1. "933E-02-35.285E-04 23.454E-04 47.087E-03 88.430E-03 13.046E-02 17.311E-02 21.668E-02 261406-02 30. 727E-02 35.413E-02 40.1346-02 45 H;161 90 509 55.,3.5 59"759E-01 45.098E-02 12.l09E-0 14. 430E-03 33.265E-03 85.174E-03 13.567E-02 18.627E-02 23.684E-02 28.743E-02 33.794E-02 39. 8 11 E-02 43.735E-02 48. 450E-02 52'. 991E-02 57.178E-02 59'19E-02 9 Ii "6 49.799E-01 41.904E-02 11.35E-02 37.039E-03 72.182E-03 13.395E-02 19.547E-02 25.611E-02 31.547E-02 37.320E-02 42.8860E 2 4. 154E-026 53.019E-02 57.3- 2 i 3'O.'I E-102 2 14 E-02':E-0 1.04E39.839E-0l1 39'. 841 E-02 10.275E-02 i5.417E-03 11.917E-02 19.319E-02 26.683E-02 33.797E-02 40.568E-02 46.914E-02 52.728E-02 57. 882E-02 62. 171 E-02 65.300E-02' 7 65".940E-02' 59. 2 5. 9416 L29.379E01 29.753E02 93 911E-03 17. 8916 E03 17.08E-0''5.597E-02 33.9946-02 41.6926-02 49.140E-02 955600E 1 1 1 7. 48E-02 68.477E-02 19.671E-02.E -4. 3702E-0 I E2 9. 19-920E101 20"794E602 64.260E-03 12.381E-02 20.692E-02 29.660E-02 38.176E-02 45.916E-02 52.691E-02 58.329E-02 62.652E-02 65.474E-02 G6.488E-02 65.408E-02 61.895E-02 55. 14E -0 43 94E-'I E9.58IE-02 10. 56SE-02 62.754E-03 1 1. 225E-02 18.104E-02 25.020E-02 31.362E-02 36.894E-02 41.459E-02 44.923E-02 47. 161 E-02 48.057E-02 47.418E-02 45.147E-02 41.097E-02 34. 921 E-0 25 E-02 0- 0. _.....__,0 0 0. 0. _ 0. 0._ 0. q__ __ 0. -0. 0 _0._ _0. 0. Vilues of V(Pr3/Gr)4 for Solution La -. _ -.._ O. 0. - 0~.0~.0~.0. ~~~~~~~~~~~ ~~~~~~-.-. —---—. -^.-.. - -.- _ ______ ___-. -. —_.. —,_.. - ^ ^Z'^PZ. 00 0*____ ________ 0.0. 0. 0. 0. 0. 0. O7.07 i0 0. 0. 0. 0. 0. 0. 0. 0 45.S15E 00-16.577E 01 -15.958E 01 60.481E 00 50.017E 00 46.139E 00 40.236E 0:0 59.146E 00 G 3 223:.E 0. Cn 433.823E 00-10.9121E 01 -10.435E 01 39.430E 00 33.181E 00 30.65TE 00 ____ 26.689E 00 39.126E'90 434.84E 00 C,1 -1 41.331E 00-80.732E 00 -76.497E 00 28.805E 00 24.714E 00 22.882E 00 19.838E 00 229.045E 00 31.S 06 -:?':i.7?2E 32. 39.839E 00-63. 519E 00 -59.610E 00 22.376E 00 19.600E 00 18.197E 00 15.783E 00 _____22.946E 00 4.22 7E 00 *. 37.847E 00 -51'OE C0 16 -48.213E 00 18.058E 00 16.165E 00 15.051E 00 613.023E 0 18.841E 00 1 O26E 05 "35.855E 00-4. "614E 00 -39.952E 00 14.952E 00 13.689E 00 12.783E 00 11.030E 00 15.878E 00 15.'34E 001 C 33.83EE'0 9' E 00 -33.649E 00 12.612E 00 11.813E 00 11.063E 00 95. 144E-6l 13.28E 00 1 2."32E 0E 31. 871E 00-32. 34E 00 -28.645E 00 10.786E 00 10.337E 00 97.055E-01 17E- 01 1 854E 00 1'13E-01 4'; 2 9.879E00-2 3.'7E H0 -24.546E 00 93.246E-01 91.410E-01 8 66.019E-01 73.413E-01 10.410E 00 L 3640E-0 -I 27.887E 00-25. 019E 00 -21.099E 00 81.312E-01 81.480E-01 76.817E-01 __ 65.262E-01 924U4EE001 -25i.:4EE -O 25.895E 00-2 221 76"0 - -1:.133E 00 71.415E-01 73.070E-01 68.982E-01 58.314E-01 81.773E-01 4 941 E-1-1 23.903E 00- 1 9.806E 00 -15.530E 00 63. 106E-01 65.822E-01 62.186E-01 52.284E-01 72.770E-01 31. 11 E-01 rr'.rE 00 21.911E 00-1 7.692E 06-13.205E 00 56.061E-01 59.478E-01 56. 189E-01 4i.96SiE-01 64.666E-01 1-5.G4E-01 6 0L 19.920E 00-15.795E 00 -11.095E 00 50.032E-01 53.841E-01 50.800E-01 42. 1'3E601___ 113E-01 14.920E-01'96 00 17.928E 00- 14. " 0 4 -91.553E-01 44.816E-01 48.746E-01 45.858E-01 37.8'14E-01 49.779E-01 9.214."496 O00 15.936E 00-12.378E 00-9. 656E-01-73. 530E-01-16. 648E-01 40.233E-01 42.139E-01 44.044E-01 42.625E-01 41.206E-01 37.439E-01 33.671E-01 38. 007E- 01 42.344E-O11_1' 07E-1 -41 -rr 5 -0-. "'46-1- 1" "9 i7E0 14.940E 00-15.7930E 00-19.669E 00-60. 796E-01 35.720E-601 47.853E-01 44.5 56E-01 4. 43.004E-01 41.061E-01 39.187E-01 36'.999E-01 33.795E-01 29.555E-01 C400 S. 01 5-01-13 1 E01 -1 7'0 1 3.944E 00-1 5. 01 3E 00-17.599E 00-50. 374E-6 33.962E-01 45.318E-01 42.286E-01 40.763E-01 38.860E-01 36.943E-01 34.711E-01 31. 620E-0 1 7.539E-01 25.173E-01 273E- 620J. 3E - 01-I4. 01 1'0 12.948E 00-14.283E 00-15.572E 00-40. 132E-01 32.373E-01 42.757E-01 40.048E-01 38.540E-01 36.662E-01 34.694E-01 32.407E-01 29.3736E-01 25.396E-01 22'.' 1125 1 76E-"''44 Lii 143E1-01 1 - 00 11.952E 00-13.434E 00-13.597E 00-30.469E-01 3Q.912E-01 40.181E-01 37.810E-01 36.305E-01 34.437E-01 32.413E-01 30.058E-01 27. 035E-01 23. 111 E-01_19 ""6E-01 19.96-il'l 5.2. 2-01-1. 46 00 109956E 00 -1 287E 00-1 1 682E 00-21.576E-60 29.530E-6 1 37.584E-01 35.540E-01 34.025E-01 32.154E-01 30.066E-01 27 37E -01 24.587E-01 "0'67 U0 16.5376-01 1 1 74 4" 49' -0240. 4-0O-01i-1. E 00 99.598E-01- 11.574E 00-98. 333E-01 -13.582E-01 28.163E-01 34.945E-01 33.196E-01 31.662E-01 29.775E-01'27.625E-01 25.114E-01 22.012E-01 18.0iS9E-01 13.3lE-01 6'. 0C 5E-' -5. 3 4 -i-l 9 46'0 9.638E-01 -10. 438E 00-8.-u60 E-01- 66.017E-02 26.734E-01 32.230E-01 30.730E-01 29.171E-01 27.260E-01 25.049E-01 22 46 I-01 19. 27E-0l 1 5.31 5E0i1 10 u 14-.'7-02-5.3 2 *S'1-'6 001 79.678E-01-91.787E-gi-63.809E-01-74.970E-03 25.146E-01 29.388E-01 28.081E-01 26.499E-01 24.565E-01 22.304E-01 19.655E-01 16. 438E-01 12.428E-01 7 6.438E - 1. 5E7-0 -7.3'49-0- 7' 4 69o.718E-01-73.092E-01-48. 169E-01 38.526E-02 23.280E-01 2S.351E-01 25.180E-01 23.587E-01 21.643E-01 19.357E-01 16.680E-01 13. 449E-01 4. 594E -02 46. 502E-02-1. 1 155E-01-.4L4E-0Li73.02-01 59. 759E-01-63.557E-01-34.002E-01 70. 803E-02 21.003E-01 23.036E-01 21.950E-01 20. 3736-01 18.4516-01 16.1686-01 13- 456 01 10- 3796-')' 659 00E0" 1 1'8.31 O i4 49.799E- 01-48. 634E-0121 7176-01 88.164E-02 18.175E-01 19.345E-01 1 8.31 E-01 16.804E-01 14.967E-01 12.808E-01 1.03006iE-l 73. 159E-02' 37. i17E-02-'1.4 3.4 5E 7 E-01 i 40 39.839E-01-34.011E-01-11.807E-01 89.699E-02 14.684E-01 15.191E-01 14.240E-01 12.872E-01 11.220E-01 92.920E-02 70.715E-02 44.580E-02 13.506E-02-23.i185E-02-710.Q77E-'12-14.45.3E-01-29.752E-01 -34.'9 16-01 -'29. 879E-01 -20.638E-01-~'47.79112 75.300E-02 10.5176-01 10.5856-01 97.8646E-02 86.821E-02 73.631 E-02 58.395E-02 41.069 -02 20.964E-02-24.1 1E-03-29.242E-02-'2. 301E-02-11. 171E-01-20.523E-01-20.'E-01 19.920E-01-97.092E-02-93.460E-03 47.462E-02 59.2716E-02 58.119E-02 52.835E-02 45.839E-02 37.600E-02 28.213E-02 17.7096-02 57. 388E-03-77. 812E-03-22. 770E-02-40.'1 2E-612-5. s 51 -0E-1C.9 76E-0I-7.'9'26-I2 -99.595E-02-25.024E-02 85.783E-04 15.-309E-02 17.852E-02 17.156E-02 15.362E-02 13.063E-02 10.388E-02 73.759E-03 40.532E-03 3.3. 03"9E-04-37. 674E-03-81.799E-03' 1 2.3E-220. 03uE-02-31.512E-02-25.924E-02 _0. 0.. _ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. __ O.

TABLE X. CONTINUED r/ro 3s2 3 4. 6 9 8 10 11 12 1l; 16 18 Values of U for Solution Lb b 0. 0. 0. 0. 0. 0. 0. 40.OOOE-03 24.227E-01 -10.530E-02 -40.850E-02' -29.028E-02 46.465E-031 53.246E-02 11.241E-01 18. 114E-01 24.273-01 80.000E-03 20.209E 00 -10.053E 00 -15.545E-01 -53.316E-02 -34.453E-03 55.645E-02 13.470E-01 35.093E-0120.0E 00 12.OOOE-02 23.438E 00 -12.252E 00 -25.066E-01 -87.027E-02 -17.023E-02 58.272E-02 16.101E-01 47.690E-01 3. E 00 16.OOOE-02 26.024E 00 -13.890E 00 -33.840E-01 -12.779E-01 -35.297E-02 60.679E-02 18.896E-01 58.181E-03 u6024EO0 _20.000E- 02 28.440E 00_0 -15.257E 00 -42.423E-01 -17.485E-01 -57.803E-02 62.744E-02 21.819E-01 67. 605E-01 40 00 24.OOOE-02 30.822E 00 -16.435E 00 -51.085E-01 -22.772E-01 -84.091E-02 64.496E-02 24.868E-01 76.509E-01 30.2E 00 28.OOOE-02 33.251E 00 -17.464E 00 -59.953E-01 _ -28.578E-01 -11.360E-01 66. 109E-02 28.054E-01 0985 E01 913 00 32.000E-02 35.793E 00 -18.360E 00 -69.066E-01 -34.819E-01 -14.558E-01 67.957E-02 31.391E-01 94.054E-01 37E 00 36.OOOE-02 38.507E 00 -19.1129E 00 -78.387E-01 -41.380E-01 -17.897E-01 70.753E-027 34.910E-01 10.324E 00 __ __ _ 07E 0J 40.OOOE-02 41.455E 00 -19.767E 00 -87.812E-01 -48.097E-01 -21.224E-01 75.794E-02.38.681E-01 11.322E 00 953 0J 44.OOOE-02 44.709E 00 -20.263E 00 -97. 153E-01 -54.741E-01 -24.307E-01 85. 428E-02 42.872E-01 1 483E 00 O. E 00 48.003E-02 48.349E 00 -20.596E 00 -10.611E 00 -60.979E-01 -26.785E-01 10. 388E-01 47.848E-01 13.964E 0 839 00 52.0003E-2 52.466E 0 0 -20.738E 00 -11.425E 00 -66.308E-01 -28.070E-01 1 3. 868E-0 1 54.361E-01 16.050E 00U52 E LL 56.OOOE-02 57.154E 00 -20.649E 00 -12.090E 00 -69.942E-01 -27. 187E-01 20.297E-01 63.90E-01 19.199E 00 57.154E 00 60.0 OE-02 62.493E 0 0_-20.272E 00 -12.506E 00 -70.628E-01 -22.498E-01 31. 905E-01 79. 195E-01 24.r066E 00 L_. 6 3E O 4 0OE3 -02 687513E 00 -19.532E 00 -12.522E 00 -66.338E-01 -11.270E-01 52.369E-0O 10.516E 00 31.494E 00 1 3E 0 68.0OE-02 759132E 00 28.406E 00-18.321E 00-15.115E 00-11.910E 00-86.443E-01-53.785E-01-21.380E-01 11.025E-01 49.288E-01 87.550E-0O 11.871E 00 14.987E 00 2.713E 00 42. 439E 00 9 00 132E 00 70.OOE-020 89 539E 00-11.031E-01-20.189E 00- 16.8E 00-11.854E 00-83.960E-01-48.725E-01-13.650E-01 22.050E-01 63.517E-01 10.601E 00 14.401E 00 18.195E 00 30.519E 00 43.416E 00 62.01E 100 39.539 00 72.OOOE-02 96.025E 00-98.872E-01-21.176E 00- 16.970E 00-11.534E 00-78.119E-01-39.696E-01-12.364E-02 38.493E-Q1 83.949E-01 13.149E 00 17.704E 00 22.315E 00.3 "E.660E 00 46.035E l0 6 2E 00 9025 74. OOOE-02 10.057E 01 -11.928E 00-21. 390E 00- 7.160E 00-10.851E 00-67.714E-01-25.283E-01 17.481E-01 62.159E-01 11.257E 00 16.612E 00 22.002E 00 27.556E 00 384' 6E 001 50.59bE 00 74.266E 00 1".0 01 76.OOOE-02 10.454E 0 -11.559E 00-20.879E 00- 6.755E 00-6. 71 1E- 1-51 1 85E-01-36.684E-02 44.569E-01 95.331E-01 15.187E 00 21.251E 00 27.559E 00 34.164E 00 4r4. 799E 00 57.25uE 0C 89493 00 10.454E 01 78.OOOE-02 10.813E 01 -10.1643E 0-19.049E 00- 15.642E 00-78.29lE-01-2.534E-01 27.467E-01 82.633E-01 14.0863E 00 20.487E 00 27.381E 00 34.688E 00 42.422E 00 5 3.24E 0'I) "'. 6.1E 09 9 003 10.l1E 01 80.0J0E-02 11.114E 01 -81.94E- 01-17.672E 00- 13.672E 00-51.152E-01 87.647E-02 71.041E-01 13.492E 00 20.225E 00 27.529E 00 35.376E J00 43.752E 00 52.6:46E 00.9- -4E 00O 77.6l66E 00C 10.404E 01 11 1143 01 82.OOOE-02 11.321E 01 -5.631E-01-14.891E 00- 10.649E 00-12.697E-01 57.876E-01 13.069E 00 20.543E 00 28.376E 00 36.747E 00 45.6673 0J 55.152E 00 L5.163E 0 00 9.1E 0" 11.563E 01 11 132 01 84. OOOE-02 11.385E 01 -28. 808E-01 - 11.207E 00-63.209E-01 40.279E-01 12'.469E 00 21.085E 00 29.898E 00 39.042E 00 48.645E 00 58.727E 00 69.297E 00 80.272E 00 9'2.470E 00 10.694E 01 12.07E 0 11.3E Li 86.OOOE-02 11.243E 01 54.3123E-02-64.845E-01-37.020E-02 11.195E 00 21.389E 00 31.676E 00 42.112E 00 52.782E 00 63.751E 00 75.014E 00 86.529E 00 98.130E 00 11.030E 1 2.36E 01 13743 I 11.433 0J 88.OOOE-02 1O.822E 01 45. 925E-01 -53.90E-02 75.739E-01 20.719E 00 33.072E 00 45.403E 00 57.744E 00 70.113E 00 82.490E 00 94.304E 00 10.6913 01 1 -0 1' l'01 14 00 F 1 4 56 0 10 223 01 90.0O0E-02 10.0.38E 01 87.621E-01 68.394E-01 17.76SE 00 33.' 033E 00 47.944E 00 62.651E 00 77.093E 00 91.185E 00 10.479E 01 11.7733 01 12.9703 01 14. 01 4.83 01 1.95 93 14 0 10.033E J 9.OOOE-02 88.170E 00I 12.644E 00 15.597E 00 I 0.130E 00 4S.070E 00 65.780E 00 82.955E 00 99.382E 00 11.484E 01 12.906E 01 14.170E 01 15.225E 01 15.999E 00 I I 3 0_ 1'I 1 I " 1 14.5843 01 881703'J 94.OOOE-02 71.173E 00 15.770E 00 24.936E 00 43.366E 00 63.993E 00 84.107E 00 10.313E 01 12.068E 01 13.639E 01 14.986E 01 16.061E 01 16.795E 01 17.094E 0I 1I. 4 7E 0l 15.811E 0 1 102'0E 0 71.1733 0 96.OOOE-02 49.729E 00 17.175E 00 31.899E 00 52.652E OO 74.197E 00 94.62'7E 00 1 1.328E 01 12.968E 01 14.340E 01 15.395E 01 16.086E 01 16.337E 01 16.075E 01 1 5.213E 13.573E 0 1 1l.63 0.7293 J0 98.OOOE-02 25.268E 00 14.175E 00 28.721E 00 45.755E 00 62.46E 00 77.759E 00 91.144E 00 10.223E 01 11.066E 01 11.612E 01 11.831E 01 11.673E 01 111.115E 0L1 l0.119E'i 85.993E 00'0.043E 00 25'2630 10.o0 0E-01'J 000 0 0. 0. 0. 0. _0. 0. 0_ 10 - Values of V for Solution 4.b__ V a l u e s o f V f o r S o l u t i on 1^-b,.._. _ _ _ __ _ _____________________________________ ________________ - _-_~ ~ _-__. ______ _ __._ _______ __ _ _ __ _.. _._~~~______________________________________ 0. 0. - - - --. 0 —. 0-. 0. o 40.000E-03 0. 0.n00 o0. o0. 0. 0. __.0 80.OOOE-03-82.550E 01 -79.469E 0 1 30. 1 19E 01 24.908E 01 22.977E 01 20.037E 01 29.454E 01 434. 54E 01;.0'0E *' 12.000E-02-54.384E 01 -51.967E 01 19. 633E 01 16.524E 01 15.264E 01 13.291E 01 19.485E 01 22. 13 01 E 4 0I 16.000E-02-40.203E 01 38.094E 01 14.345E 01 12.307E 01 11.395E 01 99.041E 00 14.464E 01 15.889 3J 01 - 1: 20.OOOE-02-33.31E 01 -29.685E 01 11. 143E 01 0097.6E 00 90.617E 00 78.596E 00 11.4207E 01 ___ 12. 065E 01 13 l 24.OOE-02-25.8E 01 -24.009E 01 89.924E 00 80.497E 00 74.952E 00 64.851E 00 93.208E 00 4.74E 14' LII 2'. "E 28.OOOE-02-21.719E 01 / -19.896E 01 74.462E 00 68.3 18E 00 63.659E 00 54.926E 00 79.071E 00 75.865E 0-. E 32.000E-02- 8.577E 01 -16.757E 01 62.807E 00 58.827E 00 55.091E 00 47.381E 00 67.868E 00 bI. 36 1E 1. 101 36.OOOE-02- 1 1,8E 00 1 -14.265E 01 53.715E 00 51.478E 00 48.332E 00 41.418E 00 499" 05 00 4"''1'5 OE Il 40.00JEO o 2E 0 E 14 ~1 -12.224E 01 46.436E 00 45.521E 00 42.836E 00 36.559E 00 51.842E 00 40.1 5'IE: - 1113 0 44.000E-02- 12459E3 1 -10.507E 01 40.493E 010 40.576E 00 38.254E 00 32.500E 00 45.9S'45E 00'.992E 3 I9E 48. E003-02- 11 064E 01 -90.298E 00 35.564E 00 36.388E 00 34.352E 00 29.039E 00 40.722" 00 o24.8G3 7'IE:4E I01 52.'OOOE-02-98. 633E 00 -77.336E 00.31.44E 006 r32. 779E 00 30.968E 00 26.037E 00 036.239E 00 1''21 3 1E Q- E 00 56.0OE-02-88.102E 00 -65.759E 00 27.918E 00 29.620E 00 27.981E 00 23.388E 00 32.203E 00 1 7513 00-'02E 0' 60.'000E-02-78.655E3010 -55.253E 00 4'.9 195E 00 __ 26.812E 00 25.298E 00 21.012E 01'.443 00 74.297E- 00 64.000E-02-69.944E 00 -45.592E 00 22.318E 00 24.275E 00 22.837E 00 18.831E 00 24.789E 00 24. 3E1-301 -6944E 0'0 68.OOOE-02-61.643E 00-49.130E 00-36.617E 00-82.905E-01 20.036E 00 20.985E 00 21.933E 00 21.227E 00 20.520E 00 18.644E 00 16.768E 00 18.927E 00 21.087E 00 99. 0102E-3 01-20. 843u2lE-01.063E 00-61. *E 00 70.000E-02-11.376E OT-10.067E 01 58.303E-01 21.940E 00 23.698E 00 21.788E 00 21.047E 00 20.145E 00 19.276E 00 18.240E 00 16.687E 00 14.6013 00 10.'083 00 1 00 CI3 L 00 1 3."3 00-'o.164E1 -l.37E 01 72.000E-02-10.564E 01-89.847E 00 64.489E-01 20.695 E 00 4073 00'0.668E 010 19.941E 00 19.058E 00 18.166E 00 17.108E 00 15.60.3 00 11.2103 00 10.4043 00 1' I Q"'-3 -1 5 00-10.43 74.000E-02-97.156E 00-79.189E 00 66.913E-01 19.441E 00 21.159E 00 19.563E 00 18.845E 00 17.973E 00 17.055E 00 15.967E 00 14.497E 010 12.550E 00 11.026rE 00 9 0.508E-0i 78. 8423E-1 -19.95 0I00 —15LE 0 76.000E-02-88.537E 00-68.817E 00 68.572E-01 18.201E 00 19.863E 00 18.457E 00 17.743E 00 16.876E 00 15.928E 00 14.806E 00 13.341E 00 11.421E 00 96.103E-01 75.6.02E-01 591.490 L41-1804E 00-88.5-37 00 78.0OOE-02-79.807E 00-58.786E 00 70.225E-01 16.983E 00 18.550E 00 17.335E 00 16.619E 00 15.749E 00 14.770E 00 13.610E 00 12.132E 00 10.215E 00 81.951E-01 58.34 4E- - 00 I' 02'5.18 0 413 L0-79.80 010 80.000E-02-70.949E 00-49.149E 00 71.886E-01 15.782E 00 17'103 00 16.175E 00 15.454E 00 14.577E 00 13.565E 00 12.364E 00 10.860E 00 89.313E-01 67.581 E-1 41.334E-01-80.913E-0-22.81E 00-'0403 00 82.003-02-61.956E O0-39.973E00 73.281E-01 14.584E 00 15.828E 00 14.957E 00 14.227E 00 13.339E 00 12.295E 00 11.055E 00 95. 196E-O 75.718E-01 52.905E-01'4.417' 01-2 00-61.956E 00 84.000E-02-52.854E 00-31.347E 00 73.990E-01 13.362E 00 14.383E 00 13.649E 00 12.913E 00 12.013E 00 10.943E 00 96.712E-01 81.090E-01 I 1.474E-01 38.011E-01 7_. 01.3 E-02-6 93E01'4. 47 090-9'3843 1100 86.000E-02-43.717E 00-23. 393E 00 73.487E-01 12.077E 00 12.848E 00 12.220E 00 11.483E 00 10.577E 00 94.925E-01 82.051E-01 66.344E-01 46.820E-01 2 3.229E 01-"8 166E —70264. 123E —1-23.'35E 00-4?. 71 73 00 88.000E-02-34.681E 00-16.273E 00 71.149E-01 10.675E 00 11.188E 00 10.635E 00 99.079E-01 90.104E-01 79.345E-01 66.608E-01 51.180E-01 32.210E -01 92. 31 4E-02-21.242E- 1-75. 054E-'l I01- -'0 00-4."813 00 90.000E-02-25.954E 00-10. 187E 00 66.116E-01 90.942E-01 93.640E-01 88.623E-01 81.6483-01 73.0363-01 62.7423-01 50.6293-01 36.0883-01 18.4343-01-28.246E-02-30.753E-n01 -77.875E01-18.90F 00-'9543 00 92.000-02-17.816E 00-53.670E-01 57.299E-01 72.719E-01 73.383E-01 68.840E-01 62.503E-01 54.723E-01 45.496E-01 34.744E-01 21.990E-01 67.535E-32-11.2'3E-01 -34.383E-01 -71. 22E-0114."3 010-19163 00 94.OOOE-02-10.655E 00-20.333E-01 43.835E-01 51.787E-01 51.125E-01 47.322E-01 42.162E-01 35.907E-01 28.582E-01 20.168E-01 10.336E-01-11.411 E -02- 449-301-00.6933E01-554EE-I01-10.305E3 00-110.655E 00 96.000E-02-49.602E-01-29.064E-02 26.174E-01 29.123E-01 28.116E-01 25.586E-01 22.283E-01 18.345E-01 13.809E-01 86.898E-02 28. 230E-02-38. 265E-02- 1. 21 7E-01 -1 9. 982E-01-32. 3850-'01 -99 030"02E-L 98.000E-02-12.704E-01 89.487E-03 82.079E-02 87.680E-02 83.168E-02 74.538E-02 63.587E-02 50.724E-02 36.108E-02 19.869E-02 15.882E-03-18.596E-02-40. 379E-02-65.37 E-02-99. 11 5E-02-I1577001 1'7043-01 10.000E-01 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0 _

TABLE X. dONTiNyUE J=2 3 ~ ~ ~~~6 8 10 1112 U16 Values of U(Pr/Gr)i for solution 4b - 0. ~~~~~~~~~~~~~~~~~~~~0. 0. 0. 0. 0. 0.0 47.B07E 00 ~~~~~~I7*. 19L L14 ~ ~ 4'-.462E-05 -16.472E-04 -11.705E-04 18.736iE-051 21. 471 E-04 45.327E-04 30E-49.3-4 45.815E Ou'1 - ~~~~~~~~~~~~~40.537E-03 -62.683E-04 -21.499E-04 -13. 893E-051, 22. 438E-04 54.315E-04 14.151E-03 14E-. 43.823E 00 94 91' —E -. E-3-10. 107E-03 -35.093E-04 -68.643E-05 23. 497E-04 654.924E-04 19.231E-03 452,0 41.831E 00 11' ~~~~~~~~~~i''.O-6-11E-03 -13.64GE-03 -51.528E-04 -14.233E-04' 24.468E-04 7.9E0 341-31.9E0 39.839E LIII 1 -~~~~~~~~~~~-.i -6.522E-03 -17.107E-03 -70.507E-04 -23.308E 04, 25.301E-04 87.981E-04 27.261E-03 1.6E0.37.847E 00I'.49-2 —6273E-03 -20.600E-03 -91.825E-04 -33.909E-04 26. 007E-04 10.028E-03 30.852E-03 1.2E0 35.855E 00 13.4"E LI 70.420E 09 -24.175g-0. -11.!524E-0 -45.809E-04 26. 658E-04 11. 31 3E-03 34.371E-0' 1340.4: ~ ~ ~ 3E ~-27.850E0 -14.040E-03 -58.705E-04 27. 403E-04 12.65SE-03 3792E3 31.871E 00 19. 9" L7E' 77135E 03 -31.609E0 -16.686E-03 -72.168E-04 28. 530E-04 14.077E-03 19 9L — ")E- 2.7E00 C.-'iE-0'1 9*70E-03 -35.409E0 -19.394E-03 -85.582E-04 30.563E-04 15. 598E-03 45S.655-03 2 7.88 I EOQ 0 - ~2 -3.70EQ 0 -.39.176E0 -22.074E-03 -98.014E-04 34. 448E-04 17.288E-03 50.335Efl9 ___ 25.89E L0IO 14 -D E -I 0 -053E 03 -42.789E0 -24.589E-03 -I 0. 801 E-P 41.889E-04 19.294E-03 5.01E01 23.03E ID C2. 1i5 GrE - 9' I -832E 03 -46.071E 3-26.738E-03 -11.319E-us 55. 923E-04 21.920E-03'470E-0. 21.911E 00 23.'4" E -I02 "9.3 2E-03 -48.751E 3-28.203E-03 -10.963E-03 81.846E-04 25.763E-03 77418'E-03 4 1 9 2 0 E0'O 5 E _-__2 _ 1.46E 03 -50.428E0 -28.480E-03 -90 722E-04 12.866E-03 31-934E-0304'0 17'I"E I..LE 03 ~~~~~~~~~~~~~~~~~~~~~~~~~~-50.493E0 -26.750E-03 -45.444E-04 21. 117E-03 42.404E-03 I.70OE0E U 1593E00 90 29 CE -0' __2 1144,.j —.-.76E 03 0 O'-.51E-03-48.026E 03 34 857E 03-21.688E-03-86.212E-04 44.459E-04 19.875E-03 35.304E-03 47.869E-03 60.435E-03 11 57SE-02 1 0113E'0 --'3 14.40E 00 0-.'-. 109F-0"-'-44.4" VE 0-8.411iF 03 I69.GOIE-03-47.799E 09393385bE 03-19.648E-03-55.041E-04 88.915E-04 25.613E-03 42. 748E-03 58.069E-03 73.371E-03 1-2f3lE I2 17.0'E- 259Il. 1I, 13q4E'0 3"n..,' "-IFCl' 99 99219-'- 99 9'O-785.38E-03-68.431E 03 46.511iF 03 91 S0lE 03-16.007E-03-49.858E-05 15.522E-03 33.852E-03 5.3. 023E-03 71 391E-03 89.983E 09 17.73-0 13.56I17E-02 27. -' F "SEC- 04i~.5"~)E0~48.099F-L03-0-9 -, 4E-u3-69. 197E 03'4'95E 03- 21.305E 03-10.195E-03 70.489E-04 25.065E-03 45.394E-03 66. 98GE 03 88.720E 03 11.112E-02 19 4`3E-02'0.39IOE 02 "D4..F049''O —'l-:E0~-4'- i-llE-II —84 107~E LI9 67.5b2E j39399 98E 03-20.I40E 03-14.792E-04 17.972E-03 38.441E-03 61 28-38.9E0 111E0 37C I "0,9E 0''n 41E-09-91 1~~1E 04'3 33'E 03 52.698E-03 82.838E-03 11.442E-02 14.818E-02 18.415E 02 22 2s)E 02 263270 E 0' - -.' E-v 41. -6 E9- 0'.' F 2E 49C I -1I1~-'EI,2 9Ej ~'' 351'9 38.2E0 206-21.4E0 9.238E-0 85.690E 021'7193E 02923677E 0' " "' E 99 Ii9 E-01 —8 E0-3 023F- 7-7 9'2OE 03'5.13IC4E 0' 3 390 —GE-03 19 343E-04 25.264E-02 54.408E-03 36.769E-02 11-IOIE-02 147.45E I0' 5'- 299E 02-7- E -34E )710 31E-LI 10I 7'a E- 0 I 9 GEI03-4. 1993E-0'3 1' 458E 0' 16.242E 0' 210229E 02 85.051E-03 40.075E-02 16.739E-02 19.042E-02 57.814E 02 27l939E 02 642539E 02 37"~ 7E 4712E0F I591. - 9.8 -O'R.7 i00E' 0'2- f-3 59'0E I'- 10. 055E 0' 17.487E 02 29 "04E-0 J 3.9915E 02 41.585E-02 48.662E-02 54.999E-02 60.429E-02 64.765E0 6.724E 02 68.9289E 02'- 19 Cl -, 1 -3 -0 G9 2.58E -O. 12.-9C93E -0- 2 21.2'31E-0 29 91 9HE-f119'9 157E 02 45.679E-02 52.294E-02 57.823E-02 62.0783E-02 64.865E 02 69 876,E 02 64.P I1'E-02 1.'F14'- II1''1.i9 D 115SSr- 2 1" 3 9E -1' 22 I 7. F-,: E - I 11.581E-0' 18.450E 0' 29 187E 112 91.355E 02 36.753E-02 41.222E-02 44.624E-02 46.825E-02 47.707E-02 47.LI E 02' 44.821E 02 4L -. S 0 l`5 7.-;T', 2'' 0- -j. o. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. LI LI Values of V(Pr3/Gr)'* for Solution t4b 0. 0. 0. 0. 0. 0. 0. 0. 41 740- C E 0 0 Q 0. 0. 0. 0. 0. 0. 0. 4 9 1-I5' 0 0 - 9 l 7 E 021 -15.958E 0 1 60.481E 00 50.017E 00 46.139E 00 40.236E 00 59.146E 00 611. 323E Q9I 4' 3 E 00 I1'I'1E Ill -10.435E 0 1 39.4.30E 00 33.181E 00 30.651E 00 26.689E 00 39.126E 00 ------ OQ (I 4.3E00-8017''E 0j0 -76.497E 00 28.805E 0 0 24.714E 00 22.882E 00 19.888E 0 0 29.045E 00 31.894E 00 E0'09 39 LO-6-7.51RilE 00 -59.610E 00 22.3761E 00 19,600E 00 18.197E 00 15.783F 0 0 22.946E 00 24.227F?- Q00E 37.847 00-51 )950E On -48213E 00 18.058E 00 16.165E 00 15.051E 00 13.023E 00 18.841E 00 I O'l.2.E OCIO9 9E395SF55 00-43 I'l,4E 00 -3.952E 00 14.9,52E-00 13.6i89E 00 12.78JE 00 11.030E 00 15.878E 00' 5 I3 I -0 -l 33 SG3E00 3.-903 F0 -33.649E 00 12.612E 00 11.813E 00 11.063E 00 95.144E-01 13.628F 00 I' 2E0'1.871E 0 0 - 2".- 4 6E 011'8.64SF 00 10.786E 00 10.337E 00 97.055E-01 83.170E-01 11.354E 00 9-).1 X 0 1 I 29.879E 00-2' "7' 00 -24546E 00 93.24SE-01 91.410E-01 86.019E-01 73.413E-0l 10.410E 00 SOII 640EF011 2.7E00-'9 ul9E 00 -2.099E 00 81.312E-01 81.480E-01 76.817E-01 65.262E-01 92.OGOE-01'4" 2''10 00.95 oo2'2 17E 00 -18133E 00 71.415E-01 73.070E-01 68.982E-01 58.314E-01 81.773E-01 4').91F E-01 23.903E 00-10 806E 00 -1.530E 00 63.106E-0l 65.822E-01 62.186E-01 52.284E-01 72.770E-Ol li'7211E 91.9liE 00-17.L.O2E Lb -.205E 00 56.OGIE-01 59.478E-01 56.189E-01 46.96GE-01 64.666E-01 290 G 4 E — Il 1I9.9OF 00-15S71SF 00 -1.095E 00 50.032E-01 53.841E-01 50.800E-01 42.193E-01 57.113E-01 0' lI2CE- Il1'09 17.9'8E 00-14.045E 00 -9.553E-01 44.81GE-01 48.746E-01 45.858E-01 37.814E-01 49.779E-91 9"2 G' 14 l-F 0 15.93bE 00-12. 3-8E 00-98.b56E-L0l-7.530F-01-16.648F-01 40.233F-01 427. 139E-O1 44.044E-01 42.625E-01 41.206E-01 37.439E-01 33.671E-01 38.007E-01 42.344E-91 1' 070rE'11 -41' l' " 14.940E 00-22.845E 00-20.219F 00 11.708E-01 44.056F-O1 47.587E-01 43.751E-01 421.263E-01 40.452E-01 38.708E-01 36.628E-01 33.508E-01 29..335E-01 28. 108F-1)1 2SL"39-)I?.'2 —1 1" S.I, 0 "9 13.944E Ou-2l'13E 00-16 0422E 00 12.950E-01 41.556E-01 45.O55F-0l 41.503E-01 40.044E-01 38.270E-01 36.479E-01 34.353E-01 31.344E-01 27.330F-01- 25.049E-91 -2. I' -l~ 0 1' 1 02 1'9'i 11' i 12.94FF 00-19.510E 00-15.902E 00 13.437E-01 39.039E-01 4'?487F-O1 39.284E-01 37.842E-01 36.092E-01 34.248E-01 32.064E-01 29. 112E-01 25.202E-01 22.141FE121 1' "'F'0-1 I' 4,.8 -III-, 49I -l i 91,F 1C 11.952E 00-17.779F 0Q-13.819E 1)0 13.770E-01 36.549E-01 39.887F-01 37.064E-01 35.628E-01 33.887E-01 31.985E-0l 29.732F-01 26.790E-01 22..934F-01 19.298F-01 19 1"2'Ei 1 I 4" 3'11 Ol - 9' 31 1'"7 10.956F 00-16.026F 00-11.805E 00 14.102E-01 34.103E-01 97.250F-01 34.810E-01 33.371E-01 31.626F-01 29.660E-01 27.330F-01 24.361E-01 20.513E-01 lb 457E-01 I11.7IL' 6E,- 1 5-.711-il'- ll, 99.598F-01-14.247F 0O-98.696E-01 14.49SF 01 31.692F 01 "74.59E"-01 32~.483E-01 31.033E-01 29.272F-01 27.240F-01 24.829F-01 21.807F-01 17.935E-Ol 13.571E-01 __7 c HCll - 1- ICC4-li-i,'4'4F' 89.638E-01-12.441E 00-8-O.263-)E-01 14.71SF 01 29'87FE101 31.784E-01 30.034E-01 23.569E-01 26.785F-01 24.690F-01 22.200F-01 19. 1 16E-0lI 15.205F-01 10.U24E-11 4 S.0 E" 0' -C 508IiE It- 2 79.678F-01-10.614E 00-62.947E-01 14.85SE-01 26.833E-01 2.8 2E-01 27.408E-0l 25.930F-01 24. 122E-01 21.974F-01 19.421F-01 16.284F-01 12.344E-01 6. 330E-3' 195 I~1' I - 12-94. 2 4 5 -I- 0.2 8-'-1i0 69.718E-01-.8q7.786F-01-46.976$E-Ol 14.757F-01 24.251E-101 25.99E-0-O 24.539E-01 23.058F-01 21.239E-01 19.062F-01 16.476F-01 13.322E-01 94.018E-02 4i 2.64 F - I 2 —15.G6E —112 12; 0I-47 -59.'759E-01-6-9.643F-0-1-32.678E-01 14 287E-01'1.436F- -01 221-.467E-01 21.356E-Oi 19.896E-01 18.094F-01 15.933E-01 13.375F-01 10.277E-01 64.680F-02 IFS 537E-0' 4''-5F I 1 O 7179 -F01- i4- - 9-11-" 49.799F-01-52.117F-;01-20.457E-0.1 13 277E-01 18'I262E01 18 8304E-01 17.796E-01 16.396F-01 14.666F-01 12.599F-01 10.167F-01 72.467E-02 37.016F-02 56. 7'22OE-03-61. -i-,I- "lIE501 11 F Ill. 39.839F-01-35.776FE-01-10.777E-0OIll1SOLE-01 1.03E-01 14.73bF-01 13.824E-01 12.551E-01 10.989E-01 91.359E-02 69.769F-02 44.157F-02 3056F0 -.*9E-14 0J~11 9.0 0l-. -.'' 1 29. 879F-01-21.-395F-01-40.830E-02 88.024F 02 10.399F 01 10.2I6FE-01 95.026F-02 84.665E-02 72. 105F-02 57.394E-02 40.498E-02 20. 756F-02-22.91 5E-03 —8. 31 4E-L-61i 39F'I-11.07'4E-LI1-210V4-0 1"9F0 19.920F-01-99.604F-02-58.363F-03 52 560F 02 58.481F 0' 5b 458F-02 51.379ErQ02 44.745E-02.36.839E-02 27.729EF-02 17.450F-02 56.688E-03-76.840F-0322. 524FE1)2-4. E-2 032'F -0'211 I E 110"4 99.598E-02-25.510E-02 17.970F-03 16.482F 02 17.bO7F 02 16.701E-02 14.968F-02 12.769E-02 10.186F-02 72.508E-03 39.899E-03 31.893E-04-37. 343F-03-81.084FE113 13. l'7F 0'2 1"-) 90-g 31o.~FL 0. 0. 0. 00 0. 0. 0. 0. 0. 0. 0. 0.__I

TABLE X. CONTINUED r/ro J=2 3 _ 6 7 8 9- 10 11 121317 18 Values of (T-T )/(TH-T,) for Solution 1 0o 0. 0 0. o. 0. -- 401 OOE-031 4.818E-07 -15. 495E-03 -22. 139E-03 -1 6.790E-060. 0. 0E 24. 17 S 0. 0OOE-03L21- -2 95E-07 -28.306E-03 -48.643E-03 -21.394E-06 0. 0. 0437. 4 10219E-0 12.00OE-02 —3.374E-07 -29. 772E-03' -52.021E-03 -40.296E-06 0. 5.6 E 112.233E-11 14'.327E-0 I'iOUIE-02 —2 33E -Oi53 -30.901E-03- -54.98E-03 -99.332E-06 0. 42.550E- l 40.381E-10 7iS.0 20. C0 E-0 2-90. 95E- -32. 152E-03 -58.067E-03 -23.569E-05 0 22.417E-09 _ 221. 63E-09 361570 2.4. 00OOE-021-27. 2`97E-05 -33. 591E-03 -1.4u38E-03 -47.439E-05 24.150E-10 98.065E-09 10.556E-08 14 32S. 0 0 E -0 2 - 9 4.-2 E -D 0 -35.272E-034 -65.120E-03'3 -80.611-5 14.092E-09 37. 344E-0 42.65E- 9E-07. 0 15.5 -04 - 37. -2 E-03 -S.988E-03 -11.977E-04 63.753E-09 12.723E-07 1'. 383E-07 I 3i0. OCE -02- 2,. 755E-04 -3 E633E-3 -73.003E-03 -16.258E-04 25.21 1E-08 39.516E-07 49.9220E2.5-E04 40.0QOE-02-52.0.2E-04 - 42.485E-03 -77.054E-03 -20.957E-04 89.582E-08 1 1. 344E-06 1U5.2 1u3. 44.000E-02-83l.515E -037-80.958E-03 -26.332E-04 28.955E-07 30.420E-06 ___ 4.30E-06 34.192 4.00 0E-02- 12. 543E-03 -50.187E-03 -84.445E-03 -32.789E-04 85.897E-07 7S.842E-0-_ 11454E5 74E-0 200 5E-0 2 —1 -9 E- -55. 131 E-03-87 128E-03 -40.870E-04 23.596E-06 18.41 1E-050 28.804E-05 17. 900E-1 - - -1i4 5.S OOO E - 0 2 -724i 9 I 8 3-61 2'41 E-03 -88.478E-03 -51.320E-04 60.668E-06 42.083E-0568.924E-05 1O.OE-02-3 3.395E-03 -8. (619E-03 4-89.213E-03 -65.217E-04 14.816E-05 ___ 92.2G7E-05 1 5.747E-4 9 72.5 -4 44.000E-02-45. 4 52E-03 -77.443E-03 -90.253E-03 -84.172E-04 35.086E-05 1 9.508E-04 34.4037E-04 E - l000E-C4 - -. 417E-0-7D. 10J1E-31-8 7 78 E-03-3.674E-0 3 -91 53E-03-51 3 30E -03- 11 053E- 03-51 125E-504 82.814E-05 24.148E-04 40. 014E-54 55.966E-Q4 71.918E-04 15.215E4-03 2 3 7 4444iii-~2-..32i -.33. 47 -3 10 12-1E-02 10.787E- 025 I 10. 7757 -02-69. 782E-03-1 3.3875-03 47. 446E-04 13.317E-04 33 8135 04 53.25E-0 73.77E0 95.2E0 11'l ^5 -:. 5 S5 -:? 4 15 -036;.3i.7.010 E - 0 270 -' 5 7 E - 5 E - 0.3^ - 1 31 1 2 2 E - 02- 1. 6-1E- E -79. 8.9E-03-1 5 00E-03-14. 45E-04 21.891E-04 48.952E-04731 2'S7E-04 9.77E-3-04 -1.99E-03 20. E -0 74 10':E 2-7 4 -:I5-:i3- 1' I 02 -2-1 2 4. 4 47-I12- 1.84E-2 1 01 OSE-02-85. 079E-0301 19 (41E-03-36,.951 6E04 35.891E-04 71.287E-114 10. 259E0 1.2E03 1" 7.40E15 5.'' -:33.24 -:? 2-F-3'74.E 7 i'.OOE-':2-1 0. 22E -02- 2. 43DE-I2-1 3.' 496E-0'-1 3. 471E- 2-I1 —. 2I1D E-G -87 1 195E-43-1. 434E-03-27. 3.38E04 58.185E-24 10 —0.5 3E-03 14.594E-03' 9.I33E-03 24.497E-03 3L. 0E74.010 -0 - 2.6E-2 3.7ED -459 -2 13.92E-2 222E0"2''9'-8.291E-03-24.39E03-11 5GE-04 93. 191E-04 15.508E-04 20.856-0-3- - 26.57- 033.9EI -.*:3-! *.J 35 -i5 O;^0.OiE' aO.CiOEC-I 2-1 4. 14 5E-02 I. E-2- 1 5. 427E-02- 1 42. 1 12E-02-11. 07EE-03-2 1 54E-03 18. 294E -04 135.73OE-03 22. 873E-03 29.82!SE-3 37. 24-7 l. 4-. 1 91: -0E 3 5 - 7 E-.37 32:E-2-1 E 4 -3C-02- 7. 54E-2- 1 uE 5 0E —-1. I19(E-02- 1 1.T IE —2-73. 82-032E 201'S3 -5 1 4095E04 23. 271E-03 39. G34E-014 2.5167E-03 51 7.32E-04 E' 2.?'3 77 "'4E1 3'. 0 3. 1 34. I -02- 1, 3. e?5E-02- 3. 7E-!2-1 0. 1 21E -0[ 12 10 05 3 -1I2-. 7 SE -0-59. 9iiE -03 -1 32.3' E-03 17 7. 2E -03 36.382E-03 49. 21 9E-03 53. 30!E-!:i3. 75E-03 815. 1 2E-3 1. - I 3b. 1 OD 2 -0 — 4'2E-02- 9505 0'407. 1''5 4 E-02- 1. 389E-02-74.1794E-03-3.860E-03249.3E-04 35.765E-03 56.290E-03 71.40E-03 4.782E-3 98.338E-03 1. 21. 3 EE-2 1. 3 E-.2 3 3-I. C3 4E-':I2 3S. OOE-D2-2 31 5E-12- 1. 214E-Fi2-1 3. 55E-02-83. 1 I 4E-I3-4 I. 44 -i3 - 73E-04 I4.4E-03 1 4041.5 4E705 14.625E-03 102263E-02 1 1. 77-E-02 13.305E —2 15.0E-1 2 I5. -O^ 7 5,5E 2- 7. 2-9. 9-3-35. S - 3 92. 4E- E45.413E70378.245E-03-10.548E-02 12.98E-02 14.474E-02 13S.0 J 9E 1 -.E5 0' 2. 2E 3 - 1792.DOD -022':. 39E02-2. 33 -:i-3I.-9 1'9-03 -33. 54EI37 33 -':3 1 1.01' E-2 1.3 -02 16.222E-02 18.205E-02 19.97IE-02 21.472E-5 23.038II E-02 24 4; -:22.1'i 3^0. *'7 -22;.33FE:: 4. OE - 5 0 5445E-03 1268 0 i 4- - i 91'11E-0221.475E-02 23.405E-02 25.059E-02 23'515E-02 27.3285-0' 19.4217E-02 E- 130 1 3- 02' 2 3 E - ID. *' 1 0 - -2; 9 1.E-02-2. C.' 7.3-5E — 49 2 2 D 3E0.'4E -' 2I4E0'0.446E-02.31.799E-02 32.976E-02 34.0265E-02 3 007E-02 3"19.1 31 1 E -. 93..91E-12 27 IE' 3 1 3.9.E-02 38.361E-) 39.35E-I2 40.1705-02 40.857E-02 41.459E-02 4.9 E-02 4'.5 52E-02 4.3. 2E -02 43.5 E -2 4.'?7 - - S 5. CO 1E-01 90 0. 5 00 E - 5.0EE 5. 0E-02 5 0100E-0 50L0J0E-2 50.000E-02 50.000E-02 5p.OOOE-0 2 0 5091 00E- IE 91 ) 1. 90. 3,4 - 0 2- 0 Values of (T-T^dHT)frSlto ____________________________________________________________________ _________ _ ________ H Tj9/(T T)orouto3_____________________________________ __-__________ _________ 00D DE 2. 30E.- I 0 0. 0. 0. *2.000E-32 0. 0._____ 0 ___ _ _ _ 0. 0. 0. 0. __ _100 06 IlbOE 02-32. 850E-0 -28.979E-03-16.66OE-030-8 16-110. 0.- - - _ -?23fOEOr 24.00llS-02-12.8 E-5'30 439E 0- -20. 190E-03 -56. 107E-10 0. ~0. 070.2..'-E*7 2 3. 00 0 E - 0 -^^ _13- 075- 1__ - 0.911 1E 03' _____ 22 0615E-03 _____ -61.029E-09 _______ 0.___________0.___________ _ _ _ _ _ _ _ __L_____. ___ 7 1.E''.3 i'. OOOE0- 0 -' 0-05 —3.1 11-8E-03-24228E-03 -52.203E-08 0.O0.0~- -- 1 — i,. 44LE1 0*' I* I'-'L:.31 1.1050 0 E- 0'2 -1 4.713 45 E - 0 6 -31. 34405 I 735E-Q3 ______4 E0-35.464E-07 ____________ 0_.________________ __79. 1 E8E- 1 __0 1' 12_98^0 14 -21 061E 1 0. 3 4 i E -I'' 0 l*14. *: 3 E - *:iEE 40.00011E-2.. 31' 2E-0 -31 471E-031 -9634E-03 -19.123E-06 0. 48. 1045^ 1'8-E1 2 2: -9^.^^ —. 44J.000E02-8S. 4.1 EIl - E 31 - 93E-03 -81. 246E-06E —' 17 -0.23.107035 0 _40.2E- "1E " - 1 E -: 4" OOO5E-02-2ID 1 1 3E- 31.5215 03 5 -3S.39E-033 -26.8845E-05 -10.I109-10 10.599EE-0 120E2 4E5 2. 1-10 E -02-4.1 E.-1 5 ___2__2E -31.4401E-034 _2______E - 41. 244E-03 __ - ID -68.2791E-05 ____2_E_2_ -68.277E-10_4_02 4_259 244.2027'E- 014 22912 ~7.40 11 E-7* 5'S 014101E - 0 -1 0. 9E -0444 05E-03 91J-03-13.839E-04-37.667E-0917. 3L4E-0.30.'9-0714 3. -. OOO~E-04 -22 1' 41E-4 -33 174 7E 03 -5 979E-03 ______ -23.084E-04 ______ -18.893E-08 ____ _64.6315-0 III _ _ 1 1. 34E-0 51.7 5E-Ill.l45 -F -'4.040015-02-43.0145-04 -35. 13E0 -45-.l215-03 -33.8595-04 -86.458E-08 22896E-0-$ 41.245-61.7E0*3*1:E:438.10E10-78.S44E-014-3 1 2150E-4338 3755-03-50. — 96E-03-2. 2115 04-03133 4405-01-46. 6395-04-23. 337E-04-35. 182E-07 31'.9835-01'7.4835-U — 10.-926E U-0 14. 103E5-05 34.495E-05 54I. 70E042. 05E074;~~-; 7.0 400E-02-81.7005-42 39E033.41E 4- - 77E-03 —7. 2145-03-29. 291'-03 43.3345-04-21.947E-04-58. 3915E-08 49. 116E-01'95.3815-Ub 113.889E05 1 ~.8E0 2 - E0.-.74-0 4 ^3- -,!i-:71 Z005- 0'1 2 -9.' 74-014-22. 1 1'3E-Q-2 01 E-3-1.' 59E031 0. 272E-02-25. 47:85-03-40. 1 09E-04-20. 5395-04 74.6665-07 72.14325-06 134..498-5 )91'19.105-09'28."99-53.5 E0 90.01E04.13E0- ^: 74.11111 144 852E -03'4. 930El1-03-46. *-353-03-719'933E-031 10. 4-15-02-271 8951 E013-37 119 5E04-1 9. 070E-04 28.1945-06 12.480E05 2 1.9581-5 LI 3 1.964E-09 41.535E0 *. 3-:51 37 -44 40!-0 10.*?E..; 7i. 5OO-0'1 1' 93E-0-l 37 -0 -2 670E5-03'-8. 1445-03-10. 948E05_LI4.11 94E-01-34. 7Q9E- 04-1 7. 4405-04 79.2315-06 23. 538E-0 389 1' 1 2E-0 5.^'195E-09 1. 344 -5 1. E0!39 -4S.25- -3E — 7.00QE-I02-13. 949 - "3- 7. 240E-031-l0. 570lE-03 —-5 — 2uE-0 — 10. 7025E-02I-3.6 1 5-03-333 6115 E04-1 5. 434E-04 20.008E-05 4b.516E-5 0911. 400Eu510 U 2 19 1E14 14.7035-42.; E0 570Em3 2-0l.^E 80. OO-22.78.13U14-47. 444OE I-0-7.0 - SS3-03'-93 734E-03-10. 888E-02-40. 4395-03-3. 1 335-04-1 2.61 85-04 47.5795-05 93.898E-09 13.756E5-U4 1'9 1415 0-0' 26. 195E-44. E0 031-1*.4I3- *;."'; — 821OOE02-38-,. -.I4:.5-I1 -4-1. 045E-03-83. 1495-0310. 045-34-11I.095E-02-44.5435-04-41.4495E-04-81.8585-05 10.8255-04 18.9795-04 2I,.7395 U4014361195-04 44.348E-04'1. 3E-04 1' 4rE:39.7EO3 1:' ** 3 44.050E-Q-54.41 417-0-73. -569E013-98.0275-03- 11.5 12E02- 1 4.287-02-53. 74E-01-55. 743E-04-70. 704E-06 23.729E-04 38.1959E04 91.8905-04 698.0185-04 90.079E-054 12.3I I1'-4317.3E03.3E-i *:17-' 4 I;II0I 0 F 2-~77;0 4 E -I 0310. 03~5E I-0 - 1. 4885-02-12.U045E-012- 1.'905-0'-60. 1735-03-81.787E-04 12.366E-04 50.4025-04 76.0063E-04 10)08E-0 12.345-00 1.195E-1-. 21. 5E9-032.4E0Il-7E-0;4: 8800E-2-0.16447l 54E-212.L' 60E02-13 2255-02-12. -895E-I)2-1 1.27E,-0-65 969E-15Q04142 0815-03 37.6755-04 10.4765-03 15.0245E-0 19.1395-04 23.617E-03'29.8J25 1'3'715 4:-0 491.945-037.. E01u 7E-: 90.000E-02-'~14.432E-02-19. 11,E050141,405-0' 2-f9SE 02-10.495.4402-62.907504-13.0335-03 10.2475-03 21.6965-03 29.4705-04 46. 2555-01 43.3'45-0' 51 941u- 3"'13.4 7E-0 3221- 1 41.- -41 11 4.*3E1 924.43 OOE02-18.'950E-0-17.6735-Q2-14.859E-02- 11 5755-02-81 -612E-03-41.520E-03 12.'185-04 28.8025-03 45.5205-03 57.5535E-0167.8815-03'9.3155-0' 90.449E5-03 144. "1 9E4'.13.03SlE-01';.i5- 13. 15E 94.00:0E-02-23.270E-02-17. 831E-02-1.92E-02-67. 905E-03-24.976E-03 13.2285-03 48.6165-03 75.702E-03 95. 4325-03~ 1 1.0945-021T2. 4545-02 13.8025-02 15.320)E-02 1'7;288E-024 191.9J3E05-i>:2334- 23:7E96.J00E-02-25.654E-02-11.715E-Q2-19.273E-03 45.319E-03 35.869E-03 12.S5E-02 14.892E-02 17.118E-02 18.926E-02 20.456E-02 21.834E-02 23.199E-02 24.718E-0'2' 2-545E-D2 23.797E-231.55E- 25.-E-d-2 98.oaOI34YE-0F2-203.6I7E-02 88.69E-03 19.665E-02 24.615E-02 27.449E-02 29. 3995E-02 30.935E-02 32.2045E-02 33.2775E-02 34.215E-02 35.072E-02 3C.919E-02 36.840E- 31. 1455-0 0 44E 2-37E-0 10.000E-01 0. 50.00E-02 50.000E-02 50.000E-Q2 5.0OOE-2 50.000E-02 5Q.OOQE-02 50.0005-02 50.0005-02 50.0005-02 5Q.OOOE-02 50.0QOE-02 50.000E-02 950.15OOE-02 5.000E-0250.000E-2 0LI.ll

TABLE X. CONTINUED r/rpo j.-2 3 _ JL -6 9810 11 12 3 16 Values of (T-Ti)/(TH-TC) for Solution i a 0. 0. 0. 0. 0. 0. 0. 0. 0.. 40.000E-03-34.134E-04 -35.088E-04 -25.503E-04 -16.583E-04 -82.400E-05 73.076E-07 91.402E-05 20..023E-04 34.134-4 80.000E-03-15.632E-03 - -10.541E-03 -26.360E-04 -16.571E-04 -82.326E-05 54.802E-07 91.183E-05 21.293E-04 15.632-03 12.OOOE-02-16.160E-03 -10.689E-03 -26.868E-04 -16.535E-04 -82.149E-05 28. 278E-07 90.729E-05 22.061E-04 16.160 -3 16. OOOE-02-16.732E-03 -10.850E-03 -27.180E-04 -16.470E-04 -81.838E-05 -34.775E-08 90.045E-05 22.547E-04 16.732-03 20.OOOE-02-17. 376E-03 -11.034E-03 -27.351E-04 -16.372E-04 -81.362E-05 -38.523E-07 89..133E-05 22.838E-041 7. 376E-ii3 24.OOOE-02-18.106E-03 -11.241E-03 -27.406E-04 -16.238E-04 -80.690E-05 -74.941E-07 87.989E-05 22.972E-04 18.106E-03 28.OOOE-02-18. 938E-03 -11.472E-03 -27.356E-04 -16.061E-04 -79.786E-05 -11.043E-06 86.6 10E-05 22.968E-04 18.938-03 32.OOOE-02-19.894E-03 -11.730E-03 -27.203E-04 -15.838E-04 -78.615E-05 -14.212E-06 84.993E-05 22.830E-0419.894-0 36.000E-02-20.998E-03 -12.017E-03 -26.948E-04 -15.564E-04 -77.139E-05 -16.650E-06 83.141E-05 222.554E-04 20.998-0 3 40.000E-02-22.287E-03 -12.336E-03 -26.588E-04 -15.234E-04 -75.322E-05 -17.982E-06 81.058E-05 22.129E-04 22.27-03 - 44.000E-02-23.806E-03 -12.691E-03 -26.119E-04 -14.845E-04 -73.129E-05 - 17.908E-06 78.752E-05 21.529E-0423.806-03 48.000E-02-25.617E-03 -13.087E-03 -25.540E-04 -14.393E-04 -70.534E-05 -16.412E-06 76.222E-05 20.705E-04 25.617-03 52.000E-02-27.809E-03 -13.529E-03 -24.847E-04 -13.876E-04 -67.532E-05 -14.089E-06 73.431E-05 19.553E-04 27. 09-03 56.000E-02-30.505E-03 -14.024E-03 -24.042E-04 -13.294E-04 -64.153E-05 -12.540E-06 70.260E-05 17.-893E-043.505-03 60.000E-02-33.886E-03 -14.581E-03 -23.1 28E-04 -12.649E-04 -60.473E-05 -14.639E-06 66.428E-05 915.786E-04 64. 00E-02-38. 225E-03 -15.208E-03 -22.118E-04 -11.948E-04 -56.636E-05 -24. 327E-06 61.424E-05 15.229E-04 33.225-3 68.000E-02-43.910E-03-29.913E-03-15.916E-03-90.099E-04-21.034E-04-16.118E-04-11.201E-04-82.477E-05-52.944E-05 -28.747E-05-45.503E-06 25.002E-05 54.554E-05 15. 293E 04 25.131 E-04 23.212-3 43910-03 70. 000E-02 —37. 673E-03-830.613E-03-83.160E-04-82.129E-04-20.686E-04-15.860E-04-11.110E-04-82.278E-05-53.540E-05 -30.073E-05-68. 856E-06 20.717E-05 48.689E-05 12.6'E0- 4 20. 95-4 70. 70 —4 37.73E-0 3 72.OOE-02-41.963E-03-86.092E-03-72.0 57E-04-74.808E-04-20.349E-04-15.628E-04-11.042E-04-82.388E-05-54.556E-05 -31.860E-05-97.955E-06 15.613E-05 41 441E-05 10. 150E-04 1i..752E-A4 46.450- 04 41. n3-03 74.000E-02-48.459E-03-90.095E-03-69.327E-04-68.332E-04-20.036E-04-15.437E-04-11.009E-04-82.953E-05-56.162E-05 -34.283E-05-13.450E-05 95.729E-06 32.768E-05 77.07-E-05 2. 592- 04 55. 864E-04 4:. 4E -n3i 76.000E-02-56.605E-03-94.276E-03-68.191E-04-62.816E-04-19. 76E-04-15.305E-04-11. 034E-04-84. 152E-05-58.539E-05 -37.515E-05-17. 972E-05 26.256E-06 23.013E-5 54.8 2:3E-05 90. 59- 05 0. 210-04 5.0. 5E-03 78 00u0E-02-66.61 i1E-03-98.554E-03-67.858E-04-58.327E-04-19.568E-04-15.257E-04-11.1GOE-04-86.134E-05-61.764E-05 -41.557E-05-23.167E-05-46.029E-06 1 3.680E-05 3. 11 0E-I5 76. 421 E-05 12.019-03 6..61E80. 000E-02-78.854E-03-10.274E-02-68.512E-04-54.916E-04-19.492E-04-15.306E-04-1 1. 386E-04-88. 680E-05-65.314E-05 -45. 527E-05-27.51 5E-05-93. 979E-06'7. 636E- 36.- 6E-09 12. 247E-04 18. 239-03 73. 84E-'3 - 2.OOE02-93. E3-1 0 57-2-70.980-04-52.607-04-19.672E-04-15.394E-04-11.577E-04-89.785E-05-66.206E-05 -45.186E-05-24.819E-05-23. 142E-06 26.440E-05 77. 433E-05 32.579E-04 27.726-03 3.771E -3 84. OOOE-02-11 178E-02-10.960E-02-77.7E-04-51 253E-04-20. 523E-04-15.115E-04-11.052E-04-80.455E-05-52.555E-05 -24.852E-05 60.489E-06 46.075E-05 10.668E-04 23.421E-04 82.2254 4' 41.861 E-03 11 1'.35-2 86. OOOE-02-1 3.315E-02- 1 12E-02-97. 641E-04-50. 1 18E-04-20. 564E-04-12.493E-04-71. 360E-05-27. 421E-05 16.750E-05 66.604E-05 12. 969E-b'4 21.931E-04 36. 293E-04 69. 561 E-04 18.484E-0.3 470E-00 13.31E-6 88.000E-02-15.774E-02-10.988E-02-14.828E-03-47.173E-04-13.291E-04 22.027E-06 94.679E-05 17.844E-04 26.934E-04 37.924E-04 52.430E-04 73.179E-04 10.629E-03 18.2iE-03 37.555E-03 91.740E-03 1.774-2 90. OOOE-02-18.44,E-02-10.357E-02-18.660E-03-26.008E-04 23.590E-04 48.548E-04 67.329E-04 85.462E-04 10.573E-03 13.040E-03 16.249E-I'3 20.687E-03 27.757E-03 41.931E-03 69.918E-03 13.194E-02 18.445-02 92.000E-02'-21.017E-02-87.416E-03-12.723E-03 79.076E-04 15.725E-03 20.385E-03 24.182E-03 27.944E-03 32.123E-03 37.088E-03 43.310E-03 51.530E-03 64.541E-03 8 5.-607E-03 12.035E-02 18.4,7-2 21.017E-02 94.000E-02-2'2.7'1E-02-50.505E-03 19.470E-03 43.456E-03 54.889E-03 62.629E-03 69.238E-03 75.785E-03 82.887E-03 91.024E-03 10.078E-02 11.319E-02 13.112E-02 15.599E-02 1.200-02 25. 1 -2 22. 9771 E-02 - 96.000E-02-22.133E-02 32.624E-03 10.597E-02 13.230E-02 14.622E-02 15.611E-02 16.461E-02 17.287E-02 18.155E-02 19.113E-02 20.213E-02 21.571E-02 23.293E-02 25.477E-0 240Q2E-02 32. 98.OOOE-02-15.891E-02 20.863E-02 27.434E-02 29.520E-02 30.573E-02 31.296E-02 31.900E-02 32.472E-02 33.057E-02 33.686E-02 34.386E-02 35.211E-02 36.199E-02 37.410E-02 33. 9'8'79E-E020 41 -4 10.OOOE-01 0. 50.O0OE-02 50.OOOE-02 50.OOOE-02 50.000E-02 50.OOOE-02 50.OOOE-02 50.OOOE-02 50.OOOE-02 50.OOOE-02 50.000E-02 50.OOOE-02 50.OOOE-02 50. 00E-026 5.0.OOIJE-0'2, 5 E0. 02 0 Values of (T-Ti)/(TH-TC) for Solution 4b ___________________________________________ 00.'. O0. 0.. 0.,.. 40.000E-03-34. 134E-04 -35.088E-04 -25.503E-04 -16.583E-04 -82.400E-05 73.076E-07 91.402E-05 20.2-0434. 134-:':0.IOOE-03-15.632E-03 -10.541E-03 -26.360E-04 -16.571E-04 -82.326E-05 54.802E-07 91.183E-05 21.29E- 4 15..3-E-0 3 12.000E-02-16. 160E-03 -10.689E-03 -26.868E-04 -16.535E-04. -82. 149E-05 28. 278E-07 99.729E-05 __2.0: 1- - E4 1-. 1.-3 16.000E-02-16.7.32E-03 ~ -10.850E-03 -27.180E-04 - -16.470E-04 -831.838E-05 -34.775E-08 90.0 45E-05 22.57-4 1. 7 - 3 20. OOOE-02-1 7. 376E-03 -11.034E-03 -27. 351E-04 -16. 372E-04 -81.362E-05 -38. 523E-07: 89.1 33E-05 22. E-4 1 7. 7,-.E-3 24. 000E-02-18.106E-03 -11.241E-03 -27.406E-04 -16.238E-04 -80.690E-05 -74.941E-07 7.'39-05 7..'-,72E-04 1. 1, _E-0 28.OOOE-02-18. 938E-03 -11.472E-03 -27 356E-04. -16.0Q61E-04 -79.786E-05 -11. 043E-0r86. 10E-0A25 E:?93:- 4.1.933-3 32. 000E-02-19.894E-03 -11.730E-03 -27.203E-04 -15.838E-04 -78.615E-05 -14.212E-06 4. 93-05 2E. E- 1'-. 3:' +E-: 36. OOOE-02-20.998E-03 -12.017E-03 -26. 948E-04 -15.564E-04 -77. 139E-05 - 16. 650E-06 83. 141 E-05 22.554- 20.'' - 3 40. 000E-02-22.287E-03 -12.336E-03 -26.588E-04 -15.234E-04 -75.322E-05 -17.982E-06 S1.058E-05 22.29- 22. 2::7- 44.000E-02-23.806E-03 -12.691E-03 -26. 1 19E-04 -14.845E-04 -73.129E-05 -17.908E-06 7O8. 752-5 2152E- 23.E -::-.E-n 3 48.000E-02-25.617E-03 -13.087E-03 -25.540E-04 -14.393E-04 -70.534E-05 - 6. 412E-06o 76.222-05 20.705E-i 25.-1 E-0 52. OOOE-02-27.809E-03 -13.529E-03 -24.847E-04 -13.876E-04 -67.532E-05 - 14. 089E-06' 73. 431 E-05 1. 553-027. _:0'E-,' Q 4 56. 000E-02-30.505E-03 -14.024E-03 -24.042E-04 -13.294E-04 -64.153E-05 - 2.540E-06 70.260E-05 17. 9'E-04. 55-0 60. 000E-02-33.886E-03 -14.581E-03 -23. 128E-04 -12.649E-04 -60.473E-05 -14. 639E-06: 66.428E-05 15.7 -04 33. 7':.- - 3 64.000E-02-38.225E-03 -15.208E-03 -22.118E-04 -11.948E-04 -56.636E-05 -24. 327E-06 61.424E-05 15.22''E- 3. 225- 68.000E-02-43.910E-03-2-9.913E-03-15.916E-03-90.099E-04-21.034E-04-16. 1118E-04-11.201 E-04-82.477E-05-52. 944E-05 -28.747E-05-45. 503E-06 25.002E-05 54.554E-05 1 5. 2'3E- 4 25. 11E-4 2.221-03 4 3''E- 70. 000E-02-49.969E-03-37.674E-03-13.931E-03-83.357E-04-20. 684E-04-15.856E-04-11.109E-04-82.267E5-05-53. 533-05 -30. 0696-05-8. 8045-06 20. 730 -05 4:7-4. 7 1E- 2 3I 2.100E-02-54.817E-03-37.499E-03-12.360E-03-76.863E-04-2.344E-04-15.619E-Q4-11.038E-04-82.358E-Q5-54.531E-05 -31. 837E-05-97. 648E-06 15.669E-05 41.541 E-05 1. 17 1E-4 1f.:': E — 4.'572 — 4::17-03 74.000E-02-61.052E-03-37.215E-03-1 1. OOOE-03-70.792E-04-20. 027E-04-15.422E-04-11.002E-04-82.892E-05-56. 10c-05 -34.222E-05-13.371E-05 96.992E-06 32.970E-05 71.4750-'5 1 2.?6E-4 0 3.2':;E-C EE- 76.300E-02-68.692E-03-37. 000E-03-98.031E-04-65.297E-04-19.750E-04-15.284E-04-11.022E-04-84.045E-05-58.42...-05 -37. 397E-05-17. 826E-05 28.384E-06 23. 324-05 55. 5-5 2. 505 -5 -:4::37-:;4.-: 2-3 78. 000E-02-77.967E-03-36. 775E-03-87.755E-04-60.518E-04-19.541 E-04-15. 228E-04-11.139E-04-85. 966E-05-61. 585E-05 -41. 371 E-05-22. 954E-05-43. 257E-06 14.036-05 3::. -'97' E- 5 7,-. 21 -05 1 2.535- ~7 E-: 80. 000E-02-89.230E-03-36. 482E-03-79.405E-04-56.5855E-04-19. 445E-04-15. 267E-04-11. 357E-04-88. 463E-05-65. 099E-05 -45. 322E-05-27. 31 4E-05-91. 954E-06 9. 1 86E-_: 3,_'. 4_*: E -1:5 11.727- 1:.:2-0'E-, 82.000E-02-10.289E-02-36. 078E-03-73.219E-04-53.589E-04-19. 577E-04-15. 3466E-04-11. 549E-04-89. 650E-05-66. 138E-05 -45. 206E-05-24. 965E-05-26. 6-0E-06 25.:66E-05 725.44'3-05 3. 30 2-' — 2::. E:-+'E 10 2 4' —084. OOOE-02-11.934E-02-35. 535E-03-69.234E-04-51.338E-04-20. 272E-04-15.066E-04-11.062E-04-80.922E-05-53. 304605 25 931 E09 4940-'E-02 479-0 101 10- 4 3 86.OOO0E-02-13.886E-02-34.873E-03-66.655E-04-48.280E-04-19.977E-04-12.457E-04-72.926E-05-30.062E-05 13.2416-05 6 2. 127- 05 12.394E-04 21. 5169-04 35.211 -04 5:-'.:-3.0E-03 1 _:;3- 0 88. 000E-02-16.135E-02-34. 148E-03-61.7755E-04-38.058E-04-11 360E-04 36.484E-06 89.683E-05 17.009E-04 25.850E-04 36.595E-04 50.809E-04 71.167E-04 1 0. 360E-03 17.33-3 3'. --. -- 1-135-2 90.000E-02-18.579E-02-31.337E-03-42.174E-04 63.911E-07 30.014E-04 49.640E-04 66.397E-04 83.511E-04 10.311E-03 12.721E-03 15.873E-03 20.243E-03 27. 183E-03 41.1~9-3 -.'E- 13.211-02 13?-0 92.000E-b'2-20.924E-02-21.898E-03 60.403E-04 13.075E-03 17.366E5-03 20.8226-03 24.1056-03 27.6025-03 31.6196-03 36.469-03 42.595E-03 50.751-03 63.584 2E-2 1.43E-02 20.2E-02 94.000E-02-22.473E-02 48.792E-04 39.794E-03 51.041 E-03 57.871E-03 63.649E-03 69.300E-03 75.329E-03 82.125E-03 90.07: E-03 99.695E-03 1 1. 199-02 2. 984-02 1 5. 4:'.-E-2 1.-'. 77-2 25.07-2 22. 73-02 96.000E-02-21.703E-02 74.382E-03 12.364E-02 14.018E-02 14.971E-02 15.741E-02 16.473E-02 17.233E-02 18.064E-02 19.002 —.-02 20.090E-02 21.443E-02 23.165E-02 25.356-02 2.27-2 32.:24-02 -'1.703-02 98.000E-02-15.508E-02 23.264E-02 28.445E-02 29.981E-02 30.771E-02 31.357E-02 31.887E-02 32.419E-02 32.983E-0. 33.602E-02 34.300E-02 35.126E-02 36.120E-0 7. E-2 37.33E3 —:2 34. 3027E -02 4.0:,2 15.509E-02 10. E-01 O50.000E-02 50.00E-02 50. QE-02 50.000E-02 50.000E-02 50. 00E-02 50. 000E-02 50. 000E-02 50.000E-02 50. -02 00-02 50. -02 50.00E-02 50. 006E-02 50. 00OE-032 50.00,3E-02 50. 300-E-02 0.

l!,;~ ~ ~l iiiiiii!!! iiiiiiiriii - 1-~~A 1.0 0) Cc, cc, -4 -.1 17-1.1-1. G', LrI Ln 4-,' 4 —t -P- IW W I`- -'- I` —_`I r-.' — - CC,.P. 0: L.;-: z,,z, _,~ *.. *.. ~..,,:.~.'~.... ~ 0 -w 1~~~~~~~~~~~~~ Xcc!.-. X-l4.- T._lc ~ 0 - j 6,~:, - - c f.. c,. I C:' -'O,:,_;0 ~: 0..C, C;,.O~...,D....... a;,, -.. P;. - C:J ~ -;"~~~~~~~~~~~~~~~~~~~~~~~ ~, — C."...... ao,,a: I~ ~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I M. M M M MIM M M M M M M~~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ M lim m r.....m I ~~~~~~~ ~ ~ ~~~~~~~ ~ ~ ~ ~ ~~ I.....I I I I I.. IL~ I~ I I~ I I'~ I I I I I I I.1 I I,:I-,I - I I 1..)`j 1.. i...'`_ ". Ii..) 1 II 1'.. r- r.. t-,: f.) r~I t.':. C: t....... _ _ _, _,: - _, C, I~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~ ~~~ I I III* I IIl I II I~[ I I- II I I I I I. I' Al' -- ~~~~~~~~-,~~~~~~~~~~~~:-C4 -l: -~~~~~~~~~ [: -- -1~ i::i C _i...Z. ~,,:D.._,, " -~C,.~,...,...-! 0 I," 0 _ -. %: C D O 1......,.. cc "':....... -— ~~ ~~ ~ ~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~ ~' - 1 CC. ]T'_ 4' ~C., ~' ~~~~~",~~~~~~~~~~..:,.._......9::1 119~~fn rn m m, ri mrn m m rn rri. rii r, m'i [ ['C'I.... I....::, -.~~:,:D C~ C I.:,.!,Z) C. t, C. C::,.:.,oiC:_' c - C.~~_. rj,.-i r5 t-j- t.. -, r., r_ ll-i ll'i ill_ I,-:.- ij fj -I.. - 1-.A C: T-,:, I.~~~~~~~~~~~~~~~~~~~~~~~~~~~ i I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I, -~~~~~~~~~-,. -...........,. "I ~~~~~~~~~~~~~~~~~~~~~~~~~~~I'''',,,l,::c:,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~....'

TABLE XI STEADY STATE HEAT TRANSFER RESULTS FOR THE CYLINDER Solution 1 Solution 2 ~ Solution 3 1 Nu Nu/GrPr)'4 Nu Nu/(GrPr) Nu Nu/(GrPr)4 2 11.980 0.4677 3.954 0.2968 20.787 0 4022 3 22.524 ~o.8793 41,131 0.7996 4 15.689 0.6125 8.890 0.6673 30.335 0.5897 5 13.094 0.5112 25.385 0.4935 6 11.639 0.4544 6.131 0.4602 22.551 0.4384 7 10.635 0.4152 20.601 0.4005 8 9.830 0.3838 5.068 0.3804 19.065 0.3707 9 9.143 0.3569 - 17.796 0.3460 10 8.541 0.3334 4.387 0.3293 16,723 0.3251 11 8.002 0.3124 15 -. L5785 0.3069 12 7.498 0.2927 3.813 0.2862 14.928 0.2902 13 6.998 0,2732 14.081 0.2738 14 6.473 0.2527 3.206 0.2407 13.160 0.2558 15 5.897 0.2302 - 12.107 0.2354 16 5.232 0.2043 2,70 0.2057 10.855 0.2110 17 4.462 01L742 - 9.356 o.819 18 -11.980 -0.4677 -30953 -o02968 -20.687 -0.4022

TABLE XI (cont.) Soution Slution 4c Soutio 4 J Nu Nu/(GrPr)t Nu Nu/(GrPr)2 15.508 0.3114 15.728 0.3158 3 26.736 0.5369 28.040 0.5631 4 21,555 0.4328 22,263 0o4471 5 20.019 0o.4020 20.498 0.4116 6 19.229 0.3861 19,578 0-3931 7 18.643 0.3744 18,907 0.3797 8 18,113 0.3637 18.317 0.3678 9 17.581 0.3530 17,741 0,3563 10 17.017 0.3417 17,143 0.3442 11 16.398 0.3293 16.497 0.3313 12 15.700 0.3153 15.777 0.3168 13 14.874 0.2987 14,933 0.2999 14 13.880 0.2787 13a925 0.2796 15 12.662 0.2543 12.692 0.2549 16 11.073 0.2224 11.092 0.2227 17 8.694 0,1746' 8716 0.1750 18 -15.508 -0.3114 -l15o728 -0.3158

- 15 1TABLE XII VELOCITIES FROM MARTINI AND CHURCHILL (21) Experiment 4 (1-r/ro) (GrPr)4 5.31 4.25 3.19 2.13 u, in./sec. 0 0 75 2.21 3.40 (uro/-v)(Pr/Gr)2 0 0,063 0.183 04282 Experiment 8 (l-r/ro) (GrPr)4 6.65 4.22 2.67 2.14 1.61 1.07 0.54 u, in./sec 0 0.50 1,20 1.55 1.80 1.90 1.80 (urO/v) (Pr/Gr) 0 0.178 0.428 0.555 0.642 0.678 0.642 Experiment 9 (1-r/ro)(GrPr)4 5.45 4.35 3.26 2.18 1.09 u, in./sec. 0.45 1.02 3.44 3.58 2.32 (uro/V) (Pr/Gr)2 0,032 0.074 0.247 0.257 0.167 Exeriment 10 (l-r/ro)(GrPr) ) 5.58 4.46 3-34 2.23 1,12 u, in./sec. 0.20 1.38 2.87 2.22 0,88 (urO/-v)(Pr/Gr) 0.012 0.078 0,162 0.125 0.050

-152TABLE XIII TEMPERATURES FROM MARTINI AND CHURCHILL (21) Experiment 4 (1-r/ro)(GrPr)4 0 0.60 1.31 1.91 4.88 9.47 T, ~C 28.5 31.8 35.5 40.1 44.3 43.6 (T-Ti TH-TC) -0.500- -0.411 -0.311 -0.187 -0.073 -0.092 Experiment 8 (1-r/ro)(GrPr) 0 0 0.310, 0.56''0.87 1.,49'2.49 4.05 7.16 T, ~C 15.97.16.07 16.23 16.26 16.58 16.95 17.25 17.37 (T-Ti TH-TC) -0.500 -0.453 -0.381 -0.368 -0.224 -0.058 0.076 0.13( Experiment 9 (l-r/rO)(GrPr) 0~ 0.63 1.14 1.77 3.04 5.06 8.22 14.55 T. C 26.5 29.5 34.6 38.8 44,5 46.0 44.0 44.0 (T-Ti TH-TC) -0.500 -0.430 -0.312 -0.215 -0.082 -0.048 -0.094 -0.094 Experiment 10 (1-r/rO)(GrPr)4 0 o.65 1.17 1.82 3.11 5.19 8.43 T. OC 37.2 44.5 55.0 61.6 69.0 69.0 66.8 (T-Ti)/(TH-TC) -0Q500 -0.400 -0.255 -0.165 -0.063 -0.063 -0093

APPENDIX D COMPUTER PROGRAM The computer program used for the calculations on the unsubdivided grid is given on the following pages. The symbols U and V in the program are the same as in the text. The meaning of the principal symbols which are not defined in the program are given below. P dimensionless temperature, (T-Ti)/(TH-TC) T1,T2 temporary locations XNU local Nusselt number S sine of the angle DT time increment DR radial increment DTH angular increment I integer denoting time step IEND maximum number of time steps G Grashof number PR Prandtl number PS either the Prandtl number or 1, whichever is smaller ANG angle L,Y integer denoting radial position J integer denoting angular position R dimensionless radius, r/r -1535-

1 DIMENSION U(18,26),V(18,26) P(18,26),T1(26).T2(26)-XNU(18),S(10) 2 DT1. OE-6 3 R= 5 —------ 4 DTH= 3.1415927/8 READ INPUT TAPE 7,5 IEND,GPR 5 FORMAT(I5,2P2E14.5) IF(1O0-PR) 501.501i502 501 PS=1.O GO TO 503 502 PS=PR__ __________ ___ _ ______ 503 CONTINUE 601 TIMEO 602 =0 ANGn0, DO 603 J=3,9 ANG=ANG+DTH 603 S(J)=SIN(ANG) S(2)=0 S(10)=0, 12 DO 13 J=3,9 13 P(J,26)=0.5 14 READ INPUT TAPE 7,1069 C (U(J,L) L=2.25) Jz2,10), C( V(JL),L=225),J=2,10), __ C((P(J.L),L=2925)tJJ=210).....C'( T( P 8(JL),'L =2, 2-5-)'-J=2,iO -)-_ ______________ GO TO 82 7 Au DT/(DR*DR) AS=A/PS 8 AP= A/PR 9 B= DT/DTH 10 C= DT/DR 11 GDT = G*DT D=DT/ r-RTHDTH )- - - 65 DO 69 L=2.25 __ 651 V ( 1 t L) =V (9 -— L- - ----- 652 V(11.L)=V(3_L) 66 U(1,L)=U(9,L) 67 U(11,L)=U(3,L) 68- P(1,L)=-P(9.L) 69 P(11,L)=-P(3,L)_ _ _ _______ 185 I=I+1 186 TIME=TIME+DT DO 19 J=2,10- - ). 19 T2(J)=P(J,26) Y=26. R=lo _____ _ ____ L126 20 LaL-1.. R=R --— DR Y=Y-1. DEL=D / (RR*PR ) IF(L-2) 52,22,22 22 DO 48 J=2,10 -- BET=ABSF(U(JL) )*B/R _ GAMS=V J-, L )-'C —--------------- GAM=ABSF ( GAMS) CI1='I-BET -GAM-2 *AP+AP/(Y-1e )-Ze*DEL — 30 IF(U(J.L)) 34,32,32 32 C2=DEL+BET -154

CSDEL G O TO 36 34 C2uDEL CS 5DEL+BET36 IF(V(JUL)) 40~38,38 38 ~C'-A-A-P / ( Y-1 ) + — GAM C4=AP GO TO 42 40 C3-AP-AP/(Y-1.) __ C4=AP+GAM 42 T1(J)CCl*P(JL+C2*P+C2PJ-1 +C3*P(JL-1)+C4*P(JL+1 )+C5*P(J+1~L) XA=ABSF( T1(J)) - - )- - -- IF(XA-1.OE-10) 44,44,46 44 T1(J)=O. 46 P(JL+1)=T2(J) T2(J)=T1 J) 48 CONTINUE 50 GO- TO — 20 — 54 P (J2),T2(J) 70 DO 76 J=210__ __ 71 L226 72 L=L-1 ____ _ _ 73 YuL-1 74 V(JtL) (( U(J+1L) - U(J-1,L) + U(J+1,L+1) - U(J-19L+1)) C /(4.*DTH) + (Y+1.)*V(JL+1))/Y 75 IF(L-3) 76~76,72 ____ ___ ___ 76 CONTINUE DO 160 J29 10_ __ 160 T2 (" J0) -.. Yz26* Rsl. LY26. _______._____________________.__ ___ __ __ _____162 L=L-1 RwR-DR'YY-1.. DEL~D/(R*R) U(11L+l)zT2(3) IF(L-2) 194164 164 164 J311 166 J~J-1 IF(J-2) 162,168-168 168 BET=ABSF(U(JtL) >*B/R GAMS=V(J L *C GAM=ABSF(GAMS)_ ____ C11~-BET-G AM-2**DEL-A*R*(l1/(R+O 5*DR.) + 1/(R-OeS5*DR)) IF(U(J.L)) 172,t170170_ 170 C2=BET+DEL CS=DEL GO TO 174 172 C2=DEL _____ C5=BET+DEL 174 IF(V(J.L)) 1789176.176 176 C3=GAM+ A*(R-DR)/(R-0S5*DR) C4= A*(R+DR)/(R+O-5*DR), GO TO 180 178 C3= A*(R-DR)/(R-O0S5DR) C4" GAM+ A (R+DR)/(R+ 0 5DR) -155

180 CONTINUE 184 T1(J) (Cl*U(JL )+C2*U(J-1,L)+C3*U(JL-1 )+C4*U(J*L+l )+C5*UIJ+1,LT C+GDT*P(JtL)*S(J)) )__ XA=ABSF(T1(J)) IF(XA-1.OE-10) 18891889190 ____ 188 T1(J)=0O 190 U(J,L+1)=T2(J) T2(J)=T1(J) GO TO 166 ________ 194 Do 196 J=2t10 196 U(J,2)=T2(J) __ ______ 198 SC=O, Y=O, DO 28 L=2,25 Y=Y+1 R=Y*DR DEL=(D/PS)/(Y*DR)**2. DO 28 J=2,9 BET=ABSF(U(JL) )*B/R'SAMS=-V(J, L)*C GAM=ABSF ( GAM S) X=BET+GAM+2 *DEL+2 **AS 26 IF(X-SC) 28928,27 27 SC=X 28 CONTINUE DT=DT/SC 199 IF(I-3) 82,82,200 200 DO 201 N=1,10 NN 40*N_ __ _ ___ IF(I-NN) 201,82,201 201 CONTINUE iFiI -IEND+1)) 7,82,105 82-WRITE OUTPUT TAPE 6,85,I,((U(JL),J=2,10)L=1,26) 83 WRITE -OUiTPUT — TAPE- 6 8, 8, ( ( V( j L(J L —J2- 10i= Lr- 0 —26 L —6 —84 WRITE OUTPUT TAPE 6,85,I,((P(JL)*J=2,10),L=1,26) _ 841 WRITE OUTPUT TAPE 6,842, TIMEDT - -- 842 FORMAT (6H TIME 2P2E14.5) 85 FORMAT(5H6 I I5/(2P9E13.4)) DO 851 J=2,10 851 XNU(J ) =5-0.* ( P-J*26 )-P(JJ-25- ) WRITE OUTPUT TAPE 6,852,(JXNU(J),J=2,10) 852 - FORMAT ( —14H-6 — J AND NU ( J ) f -5 *-2P1-E-14.5 ) ). —. - - -. — DO 853 NN=2,1U 853 X=X + XNU(NN) DO- 854 —NN=3,9 854 X=X + XNU(NN) 8O - 85X 5 -uNN 3,92-.-2 -- 855 X=X+2.*XNU(NN) X=X/24. WRITE OUTPUT TAPE 6,856,X 856 FORMAT (18H6MEAN NU ONE- SITD-E PE14.5 — -- IF(I-IEND) 79105,105 1 —C OPUN CH -16, - - C((U(JL),L=2,25),J=2,10 ), — C ((V( J L *L=2,25 Z Z ) J 2, 10 J ) --- C((P(J,L),L=2,25),J2,10) I06 FoRMAT(5E14.7 )-156