THE UNIVERSITY OF MI CH I GAN College of Engineering Department of Mechanical Engineering Heat Transfer Laboratory Technical Report No. 5 INFLUENCE OF FLOW AND ROTATIONAL OSCILLATIONS ON THE MECHANICS OF TWO-DIMENSIONAL LAMINAR BOUNDARY-LAYER FLOW PAST CYLINDERS, INCLUDING UNIFORM SUCTION OR BLOWING Wen-Jei Yang John A. Clark ORA Project 05065 under contract with: AEROSPACE RESEARCH LABORATORIES, OAR AERONAUTICAL SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND WRIGHT-PATTERSON AIR FORCE BASE, OHIO CONTRACT NO. AF 33(657)-8368 administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR September 1965

ACKNOWLEDGMENTS The authors wish to express their appreciation to Dr. Max. Scherborg of the Aeronautical Research Laboratories for his assistance and encouragement and Dr. Eiichi Hori of the Central Research Laboratory of the Hitachi Ltd., Japan for his cooperation in providing information on the programming for a digital computer. Their gratitude is also extended to Messrs Tomokita Izumi, R. Mutyara and Hsu-Chieh Yen, all graduate students of Mechanical Engineering, who analyzed the problems of fluctuating circulation and rotational oscillation as their graduate research works in ME 600. The financial support of the Aeronautical Research Laboratories is gratefully acknowledged. iii

TABLE OF CONTENTS Page LIST OF TABLES vii LIST OF FIGURES ix NOMENCLATURE xiii ABSTRACT xix I. INTRODUCTION 1 IIo THEORETICAL ANALYSIS 3 Part A. Oscillation of Free Stream 3 1. The Fundamental Equations 3 2. Solutions 5 a. Ordinary Blasius Series Solutions for a Symmetrical Blunt Body 9 b. Generalized Blasius Series Solutions for a Sharp-Edged Body 15 35 Solutions up to Second Order 25 4. Permanent Alterations in the Wall Shear Stress and Heat Transfer Rate Caused by Fluctuations in the Stream Velocity 27 Part B. Fluctuating Circulations of Free Stream 28 Part C. Rotational Oscillation of Cylinder Surface 31 III. CONVERGENCE OF THE BLASIUS SERIES 36 IV. NUMERICAL RESULTS AND DISCUSSION 37 V. SUMMARY OF RESULTS 65 VI. REFERENCES 66 APPENDIX A. THE COMPUTER PROGRAM FOR FLOW OSCILLATION IN FREE STREAM 73 APPENDIX B. THE COMPUTER PROGRAM FOR FLUCTUATING CIRCULATION IN FREE STREAM 78 APPENDIX C. THE COMPUTER PROGRAM FOR ROTATIONAL OSCILLATION OF CYLINDER SURFACE 84 v

LIST OF TABLES Table Page I Equations and Boundary Conditions Which Define an Ordinary Blasius Series Solution for Velocity in Oscillating Flow Past a Symmetrical Blunt Body 10 II Equations and Boundary Conditions Which Define an Ordinary Blasius Series Solution for Temperature or Concentration in Oscillating Flow Past a Symmetrical Blunt Body 11 III Equations andBoundary Conditions Which Define a Generalized Blasius Series Solution for Velocity in Oscillating Flow Past a Sharp-Edged Body 17 IV Equations and Boundary Conditions Which Define a Generalized Blasius Series Solution for Temperature or Concentration in Oscillating Flow Past a Sharp-Edged Body 20 V Equations and Boundary Conditions for the Universal Distribution Functions of Velocity for Fluctuating-Circulation and Rotational-Oscillation Cases 52 VI Equations and Boundary Conditions for the Universal Distribu-, tion Functions of Temperature or Concentration for FluctuatingCirculation and Rotational-Oscillation Cases 54 vii

LIST OF FIGURES Figure Page 1o Oscillating flow past. a sharp-edged body. 4 2. Profiles of first-order velocities for oscillating flows past a circular cylinder. 38 3. Amplitude and phase of fluid velocity for oscillating flows past a circular cylinder 39 4. Profiles of first-order temperatures or concentrations at stagnation point for oscillating flows past a circular cylinder for a fluid with Pr = 0,7 or Sc = 0.7. 41 5o Amplitude and phase of fluid temperature or concentration for oscillating flows past a circular cylinder. 42 6. Streamline pattern of the steady secondary motion produced by oscillating flow past a circular cylinder at quasi-steady state. 43 7. Distribution of the steady secondary component of temperature or concentration produced by oscillating flow past a circular cylinder for a fluid with Pr = 1 or Sc = 1 at quasi-steady state. 44 80 Effects of flow oscillation on the local wall shear stress and Nusselt number for flow past a circular cylinder at quasisteady state, 45 9o Profiles of first-order velocities for flow past a circular cylinder with fluctuating circulations. 47 10. Amplitude and phase of fluid velocity for flow past a circular cylinder with-fluctuating circulations. 48 11o Profiles of first-order temperatures or concentrations for flow past a circular cylinder with fluctuating circulations for a fluid with Pr = 1 or Sc = 1. 49 12. Amplitude and.phase of fluid temperatures or concentrations for flow past a circular cylinder with fluctuating circulations. 50 ix

LIST OF FIGURES (Continued) Figure Page 13. Streamline pattern of the steady secondary motion produced by flow past a circular cylinder with fluctuating circulation of amplitude EU1 = e at quasi-steady state. 52 14. Streamline pattern of the steady secondary motion produced by flow past a circular cylinder with fluctuating circulation of amplitude eUl(x) = E(l+x2+x4) at quasi-steady state. 53 15. Distribution of the steady secondary component of temperature or concentration produced by flow past a circular cylinder with fluctuating circulation of amplitude eU1 = E for a fluid having Pr = 1 or Sc = 1 at quasi-steady state. 54 16. Distribution of the steady secondary component of temperature or concentration produced by flow past a circular cylinder with fluctuating circulations of amplitude ~Ul(x) = e(l+x2+x4) for a fluid having Pr = 1 or Sc = 1 at quasi-steady state. 55 17. Effects of fluctuating circulation on the local wall shear stress and Nusselt number for flow past a circular cylinder. 56 18. Profiles of first-order velocities for flow past a circular cylinder in rotational oscillations. 57 19. Amplitude and phase of fluid velocity for flow past a circular cylinder in rotational oscillations. 58 20. Profiles of first-order temperatures or concentrations for flow past a circular cylinder in rotational oscillations for a fluid having Pr = 1 or Sc = 1. 59 21. Amplitude and phase of fluid temperature or concentration for flow past a circular cylinder in rotational oscillations. 60 22. Streamline pattern of the steady secondary motion produced by flow past a circular cylinder in rotational oscillations with amplitude eUl = E at quasi-steady state. 61 23. Distribution of the steady secondary component of temperature or concentration produced by flow past a circular cylinder in rotational oscillations with amplitude eU1 = E for a fluid having Pr = 1 or Sc = 1 at quasi-steady state. 63 x

LIST OF FIGURES (Concluded) Figure Page 24. Effects of rotational oscillations on the local wall shear stress and Nusselt number for flow past a circular cylinder. 64 xi

NOMENCLATURE Symbols a: Coefficient depending on the geometrical configuration of the body, dimensionless. b: Coefficient depending on the nature of flow oscillation, dimensionless. Cw-C C: Dimensionless concentration, = -. Cw-Co * * * C: Concentration, Ibm - mole; Cw at the wall; Cm of the free stream. F: Functional coefficient or universal distribution function of temperature (or concentration), dimensionless; Fok for the zeroth-order approximation; Fl1k for the first-order approximation; F21jk for the second-order approximation. f: Functional coefficient or universal distribution function of velocity, dimensionless; fok for the zeroth-order approximation; flik for the first-order approximation; f2ejk for the second-order approximation. i: ( 1)l/2 k: Integer, dimensionless. L: Characteristic length, ft. = 2R for a circular cylinder. 2: Integer, dimensionless. M,m: Constant, dimensionlesso Nu: Nusselt number, dimensionless, = (I ) 0 n: Constant, dimensionless. Pr: Prandtl number, dimensionless. q: Rate of heat transfer, BTU/hr-ft2. R: Radius of a circular cylinder, ft. xiii

NOMENCLATURE (Continued) Symbols Re: Reynolds number, dimensionless. Sc: Schmidt number, dimensionless. Sh: Sherwood number, dimensionless, = ( Ty =-T T: Dimensionless temperature, = TW oo T*: Temperature, ~F; Tw, of the wall; To, of the free stream. t: Dimensionless time, = O L t*: Physical time, hr. U*((x,t) U(x,t): Velocity of potential flow in dimensionless form, = - U0o Uo(x): Time-average or steady-velocity of potential flow in dimensionless form, = U (x Uoo Ul(x): Oscillation amplitude of potential flow or cylinder surface in dimensionless form, = (x) U00 U0: Velocity of potential flow at infinity, ft/hr. U (x,t): Velocity of potential flow, = Uo(x) + EcU(x) cos ot for parts A and B, = Uo(x) for part C, ft/hr. Uo(x): Time-average or steady velocity of potential flow, ft/hr. U*(x): Oscillation amplitude of potential flow or cylinder surface, ft/hr. U u: Dimensionless velocity in x-direction, = -; uo for the zerothU0 order perturbation, = y; ul and uls for the first-order perturbation, ul =- 'y; u2 and u2ij for the second-order perturbation, u2ij = 2y. xiv

NOMENCLATURE (Continued) Symbols u*: Velocity component in x-direction, ft/hr. V* V: Velocity of uniform suction in dimensionless form, = - Uo0 V*: Velocity of uniform suction, ft/hr. * v: Dimensionless velocity in y-direction, = v-; vo for the zerothUoo order perturbation, = -; vl and vlg for the first-order perax turbation, vi = -; v2 and va2j for the second-order perturbation, V2aj = - 2 6x v*: Velocity component in y-direction, ft/hr. x: Distance measured along the wall in dimensionless form, = L x*: Distance measured along the wall, ft. y: Dimensionless distance measured in the direction perpendicular to y* the wall, = L y*: Distance measured in the direction perpendicular to the wall, ft. y: Integer, dimensionless. E: Small constant parameter, dimensionless. I: Dimensionless distance measured in the direction perpendicular to the wall; = y(a Re)l/ xm/ 2 for generalized Blasius series solutions; = y(aiRe)l/2 for ordinary Blasius series solutions. 0: Dimensionless temperature; Go for the zeroth-order perturbation, = To; @1j for the first-order perturbation, 2iaj; for the secondorder perturbation. it: Absolute viscosity, lbm/hr-ft.,... xv

NOMENCLATURE (Continued) Symbols T: Wall shear stress in dimensionless form, = or =( Uoo ayy = O) _^ *Y T* Wall shear stress, = - ib (-)_, lbf/ft2. 4: Stokes stream function in dimensionless form, =.V; '0 for the LUco zeroth-order perturbation, =; toa for the first-order perturbaLUoo tion, = *iz Uo /; 't22j for the second-order perturbation, i* R-1 ~+1 = '2VajUoo /L~ V*: Stokes stream function, ft2/hr; 'V for the zeroth-order perturbation; V1R for the first-order perturbation; 'Vijk for the secondorder perturbation. *wL c: Frequency of oscillation in dimensionless form, =. U0 co*: Frequency of oscillation, rad/hr. Superscripts ',, "s: First, second and third derivatives with respect to Tr respectively. Subscripts for Functional Coefficients f and F o, 1, 2: Zeroth, first, and second-order perturbation. j: Refers to either steady- or transient-state in the second-order perturbation.:' Order of approximation in term of the frequency of oscillation, (iou>). k: Order of function. s,t: Steady- and transient-states in the second-order perturbation respectively. xvi

NOMENCLATURE (Concluded) For Example Subscript Order of 1st 2nd 3rd 4th 5th Perturbation Exponent Order of Zeroth Order of E of x Function Order of Exponent Order of 1st Order of E l ir of x Function Order of Steady- or Exponent Order of 2nd Order of E Transient- Function Stateof x Funct State xvii

ABSTRACT A theoretical investigation of the influence of flow oscillationfluctuating circulation and rotational oscillation upon the transfer of momentum, heat and mass in two-dimensional laminar boundary-layer flow past cylinders with or without uniform suction. The boundary-layer equations for flow, temperature and concentration are linearized by means of a perturbation procedure and the first three terms retained. The solutions of the velocity, temperature and concentration components are obtained in the forms of an ordinary Blasius series for a symmetrical blunt body and a generalized Blasius series for a sharp-edged body. Theoretical results include the frequency response of fluid velocity, temperature and concentration, the streamline patterns of the streaming, the distribution of the steady second-order temperature and concentration, and the alternations in the shear stress, rates of heat and mass transfer. For flow: around a circular cylinder numerical results show that the permanent alterations in the skin friction and heat transfer rate induced by the flow oscillation, fluctuating circulation and rotational oscillation are very small in the range of small amplitude and low frequency. xix

I. INTRODUCTION In recent years, considerable attention has been focussed on the problems of periodic boundary layers. Since fluctuations in a stream incident upon a body is known to occur, it is important to understand how the boundary layer responds to the oscillations of the stream. For instance, in the occurrence of flutter on aircraft, the boundary-layer effects may be considerable. It is generally observed that a steady flow, which is known as secondary or streaming flow, exists in an oscillating fluid or is generated in a quiescent fluid where solid boundaries oscillate. This phenomenon is also known to occur when oscillating acoustic waves interact with a stationary object. For periodic boundary layers in the absence of a mean flow, Rayleighl has given a unified theory for the steady secondary motion, valid both inside and outside the boundary layer, in connection with certain acoustic phenomena 2 of Kundt's dust tube. Schlichting has applied his theory to the periodic flow generated by a circular cylinder oscillating along a diameter. The existence of the steady streaming flows which persists both inside and outside the oscillatory boundary layer was mathematically established. Photographic evidence of the streaming was observed in air by Andrade3 and Holtsmark, et al and in water by Schlichting.2 The Reynolds number is defined as U2/wv, where U. cos wt is the cylinder velocity. For large values of the Reynolds number, as has been pointed out by Stuart,5 there exists a second, outer boundary layer. The nonlinear inertia terms are not negligible within the outer layer at the edge of which the steady velocity component along the surface tends to zero. Other related works include West,6 Westervelt,7 Andres and Ingard,8 Nyborg9l10 and Segal.11 For the periodic boundary layers in a fluctuating flow, Lighthilll2 has studied the response of skin friction and heat transfer rate to small oscillations in the main stream, Lin13 and Lighthilll2 have independently treated high-frequency oscillating flows by means of the theory of differential equations.' More general case in which the stream fluctuates both in magnitude and 14 in direction has been investigated by Gibson.l The method of series expansion was used by Horil5 to solve various oscillation problems. The exact solution for the fluctuating flow past an infinite flat plate with uniform suction is obtained by Stuart 16 Unsteady Blasius flow has been treated by Moore,l7 and Cheng and Elliottl for low frequency case and by Illingworth19 for compressible flow of both high 20 and low frequencies. For the flow near a stagnation point, Wuest20 has investigated the response of velocity in the boundary layer when a flow impinges on 1

a wall which is oscillating parallel to the stagnation line. Glauert2 and Rott22 studied the two-dimensional flow against an infinite flat plate making transverse oscillations in its own plane. Solutions are obtained for small and large values of the frequency as well as for the whole frequency range. Later work pertaining to the effects of periodic boundary layer on both natural and forced convective transfer phenomena include Jackson, et al.,23 Fand, et al.,24 Byley, et al.,25 Kesten, et al.26-28 Clark, et al.,-2 Eshgy33 and Na.34 The present work, which consists of two parts, is devoted to a study of problems pertaining to the effects produced on forced convection flows from harmonically fluctuating stream and from rotational oscillation in two-dimensional laminar boundary-layer flow past cylinders. The perturbing effects are respectively from a potential flow and rotational oscillation of small fluctuating amplitude and low frequency. Both induce fluctuations of velocity, temperature and concentration as well as the secondary or steady-state alternations in the forced convection boundary layer. The governing differential equations are linearized through the use of the perturbation technique with the first three terms retained. The first terms being the case of steady-state forced convection are the classical problem. The second terms are the frequency response of the fluid velocity, temperature and concentration. The third terms consist of two components, one which is harmonic with twice the frequency of flow oscillation and one which is time independent and gives rise to a net change in the steady-state values of the shear stress and the rates of heat and mass transfer at the wall. To solve the transfer equations for low frequencies, Lighthill12 used a Karman-Pohlhausen method and Hori15 used the method of Blasius and Howarth. At high frequencies, since viscosity is only effective for oscillation within a very thin shear-wave boundary layer closed to the wall, the theory of differential equations with a large parameter for high frequency approximation was applied independently by Lighthill12 and Lin.13 For the present investigation, the method of Blasius and Howarth is used. By means of this method, the transfer equations may be solved by power-series development from the stagnation point, without any arbitrary assumptions regarding the velocity, temperature and concentration profiles. Furthermore, the solutions as expressed in terms of the coefficients representing both the geometrical configuration and the nature of the flow or rotational oscillations and the universal distribution functions may be applied to any two-dimensional flow. Also treated are the effects of uniform suction or flowing imposed on the cylinder surface. Theoretical analyses include the shear stress, and the rates of heat and mass transfer for a.symmetrical blunt body and a sharp-edged body. Numerical results are obtained for flow around a circular cylinder with flow oscillation, fluctuating circulations and rotational oscillations. 2

II. THEORETICAL ANALYSIS Part Ao Oscillation of Free Stream 1. THE FUNDAMENTAL EQUATIONS The physical, system, consists of a heated cylindrical body around which flows fluctuate harmonically with time. A coordinate system x*, y* is fixed at the forward stagnation point, with x* measured along the cylindrical surface and y* in the direction perpendicular to the surface as illustrated in Fig. 1 for a sharp-edged bodyo The analysis is restricted to two-dimensional, incompressible flow in the x*-y* plane. The external potential flow is represented by U*(x*t*) = U*(x*)+eU*(x*) cos cL*t*o Where U*(x) is the time-average velocity, Ul(x) is the amplitude of oscillation, c* is the frequency of oscillation, and t* is the physical time. * The cylinder with surface concentration of Cw is heated to a uniform temperature, Two It is maintained in contact with a fluid at temperature Too, and concentration C, which otherwise would flow with a constant velocity Uo at infinityo The following assumptions are imposed on the analysis: (a) The velocity components are small compared to sonic velocity so that the compressibility effects are negligibleo (b) In the temperature boundary-layer, the viscous dissipation and the heat generated by change in pressure may be neglected. (c) The differences in temperature and concentration are not so large that the physical properties of the fluid vary from point to pointo (d) For problems involving suction or blowing, the flow through the surface is assumed to be wholly normal, since the pressure gradient through the surface is usually large. With these assumptions, the boundary-layer equations for flow, temperature and concentration read (the fundamental equations for parts A, B and C are all presented here, although the statements of the problem for parts B and C will be given later). 3

Fig. 1. Oscillating flow past a sharp-edged body. 4

xu av Tx + =.. ax ay for both flow oscillation (Part A) and fluctuating circulation (Part B) of free stream au &~ au i a2u - + u + v -.2 +< 6t 6ax 6ay Re 6y2 Uo O dx for rotational oscillation of cylinder surface (Part C) at + uT *t ux at aX aT 1 62T + V- y ay RePr ay2 6c 1 2c + y ReSc ay The boundary conditions are 0 for Parts A and B y = O: u = LUl(x) cos wt, r for Part C U for Parts A and B y = ~o for rt C L(x) for Part C- r r = C = 0,.without suction or blowing V = V positive for uniform blowing and negative for uniform suction = C = 1, where the external potential flow is U(x,t) = Uo(x)+~Uz(x) cos wto Because of the identical form of the equations as well as boundary conditions for temperature and concentration, both may be treated simultaneously, In the following analysis one may, therefore, interchange the quantities T and C, Pr and Sc, and Nu and Sho 2. SOLUTIONS Considering the nature of the velocity of the potential flow U(x,t), which is imposed as perturbations, the following forms may be assumed for 5

the functions involved: u(x,y,t) = Uo(x,y) + Eul(x,y,t) + 2u2(xy,t) +... v(x,y,t) = Vo(x,y)+ EVl(xyt)+ ev2(x,yt) +.. T(x,y,t) = To(x,y) + eTl(x,y,t) + e2T2(x,y,t) + o. By substituting these expansions into the governing equations and boundary conditions and by separating terms according to the powers of en, a set of simultaneous, linear differential equations and boundary conditions are found as follows: Zeroth-order perturbation, e0 +vo = o 1 ox +y buo auo O v6 O a = 1 o 2uo dUo Re ay2 dx aTo 6To x= 1 62To u -+ - = u0 x + V ay RePr ay2 with the boundary conditions >(1) y = O; uo = To = 0 0 and vo = V without suction or blowing positive for uniform blowing and negative for uniform suction y = o; uo = Uo(x), To = 1 6

First-order perturbation, E1 ax + ay = 0 ~x ~y ur U+ Ul at. ax + U0 - + vo ax + v auo ay dUo. \z dx + Uo l cos dx/ + Uo. + Uj ax cut - wUU1 sin ut (2) aTo -- + V 6x 6TO V 6 1 a2Ti RePr ay2 with the boundary conditions y = O; ul = vl = T1 = 0 y = o; ul = UI(x) cos w0t, T1 = 0 Second-order perturbation, 62 u2 V+ 2 ' 6x by au2+ U2uO 2 ut+ u2 - + + a-V ~ x:..~ — u1 a + v2 a ax ay au2 +0 ' v i1 ay _ dU (1 + cos 2ct) 2 dx 1 62u2 Re 3y2 i(3) T2 + ' 2 + at ' x ax + + + U - + V x 0 +To + V2 cT: ay + T, + V1 - ay 1 2T RePr ay2 with the boundary conditions y = O; u2 = v2 = T2 = 0 y = o; u2 = T2 = 0 Here the value of e is chosen small such that the first three terms of the 7

expansion will approximate the physical problem, The best analytical method available at present for the solution of the nonlinear partial differential equations of the boundary layer such as Eqs. (1), (2), and (3) is that of the power series. Frossling (1938) has discussed the series solutions of the steady boundary-layer Eq. (1) for several types of outer velocity distribution Uo(x). For a flow over a symmetrical blunt body, Blasius (1939) has introduced a symmetrical velocity distribution for Uo(x) as 00 Uo(x) = a2k+ x (4) k=O Later Howarth (1940) has extended the Blasius series solution to an unsymmetrical velocity distribution. 00 Uo(x) = ak x (5) k=O For a sharp-edged body, Gortler (1941) has adopted 00 m V k(m+l) Uo(x) = x ak x (6) k=O as the external flow. The corresponding solution of the heat transfer problem with the G'ortler series development has been given by Sparrow (1942) and Wrage (1943). The Blasius technique was further generalized by Frossling (1938) using the very general velocity distribution 00 Uo(x) = xm ak xk (7) k=O for a two-dimensional velocity field, where m and M are arbitrary numbers. The velocity distributions described by Eqs. (4), (5) and (6) are special cases of this general form. The outer velocity distribution of Eq. (7) represents approximately a wedge flow Uo(x) = aoxm in the vicinity of x = 0 when the wedge angle is 2mc/m+l. In the following the solutions of the boundary-layer Eqs. (1), (2) and (3) are obtained in two forms: an ordinary Blasius series for a symmetrical blunt body and a generalized Blasius series for a sharp-edged body. 8

ae Ordinary Blasius Series Solutions for a Symmetrical Blunt Body 1. Solution to the Zeroth-Order Perturbation The zeroth-order perturbation is the case of steady-state forced convection, Frossling (1957). The dimensionless distance from the wall, defined as r = (alRe) /y,y is selected as the similarity variable. The equation of continuity is satisfied by the introduction of the stream function defined by u = 6*/6y and v = -88/3x. This definition of the stream function is employed for all other orders of perturbation. Using Eq. (4) as an expression for the external velocity distribution around a symmetrical blunt body, the solutions for the stream function and temperature are obtained as: 1/2 o(x, n) = (a, O~~ii Re (foi x + 4 as fo3 X3 al + 6 a5 fo5 x5 + ai (8) go(xB) = Foo + 4 2a Fo2 x2 + 6 a5 Fo4 4 +.., al al where 2 fo5 = fo51 + a3 fo52 a!as fo7 = fo71 +.as fo72 + 23 a 73 ala7 ala7 and 2 Fo4 = Fo04 + -3 Fo42 alas 2 F07 = F071 + 3a5 Fo2 + -3 F ala7 ala7 A system of simultaneous ordinary differential equations and appropriate boundary conditions for the functions f and F described presented in Tables I and IIo 9

TABLE I EQUATIONS AND BOUNDARY CONDITIONS WHICH DEFINE AN ORDINARY BLASIUS SERIES SOLUTION FOR VELOCITY IN OSCILLATING FLOW PAST A SYMMETRICAL BLUNT BODY s 0 For f Equation for f,,+,,f,f'2 = -1 0.01 01 0101 01 fo3 fo3+f01fo34folf o 03fof = -1 f051 f ol+folfosl-6f&foiS+5flfosi = -1,51 1 5 0 f os2 f-6f os o = + 8( 63-fo3) 81n fin ti^ Wl -^ol ii-^i+f f foi + 2 r oli 1 f f,,, +f f,, _4ff f',+5ff, ft +f (3 f f i4ft ff +If f 1), 113 113 01 113 113 113 2 03 03 03 111 03 111 03 111 4 'r1l2 f12i fof'I+f01f"-2f' fo ll+fol f- f1 1 01 121 01 121i 111 20t fl210 6 210 1 1 01 20 01 201 01 20 1 0l1 1 fll f f I" +f f " -4fo fa +Sfo fs = -2f +f t -+ f ' ( 3ff 21tf 20St 2ts 1ol 01 2 0 2S1 01 2S1 201 11 01 1 2 01 01 1i f f "' +f f " -4f' f ' +3f f = 2f f ' + "f f ' "(f '+ ff )+3f ' (f' + 1 f" 2tS3 2ts3 ol 2it3 ol 2ot3 01 2tS3 20o3 03 2tl 11s1 03 2 03 113 01 2 0 1 - fo11 )(fof ) f3 (f0o + l) - 03 + Tlf'3) - +f.f2,,,.,I, f03 _ t 4 4 Vr f f+ f +f f -2f f +f' f " 2f ft 1 f+f f ) f ft 1 (f _+Tf () f f 22t 221t 221t 01 22tl 01 21l1 01 221i 12 1 01 2 0 2 2 1 01 + 2 i + f21( f f f "' +f f " -4f' f ' +3f f = 2f ' +4f f ' +f - f + " f ' (f ' + t f" ) 22t3 22t3 01 22t3 01 22t3 01 221t 3 03 2 1tl03 1 03 22 1 121 03 2 03 123 01o 2 111 713 1 2 1 0l 3 0f3) 2 123 ol) 2fl23 (f01 0fl) + f2f (* ~~ + ^ ft ) + f f + Ti f f 2ltl 2 01 i 1 01 f f '" +f f" -4f f ' +3f "f -4f 'f -3f f -f" f +4f ' (f '+ f ") 22t3 22t3 01 22t3 01 22t3 01 22t3 03 22tl1 03 22tl 03 22tl 121 03 2 03 + 4f I (ft + f" ) +4f I f I - - f ( + f 0 3) - f 1 ( ' + o if ) - f (3 f i i) - 01 (3f" + if'"1 ) - 5f 12 f -+f f " f 2 03 03 121 123 121 123 2lt3 2 Boundary conditions for f f(O) f(O) f'(X) Oor~V 0 1 o o 1/4 o 0 1/6 O 0 0 O 0 0 00 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 Equation Number Used in the Computer Program 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 0 0 0 0

TABLE II EQUATIONS AND BOUNDARY CONDITIONS WHICH DEFINE AN ORDINARY BLASIUS SERIES SOLUTION FOR TEMPERATURE OR CONCENTRATION IN OSCILLATING FLOW PAST A SYMMETRICAL BLUNT BODY For Equation for F Boundary conditions for F F(O) F(oo) 0 1 Equation Number Used in the Computer Program 15 H H Qo Foo -foiFoo = _Fh Pr Fo02 2foF02 - 3fo3Foo - folFo2 = P Pr Fo41 -foiFo41 + 4folF041 - 5fo41Foo = P Pr F042 -folFo42 + 4folFo42 - 5fo42Foo + (2fo3Fo2-3fo3o2) = 3 Pr 811 F110 -folF{lo - (111 - o = F112 2folFll2.+ 2fflF02 - 3fosFilo - folFo2 - 3flls3Fc - fllFo2 + F 2 = PF Pr 2 Pr F Fr (12 F120 -f121Foo - folFI20 + F1lo = io Fl22 -3fo03Fl0 - 3f123Fo - fl21Fo2 + 2folF122 + 2fl2Fo2 - fo 22 + F12 = ~~~~~~~~~~~~4 ~~~Pr 92os F2oso -folF0oso - 2 (Foo+'-Foo)(foi+1foi) - f20szF6o = FPr F20S2 2folF20S2 + F02 (1 + f + 2f20sFo02 - 3fo3F2oso - folF20S2 - 3 (FoO+'IFoo) (fO3+rlf3) - -r - t F20S2 - (Fo2+1lF02)(foi+qrfoi) - f20S3Foo - f20SlFo2= s 921t F2ito - _. (Foo+IFoo) - _m (fol+ifoi) + %Foo - f2ltlFoo - folF2ito 2 2 Pr rF2t2 2foiF2it2 + 2f2it. Fo2 - ifoFlto f- t - f Foo - f2ltlFo2 + 3foiFo2 + 2 o1 + 0 fo) F112 fis(F~o+=F'&) ~1 fll(F.m+nFm) (3~ L 2 F'l 21120000 2l - fl - l(F2+TF2) (fo3+lo3)Fllo - (fol+rlfo)Fo2 + lFo2 = PF 2 2 2 2 Pr 1 F F Fso 922B F22SO -folF22SO - f22S1FFo - fl2(1 o) (foF0l+ro)Fl20 + flF = c2 2 Pr F22S2 2f22slFo2 - 3fosF2280 - folF22s2 - 3f22s3FOO - f2aslF02 + 2foiF22S2 + 2 foi + -1 fol F;12 - 2fillFl2 3 f,,,(FCF,+rlFoo)- t 1F " 2S 2 fil(Foo+Foo) f- L (Fo2+Fo -2(fo3+lfo3)Foo - (fo+rlfol)Fl22 + 3fll3F10o + flllF112 = F 2 3FIl ) 2 2 fo+fF1 lF Pr I 't 1 F22tO 922t F22to -folF22to - f22tlFoo - - fl2iFooo+TFoo) - (foi+Tlfol)Fi20 - f111iF1 + F2lto = F 2 2 2 Pr F22t2 2f22tlFo2 - 3fo3F22to - folF22t2 - 3f22t3Foo - f22tlFo2 + 2folF22t2 + f + flF 2 +2 + Il fol F122 + 2fllFii2 - 2 fos(Foo+TlFoo) - fl2 (Fo2+rFo2) - 2(fo3+lqfo3)Fl20 - (fo+ fo )Fl22 - fll3 - f 2 + F2 = - 3fllsFilo - fillflm2 +- F21t2 2 Pr 3 0 0 0 0 0 0 0 sO O 0 'O 0 0 0 O 17 18 19 20 21 22 0 0 23 24 0 25 26 0 0 0 0 27 28

2. Solution to the First-Order Perturbation In the first- and second-order perturbations, most terms depend upon the functions of solutions of the zero-order perturbation. For low frequencies, the velocity and temperature components are expanded in terms of frequency as follows. It is convenient to adopt complex notations and to write ul(xyt) = real [[(ulo(x,y)+(ia)ull(x,y)+(ica)2Ul2(x,y)+.. ]eic)t] v(x,y,t) = real ([(vlo(x,y)+(icD)vlI(x,y)+(i)U)2V2(xy)+. ]eit} (9) Tl(x,y,t) = real-([(Glo(x,+(i)11(y)+(i)(x,y)+(i)2g12(x,y)+... eict The new functions ulo, u o ejlo0.o0 are now complex quantities and independent of timeo Substituting Eq. (9) into Eq. (2) and observing that the symbol 'treal" appears in front of every term, it is disclosed that since "real" is a linear operator, it can be dropped out of the equations. All terms are now linear in eilt which can also be dropped outo In this manner, time dependency is omitted from the differential equationso Arranging the time-independent equations according to the powers (~ = 0,1,2, o etc.) or io), one obtains: 0th order approximation, (icu) ~u — + o ax 6y uo -U-s _ + + uv. + V A + v.x ax by by Re by2 Uo dUa. + U! E for = dx dx +< U - uo1 for, = 1 (10) u- (g.i) for ~ > 1 ___ + aG _" E__ le _ ___ 1 - r2O, for =O I x +v. + Voy. y RePr ~y2 -( ), for > 12

with the boundary conditions I= O; U1 = v1i = Q,1 = 0 rl = 0; ule = Ul(x) for O = 0 0Q1 = 0 0 for S > 0 where ~ is the order of approximation, (ic). Let us consider a particular case of two-dimensional flow about a fixed symmetrical blunt body, when the fluctuations in the external flow are produced by fluctuations in the magnitude but not the direction of the velocity Uo of the oncoming stream relative to the body. The latter, which is called the fluctuating circulation,l5 will be treated later. If the fluctuation of the oncoming flow velocity is U0(l+Eeict), then the external flow about the cylinder will fluctuate by the same factor and may be expressed as U(x,t) = Uo(x) (l+eeiWt). (11) This suggests that one is concerned with the case Uo(x) = U~(x) in this section. The solutions to the first-order perturbation are obtained: f2 o +y for =0 J - < 1 (aRe)1/2ajK x + 4 a3 fl,3 X3 + 6 a- fjgs x5 +), a1 a l (12) 1 Y _ for I = O QUl2 =y o 2 - F ^o + 4 3 FY2 x2 + 6 5 Fai4 x al where fjl and F11 are universal functions as in Tables I and II, respectively. 3o Solution to the Second-Order Perturbation In the second-order perturbation for u2(x,y,t), v2(x,y,t), and T2(x,y,t), the convective terms of the governing equations will contribute terms with 13

cos2 wt. These, in turn, can be reduced to terms with cos 2ut, steady-state, i.e., time-independent terms, Schlichting (1935). u2, v2, and T2 for low frequencies may be expanded as follows: sin 2cot and Therefore, u2(x,y,t) = 1 real (u2os(xy) + (iM)u2us(xy) + (iu) 2U22S(X,y) +.0. 2 + [uaot(x,y) + (io)u21t(x,y) + (ic)u 22t(xy) +... ei2t ] v2(x,y,t) - real (v2os(x,y) + (ico)v21s(x,y) + (iw) V22s(,) +. 2 >(13) + [v2ot(x,y) + (i()V21t(x,y) + (i()2V22t(x,y) +... Jei2t) T2(x,y,t) = real (G2os(x,y) + (ico))21s(x,y) + (icn) 22s(x,y) +... 2 + [Q2ot(x,Y) + (ic)021Lt(x,y) + (i(o)2@22t(xy) +... ]ei2t The equations of continuity, momentum and energy, which are obtained by substituting these relationships into Eq. (3) and by separating according to frequency- and time-dependency, are as follows. Although the analysis has been extended to include the first- and second-order approximations, only two resulting time-dependent (t) and time-independent (s) equations corresponding to the zeroth-order approximation, (io)) are presented hereo They are: (j = t or s) 6u2oj aV2oj +... - _ O ax ay au2oj aulo Uo ~ —Z- + UIo -X — 1 Re O~ 20j 6o0 uo -- + ulo ax ax auo 6u20j au-o auo + u20j - + Vo -y- + v ~ + V20j X 6"j ~ ay 6y Vy 2U20 j ay2 dUl + U1 -y dy 6ao ao20j ao10 aGo + U20oj -+ vo + Vo V20+ V - ax a; ay 6' 6 1 a2 yoj RePr ay2 (l14) 14

with the boundary conditions 1 = 0: U20j = V20j = 920j = 0 T = o U20j = 20j = 0 where j refers to either the steady or transient state. The solutions to the second perturbation are as follows; (aalRe)l7/2a (2 jl x+4 23 x3 + 6 as f2ej5 X5 +. j = 1 (15) =2j - F2ejo + F2j2 X2 + F2 j4 +...a a~l a, where the functions f2ij and F21j are defined in Tables I and II respectively. b. Generalized Blasius Series Solutions for a Sharp-Edged Body The Blasius technique, as demonstrated above, may be applied to obtain the solution of oscillating boundary-layer flow problems for symmetrical bodies with blunt noses. It is now desired to produce a more general solution from which several special cases of interest could be obtained. For example, the body has a sharp-edged nose or the fluctuations in the velocity of oncoming stream is both magnitude- and direction-dependent such that Uo(x) is different from Uj(x). In a paper by Hori15 concerning unsteady boundary layers, an external flow 00 Uo(x) = x ak k=O superimposed by an oscillating component with amplitude 00 U1(x) = xn bk x k=O has been studied for a two-dimensional velocity field by a Blasius technique and several universal functions have been defined. In this section a more general velocity distribution as expressed by Eq. (7) superimposed by a fluctuating component with amplitude 15

00 \-I Ul(x) = xn bk xMk k=O (16) will be considered, wherefore the coefficients bk depend upon the type of flow oscillation. The solutions thus obtained for fluctuating velocity and temperature in the unsteady boundary layers will be in the form of a generalized Blasius series. Since the velocity distribution, Eq. (7), represents approximately a wedge flow in the neighborhood of the forward stagnation point when the wedge angle is 2mg/m+l, the solutions may be applied to fluctuating flow problems about a sharp-edged body. Solutions for the velocity and temperature components are obtained following the same procedure as presented in the preceding section for a symmetrical blunt body. The zeroth-order perturbation is the case of steadystate forced convection[ Frossling.38 The dimensionless parameter, defined as 1 = y(aoRe) l/ xm1/2, is selected as the similarity variable. The solutions of equation are obtained as ro(x n) = 1 x (aoRe) l2 00 m+1/2 kM ak fok(2)x, k=O Mk x Fok.. (17) T = - o 00 k=-O The functions fok and Fok, fi.k and Fljkj and f2ejk and F2ejk as defined respectively by Eqs. (17), (18) and (19) must satisfy the equations and appropriate boundary conditions given in Tables III and IV. The solutions to the first-order perturbation for U1.(x) defined as Eqo (16), are obtained as 2n+(Ua+l)(1-m) 1 = (aoRe)1/2aoa x 2 and 00, ~ bk f ik x k=O (18) 00 bCO Y~1 F~ekn-m+ (l-m)+Mk iQ = b 7a FFl ik x k=O

TABLE III EQUATIONS AND BOUNDARY CONDITIONS WHICH DEFINE A GENERALIZED BLASIUS SERIES SOLUTION FOR VELOCITY IN OSCILLATING FLOW PAST A SHARP-EDGED BODY For f Equation for f l+1 " 12 To fol fo+ -fol- f - m 2 2 fo2 foi + m+2 folfo - (2m+l)fofo2 + - fofos - (2r+2) fo0l '3o + - folfo3l - (3m+2)folfo3l + 5 ofo = - (2m+2) ~~~~~~~-m m+1oxfos+5 - +M -2 +1 2 fo32 fo2 + folfo2 -(2m+)flfo32 + +5- fo - (m+l) + m f m+3 ftio + - folfoi fi + -fOo2 211 - l + fo o = ( *10 flol fl + 1fo f 1- (n+l)folfo, + 2n+l-mfo foi = - (m+n) -1 flom1 f102o f1031 f1032 f1033 f1034 *11 fll f1121 1122 fll31 ~1131 f1132 f1134 fll34 *12 p121 f1221 f1222 f1231 f['oz1 m+l " ( ' ' + 2n+3-m " fi022 + I+l.I.... 1031 + — 1 folf10o21 - (n+2)fOlfl021 + 2n3r fOlflo21 = floo3 + 1 flfo223 - (n+2)folflo03 + 2n+ folf = 2 2 ft" + m" 01ffl - 2n+5-m rfof +o f1032 + folf12031 - (n+3)fofl~31 + 5 folflO31 = flo32 + folflo3 - (n+3)foflo32 + 2-5- folflo2 fli03 + 2 folf1"031 - (n+3)foifl032 + fj l fif.03 l = 2 2 1 o3 (n+4)foif^32 + 2n-o-m f1033 + 2 f0lf1033 - (n+3)folf0343 + 2- folfo134 = 2 2 flt+ 0 - (n-,m+2)f fll4, + n m f2 lff10 +2 1 folfl034 - (n-m+3)fofo a + 25- folf13 = f1'21 + mj1 folfi'22 - (n-m+3)foif2ll + 23 5 foifi22 2 7 folfll2 + (n-m+ fo " 2n+5-5m " flo31 + folf1134 (n- +3)flfo34 + 2-m fOlf34 fl~l + 2folfll2- (n-3m+3)folfl~ nm5,fll fl122 + m-1 foifl.22 - (n-2m+3)fOlfla22 + — 5m-7 folfl..22 2 2 ~ m +l ' 2n-3m+7 5 113 flz23 + folf 112 - (n-m+4)f' lfl + 2n f113 + 0 133 - (n-+4)fozfl3 +..2 folfl2 2 2 ', m+l " t I 2n-3m+7 f113 + -~- folf133 - (n-m+4)fOlfll33 + f01lf113 2 2 + "1 " I 2n-m7 f1131 +- fof1121 - (n-m+4)folf13 + 2 M4 22 2 2 f1of2 + 2n.folf121 - -m+37"folf,21 + fOlf121 I ~~2 (n ~2 '"( m+l "'2n-m+ " f1133 + -~- folffll3 - (n-m+4)fojf113s + fol2n-m+ 2 2 '" m+l "2n-m f'" f12z4 +-2-folf1224 - (n-m+4)folfl224 + 2n —2+7 f12i +"M-folfz231- (n-2m+5)folfl1+ 2-31 m+ fo11231 2 2 - (n+m+l) 2n+l-m,f - fo2fio0 + (n+2)fo2f0o fo2flo- - (n+m+l) 2 2 - (m+n+2) - 1 fo2flo2i + (n+3)fo2flo22 - 2n+-m f02fl021 - (n+m+2) 2 2 - -+ fo2fl022 + (n+3)fo2flo22 - 2nm fo2fl022 2 2 - fo03fo + (n+3)fo3fzoi 2n+l m fo3flol - (n+m+2) 2 2 101 - 1 = f102 - 1 = _- m+ fo2f'll + (n- m+)fo21 - 2n-3m+3 f M+3~'s - - 5 — 2lll = 113 - 1 f2f1121 + (n-m+4)fo2fl21 - 2n-m+5 fo2 =- m+ fo2f1122 + (n-m+4)foaf122 - 2n-m+5 fo2f1122 2 2 m+3 " I2n-3m+5 - -- fo3flll + (n-m+4)fo3fll - 2n+ fofll3 2 2 fill = - fll2 = -+ fo2fi21 + (n-m+4)fo2fl2l - 2n-r+5 fof 2= fl2 -123 Boundary Condition f(O) f'(0) f 1() O 0 1 O 0 1 O 0 1 O 0 0 O 0 1 O 0 1 O 0 0 O 0 1 O 0 0 O 0 0 O O 0 O O 0 O 0 0 O 0 O O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 O O O 0

TABLE III (Continued) For Equation for f Boundary Condition f(0) f'(O) f'(o) f1232 f1233 f1234 I20s f20s8 f20821 f20s22 f20S31 f20 32 f20833 f20s34 f20o35 I21t f2ltl fl22 + 2 folfl232 - (n-2m+5)folfl232 + 2n- f+ 232 = 2 2 f1233 + m+ folf1233 - (n-2m+5)fOlfl233 + 2n- 2+9 33 = +o'" m+l "(-m-)xfs.+ 2n-5m+9 fl234 + -- folfl 3234 -(n-2m+5)fOlfl234 + folfl234 2 2 8 + o (2n-m+ l + n ff20 "' m+ ' ' 4n-3m+2 " f20s82 + -- folf20s21 - (2n-m+2)folf20os2 + 2 folf20821 f2'S22 + 2 1 oif20822- (2n-m+2) foif20s22 + 2 foif20822 f2'03 + of20 (2n-m+)to2 +4n-3m+ f23 f20s32 + l folf2032 - (2n-m+2)folf20s32 + 2 fo0f20832 f20533 + - foif20s33 - (2n-m+3)fof120s33 +- foif0s33 2 2 +1 it " 4 4n-3mn+5 f f20ss3 + 2 folfs2083 - (2n-m+3)folf20s83 + 4n- folf2os3 2 2 "' + " '4n-3m+5 f20S3s + -+ folf203s - (2n-m+3)folf20s3s + - foif2083s 2 2 '"+1 "' ' 4n-3m+5 f2o3s + +l folf2ossr - (2n-m+3)foifzoss3 + 2 folf2oss 2 2 m+l fO n-2+ ' ff 4n-3m+5 fost + 2+l folfost35 - (2n-m+3)folfosst + 2 fOlf35 faltl + --- folf2ltl - (2n-2m+l)f 'f'lt l + 4n-Sm l " m 2n-5m+7 f"" - m+ fo2fl221 + (n-2m+5)fO2fl221 - 2 f2231 2 2 m+5 fo3l'21 + (n-2m+5)fo3fml2 - 2n-5 fo3fl2 2 2 - m+ fo2f222 + (n-2m+5)fo2fl222 - 2n fm 222 2 + 2 f~inf 1222 -mflol + nflolf0lo - n 2 - m3 fo2f2os1 + (2n-m+2)fo2f20s1 - n-3+ f2f20 2 2 2n+l-m "2n+3-m 2n+ fo02 + (2n+l)flo0fio2 - 2+ flolflo2 - (2n+1) - -m fo2f20s82 + (2n-m+3)fo2fo20s2 - -3 fo2f202 2 2 = 23 fo2f20s22 + (2n-m+3)fo2f20s22 - n-3 f2f222 2 2 - - fo3f201s + (2n-m+3)fosf2osl 4n-3m+1 3f2 2 051 2 2n t2n-m+3 "' _2 - ml fioif3o0 + (2n+2oifi03 _s- 2nm flolflo03 - (2n+2) 2n-m f102f102 + (n+l)flo2flo2 - (n+l) 2 ~ 2n-+l-m ff 4n-3m+3 ' ' 2n-3m+3 " ' 2f2081- 2 flOllll + llfll flofll 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O O O O 0 0 0 0 0 0 0 0 0 0 O O 0 0 H Go fit 4n,-5m+3 f m+ fo1f1ltl+ (In-2m-2)fozfzltl f2lt2l f2lti2 + ml folf2lt2l - (2n-2m+2)folf2at2l + n 3 folf2lt2l - 2f20o2 - +3 fo2f2ltl + (2n-2m-2)fo2f2it 2 2 2 4n-5m+l ' t 2 ~ 2 fo2f2ltl "' m+l 4n-5m+3 2n+5-m 4n+5-m 2n+5-5M f2lt22 f2lt22 + folf2lt22 - (2n-2m+2)folf21t + l2t22 + 2 folf 2 = - fllf-2 + 12 2 2n+3-m 4n+5-m' 2n+3-3m 2 fll2f121 + fl2fl2l fll2fl2i 2 2 2 m+l " I 4n-5m+5 t ' m+3 "' ' f2lt3l f2lt3i + m- folf2it3i - (2n-2m+3)folf2it3i + 2 nm folf2t3l = 2f2o3 - - fo2f2lt2l + (2n-2m+2)fo2f2lt2 4n-5m+3 fo2f21t2l 2 f2lt32 f2lt32 + - folf2lt32 - (2n-2m+3)folf2it2 + 5 flf2t32 = foif2lt22 + (2n-2m+2)f2f2lt22 _ 4n-5 2fm+ 2222 2 2 2 2 Irt m+1 ' n-5m+5 m+5 of2t + fit 4n-5m+l fsfit f2lt33 f2'l't22 + — 2 folf2lt33 - (2n-2m+5)folf2lt33 + n-m5 - fof2t333ltl + (2n-2m+3)fo3f2ltl 32tl fnt ftii + 2 2 20 2 m+l ' 4n-5m+5 2n+l-3m +-3m f2lt34 f2't34 + - foif2lt34 - (2n-2m+J)folf2lt34 + folf2lt34 = - 2 flolfl23 + 4 f olfs13 - 2 flOlf113 2 2 2 2 2 2n+5-m 4n-7-3m 2n+3-3m " f113f121 + - fll3fl2l- -23- fll3fl2l 2 2 2 fzzt35 '"s m+l " I I +4n-5m5 " 2n+5-m ff 4n+7-3m ' 2n+5-3m flf f2lt- f2t35 2 + folft3S -(2n-2m+3)folf2lt S2 + 4n-Sm+5 f flofll f12 2 2 2 2 2 O O0 O O0 O O0 O O0 O O0 O O0 0 0 0

TABLE III (Concluded) For Equation for f HO - ip'" fly -m+fl p" 4n-7m+5 " 2n+l-m 4n-5m+5 ' ' 2n-5m+5 f22S f22Sl f22sl + — folf2 - + 221 f11f1o 2 +2 + flolfll - fo11f121 ~~~~~~~~2 2 2~~~~~~~2 2+ - f-5mi, - 2n+3-3m I 2n+3-3m fl21fl2_ f-21f 2282l 2 + fof2221 - (2n-5m+4) foif22s21 + n 7 fof22 = 221 + (2n-5m+4) fo2f22l - 4n275f5 fo2f22Sl f22S22 f22S22 + - folf22S22 - (2n-3m+4) folf22S22 + 2 folf22S22 = - f1 22 + fo f fl22 l223 f2 + "2 1 223n- - ( 2n-3m+5)fo + 4n-5m+75) 2222 - 7 fo2f2n22 f22S32 f22S22 +- -folf22322 - (2n-5m+5)folf2222 + folf22s233 + (2n-m+ 8 fo3f22 f22S34 f 2 - fo 2 2 + 2 _ 2n+ -m 4n-5m+5 2n-5m+5 2n-3m flllf3 (2n )fl 3 2n5 f22S31 f22S31 + fof223 - (2n-3m+5)ff223 + f22S3 - f f2 2f2 2 fO2f22 + 2n+5-3m " ' "_ n-7m ' m+5 22t f2,2tl f22tl + 2 folf22Stl - (2n-3m+5)folf22tl + -7.9 fof223tl = - M. f0. flolf(2l + (2 3.5 flof2 -o3f22 flfOf22S 2n+5-m 4n- ' ' 2n-5m + 5 " 2n+ 3-3m " 2n+7-3m ' 22t21 f22t2 1 + 2- fof122t23 - (2n-m+4) fo t + = f111f113 - (2n-n-m+4)5fo2f22tl + 2 fo2f22t rn+i " t I n-7mn+9 " - 2n+3-m " 4n-5m+9 I I 2n-5r+7 f22t22 f22t22 + 2 folf223t22 - (2n-3m+4)folif223t22 + m7 folf223522 = 2 fl0lf22t2 + 5m7 flo2f22 - 52 fLlfl22 2n+5-m fl22 + - flo2fl 2 f2fl2 2 flllf2 + (2n+4-m)f f2 2 m flfl + 2ft3 f22t32 f22t32 + 2J3- foit22t32 - (2n-3m+5) fo f22t32 + 72 9 foif22t32 = _ S fo2f22t22 + (2n-3m+5) foif22t22 _ 7mi7 fo2f f22t33 f22t33 + - -foif22t33 - (2n-3m+5)fof-22t33 + 9 folf22t33 = - -5 fo3f22tl + (2n-5m+5)fof22t -4 n-+5 f22t 22t f22t3 f22t + 2 folf22t - (2n-3m+5)f olf22t3 + 7m+9 folf22t135 = 2f2t - 2 f10o21f122 + 29 fOfl2 - fo2fl22 2n+5-3m I 2n-51+5-3 2n+'-m It 2n+1-2m ~~~~~~2 1 2 22 f2 'f"s i " ' 4n-7m+7 m+3. 4n-7m+5 t f22t2s f22t2s + 2-fofs22t2 - (2n-3m+4)folf22t2s + 2 ifOlf22t2s - O2ff22tl + (2n-3m+5)fo2f22ts - 2 fof22s m nm+i 4(n-5m+ t t 2n+l-m " 4n-5m+7 ' 2n-5m+7 f22s22 f22s22 +"~ 03. n- fol3+4)s2+ -2-7Mn+7 -+)fofs1+f22t2 Lf10fsf22t2 fOlfls22 2n+3-m + 4n-5m+7 ' ' 2n-5m+5 " 2n+3-3m " ' ' 2n+5-3m " 2 f102f121 + ~2 ~ f102f121 2 f2osfm2 2 fm11f1x3 - (2n-3m)f11ifii2 2 f2 fm2 - m+ii ' 4n-7m+9 " 2 + " -m + ''nf22t3s f22ts3 + — folf22t3s - (2n-3m+5)folf22tss + foTf22t3i = fO2f22t21 + (2n-.m+5)fo2f22t2l -72 + 2if2ot3 f22t32 L22t32 + - oif22t3 - (2n-3m+5)fouf22t32 + n-r+ folf22t32 - TO2f22t22 + (2n-m+5)fo-f22t22 fO2f22t22 2 2 2 "' -i-i.. 4n-7m+9 m+5 "n-7m+5 424 2t + - i - 2- + t + n-7m+9 2n+ l-m " 4n-5m+9 ' 2n-5 m++9 t ft4 f22t. + M folf2"2t32 - (2n-3m+5)folf22t4 + 2 folf22t34 2 2101f123 - l ~fOlfl23- + f1o~ f12 2 2 2 2n+5-m ' n-5m+9 2n-5m+5 2n+3-3m 2n2m' ' 2 f 2ifi21 + m fo3f121 - 2 lOSflll - ~lll1 ll3 111113 2 lll1 ll3 "'2t35 mf 235 + LolL22235 " '2n-5m+5)ifo~f22235 + a4n-7m+9 " 2n+5-m " 4n-5m+9 2n-5m +7 f22t35 f22t35 +-M~- folf22t35 - (2n-3m+4)fomf22t3S + 2 folf22t35 = - 7 f~2f12tm + (2n-3m+4) f~2f12 - 222 - 2 2 2 f 2 2 2n+5-3o " 2n+5-3m " ff+- fof -mf112of112 + 1 f ll2ft f t 2 2 Boundary Condition f(O) f'(O) f'(o) O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0o 0 0 o 0 0 0 0 0 o 0o 0 0o 0 0 000 000 000 000 000

TABLE IV EQUATIONS AND BOUNDARY CONDITIONS WHICH DEFINE A GENERALIZED BLASIUS SERIES SOLUTION FOR TEMPERATURE OR CONCENTRATION IN OSCILLATING FLOW PAST A SHARP-EDGED BODY For F 9o Foo Foi Fo21 Fo22 '10 Floo Equation for F 1 " m+l ' Foo + - fo1Foo = 0 -- Fol + - folFol - foFol = - foFoo Pr 2 2 1 " m+l ' m+3 - Fo22 + -- fo1F021 - 2folF021 = - fo2Fo + f2Fo Pr 2 2 1 1" m+l 5 Fo22 + M+- folF022 - 2folF22= fosFoo Pr 2 2 1 ". m+1 ' f /F 0 -(.4;lo 2n+l-m flolF*o FO + -- folFloo - (n-1 ~Floo Pr 2 ~~~~~~2 Boundary Conditions F(O) F'(-) 1 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 0 Flol1 Fl012 Fl013 F1021 F1022 F1023 F1024 F1025 Flo26 F1027 11 F110 F1111 F1112 F1113 F1121 F1122 F112 Fills 1 " m+l Floll + - folFloll - Pr 2 1 " m+l Pr F1012 + - folF1012 - Pr 2 - F1022 + -2 foFl022 - Pr 2 02 1 " m+l F1021 + - folFo102 - 1 " m+l F102o + -2- folF0o24 - Pr 2 Fl025 + -2- foiFlio2 - 1 " m+l2 Fl02s + - fo1F1o02 - Pr 2 1 " m+l ' F-o24 + - fo0F10o4 - 1 " m+l ' F1025 + - foiFio25 - 1 " m+l - Flo26 + -~ folF01102 - Pr 2 1 " m+l ' F1027 + -- folF1o27 - Pr 2 (n-m+l)folFloil (n-m+l) fo1F112 (n-m+l) folFio3 (n-m+2) folFlo21 (n-m+2) folFl022 (n-m+2)folFio022 (n-m+2) folFo23 (n-m+2) foFo024 (n-m+2) folFlo2 (n-m+2)foiFo026 (n-m+2)fo1Fi0o27 2n+l-m floF + fIo Fo 2 - = fo2Floo + (n-m)fo2Floo = _ 2n+m flo2Foo 2 2n+l-m t 2= - 2 flo0Fo2 + 2flolFo2 2 = - fo2F1011 + (n-m+l)fo2Fioll 2n+3-m 2n+3-m f021Fo + fo02Fo1 = m5- fo3Floo + (n-m)fo3Floo 2n+5-m f Foo - - 2 2 m+3 2 - fo2F1012 + (n-m+l)fo2Fo021 2- fo2F10o3 + (n-m+l)fo2Flol3 2 1 - 2n+3-3m5 -Floo - 2- 3 flllFoo 2 - 2n+5- flllFol + flliFol - Fo01 2 m+3 1 fo2F110 + (n-2m+l)f2F110o 2n+5-3m f Fo 2n+3-3m F102 o fnlFo02 * 2flllfo2 2 2 fo2Folll + (n-2m+2)fozF1ll1 - f2n+5-m 1121 fll2FOl 2 Flo + M- fo'Flo - (n-2m+l) foFlo = p- Flli + — 2- foFlmlo - (n-2m+2)foiFmi Pr 2 ~ F11l + ~ — folF111 - (n-2m+2)fozFllll Pr 2 1 F1112 + m+ folF112 - (n-2m+2)folF,112 Pr 2 LFip11 + - folF1li2 - (n-2m+2)fo1Fll1 Fills + 2 folFll22 - (n-2m+2))folFi22 Pr + 2 F121 + --- fo0F1 i - (n-2m+5)fo0Fl121 Pr 2 1 nm+l F1122 + -- folF1l22 -(n-2m+3)fo1Fp122 Pr 2 1 rm+l Fls1 + - fo0F112s - (n-2m+3)fo Fs1123 0 0 0 0 0

TABLE IV (Continued) ro H For F F1124 Fll25 F1126 F1127 @12 F120 F1211 F1212 F213 F1221 F1222 F1223 F1224 F1225 F1226 F1227 @20o F20os F2os11 F20S12 F20813 F20os21 F2os22 F2oS2S F2os24 F2os25 Equation for F F1124 + - (n-2m+3)fo F1124 = - - fo3F1lo + (n-2m+l)fo3Fllo Fr 2 2 i " m+l ' ' 2n+7-3m fzlsFoo F1125 + - folF1125 - (n-2m+3)folF1125 =- 2n+3m Fr 2 2 1 " m+l ' ' m+3' Fl126 + - folFll2e - (n-2m+3)folFl-2S = - fo2Fll2 + (n-2m+2)f2F1112 Pr 2 2 L F1127 + - fO1F-l27 - (n-2m+3)folFln27 = fo2F 0113 (n-2-2m+2)fo2F1113 L F12+ m — folFL20 - (n-3m+2)fo1F120 = F110 f121oo Pr 2 2 F1211 + - folF1211 - (n-3m-)foIF1211 = F1ll - 2 f121Fo0 + fl2F - Fl2l2 + 2- fo1F1212 - (n-3m+3)foiF.212 = - fo2FL20 + (n-3m+2)fO2FI20 p Fl222 + - folFl222 - (n-3m+4)fomFi222 = - 23 F 02l2l1 + (n-3 2F Fr 2 2 r F1213 + - folFl213 - (n-m+4)folF2213 = - f22F + 1 I m+t l 2n-5m+5 ' Pr Fzl4 + 2 folFlzzZ - (n-3m+4)fozFl224 = - -2- fo3Fl20 + (n-3m+2)fo3Fo20 Pr Fl221 + - fo0lF221 - (n-3m+4))folFI221 = - 52 f2 3Foo F1 l22 2 1 n-i+) - 2Fo P F1222 + - folFl222 - (n-3m+4)folFl222 = - fo2Fl211 + (n-3m+) o2Fl213 -"F2os + ' foF20S - (2n-2m+l)foF2os = - 2 fF + (n-m+l)foFo - m+ r F22 + - folF22 - (2n-2m+l)foF2os2 = - fo2F21 + (2n-2m)fo2F2so 202 -Fr 2 2 2foF2s3 - (2n-2m+l)foF2os3 2 + (n-m)fl02 -F20S21 + foF20S2 - (2n-2m+2)foF2s2 = 2 l02 + (n-m+2)fF2 fsFo + 2f2osFo2 1 F20S23 + " foiF20S23 - (2n-2m+2) fomF20B23 = _ 2n+ fio2Floi + (n-'+l) flo2Fo F2S24 + M folF2224 - (n-3m+4)folF2os24 =- 5fo3F20so + ( n-2m) fo3F2oso F12025 + ~ foF12025 - (n-3m+2)folF20S25 = 2-F f 2 13n1 Fr 2 2 m+l 'm+3 ' F1227 + — fo0F1227 - (n-3m+4)fo1F1227 = - - f02F1213 + (n-3m+3)fo2Fl213 1 + m+l 2n+l-m 4n-3m+l r F2oso + folF20so - (2n-2m)folF2oso 2 fllFl~~ + (n-m)floF.l~~ fnosoFoo Fr 2 2 1 " m+l 2n+l-m 4n-3m+l F20OS1l + -- folF20os - (2n-2m+l)folF2oslI = 2 flo0Fll + (n-m+l)floFil1-L f2OSIF01 + floslFol 1 _ m+3 4n-3m+3 Fr F20os2 + - fo0F2os12 - (2n-2m+l)fo1F2osl2 - -- f02F20o0 + (2n-2m)fo2F2oso - f2oszFoo 1 "l ' 2n+3-m - F20s13 + -R- folF2os,3 - (2n-2m+l)folF2os13 = - flO2F1ooo + (n-m)flo2Floo 4n-3m+3 L F20s23 + - - f0oF0osl - (2n-2m+2)fozF o5 2nfst f1oFo + Fr 2 2 2 2 fI o s F n+5 o i " m+l 2n+5-m ' - F2os25 + - foF2os25 - (2n-2m+2)fo1F20s45 = - fosFons Boundary Conditions F() F'(oo) O O O O 0 0 0 0 O O O O O O 0 0 0 0 O0 O O O0 0 0 O0 0 0 O0 O O O0 0 0 0 0 0 0 0 0 0 0 0

TABLE IV (Continued) For F Equation for F F212 F2os2e + foIF2s2 - (2n-2m+2)foF2os2 = - fo2F20812 + (2n-2m+l)fO2F20812 n3 203Foo Fr 2 2 2 1 1 n+1 ' r+ ' F20o27 - F2027 + - fO1F2os27 - (2n-2m+2)folF2o027 = - fo2F2osl3 + (2n-2m+l)fo2F20s13 Pr 2 2F 1 m+l ' 2n+l-m ' 2n+3-3m 821t F2ito 1 F21to + - folF2lto - (2n-3m+l)folF2lto = 2F20 o 2 flo1Fllo + (n-2m+l)flo1Fllo 23 fllFloo Pr 2 2 + (n-m)flllFloo - 2n f22tFoo 1 " m+l ' 2n+l-m 2n+3-3m F2itll -F21tll + 2 folF2ltl - (2n-3m+2)folF21tl 2F2o 2 m olFFl + (n-2m+2)f 2n3 Pr (m2 201 2 2 4n-5m+3 i + (n-m+l)folFlol - 2 f2ltoFo1 + f2+toF1 2 Boundary Conditions F(O) F' (m) 0 0 0 0. 0 0 0 0 1 t + l F21t12 L F2ltl2 + - foiF2it12 - (2n-3m+2)foiF2lti2 = Pr 2 F2ltis F2lt21l I) ro F2lt22 F2lt23 F2lt24 F21t25 F2lt26 F21t27 922s F22SO F2iti3 + - folF2ltl3 - (2n-3m+2)folF2ltl3 = Pr 2 1 " m+l - F2t2l + folF2lt2i - (2n-3m+3)folF2lt2l = Pr 2 4n-5m+ + (n-m+2)fl11Flo2 + (n-m+2)flloFlo2 - 2 - F2lt22 + - folF2lt22 - (2n-3m+3)foiF2lt22 Pr 2 F21t25 + folF021t2 - (2n-3m+3)folF2it23 = 1 F2lt24+ + folF21t24 - (2n-3m+3)folF2lt24 Pr 2 -F21t2 + 2 folF2lt24 - (2n-3m+3)foiF2it24 i " m+l ' - F2t26 + - folF2lt26 - (2n-3m+3)folF2lt2 = Pr 2 1 It m+l ' F21t27 + - folF2lt27 - (2n-3m+3)foF2lt27 = FsoPr 2 F22eO + -- folF22O - (2n-4m+2)folF22SO Pr 2 3+m 4n-m+5' - m fo2F2ito + (2n-3m+l)fo2F2lto - f2lt2Foo 2 2 20+5-rn 2n+5-3m 2 - 2n3-m flo2Flo + (n-2m+l)flo2Fllo - +5- 3 f112F1oo + (n-m)fll2Floo 3 f20tlFo2 + 2f2otoFo2 m+3 ' ' 4n-5m+5 ' ' - fo2F2ltll + (2n-3m+2)fo2F2itll f2lt2Foi + f2lt2Fo1 2 2 2n+-mr 2n+5-3m ' 2 flo2F2ltl + (n-2m+2)flo2Flll 2 f12Fo10 + (n-n+l)fll2Flo - 2 2 - m+ fo2F2iti2 + (2n-3m+l)fo2F2it2 m+5 - fo2F2to3 + (2n-3m+l)fFo2F2to 2 2n+5n-m ' 2n+7-3m 2 2 M+- fozFzltl2 + (2n-3m+l)fo2F2ltxz --- fo2Faltl3 + (2n-3m+l)fozFaltxs 2 flomFm2o + (n-3m+2)flolFm2o + 2n+f3-3m fmxFiio 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 I 2n-5m+5 I 4n-7m+5 - (n-2m+l)fi1,Fllo - 2 n f121Floo + (n-m) f121F1oo - 2 f22sFoo 2 2 1 m+l ' ' 2n+1-m ' 2n+3-3m F22sll F22Sl + - folF22sll- (2n-4m+3)fF22Sl = 2 flolFl21 + (n-2m+3)flo1F321 + f3lFill'- (n-2m+2)fillfn 2n-5m+5 fl2lFlol + (n-m+l)fllFol 4n-7m+5 f22SiFo + f2sFo F22S12 F22812 + 2- folF22S12 - (2n-4m+3)fiF222 = fF2250 + (2n-4m+2)fo2F22SO f22S2Foo Fs 2 2 2 1 " 2+3-r ' 2n+3-m 2n+5-3m F221s3 F22S13 + m- folF22s13 - (2n-4m+3)foiF22S3 = 2 flo2Fo20 + (n-3m+2)flo2Fl20 + 2 fl2Flo - (n-2n-m+ 7 o + (n-m)f - (n-2m+l)fll2Fllo - f122F10oo + (n-m)f122Floo 2 0 0 0

TABLE IV (Continued) For F Equation for F i F" m+l ' 2n+l-m I + 2n+3-3m ' F222s2 F22S21 + - fo1F22s21 - (2n-4m+4)foiF22s21 = fllFl22 + (n-53m4)foiF122 + fll - (n-2m+3)fliFn12 - 2' f121F102 + (n-m+2)fl2lFl02 f22sFo02 + 2f2281Fo2 Pr 2 2 2 F 12222 I 5+1 " - 'Fm+F + 4n-7+7 2222 F2222 + f01F2222 - (2n-4m+4) foF22s22 = F2s22l + (2n-4m+3) fo3F2281 2 f22sFol + f 22s2Fo F22B23 p F22s23 + 2- folF22s23 - (2n-4m+4) folF22s23 = - 23 f22 + (n-3m+ ) flo2Fl20 + 2 - (n-2m+2)fl2F11F - 2 f12lFl01 + (n-m+l)fl2lFlo, F22S24 F22B24 + - fOlF22S24 - (2n-4m+4) fF22S24 = - 5fo3F22so + (2n-4m+2)fo3F22so Pr 2 2 2 1 " m+l 2n+5-m ' ' 2n+7-3m ' F22s825 p^ F22S2s + -2- folF22s25- (2n-4m+4)fOlF2282S = - 25 m flo3FL20 + (n-3m+2)flo3Fl2O + 2 3fll3FI1o ' 2n-5m+9 ' PT 2 2 2 - (n-2m+1)fll3Fllo -2n 9 f123F1oo + (n-m)fl23Floo 1 m+l ' ' - ' F22tn p F22S2e + -- folF22s2 - (2n1-4m+5)folF22t2 = 23+t fO-F2n4lm oo+F2+ + (2n-4m+)fF2s 2 F22s27 F22S27 + 2 foIF22S2- (2n-4)o 122s27 = fo2F22sl3 + (2n-4m+3) fo2F22s 3 -(22t F22to -r F22to + 2- folF22to - (2n-4m+2)folF22to = 2F2lto - 2 f10F20 + (n-3m+2) flo2Fi20 - 2' flfFlo + (n-2m+l)f1lF _ 2n -2m5 f o121Fo00 + (n-m)f12F1oo 7m5 f2tFoo F22tll F22tll + - folF22tll - (2n-4m+3)fOiF22tll = 2F2,t2 - 2 filooFoo + (n-3m+3)flOOF121 - 2n33 flmFll 2 - (n-2m+2)filFll2 -_ 2ni5m5 f121F0l + (n-m+l)fl2lFlol - 4n-7+5 f22tlFol + f22tlFol 1 " F2 F22t2l2 - F22ts2 +2 folF22ts22 - (2n-4m+34)foF22t22 = -23 fo2F22tl + (2n-4m+2)fo2F22to - 2 f22t2Foo F22t s 1 M+1 I F- - 2n-m — f + (2n+5-3)o fF11o F22t3 F22t3 + Ft- (2n-4m+3) fF22t3 = 2 f102F120 + (n-3m+2)flo2F,2o - 23- f zl2Fllo 2n-5m+7 t + (n-2m+l)f1l2Fllo -2n-5+ f22Floo + (n-m)fl22Floo F22t2 2 2ot2 2 O1~2t2 "+ m+3 f ' I n-7m+7 Prf 2 5 ~~ 2n-'I)fol2Fl2at~ + ( 2n+2-m 2n+5-m ' F2 p Feetie + — 2 — folFeetll - (2n-4m+4)3)fo2F2 2 2n-5m+l 1 + (n-2m+2)fll2F1ll 2- f122Flol + (n-m+l)fl22Fllo F22t24 2t24 + -2 folF22F22t 24 2 3F22to + (2n-4m+2)fo3F22to ~22I24 Pr 2 - 2 Boundary Conditions F(O) F'(X) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

TABLE IV (Concluded) For F Equation for F 1 r" ' 2n+5-m I I 2n+7-3m F22t25 F22t25 + 2 fo1F22t25 - (2n-4m+4)folF22t25 = - 2n5 flo3Flo + (n-3m+2)flo3F20o 2n+ f3F Pr.2 2 2 + (n-2m+l)fll3Fllo - 2n5 fl22Foo + (n-m) fL2Floo 2 M1~~~~~~~~~~~~l~~~~~ I' 4n-7m+9 F22t2e -F22t2e + 2- folF22t2e -(2n-4m+4)folF22t2e = fo2F22tl2 + (2n-4m+3)fo2F22tl2 f22t3Foo Pr 2 2 2 F22t27 L F22t27 + - foiF22t27 - (2n-4m+4)folF22t27 = - foF222t13 + (2n-4m+3)fo2F22tt13 Pr 2 2 Boundary Conditions F(O) I'(o) 0 0 0 0 0 0 ro 4:

The functional coefficients flik and Flik are defined by the equations split up as fll, fzlll + albo fli2,aob j albp a2b a2bo l2 = aflb 2221 + a1f + flR23 + fl24 aob2 ao b2 aob2 Fij. = F2ll1 + - F3212 + b Fly13 ao bo with the appropriate boundary conditions. Similarly the solutions to the second-order perturbation are as follows: 4n-m(22+3)+(21+1) 0 ) = fIbo2 2 Nk ao~+ (aoRe) \a o/ k=O (19) 00 2 b F, (n-m+m) + (l-n) +N ao - 92ij = ao ---2 F2jk x k=O 3, SOLUTIONS UP TO SECOND ORDER For both cases a and b, the velocity components are 1/2 - / u u = uo + e[(uIo-Wu 2)2 + (Wull)2]l/2 cos 0t + tan1 ( (Du1 -L &lO -w2Ul /J 2 2 2 2 /2 + - -(U2os-tO U22s) + [(u20t-w U22t) + (cu2lt) ] x 2 cos I2ct + tan-li ( u2lt i1 U20ot-2 U22t/ J -,. (20) 25

1/2 V = Vo + e[(v1o- 2V12)2 + (v)11)2] COS g2 + - (-V2os-(0 2V22s) + [(V2ot-C2v22t)2 + 2 cos L2(ot + tan-L ( itV2Pt ) Y20t-2V22t J The dimensionless shear stress, defined as T = yO. y = O ' wt + tan-1 ( vl )1 LlO-(u V122J (av~t) 2 ]1/2 x is T = O + Uo 2 _ +l2/ cos t+tan- 1 ay LKl y yy ay f O wy ul 4YY ayU] 2+.20 2 k-y.2 L- U20t 2 au22t 2 2 Lay ay + (d) ~U2J' t X 1/2 (~ o y/ ~-E-) (21) x 6U2It cos 2cit + tan-1 ( ~" u2oot _2 u2 2t 6 ^y 6y The steady-state shear stress is Eq. (21) without the second and fourth terms on the right-hand side of the equation as the periodic terms will contribute nothing when integrated over a.whole cycle. The mean position of separation may be obtained by equating the steady-state shear stress to zero. The temperature is: 26

T = go + E[(elo_-(i2l2)2 + (g,,ll)2]l/2cos [t+tan l(Cw)j/l4ow0(- 212))] 2 ~ ~ 1/2 + 2 ((20os e @22) + [(@20t 2 22t )2 + (cD21lt)]2 (22) 2 cos [2)t + tan-l(( @2lt/( 2ot-' 22t2t)) ] The local Nusselt number, defined as Nuy y = o y =0 is obtained as: Nu o = + L(~o 2 ~ ^)fl21/2 x T o6y y 6y COS Lt + tan-l y/( - 2 ' L K V Ky 22 (23) + 2 222 aZ~t (C) 21 21l/ +2 + a L X cOS 2cwt + tan'l ae 0 l a2 ( )) ~L K y Ky 6y Again the periodic terms contribute nothing to the steady-state value. 4, PERMANENT ALTERATIONS IN THE WALL SHEAR STRESS AND HEAT TRANSFER RATE CAUSED BY FLUCTUATIONS IN THE STREAM VELOCITY Equations (20) and (23) are interesting and significant results. The former indicates that a flow oscillation induces a steady-secondary or streaming motion; the latter shows that the streaming flow resulting from the oscillation causes a change in the local Nusselt number from that corresponding to steady forced convection. The second-order contributions to the alterations in the wall shear stress and heat transfer rate are obtained from Eqs, (15), (19), (21), and (23) as follows: 27

For case a AT ( Re)1/2 (Re) /2 ANU (R)i/' 2 2alx f"(o) + 4a3 f(o) x2 +.. -(. f"(o) + f"() x2+ 2O1 2 0S1 a 03........ 22S 1 a1 22S 3 (24) 62 %aQ (0) 2.4 a/ _ Ft(o) + F3 )2TF'(o) +... '(o) + F'(o) x +. 2 20SO al 20S2 \1,- \'22SO al 22S2 (25) For case b AT (Re) 1/2 2 E 2 00 b.2 \1-O (ao)! 2 k=O 00 2 (I (-l)7 f"(~o) a6,, K \0ao.../ 2 (2) sk: a~/ 4n-m-1 4n-m1 +Mk+2y(l-m) x (26) ANu (Re)l/2 E b 2 00 00 2y k=0 7=O 4n-3m-1 +Mi+2y(l-m) 2 x F'(o) 2(2i) sk (27) Part B. Fluctuating Circulations of Free Stream This section is devoted to the study of the effects of fluctuating circulation (or free-stream oscillation with a magnitude- and direction-dependent amplitudes) on the transport phenomena. The perturbing force is that due to the flow circulation with a small fluctuating amplitude and low frequency in an otherwise forced convective field. One example is a uniform flow about a heated bluff body with a trailing row of alternating vortices. This induces oscillations in transport phenomena. - The governing differential equations and appropriate boundary conditions are identical with those of Part A for the fluctuation in the free stream. However, the time-average velocity Uo(x) is different from the amplitude of oscillation Ul(x) in the external potential flow. Only the case of flow over a.symmetrical blunt body is treated here although the analysis may also be extended to the flow over a sharp-edged body. When the circulations around a symmetrical blunt body fluctuate, the amplitude follows the formula 28

oQ U-l(x) = b2k x, k=O (28) where the coefficients b2k depend on the nature of the fluctuating circulations. Ul(x) is an even-power series, since only the velocity fluctuations of the potential flow caused by a fluctuating circulation are considered. The solution to the zeroth-order perturbation, the case of steady-state forced convection, is identical with Eq. (8). The solutions to the firstand second-order perturbations are obtained as follows: The solution to the first-order perturbation: *,I (X, ) - 1 (ai e)Vrai [bo f'.+ 3b2..f12 x2 + 5b4 fL"4 x4 +..] (29) GlQ(x,r ) = 21 1+1 [b2 a.i+ l:l'x::+,,.,2b'a F1.i2 x3,+... ] where fl 2 = flU21 + a3 fl 22, alb2 fle4 = fl41 + fl42 + a fle43 + a f f44 alb4 ab 4 a b4 1444 and = FB + a3bo F = F1b + Fb212 alb2 Fj-3 = F1S31 + 32 ' F3 + 33 + L.a F134 alb4 alb4 a~b4 Th. e.p The solution to the,.second-order perturbation: 29

*2Qj (aiRe) a Lf221x+ f2 jl xx3 +2a3 fj35 +X ( a', Re) J- 1 cal x 5. 2a j - 3bob2 al'+ + a3 F2ej2 x + 35 F2a j4 x +.] al al where f2ajl = f2Qjll + ba f2lj12 alb2 f2Qjs = f2Yj33 + boa3 alb2 f22j34 + ab f2 + a b f2j3 + a3bo a3b2 a3b2 35 and F2Qjo. = F2ejol + boa3 F2ajo2 alb2 = F2J2 + boa3 alb2 F2aj23 + a3b2 F22j22 + alb4 F2Lj24 + a3sbo a3b2 asb~ F2E j2s a3b2 The functions fiki and f2ejk, Fl1k and F2Qjk are defined by the differential equations and appropriate boundary conditions presented in Tables V and VI, respectively. The velocity and temperature profiles and the local wall shear stress and Nusselt number are defined by Eqs. (20), (21), (22) and (23), respectively. The permanent alterations in the local wall shear stress and Nusselt number caused by the streaming motion due to fluctuating circulation are obtained as AT 3E2bob2 + boa3 \ 812i abo i + 2a3 F" _ 2 N^a fJ20Sl l + bf20 s2 - 22Sl f22 + as [20s33 alb2 ajbOa2b2 alb2 ab2 Nf/Be 2 ~az ' x + as o3" + Sb f2O32 + ab 83 + fS~ 2 () (2ss alb2 asbo a3b2 a3b ab2 (31) a3bo if.... alb4 "2 asbo " ] 2 + f 22s34 + a12 f22332+ f2S31 + a5bf alb2 a3bo a3b2 a3b2 1 =0 30

and ANu 3 E2bOb2 + a3b0 F 2(w( + a3bo ' 21 F20 S0o2 I +-) F250o012 + 2SO F22so),Re 2a12l L alb2 a alb2 + 2 [F2021 F222 + ab 3 + F2os24 + a F osF2 (32) al, a3bo alb2 a3b2 a3b2 ' a3bo alb4 abo (-(a) (22S21 + b F22s22 + a0 F22S23 + J4 F22824 + F202 X2 a a3bo alb2 a3b2 a3b2 /J j Part C. Rotational Oscillation of Cylinder Surface This part is to investigate momentum, heat and mass transfer from a symmetrical blunt body (a cylinder) which performs a reciprocating harmonic motion about its own axis with small fluctuating amplitude eU1*(x) and low frequency in an otherwise forced convective field. Imposing on the analysis the same assumptions as given in Part A, the expressions for the governing differential equations and appropriate boundary conditions are obtained as shown in The Fundamental Equations of Part A. The resulting differential equations for the first- and second-order perturbations are respectively identical with Eqs. (2) and (3) without terms for Uo and U1. The corresponding boundary conditions are identical except those for ul of the first-order perturbation. They should be replaced by ul = Ul(x) cos ct at y = 0 and ul = 0 at y = co. The external potential flow Uo(x) and the oscillating amplitude Ul(x) of the cylinder surface are expanded into power series as expressed by Eqs. (4) and (28) respectively. Solutions for the components of velocity and temperature are obtained following the same procedure as described in Part A. These solutions in power series are identically expanded as those for the case of fluctuating circulations in Part B: Eqs. (29) and (30) for the first- and second-order perturbations, respectively. The differential equations and boundary conditions which define the functions f and F are given in Tables V and VI. The solutions up to the second-order perturbation are identical with Eqs. (20), (21), (22) and (23) for the velocity components, the dimensionless shear stress, tne temperature and the local Nusselt number, respectively. The expressions for the permanent alterations in the wall shear stress and Nusselt number are identical with Eqs. (31) and (32), respectively. 31

TABLE V EQUATIONS AND BOUNDARY CONDITIONS FOR THE UNIVERSAL DISTRIBUTION FUNCTIONS OF VELOCITY FOR FLUCTUATING-CIRCULATION AND ROTATIONAL-OSCILLATION CASES For 0o fol fo3 fos1 fo52 *loo floo f1021 f 1022 flooa f1041 fl042 f1043 f1044 *11 fllo f1121 f1122 *.12 f120 f1221 f1222 o20s f20 ll f20S 12 f20s31 f20o32 f20o33 Equation for f fo'l = -(fo) - folfol - 1 fO 3 = 4folo3 - 3fo1fo3 - fo01o3 - 1 fos'1 = 6foi fos1 - 5folfo51 - folfosi - 1 )2 1 fo52 = 6folf52 + 8(f3) - 5fofo052 - 8fo3f3 - folf052 - floo = folfloo - folfloo 1,t!,. 1,,. fl02 = 3folflo21 - 2folf0l21 - folflo2l 1 f1022 = 3fof1022 - 2folf1022 - fo0o1022 + 4(fo3fl00o-fo3o) 1 f1041 = 5fof1041 - folf1041 - 4folf041 - 1 "II! I I ' 11?11 if flo42 = 5foflo042 + 12fo3F1021 - folf1044 - 7.2fo3fo121 - 4fozlf044 - 4-8fo3f021 - 1 "t t I t 11 f1043 = 5folflo43 + 6fo5lfoo - folf1043 - 4fof1043 - 1 'It I I t I I I it it.. It fl044 = 5folflo44 + 12fo3f1022 + 6fOOfOs5 - fOlfo044 - 7.2fo3f1022 - 4fol0f1044 -4.8fos3fl22 II t *it t flo = folfllo - folfllO + floo - 1 I I I I II Ii f1121 = 3fOlf121 - 2folf1121 - folf1121 + fl021 - - II 11 1, I I t, 1 f1122 = 3folf122 - 2folfn22 - folf122 + f1022 + 4(fo3f1o-fo03flo) fl20 = f1o + fo0f120 - folf120 f1221 = 3fof221 - ff - fl 2fo1f1221 + f1121 rIft I I II,, I It T fl222 = 3f1olf222 - 2folfl222 - folfl222 + f1122 + 4(fo3f120-f03f120) nII t I I " 2 f20S11 = 2folf20s11 + 2f oof1o21 - folf20Sll + 2floof1021 - f1lf20o11 - ' n. I t I t, 1,1 f20812 = 2folf2os12 + 2f100f1021 - folf20s12 + 2f1oof1022 - fo1f20s12 it I' 10 I " 10 " "t 2 20S31 = folf20oS31 + ~- oo -o~20S31 - iooao141 - 3o1~2os31 - IllI2 It It 1 f20832 = 4folf20S32 + 3(f102)2 - folf20s32 - 53f1021f1021 - 3folf2s32 - "' ' ' ' ' 10 ' + I - f20s33 = 4folf2S33 + 8fo3f20811 + 3 fl00fl042 + 6fO12flo22 folf20S33 - 6fo3f20s11 - 3f021~1022 - " f 10 " " " I - 3f100f1022 - 3 flooflo42 - 2fo3f2os11 - 3folf20s33 Boundary Conditions for f f(O) f'(0) f~o) O or t V 0 1 O 0 1/4 O0 1/6 O 0 0 O 0 1 (0) (1) (0) o 0 1/3 (0) (1/3) (0) O 0 0 0 0 1/5 (0) (1/5) (0) O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 0 0 0 Equation Number Used in the Computer Program 1 20 21 3 4 5 22 23 24 25 6 7 8 9 10 11 12 13 26 27 28

TABLE V (Concluded) For f Equation for f mVtI t 10 t 1 f20S34 f20834 = 8fo3f20sl2 + 4folfoss34 + - flooflo44 + 3(f1022) - folf2os34 - 6fo3f20S2 - 3f1022 10 " " " -- f10ofo144 - 2fo3f2os12 - 3folf28s34 it I 1 10 " 10 " f20835 of208s = 4folf2ss35 + f 100f1043 folf20435 - f10ofl1o43 3fo5f20sss tt I t t I I I it i a21t f2ltll f2ltll = 2f2osll + 2folf2ltll + 2floof1121 - folf2ltll - 2flloflo21 - folf2ltll f21t12 f21tl2 = 2f20s12 + 2folf21t12 + 2f lof1m22 - fof12t12 - 2f1o0fl122 - folf21t12 it I 1, I i, t t t I t I22s f22811 f22s11 = 2folf22s81 - fo0f22s11 - folf22s11 + floo0f221 + f20of1021 - f11ofl121 - f120f1021 - 110 - f1o00f221 + f110f1121 Itt t I 1 t t * * t, i f22Sl2 f22s12 = 2folf22stl - folf22stl - fo1f22s12 + f1oofl222 + fl20fl022 - f11of1122 - f120f1022 - f1oof1222 + fllofll22 Itt t t. I, * t * I I f22t12 f22t2 = 2fo1f22tl2 - f2t - fo1f22t2 + fofo2 folf + floof 2022 + f10fl + fof22 - f120f1022 - f100oof222 - fl2fll22 + f2lt12 Boundary Conditions for f f(0) f'(0) f'(o) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Equation Number Used in the Computer Program 29 30 14 15 16 17 18 19 NOTE: The terms underlined in the f equations are for the fluctuating-circulation case only. The boundary conditions in ( ) are those to be replaced for the rotational-oscillation case.

TABLE VI EQUATIONS AND BOUNDARY CONDITIONS FOR THE UNIVERSAL DISTRIBUTION FUNCTIONS OF TEMPERATURE OR CONCENTRATION FOR FLUCTUATING-CIRCULATION AND ROTATIONAL-OSCILLATION CASES For F go Foo Fo2 Fo41 Fo42 10 Flo11 Fl012 Flo 31 F3o32 F10os2 Floss F1os034 911 Fl111 F1112 912 F1211 F1212 o20s F20sol F20S02 F20o21 F20S22 F2os23 F20o24 F20S25 Equation for F F = - foFoo Pr FP =:2foFo2 o - ooo - oFo2 Pr F4 = -folFo41 + 4folFo41 - 5fo4-Foo Pr = 4 - folFo42 + 4foIF042 - 5fo42FoO + - (2fo3F02-3fo3Fo2) Pr 5 Pr Pr oF = 5 F- foFFl0o1 - 5fo 1 2Fo Pr 191 = 2fo3F1011 + 501F122 + 6f021202- 6fo0F011- 0 032- 6f-02F- 5f104200 Pr Fl = 2folF102 + 5foF0 + 6Fo-Fo3+ -6o2loF3 - 6foFo2 -oF03 - 3f5f04loo Pr 103 = 5f01F10 + 6f1041 - 011034 - 5f1043F0 Pr Pr = folFlo1 - folF0111 - 5fl2lFoo Pr F o = fo3F112 + flF o02 2l o - 6foF1112 - 512242 Pr Pr = fFl+F211 - 1 f olFlo33 - 6flo22F2 Pr = foloFll01 - folF 20l - f2012Foo Pr - 2f21F1012 + 2fonF12l2 1 - 5f20833F00 Fr = 2ffoF20S22 ++ 4f 21F1011 - folF1o2l - 23fil2 f1F Pr 202 2foiF20S23 + f1022F1012 4F1112F02 -6f03F20S02 - foiF02023 - 2fio22Floi2 - 2f20S12Fg2 + 2fiooFio33 - 5f20SS45F~O Pr Pr = 2foF2S24 - 012024 + 2foFo31 - 3f2SloO Pr = 2foF2 02 5~ + 42f l0oFo42 - o2F02 - 33F o F0Pr =2floF20525 - f0Fol25+ f F10 - f2SFoo FoPr = 2 f ooFlo - foFos -F 2 os-Foo,. Pr =2folF2oS24 - folF2os24 + 2flooF1o03 - 3f2oS31Foo Pr = 2fo1F20s25 - fo1F20s2s + 2floOFlO34 - 3f2OssFo0 o Boundary Conditions for F F(O) F(o) 0 1 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 O 0 Equation Number Used in the Computer Program 31 32 47 48 33 34 49 50 51 52 55 36 40 53 54 55 56 57

TABLE VI (Concluded) For F F21t F21tol F2lto2 922s F22S01 F22so2 t22t F22tol F22to2 F2lto1 Pr Fitoa F22S02 Pr F22ato Pr F22S02 Pr F22to2 Pr Pr Equation for F 2 ' 2 ' ~, = flloFloll + - flooF1111 + 2F20so0 - folF2tol - f2osl-Foo 3 3 = 3 foF1012 2 + 2F2 fooFll2 + F21to2 - fo20os12Fo = f12OF1 + flooF21 - floFi - folF22so1- 3f22SlFOO = f20F1011o + flOOF1212 - f1oF1112 - folF220s2 - 3f22s12Foo = f20Fo011 + f0ooF1211 + f1loF1111 + 3F2ltol - folF22tol - 3f228llFoo = fl2oF1012 + f100F1212 + flloF12 + 3F21to2 - folF22tO2 - 3f22s12iOO Boundary Conditions Equation Number for F Used in the F(O) F(o) Computer Program 0 0 41 0 0 42 0 0 O0 O0 0 0 43 44 45 46

III. CONVERGENCE OF THE BLASIUS SERIES In a brief comparison of the Blasius and Gortler series solutions for steady forced convection, Frossling38 has pointed out that the stress in considering the merits of various techniques and varients of the method lays primarily on (a) the simplicity of application, (b) the rapidity of convergence of the series, and (c) its generality. This stress will undoubtly apply to the forced convection problems in the unsteady boundary layers. Aside from the Karman-Paulhousen integral technique which yields an approximate solution to the problems, Lighthill,12 the Blasius technique offers the best simplicity of application. This is essentially due to the fact that the desired result is directly given if the expressions for the outer velocity distribution Uo(x) and its fluctuating amplitude Ul(x) are obtained either from practical measurements or theoretical considerations. The use of a generalized Blasius series as described previously has assured the generality of the Blasius technique. The problem left to be examined is the rapidity of the series convergence. As was pointed out by Frossling,38 there is no general method available for the determination of the radius of convergence. The quality of the series has to be determined by studying the behavior of the series when the number of terms involved is varied. The convergence of the series has been examined for the ordinary Blasius series solution for a symmetrical blunt body. No effort has been given to the convergence of the generalized Blasius solution, since it involves parameters m, M and n, The universal functions f and F were evaluated by means of an IBM 7090 digital computer for all three cases using the programs in the Appendix. Their numerical reduction was obtained for Prandtl number from 0, 7 to 10 and V of 0, 0.1 and -0.1. It is observed that the series solutions for all three order perturbations give good convergence after the first two terms, especially for small values of x. A fairly good convergence is obtained near the wall for large values of x. In general the velocity solutions have a better convergence than the temperature solutions for the same number of terms included. 56

IV. NUMERICAL RESULTS AND DISCUSSION The influence of external disturbances, such as the flow oscillation and fluctuating circulation of free stream and the rotational oscillation of cylinder surface on the transport phenomena is numerically examined for the simplest and most typical case in two-dimensional flow. It is a flow over a circular cylinder with the external potential flow Uo = 2 sin(2x). By expanding the sine on the right-hand side of the equation into power series, one obtaines a1 = 4, a3 = -8/3, a = 8/15, The functional coefficients f and F for symmetrical cases, as given in Tables I, II, V, and VI were evaluated by means of an IBM 7090 digital computer. Their numerical reduction was obtained for Prandtl number from 0.7 to 10 and uniform suction of 0, 0.1 and -0.1. Numerical results are presented in graphical form in Fig. 2 to 8 for the flow-oscillation in free stream, Fig. 9-to 17 for the fluctuating-circulation in free stream and Fig. 18 to 24 for the rotational-oscillation of cylinder surface. The results and discussion on heat transfer to be presented in the following may also be applied to mass transfer by replacing temperature by concentration, Prandtl number by Schmidt number and Nusselt number by Sherwood number. Figure 2 illustrates the profiles of the first-order velocity at two locations x = 0~05 and 0.2. It shows that the velocity component u10 approaches a finite value asymptotically at the outer edge of the boundary layer and negative portion of the profile exists for the velocity component ull. All these profiles are zero at the forward stagnation point and increase in magnitude along the cylindrical surface. Figure 3 is the frequency response of the first order fluid velocity inside the boundary layer. The amplitude of the velocity 37

x I 0.20 UO - 2 0.05 3 0.20 al Ull a_ U1 4 0.05 2 5 0.20 -, u 6- - 2 6 0.05.01.10 - ~.09 - 0.008.08.8.007.07H.7:3 N-.0! o.0.,.06 h.6: 0:3.05-.04.03.002.02 -.2.001.01.1 0 OL I 4 17 Fig. 2. circular Profiles of first-order velocities for cylinder. oscillating flows past a 38

_ x wo V 1 0.2 0 0 2 0.2 0.2 0 3 0.1 0.2 0.1 4 0.1 0.2 0 5 0.1 0.2 -0.1 6 0 ANYANY I, (D 0.6 w LLU N 0i a. 2 0.4 0.3 0.2 0.I1 -0.6 LLJ Ln Q: 0 w 0) C) I a. -0.5 -0.4 -0.3 -0.2 -0.1 O I 4 1) Fig. 3- Amplitude and phase of fluid velocity for oscillating flows past a circular cylinder

profile is shown to be less affected by the oscillating frequency, but varies very significantly along the surface. However, the phase, which lags the free stream oscillation in the neighborhood of the solid surface and advances in the middle portion of the boundary layer, is a strong function of the oscillating frequency and is less affected by the location x. The removal of decelerated fluid particles from the oscillatory boundary layer by uniform suction results in a slight decrease in the first-order velocity profile in the inner half of the boundary layer. It causes an increase in the phase lag in the inner half and a decrease in the outer half of the boundary layer. The effect of uniform blowing, which supplies additional energy to retarded fluid particles in the oscillatory boundary layer, is opposite to that of uniform suction. A comparison of the first-order temperature profiles at the forward stagnation point is given here in Fig. 4 for a fluid with the Prandtl number of 0.7~ The agreement between the present analysis and the results for the Hiemenz layer obtained by Pohlhausen and Lighthill (1954) is good. The frequency response of the fluid temperature, as illustrated in Fig. 5, shows that both the amplitude in the outer half and the phase lag in the inner half of the oscillatory boundary layer increase along the cylindrical surface. These phenomena are also observed when uniform suction is superimposed. An increase in the fluctuating frequency causes a decrease in the amplitude of the temperature accompanied by an appreciable increase in the phase lag. An increase in the Prandtl number results in the reduction of the thickness of the thermal boundary layer and an increase in the phase lag. The fluctuations in stream velocity produces a secondary (streaming) flow in the oscillatory boundary layero The streamline pattern of the steady secondary motion at quasi-steady state is presented in Fig. 6. The stream lines *ocs are in the direction of the main flow. The separation of the secondary flow is disclosed at x = 0o8, about 90 degree angle from the forward stagnation pointo This separation causes a reversal in the direction for the steady secondary component of the wall shear stress as shown in Fig. 8. The flow oscillations also induce the distribution of the steady secondary component of temperature in the oscillatory boundary layer. It is seen from Figo 7 that a2os for quasi-steady state its maxima located approximately at the middle of the oscillatory boundary layer. A zero e2os line intersects with the solid surface at zero %Os at a distance of approximately 0.7 from the forward stagnation pointo Downstream from that point is the region of positive 2.os distribution in the neighborhood of the surface. This results in the alteration of the net change in the local Nusselt number from negative to positive values as illustrated in Fig. 8. In other words, the flow oscillation causes the local heat transfer rate to decrease in the forward region but to increase in the region near the separation point. With an increase in the Prandtl number the zero isothermal line in the fluid shifts toward the forward stagnation point. This provides more solid surface which contributes to an improvement in local heat transfer rate.

--- Present Analysis Pohlhausen Quartic For. Hiemenz Profiles Layer ------ Lighthill Profiles UOy a x 0.3 e8o I \ // \ m- 0.2 o< -aI 8!, 1\ a~~~ ~-a o -, 0. 0 \ 0 0 I2 3 4 71 Fig. 4. Profiles of first-order temperatures or concentrations at stagnation point for oscillating flows past a circular cylinder for a fluid with Pr = 0.7 or Sc = 0.7. 41

Pr x w V I I 0 0.2 -0.1 2 I 0 0.2 0 3 1 0 0.2 0.1 4 1 0.2 0.2 0 5 I 0.2 0 O 6 10 0 0.2 0 0.3 0.2 ILJ w a 0.1 m -5.0 — vv, n -4.0 iJ ~ -3.0 LLJ Lw -2.0 -n I -1.0 a. N6 \ i, i I ~ I mmm I I I I 1 I 0 I 2 3 4 5 ') Fig. 5. Amplitude and phase of fluid temperature or concentration for oscillating flows past a circular cylinder.

0.1 - O.c 0 +E OSWt) 4 3 2 Fig. 6. Streamline pattern of the steady secondary cylinder at quasi-steady state. I 0 motion produced by oscillating flow past a circular

x 0.3 0.2 0.1 0 x 4 3 2 1 0 a X7 Fig. 7. Distribution of the steady secondary component ing flow past a circular cylinder for a fluid with Pr = of temperature or concentration produced by oscillat1 or Sc = 1 at quasi-steady state.

0.1 I.Or Pr V Ar I ANY 0.1 72 2 ANY 0 3 ANY -0.1 4 I 0.1 ANu 5 I 0 C 2 f 6 I -0.1 7 10 0 0.9 0.8 0.7 0.6 0.08 0.071 0.06 4 -\Jl N I. '1 0.5 0.4 c|u -? N 3 2 0.05 0.04 0.03 0. 3 0.2 - 0.02 0.1 k 0.01 OL 0 0.3 0.9 -0.01 4 6 Fig. 8. Effects of flow oscillation on the local wall shear stress and Nusselt number circular cylinder at quasi-steady state. for flow past a

From Fig. 8 one observes that uniform blowing tends to enhance the steady secondary component of wall shear stress. However, it is a well known result that blowing decreases the wall shear stress in a steady flow. Therefore due to the existence of a secondary (streaming) flow in the oscillatory boundary layer, the net effect of blowing on the reduction of wall shear stress becomes less effective in the oscillatory boundary layer than in the steady boundary layer It is shown that fluctuations in the stream velocity are capable of causing permanent alterations in both the velocity and temperature profiles in the alminar boundary layer. This provides a secondary flow and permanently alters the wall shear stress and heat.transfer rate. These changes are small and are detected from the analysis only when solutions are obtained to at least the second-order approximation beyond this solution for the steady forced convection problem. The superposition of uniform suction tends to increase the wall shear stress and the heat transfer rate in steady forced convection. These effects of uniform suction are suppressed in the neighborhood of the forward stagnation point with the introduction of the fluctuations in the stream velocity. It is disclosed from the numerical studies that the uniform blowing contributes to an increase in the heat transfer rate in a steady forced convection. However, Fig. 8 shows that it causes the steady secondary component of heat transfer rate in an oscillatory boundary layer to decrease near the forward stagnation point followed by an increase along the surface to the point of separation. Therefore the fluctions in stream velocity tend to suppress the contribution of blowing on the increase in heat transfer rate in the neighborhood of the forward stagnation point and to enhance it near the separation point. A Fluctuating Circulation In order to numerically demonstrate the effect of the fluctuating circulation, two different oscillating amplitudes eU1(x) are investigated: constant and equal to e and space-dependent and equal to either e(l+x2) or e( l++X 4)o Figure 9 gives the first order velocity profileso An increase in the magnitude with an increase in x is observed for all three velocity components u10O ull and u120 The effect of the space-dependency of the oscillating amplitude is less evident. This is not the case in the frequency response as illustrated in Fig. 10o The amplitude as well as phase depend on the nature of the oscillating amplitude of the free stream. The effect of the uniform blowing is to increase the amplitude and to decrease the phase lag, although not significantly. The influence of the oscillating frequency is important to an increase in phase lag but less evident to the amplitude. Fig. 11 shows that the first-order temperature profiles 10, 11 and 912 are all zero at the forward stagnation point and are strongly x-dependency of the amplitude and phase is significant, so is the effect of the Prandtl number. An abrupt change in the phase angle near the outer edge of the boundary layer is observed especially for the low Prandtl number fluid. The streamline pattern of the steady secondary motion produced by the fluctuating circulation 46

" 1 x bo b2 I 0 ANY ANY U10 2 0.2 I 0 3 0.2 1 I 4 0 I ANY 01ull 5 0.2 I 0 6 0.2 I 2 7 0 I ANY -i 12 8 0.2 I 0.133 I, I..1 2-.11.01.09.08............ II.- - - - I I: N - 3 0.07 -.06.04 -.4.3.02 -.2.01 0.1 2 3 4 17 Fig. 9. Profiles of first-order velocities for flow past a circular cylinder with fluctuating circulations.

1.2 w LU 0, aU) w w Q: w 0 Lw U) I 1.0.8.6.4.2 x b0 b2 w V 0 ANYANY O ANY 2 0.2 I 0 0.2 0. 3 0.2 I 0 0.2 0 4 0.21 0 0.2 -0.1 5 0.2 I 1 0.2 0 0.02 0.01 0 1 2 3 4 1r Fig. 10. cylinder Amplitude and phase of fluid velocity for with fluctuating circulations. flow past a circular 48

0.032 0.028 0.024 0.020 0.016 0.012 0.008 ~ x bo b2 1 0 ANY ANY 80 2 0.2 I 0 3 0.2 I I 4 0 ANY ANY a, 8,, 5 0.2 1 0 6 0.2 I I 7 0 ANY ANY -a2 82 8 0.2 0 9 0.2 I I (N tl = co - 0 0 ~ 0 I 2,q 3 4 Fig. 11. Profiles of first-order temperatures or concentrations for flow past a circular cylinder with fluctuating circulations for a fluid with Pr = 1 or Sc = 1. 49

0.03 x Pr bo b2 _ V I 0 ANY AN ANY ANY ANY 2 0.2 1 I 0 0 0 3 0.2 I 0 0.2 0.1 4 0.2 I 0 0.2 0 5 0.2 1 I 0 0.2 -0.1 6 0.2 I I I 0.2 0 7 0.2 10 I I 0.2 0,,,, ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~.i, w LJ 0 0 - 0.02 0.01 0.03 (f) CO wJ LU < 001 I 0. r7 Fig. 12. Amplitude and phase of fluid temperatures or concentrations for flow past a circular cylinder with fluctuating circulations. 50

as shown in Fig. 13 is somewhat different from the one produced by the oscillating flow. A striking distinction is that no separation of the steady secondary flow is observed for the fluctuating circulation case. Instead, the streamlines running in the direction of the main flow get closer to the surface at downstream. For the space-dependent oscillating amplitude of E(l + x2 + x4) as given in Fig. 14, the streamline pattern is similar to that of a constant amplitude. The corresponding temperature distribution of the steady secondary component 82os as respectively presented in Fig. 15 and 16 are also similar in pattern. No positive values exist in the entire boundary layer. The isothermal lines are practically running parallel to the solid surface in case of' a constant oscillating amplitude and running closer to the surface along the stream in case of a space-dependent oscillating amplitude. The streamline patterns and temperature distribution as demonstrated in Figs. 13 to 16 are directly related to the effects of the fluctuating circulation on the local values of wall shear stress and Nusselt number as illustrated in Fig. 17. The alternation in the wall shear stress is such as to increase its value while that in the heat transfer rate is to decrease. Both alternations are proporational to the distance from the forward stagnation point except the heat transfer rate under a constant oscillating amplitude. The effect on the heat transfer is more significant for fluids with high Prandtl number. B. Rotational Oscillations The profiles of the first-order velocities, U o, u1ll and. u12 are shown in Fig. 18. ulo decreases from a finite value at the solid surface to zero at the outer edge of the boundary layer. The other two velocities ull and ul2, however, have zero values both at the surface and in the free stream and maxima in the inner half of the boundary layero All three velocities increase in magnitude with an increase in the distance along the surface from the forward stagnation point. An interesting feature is disclosed in Fig. 19 for the frequency response of the fluid velocity. The amplitude has a maximum at the surface diminishing to zero at the outer edge of the boundary layer while the phase is zero both at the surface and the outer edge with a maximum in the outer half of the boundary layer. The amplitude as well as phase lag are magnified by the superposition of the uniform suction at the surface and vice versa. The profiles of the first-order temperature as given in Fig. 20 resemble those of the fluctuating-circulation case. e10o, 11 and 812 are zero both at the surface and the outer edge with maxima in the inner half of the boundary layer. The magnitudes increase along the surface from zero at the forward stagnation point. The frequency response of the fluid temperature is also similar to those of the two previous cases. With the superposition of the uniform suction, the amplitude as well as phase lag are magnified. The streamline patterns of the steady secondary flow are plotted in Fig. 22. The streamlines in the inner boundary layer adjacent to the solid surface are running opposite to the main flow in direction. However, the streamlines in the outer boundary layer existing in the forward stagnation-point region run in the same 51

0.4 x 0.6 0.03 0.'1. Ufxt)Ue(x)+eU, (x)Cost^-p c 0.01 0 -- 03 0 Fig. 13. Streamline pattern of the steady secondary motion produced by flow past a circular cylinder with fluctuating circulation of amplitude EU1 at uasi-swteady state.st

x 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.3 0.2 0.1 0.06 0.2 0.03 x J10.01 0.1 I U (x,t)= Uo(x)+U,(x)Coswt 0 4 3 2 1 Fig. 14. Streamline pattern of the steady secondary motion produced by flow past a circular cylinder wit fluctuating circulation of amplitude EUi(x) = e(l+x2+x4) at quasi-steady state. I

x 0.4 0.5 0.6 0.7 0.8 0.9 \ d 0.3k 0.2 \J-p x 0.1 - I I I I I I I I I / / / / III I / / I I I I I;/ / I I I I I I I I I I I I I I I M I I If lop\ I O O~~~~~~~~~~~~~o I rlIIOI~~~~~~~~~~~4~~~x00 'ownON, W. 1.0 0 -- II A ~ ~n I ~ a ~l If.AV AI A.~ I _ _ 4 3 2 I 0 Fig. 15. Distribution of the steady secondary component of temperature or concentration produced by flow past a circular cylinder with fluctuating circulation of amplitude U1 = E for a fluid having Pr = 1 or Sc = 1 at quasi-steady state.

x 0.7 0.8 O.I - ^ \ \/ ^^-^:^:^^^ —1.0 0.3 0.2 04 Q3 0.110 ~ 0 o ~ 1 -0.315 ':,, -O. 120 Q0. I- I-.o5 _,-0.045 -0.040 '~ -0.035 -0.030 0 4 3 2 0 Fig. 16. Distribution of the steady secondary component of. temperature or concentration produced by flow past a circular cylinder with fluctuating circulations of amplitude EU1(X) = E(l+x2+x4) for a fluid having Pr = 1 or Sc = 1 at quasi-steady state.

Pr bo b2 b4 V w Ar I ANY I 0 0 0 0*05 2jRe2 ANY I I I.j 04: 0.5 31 0 0 0 0 Nu45 I I 1 0O 0 6- 10I I I I. - 0.1 0 0 _ 7 10 1 I I 0 o 1.0 0.2 0.15 - W - < I I. I \nr O\ I a N 0.5 U.I 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x Fig. past 17. Effects of fluctuating circulation on the local wall shear stress and Nusselt number for flow a circular cylinder.

x bo I 0 ANY U 'o 2 0.2 I 3 0 ANY 'OI Ul 4 i4 O I 2 5 0 ANY -al z6 0.2 I.07.06.05h-.20h.18-.04.161-.8.141 N NY-.03 - -.12 0 1 1 0.10-.02-.081-.06h 01 -.04-1.02 - OL.,oL I1 - 3 4 Fig. 18. Profiles of first-order velocities for flow past a circular cylinder in rotational oscillations. 57

x o AV I 00 ANY 2 00.2. -1 - 3 0 0.2 0 4 0 0.2 -0.1 5 0.2.2 0 1.I I-.6 -:.5.1 X..4 -w cr.3-.07 _ (n.06 - a.04.03 - I.02.01 00 Fig. 19. cylinder I 2 3 4 5 7) Amplitude and phase of fluid velocity for flow past a circular in rotational oscillations. 58

,-____ x bo I 0 ANY 2 0.2 I 210.2 1 3 0 ANY -a, "1 4 0.2 I - a2 5 0 ANY 1 12 6 0.2 a ~ 2.4 2.2 X o N N - I o I 0 Cb 1.6 1.4 OQ I X I 0 1 2 3 4 5 Fig. 20. Profiles of first-order temperatures or concentrations for flow past a circular cylinder in rotational oscillations for a fluid having Pr = 1 or Sc = 1. 59

x Pr w V I 0 ANY ANY ANY 210.2 1 o 0 3 0.2 1 0.2 0.1 40.2 1 0.2 0 5 0.2 I 0.2 -0.1 6 0.2 10 0.2 0 0.03 w Q 0.01 0 0.06 U) w 004 LLJ 0 I~ w 0.02 'I a 1 2 3 4 V. Fig. 21. flow past Amplitude and phase of fluid temperature or concentration for a circular cylinder in rotational oscillations. 6o

x x 4 3 2 1 0 id 77 6 0.7 0.8 0 0.6 0.7 0.8 0.9 0.4 0.5 1.0 0.3 0.2 X 0.1 0 2.. - 7 Fig. 22. Streamline patterns of flow past a circular cylinder in eU1 = ~ at quasi-steady state. the steady secondary motion produced by rotational oscillations with amplitude

direction as the main flow then turn around the direction at somwhere in the middle of the oscillatory boundary layer. Therefore an extra stagnation point is created at x = O, but in the boundary layer. In spite of the difference in the direction of the steady secondary streamline, the temperature distribution of the steady secondary component as illustrated in Fig. 23 is similar in pattern to the fluctuating-circulation case. The isothermal lines which are negative values in magnitude fall closer to the surface along the surface. From the phenomena observed in Figs. 22 and 25, the alternations in both the local wall shear stress and Nusselt number tend to increase along the surface from the forward stagnation point as illustrated in Fig. 24. It is important to note that the rotational oscillation is the only case in which the net effect on the wall shear stress is negative. Again the superposition of the uniform suction tends to enhance the alternations in both the skin friction and heat transfer. 62

K -0-.0/05 I, \\ \\. > \ '1. // -0.010 I 0.1 I 4 3 2 I 0 Fig. 23. Distribution of the steady secondary component of temperature or concentration produced by flow past a circular cylinder in rotational oscillations with amplitude U1. = E for a fluid having Pr = 1 or Sc = 1 at quasi-steady state.

.5 4 r.3-.25.2.15.1.05 Pr V | r I I 0.1 0.2 _ 2 I 0 0.2 3 I -0.1 0.2 4 I 0.1 0.2 ANu 5 I 0.1 0.2 2 Re 6 I -0.1 0.2 _ 1 0 0.2 Oc I) < I l..2 - IN w:3 I.I - 01 0.1.2.3.4.5.6.7.8.9 X 1.0 Fig. 24. Effects of rotational past a circular cylinder. oscillations on the local wall shear stress and Nusselt number for flow

V. SUMMARY OF RESULTS The system of partial differential equations governing fluid motion, heat transfer and mass transfer was solved by a perturbation technique for the cases of fluid flow around a symmetrical blunt body with the flow oscillation and fluctuating circulation in free stream the rotational oscillation of cylinder surface Flow around sharp-edged body is also analyzed for the flow-oscillation in free stream although the analysis may be extended to the other two cases with easeo Numerical results for flow past a circular cylinder include the effects of the superposition of uniform suction or blowing. It is disclosed from the analysis that an increase in the oscillating frequency results in (i) a slight change in the amplitudes of both velocity and' temperature, an increase for the flow-oscillation case and a decrease for the rotational-oscillation case; (ii) a significant increase in the phase angle for all cases; and (iii) a slight decrease in the absolute value of the steady alternations in both the wall shear stress and Nusselt number for all cases. The superposition of the uniform suction contributes to (i) an increase in both the amplitude of the first-order temperature and the phase lag of the first-order velocity and temperature but a decrease in the amplitude of the velocity profile; (ii) a decrease in the absolute value of the steady secondary alternations in the skin friction as well as heat transfer rate. An increase in the Prandtl number is characterized by a steeper temperature profile in the neighborhood of the solid surface and a decrease in the thickness of thermal boundary layero This results in an increase in the amplitude and phase lag of the first-order temperature as well as the alternation in the steady, secondary heat transfer rate. At the forward stagnation point, the net change in the wall shear stress caused by oscillations is zero. Along the surface it increases to a maximum then decreases to zero at the separation point in case of flow oscillation. However for fluctuating-circulation and rotational-oscillation cases, the net change increases monotonically in magnitude along the surfaceo These changes are positive for the flow oscillation and fluctuation circulation but negative for the rotational oscillation. The alternation in the local Nusselt number is a negative finite value at the stagnation pointo It decreases in magnitude along the surface to zero before the separation point and then changes to positive in the first case. For the other two cases a monomatic increase in magnitude is observed. 65

VI. REFERENCES 1. Lord Rayleigh, "On the Circulation of Air Observed in Kundt's Tubes, and some Allied Acoustic Problems", Phil. Trans., A, 175, pp 1-21 (1883). 2. H. Schlichting, "Berechnung ebener periodischer Grenzschichtstromungen", Phys. Z,, 33, pp 327-35 (1932). 35 E. N. Andrade, "On the Circulations Caused by the Vibrations of Air in a Tube", Proc. Roy. Soc., A, 134, pp 445-70 (1931). 4. J. Holtsmark, I. Johnsen, T. Sikkeland and S. Skaulein, "Boundary Layer near a Cylindrical Obstacle in an Oscillating Incompressible Fluid", J. Acoust. Soc. Amer., 26 pp 102 (1954). 5. J.T. Stuart, "On Double Boundary Layers in Oscillatory Viscous Flow", (1960). 6. G. D. West, "Circulations Occurring in Acoustic Phenomena", Proc. Roy. Soc., B, B 64, p 483 (1951). 7 P. J. Westervelt, "The Theory of Steady Rotational Flow Generated by a Sound Field", J. Acoust. Soc. Amer., 25, pp 60-67, (1953). 8. J. Mo Andres and U. Ingard, "Acoustic Streaming at High Reynolds Numbers'', J. Acoust. Soc. Amer., 25, pp 928-32 (1953). 9. W. L. Nyborg, "Acoustic Streaming due to Attenuated Plane Waves", J. Acoust, Soc. Amer., 25 pp 68-75 (1953) 10. W. L. Nyborg, "Acoustic Streaming Equations: Laws of Rotational Motion for Fluid Elements", J. Acoustic Soc. Amer., 25, pp 938-44, (19535) llo L. A. Segel, "Application of Conformal Mapping to Viscous Flow Between Moving Circular Cylinders", Quart. Appl. Math., 18, pp 335-53 (1961). 12. M. J. Lighthill, "The Response of Laminar Skin Friction and Heat Transfer to Fluctuations in the Stream Velocity", Proc. Roy. Soc., Sero A, vol. 224, pp. 1-23 (1954). 135 Co C. Lin, "Motion in the Boundary-Layer with a Rapidly Oscillating External Flow" Proc. 9th Into Cong. Appl. Mech., Brussel, vol 4, ppo 155-167 (1956). 66 -

14. W. E. Gibson, "Unsteady, Laminary Boundary-Layer Flow," Ph.D. Thesis in Mathematics, M.T.T. (1957). 15. E. Hori, "Unsteady Boundary Layers", Part I, Bulletin of Japan Soc. Mech. Eng., Vol. 4, No. 16, p. 664 (1961); Part II, Vol. 5, No. 17, p 57 (1962); Part III, Vol. 5, No. 17, p. 64 (1962); Part IV, Trans. Japan Soc. Mech. Eng., Volo 27, No. 183, p. 1731, (1962). 16. J. T. Stuart, "A Solution of the Navier-Stokes and Energy E Equations Illustrating the Response of Skin Friction and Temperature of an Infinite Plate Thermometer to Fluctuations in the Stream Velocity", Proc. Roy. Soc. Ser. A, 231, pp 116-30 (1955). 17. F. K. Moore, "Unsteady Laminar Boundary-Layer Flow", NACA TN. No. 2471 (1951). 18. S. I Cheng and D. Elliott, "The Unsteady Laminar Boundary Layer on a Flat Plate", Trans. Amer. Soc. Mech. Eng., 79 pp. 725-33 (1957) 19. Co R. Illingworth, "The Effects of a Sound Wave on the Compressible BoundaryLayer on a Flat Platet", J. Fluid Mech., 3, pp 471-93 (1958). 20. W. Wuest, "tGrenzschichten and zylindrischen Korpern mit nichtstationarer Querbewegung", Z. angew. Math. Mech., 32, pp 172-78 (1952). 21. M. Bo Blauert, "The Laminar Boundary Layer on Oscillating Plates and Cylinders", J. Fluid Mech., 1, pp 97-110 (1956). 22. N. Rott, "Unsteady Viscous Flow in the Vicinity of a Stagnation Point"', Quart. Appl. Math., 13, pp 444-51 (1956). 23. T. W. Jackson, K, R. Purdy and C. C. Oliver, "The Effects of Resonant Acoustic Vibrations on the Nusselt Numbers for a Constant Temperature Horizontal Tube", 1961 Intern. Heat Transfer Conf., II, pp 483-89 (1961). 24. Ro Mo Fand and J. Kaye, "The Influence of Vertical Vibrations on Heat Transfer by Free Convection from a Horizontal Cylinder", 1961 Intern. Heat Transfer Conf., II, pp 490-98 (1961). 25. F. Jo Bayley, P. A. Edwards and P. P. Singh, "The Effects of Flow Pulsations on Heat Transfer by Forced Convection from a Flat Plate," 1961 Intern. Heat Transfer Conf., II, pp 499-509 (1961) 26. J Kestin, P. F. Maeder and H. Ho Sogin, "The Influence of Turbulence on the Transfer of Heat to Cylinders near the Stagnation Point", Zert. angew Matho and Physo, Vol. 12, p 115 (1961). 67

27. J. Kestin, P. F, Maeder and H. E. Wang, "Influence of Turbulence on the Transfer of Heat from Plates with and without a Pressure Gradient", 1961 Intern. Heat Transfer Conference, ASME Part II, pp. 432 (1961). 28, J. Kestin, P. F. Maeder and Ho E. Wang, "On Boundary Layers Associated with Oscillating Streams", Appl. Sci. Res., Sec. A., Vol. 10 (1961). 29. R. J. Schoenhals and J. A, Clark, "Laminar Free Convection Boundary Layer Perturbations Due to Transverse Wall Vibration", J. of Heat Transfer, Trans. ASME, Ser. C, Vol. 84, p. 174 (1962). 30. V, D. Blankenship and J. A. Clark, "Effects of Oscillation on Free Convection from a Vertical Finite Plate", ASME Paper No. 63-HT-31. 31. V. D. Blankenship and J. A. Clark, "Experimental Effects of Transverse Oscillations on Free Convection of a Vertical, Finite Plate", ASME Paper No. 63-WA-123, Presented at the 1963 Winter Annual Meeting of the ASME, to be published in the J. of Heat Transfer ASME, Series C. 32. V. D. Blankenship and J. A. Clark, "Laminar Free Convection from a Vertical Infinite Plate Subjected to Transverse Oscillation," ASME Paper No. (1964), 335 So Eshghy, "The Effect of Longitudinal Oscillations on Fluid Flow and Heat Transfer from Vertical. Surfaces in Free Convection", Ph.D. Thesis, Department of Mechanical Engineering, Univ. of Mich. (1963). 34. T. Y. Na, "Influence of Localized Oscillations on Laminar Flow Over a Flat Plate," Ph.D. Thesis in Mech. Engr., Univ. of Mich., (1964). 355 Ho Schlichting, Boundary Layer Theory, 4th Ed., McGraw-Hill (1960). 36. L. Rosenhead, Laminar Boundary Layer, Oxford Press, (1963). 37. No Frossling, "Evaporation, Heat Transfer, and Velocity Distribution in Two Dimensional and Rotationally Symmetrical Laminar Boundary-Layer Flow " NACA TM 1432 (1958)o 38. N. Frossling, "Problems of Heat Transfer Across Laminar Boundary Layers," Theory and Fundamental Research in Heat Transfer, Edited by John A. Clark, Pergamon Press, New York (1963) pp 181-202. 39. H. Blasius, "Grenzschichten in Flussigkeiten mit Kliner Reibung," Z. Math. u. Phys., Vol. 56 (1908) and NACA TM 1256. 40. L. Howarth, "On the solution of the laminar boundary layer equations," Proc. Roy. Soc. London, Ser. A, Vol. 164, p. 547 (1938). 68

41. H. Gbrtler, "A new series for the evaluation of steady laminar boundary layer flows," J. Math. and Mech., Vol. 6, p. 1 (1957). 42. E. M. Sparrow, "Application of Gortler's series method to the boundarylayer energy equation," J. Aeronaut. Sci., Vol. 25, p. 71 (1958) and "The thermal boundary layer on a non-isothermal surface with nonuniform free.stream velocity," J. Fluid Mech., Vol. 4, p. 321 (1958). 43. E. Wrage, "Ubertragung der Gorthler'schen Reihe auf die Berechnung von Temperaturgrenzchichten, Teil I, DVL-Bericht Nr 81, (1958). 69

APPENDIX

A. THE COMPUTER PROGRAM FOR FLOW OSCILLATION IN FREE STREAM $COMPILE BEGIN SW6 SW10 SW2 START MAD, EXECUTE.PRINT OBJECT, DUMP, PUNCH ObJECT PRINT COMMENT $ THE COMPUTER PROGRAM FOR OSCILLATING FLOW $. DIMENSION Y(3)t F(39 Q(3) INTEGER J. It. N M, -K Z. T. ZET, PRI VECTOR VALUES DEL=$1H.F6.4,4F15.8*$ VECTOR VALUES ROH=$1H,F6.4,3F15*8*$ VECTOR VALUES ALP=$1H tF6.4,3F15.8*$ VECTOR VALUES BET=$1H,F6e4,12F10.6*$ VECTOR VALUES EPS-$1H 91F50.4*$.DIMENSION P(45), Y1(45), Y2(45), F3(45), R(55),Y3(45), IUS(45)TUT(45)TUT(45),UU(41)Q1(45),Q1S(45)Q2(45 )Q2S(45) AMQS(45), 2AM.Q(45)TANQ(45),PHAQ(45),TS(45 ) TT(45),TU(45) S1(45),S1S(45), 3S2(45)TS2S(45) AMSS(.45) AMS( 45),ANS(45) PHAS(45), YP3(10), 4XY.( 10) WAt10).YP2( 10.) *NTAU ( 10 ),W(1.)N ) NNUS( 10 ),5SR 4 ) 5SST(45) DIMENSION A(12.6.0.AD),B(1260,bD),C(1260CD),D(630,DD) VECTOR VALUES AD=2t,145 VECTOR VALUES BD=2,t145 VECTOR VALUES.CD-=21,45 VECTOR VALUES DD.=2.145 READ AND PRINT DATA EXECUTE SETRKD. (3. Y(1)J F(1), Q. Xs STEP) T=1 ZET=O. ZET ZET+1 ^WhEiNEVYER (-..-1,.P.ZE T-L a. Y.DP+ZE T.- WHENEVER ((-1).P.ZET).GOt YX=YDP-ZET+1 R ( O ) = 51_. R( 1)=1 N=1 Q=CF*(R(N)R(N-1 )-) WHENEV.R.Qr.G7,? 2-Qs=_ F WHENEVER Q.L. (-87.20)* Q=-87,20.DXT-Y. (L..12. 5 o 4 / L EXPEAIQJ.-.EXPA... -.QJ.)J.+1 DY=DX.N=N~+1 YX=YX+DY WH.E./EVER T. L TMA X Y(3)=YX.-QT..ERW.I SE Y(2)=YX.END.OF CONDI-TIONAL J=O X=0Q WHENEVER T E.*1Y(1)=CONST WHENEVER T.Gl.Y(i)=0 WHENEVER T.L.TMAX, Y(2)=0 J=J.+1 P'(J) =X Y1(J)=YU1) Y2(J)=Y(2) WHENEVER T..L. TMAX Y3(J)=Y(3) F31J.'. FU3 ) OTHERWISE F3 (J )=F.l2) END.OF CONDITIONAL FWH.EVER )TM.Y( 2 ) WHENEVER T.L.TMAX, F(2)=Y(3).SW1.

WHENEVER T.E.1, F(3)=(Y(2).P.2)-Y(1)*Y(3)-1 WHENEVER T.E.2. F(3)=4*B(1,J)*Y(2)-3*C(1,J)*Y(1)-A(1J)*Y(3)-. 11 WHENEVER T*E.3, F(3)=B(1,J)+0.5*X*C(1,J)-l-A(1,J)*Y(3)+2*B(1, 1J)*Y(2)-Y( 1 )*C( 1 J) WHENEVER T.E.4~ F(.3)=0.5*X*C(2,J)+B(2J)-3.*A(2,J)*C(3,J)+4*B( 12,J)*B(3,J)-C(29J)*A(3,J)-0.25-A(1,J)*Y(3)+4.0*B(1,J)*Y(2)-3. 20*C( 1JD*Y(1) WHENEVER T*E.59 F(3)=2*B( 1J)*Y(2)-A( 1J)*Y(3)-C(1J)*Y( 1)+B( 13,J) WHENEVER T.E.69F(3)=B(4,J)-3.*A(2,J )*C(5,J)+4.*B(2,J)*B(5,J) 1-A(5,J)MC( 2J)-A(1J)*Y( 3)+4.*B(1~J )*Y(2)-3.*C( 1J)*Y(l) WHENEVER T.E.7, F(3)=2*B( 1,J)*Y(2)+(B( 1J)+0.5*X*C(1,J).).P.2 -1A(1,J)*0(3)-0.25*(A(1,J)+X*B(1,J))*(3.0*C(1,J)+X*D(19J))-Y(1) 2*C(1 J) -1 0. WHENEVER T.E.8* F(3)=B(2,J)*B(7,J)+3*B(1,J)*Y(2)+4*(B(1'J)+0. 15*X*C(1,J))*(J)B(2J)+0.5*X*C(2J))+B(1J)*Y(2)-3*A(2,J)*C(7,J) 2-A(1,J)MY(3)-075*(3*C(1,J)+X*D(1,J))*(A(2,J)+X*B(2,J))-0.25 3*(A(1,J)+X*B(1,J))*(3*0*C(2,J)+X*D(2,J))-3.0*C(1,J)*Y(1)-A(79 4J)*C(2,J)-1.+3.*B(2,J)*B(7,J) WHENEVER T.E.9, F(3)=2*B(7,J)+2*B(1 J)*Y(2)+B(3,J)*(B(1,J)+0. 15*X*C(1TJ))-A(1,J)*Y(3)-0.5*(A(1, J)+X*3-(liJ))*C(3,J)-C(1,J)*Y 2(1) WHENEVER T.E.10, F(3 )2*B(8,J)+B(2,J)*B(9,J)+3*-B(1,J)*Y(2)+B( 13,J)*(B(2,J)+0.5*X*C(2,J))+3*B(4,J)*(B(1,J)+0.5*X*C(1,J))+B(1 2,J )*Y( 2+3*B (.2 J)*B (9 J)-A (1 J )*Y(3)-3*A (2 J )*C (9,J )-0.5C (4 3J)*(A(1i J)+X*B(1lJ))-1.5*C(3,J)*(A(2,J)+X*B(2,J))-C(2,J)*A(9, 4J)-3.0*C(1iJ)*Y(1) WHENEVER T.E.ll, F(3)=2.0*B(1,J)*Y(2)-A(1,J)*Y(3)-C(1,J)*Y(1) 1+2,0*B(5,J)*(B(1,J)+0O5*X*C( 1J))-( (B(3JJ ).P.2 )-0.5*C(59J)*( 2A( 1J)+X*B( 1 J) )'-0,5*A(5,J)*(3.0*C( 1,J)+X*D( 1 ) )+A(3,J)*C(3, 3J) WHENEVER T*E.12, F(3)=4*B(2,J)*B(11,J)+4*B( 1,J)*Y(2)-3*A(2,J) 1*C(11,JD-A( 1 J)*Y(3)-3*C(1J)*Y(1)-C(2,J)*A(i1,J)+4*B(6,J)*(B 2(1,J)+0.5*X*C(1,J)9J)+4*(5,J)*(B(2J)+0.5*X*C(2,J))-4*B(3,J)*B 3(4,J)-1C5*C(5,J)*(A(2,J)+X*B(2,J))-0.5*C(6,J)*(A(1,J)+X*B(1,J 4))-1.5MA(6,J)*(3.0*C(1,.J) (1J+XD(J))-0.5*A(1J)*(3.0*C(2,J)+X* 5D(2,J))+3.0*C(3,J)A)A(4J)+A(3,J)*C(4.J) WHENEVER T.E.13, F(3)=2'.0*B(1,J)*Y( 2)-A(1,J)*Y (3)-C (1,J )*Y (1) 1+2.0*B(5,J)*(B(1,J)+0*5*X*C(1,J))+((B(3,J)).P.2)-0.5*C(5,J)*( 2A(1,J)+X*B(1,J))-A(3,J)*C(3,J)+2.0*B(9,J)-0.5*A(5,J)*(3.0*C(1 3,J)+X*DU1,J)) WHENEVER T.E.14. F(3 )-A(',J)*Y( 3)+(2,J)*B( 13,J)+4*B( 1 J )*Y( 12)+3*B(2,J)*B(13 J)-3*A(2, J)*C( 3,J )-3*C(1 J)*Y(1) +4*8(5 J)* ( 2B(2,J)+0.5*X*C(2,J))+4*B(6,J)*(B(1IJ)+0.5*X*C(1,J))+4*B(3,J)* 3B(4,J)-1.5*C(5,J)*(A(29J)+ X*b(2,J))-0..5*C(6,J)*(A(1,J)+ 4 X*B(1,J) )-1.5*A(6J)*(3*C(1,J)+X*D(1,J)')-0.5*A(1J)*(3*C(2, 5J)+X*D(2,J))-3*C(9,JJ*A(J)-A(3J)*C(4,J)+2*B( 10J) WHENEVER TE.l15 F(2)=-PR*A(1J)*Y(2) WHENEVER T.E.16, F(2)=PR*(2.0*6d(1J)*Y(1)-3*.*A(2,J)*b((15J)1A(1,J)*Y(2)) WHENEVER T.E.17, F (2) —PR*(A(1,J)*Y(2)+A(3,J)*.o(15 J) -0.5*X*B 1(15,J)) WHENEVER T.E.s18 F(2) =PR*(20*B(1 J )*Y(1)+2.0*A(16 J)*(3 J )13.0*A(2,J)*B(17,J)-B(16,J)*A(1,J)+0.5*X*B(16,J)-3.0*A(4,JJ)b( 215,J)*A(3,J)) WHENEVER T.E.19, F(2)=-PR*(A(5,J)*b( 15J)+A(l.J)*Y (2)-A(17,J) 1) WHENEVER T.E.20, F(2)=-PR*(3.0*A(2,J)*B (19J )+3.0*A(b J) b(15 74

19J)+B(16,J) *A(59J)-2 0*6B( 19J)*Y(1)-2*0*B(59J)*A(16,J)+A(1.J) -) 2Y(2)-A(18,J)) WHENEVER T.E.21, F (2)=-PR*(A(1,J)*Y(2)+025*(b( 15,J)+X*C(15,J 1))*(A(1TJ)+X*B (1' J))+A(7, J)*b3.( 15 9J)) WHENEVER T.E.22, F(2 )PR*(2.0*B(1, J)*Y(1)+B(16,J)*(B(1,J)+0.5 1*X*C( 1, J.) )+20*A( 16,J)*b(7,J)-3.0*A( 2, J)*B(21 J)-A(1J)*Y(2)20.75*(B( 15,J)+X*C( 15 J) )* (A(2 J)+X'B i(2,J) )-025* (( 169J)+X*C( 316,J))*UA(1,J)+X*B(1,J))-3.0*A(8,J)*B(159J)-A(7,J)*B(16.J)) WHENEVER T.E.23, F(2)=-PR*(05*A(3,J)*( (15, J)+X*C(15,J) )+0.5 1*B ( 17JD*(A( (1J) +X*.( 1,J) )-X*B( 15,J)+A(9,J)*B (15J)+A(1J)*Y( 22)) WHENEVER ToE.24, F(2 )-PR*(-2*b 1 J )*Y(1)-2*B(9,J) A(16J)+3* 1A(2,J)MB (23,J)+A( 1,J )*Y( 2) +3*A( 109J ) *-( 15,J )+A ( 9,J) * ( 16,J )-X 2*6(1,J)M.B(16,J)-2.*A( lb'J)*(B(1,J)+O.50*X*C( 1,J))+'15*A(4,J)*( 36(159J)+X.*C(15,J))+O0 5*A(3,J)*(6(16,J)+X*C(i16J) )+1.5*B(17,J) 4*(A(2,JD+0.5*X*B(2,J) )+0.5*B(16,J)*(A( 1 J)+05*X*B(1J) )-X*B( 516,J)) WHENEVER T.E.25, F ( 2 ) =-0*5*PR* (2.O*A (1 J-) *Y (2 )\+2 0*A ( 11 J )*B( 115,J)+A(5,J)*(B( 15,J)+X*C( 15,J) )+(A ( 1J)+XB(( lJ) )*'B(19J)-2*.20*A(39JD*B (17J) ) WHENEVER T.E.26, F ( 2 ) =PR* ( 2 0*8 ( 11 *J ) *A (16,J ) -3 *O*A (2.J ) *B( 25 1,J })-A( TJ)*Y (2)-30*A ( 12,J )*B (15,J )-A ( 11 J) * ( 16 J )+2.0*B(1,J 2)*Y(1)+2.O*(B( 1J)+0O.5.X*C( 1J) )'*B(20,J ) -2.*B ( 3,J ) *B ( 18,J)31.5*A(6TJ')*(B(15.J)+X*C(15,J) )-0*5*A( J)* (B ( 16J) +X*C (16 J ) ) 4-2.0*(A(2,J)+X*B(2*J) )*B(15,J )-O.5* (A( 1J)+X*B( 1J))*B(20J)+ 53.0*A(4TJ*B)B(7,J)+A( J)*B(18,J)) WHENEVER T*E.27, F(2)=-055*PR*(A(5,J)*(B-(159J)+X*C(15,J))+(A( 11,J)+X*B((1,J) )*B(19J)+2O'*A(3,J)J*b(17,J)-A(23,J)+20O*A( 1J)* 2Y(2)+20*A(11, J)*B( 15,J)) WHENEVER T.E*28, F(2 )=-PR* (-2O*B( 1, J)*A( 16,J)+3*0*A(2J )*B( 127,J)+A(,J)*Y(2)+3.0*A(12 J) *b( 15,J )+A(11 9J ) b ( 16J )-20 * -( 1 2 J)*Y( 1D-X*B( 5 J )*B( 16,J)- 2 O - ( t( 1.,J ) + 0.5-X-C(1,J') ) - (20,J)-2 3-O0*B(3,J)*B( 18J)+1 5*A(2,J)*(B( 15 J )+X*C(1 5J) )+0. 5A ( 5,J ) ( 4iB(16,J)+X*C(16J) )+3O0*A(4,J)*B(17, J)+A(3 J)*6( 18J)-0.5-^A(24 5 9J)+ 0 5*(A( 1J)+X* ( 1,J))*b(20,J) +2.0* ( A ( 2 J )+X* ( 2,J ) )* ( 19 69J)) CALC S=RKDEQ (0.) WHENEVER S~.E1..O. TRANSFER TO SW1 WHENEVER J.E.GRSIZ9 TRANSFER TO SW3 TRANSFER TO START SW3 WHENEVER T.E.i1 TRANSFER TO SW8 WHENEVER T.E.2t TRANSFER TO SW11 WHENEVER 2.L.T.AND. T.L.TMAX, TRANSFER TO SW7 WHENEVER T.E.TMAX. TRANSFER TO SW12 WHENEVER ToGoTMAX, TRANSFER TO SW13 SW7 WHENEVER *ABS.Y(2).(6.0001.AND. N.L.NMAX R(N)=Y(2) TRANSFER TO SW2 OTHERWISE TRANSFER TO SW9 END OF CONDITIONAL SW8 WHENEVER.ABS (Y ( 2)-1 ) *.G*0001..AND. N.L.NIMAX R(N)=Y(2)-1.TRANSFER TO SW2 OTHERWISE TRANSFER TO SW9 END.OF CONDITIONAL.SW11 WHENEVER.ABS.(Y(2)-0*.2500).G.O.001.AND. N.L.INMAX R(N)=Y(2 )-0.2500 75

TRANSFER TO SW2 OTHERWISE TRANSFER TO SW9 END OF CONDITIONAL SW12 WHENEVER ABS. (Y(1)-1 )*G0.001 *AND. No.LNMAX R(N ).=Y ( 1.)-1 TRANSFER TO SW2 OTHERWISE TRANSFER TO SW15 END OF CONDITIONAL SW13 WHENEVER *ABS.Y(1)G.0O.001 *AND. N.L.NMAX R(N)=Y(1) TRANSFER TO SW2 OTHERWISE TRANSFER TO SW15 END. F CONDITIONAL SW15 WHENEVER.ABS. Y(2).G.0O001.AND. ZET.L.ZMAX, TRANSFER TO SW1 1.0 SW9 WHENEVER.ABS. Y(3).G.O.001 *AND. ZET.L*ZMAX, TRANSFER TO SW1 10 PRINT RESULTS T PRINT RESULTS ZET WHENEVER ZET.E.ZMAX, PRINT COMMENT $ EQ. CAN NOT BE MADE TO S 1ATISIFY BOUNDARY CONDITIONS $ WHENEVER PR.G~.5,TRANSFER TO SW40 TRANSFER TO SW41 SW40 WHENEVER N.E NMAX GRSIZ=GRSIZ-5 WHENEVER N E.NMAX.AND.GRSIZ.L.25 TRANSFER TO 5W14 SW41 WHENEVER N.E.NMAX.TRANSFER TO SW14 PRINT.COMMENT $ R(M) $ THROUGH SW5, FOR M=2, 1. M.G.N S.W5 PRINT FORMAT EPS, R(M) THROUGH SW4, FOR I=1,1 I.G.GRSIZ A(TI)=Y1(I) B(TI)=Y2(I) WHENEVER T.L.TMAX C(TI)=Y3(I ) D(T.I)=F3(I) OTHERWISE C(TI)=F3(I) END OF CONDITIONAL WHENEVER T.L TMAX PRINT FORMAT DEL, P(I),.Y1( I), Y2(I), Y3(I), F3(I) OTHERWISE PRINT FORMAT ROH, P(I.), Y1(I), Y2(I) F3(I) SW4 END OF CONDITIONAL WHENEVER T.L.TMAX PRINT FORMAT DEL. X. Y'(1) Y(2). Y(3), F(3) OTHERWISE PRINT FORMAT ROH, X, Y(1), Y(2.) F(2) END OF CONDITIONAL Z=GRSIZ A(T*Z)=Y(1) B(T Z) =Y (2) WHENEVER T.L.TMAX C(TZ)=Y(3) D( T. Z)=F(3) OTHERW I SE. C(T Z)=F(2)

END OF CONDITIONAL WHENEVER T.E.TMAX. EXECUTE SETRKD. (29 Y(1), F(l)t QU X, STEP 1) - WHENEVER T.LE.TEND*TRANSFER TO SW6.TROUGH.-SW2L FOR J=1,1J*G.GJMAX THROUGH' SW21,FOR K= 1 1.K. GKMAX ]HIROUGH SW20*FOR I=1 *.1.1G*6.GRSIZ YP3(J)=4.*A3*(XY(J).P3) S=STEP* ( -1 ) P( I )S US( I).=Al*XY(J ).*(( 1 1)+.05*S*C ( 1 1.) +YP3 (J)* ( ( 21 I) +O.5*S*C(2 1,1)) UT. (1.).=AL*X.Y ( J 3 ) *. 3, I) +YP 3 ( J.) *B (4, I) UU(I)=A1*XY(J)*B(5 I)+YP3(J)*B(6, ) WA (K) W.U).*W (K)../ (Al*A ) Q1 I )=WA(K)*UU( I )-US I ) Ql.S..(i) Q I )*Q ( I) Q2(I )W(K)*UT( I)/A1 Q2S:( I..) -2 ( I )*Q2. AMQS( I )Q1S( I )+Q2S(I ). AMQ (.1..)SQR T. (AMQ. I ) ) ANQ(I)=Q2( I)/Q1( I) PHAQ11.1=.ATA N (ANQ( I) ) *57 2958 A1S=0.5*SQRT. (A1) YPZ).J. )J..A..*A3*XY (J) *XY (J ) /Al SSR'(I)=2.*A1S*XY(J)*(A(7, I )+YP2(J)*A(,I)) TS( 1= QCS B (1.5, I) +YP2 ( J ) *b (16,1 ) ) TT( I )A( 17 ' )+YP2 (J )*A( 8 I ) T U J..=.AU19...I.).+YP2 ( J ) *A.(20., I) Sl( I)=WA(K)*TU( I )-TS( I) SlS.L..I)=S1(I )*Sl(l. ) S2(I )=W(K)*T (I)/A1 52S( I).=52.(..) *S2 ( I ) AMSS( I )S1S(I )+ZS (1) AMS.I ) = SQR T. (AMSS( I) ) ANS(I ) 52( I') /51 ( I ) PHAS( I )ATANl'(ANS( I )*57.2958 SST(I )=A(21,I)+YP2(J)*A(22,I).iW/20 ' PR.INT FORMAT BET P( 1),US( I.),UT( I),UU( I ) 'AiM ( I) tPHAW( I),SSR( ) 1 TS( I ),TT( I.)' TU( I ) AMS( I ) PPHAS( I ),SS ( I ) NT AU.K) 3A.S*A1*XY ( J ) *C ( 7 1 ) +YP3 ( J ) *C ( 81 ) -WA ( K ) *(Al*XY ( J)* iC(ll,l)+YP3(J)*C(1i21)))..NNUS5(K)3Al1*(B(.21,1)+YP2 (JL*B.(22,1)-WA(K)*(B(2.lb,1)+YP2(J)*B( 126.1))) S.W21 PRI.N.T..EORMA.I ALP,X.Y.i J ). W ( K..:NTAU ( K) sNNUS ( K) TRANSFER TO BEGIN S14L.END OF PROGRAM $DATA.ZMAX 6., T.MAX = 15 T.El~D=28 *- CF = 1, NMAX: 50 9 T LP.0 1,GR 1L Z =45, PR - l., DYOOOY001YDP=lOO000CONST=-O.l Al=4. A3=-2.666,XY ( 1 ) O,XY ( d ) OO XY(Y 1QXY =O XYL.1.=o ( 5) = O 3 X Y (6) = O 4XY( 7 ) 0 *5,XY ( 8 ). G. 6,JI AX=6 W(1)=0KMAX=1' * 77

B. THE COMPUTER PROGRAM FOR FLUCTUATING CIRCUIATION IN FRE STREAM $COMPILE BEGIN SW6 SW0. SW2 START SW1 MAD, EXECUTE, PRINT OBJECT, DUMP, PUNCH OBJECT PRINT COMMENT $ THE COMPUTER PROGRAM FOR FLUCTUATING CIRCULAT ION $ DIMENSION Y(3) F(3), Q(3) INTEGER J, I, N, M, K, Z, T, ZET, PRI VECTOR VALUES DEL=$1H,F6.4,4F15.8*$ VECTOR VALUES ROH=$1H, F6.4, 3F15.8*$ VECTOR VALUES ALP=$1H,F6b.4 3F15.8*$ VECTOR VALUES BET=$1H,F6.4,12F10.6*$ VECTOR VALUES EPS=$1H, 1F20.4*$ DIMENSION P(45), Y1(45), Y2(45), F3(45), R(55), Y3(45), 1US(45)TUT(45),UU(45).,Q1(45) Q1S(45) Q2(45),Q25(45),AMQS(45), 2AMQ(45)TANQ(45),PHAQ(45),TS(45),TT(.45)*TU(45),S1(45),S1S(45)', 3S2(45) 52S(45) AMSS(45) AMS(45)oANS(45.) PHAS(45) YP3 (10) 4XY(10 ) WA(10),YP2 (10),NTAU(10),W(10),NNUS(10),SSR(45) 'A2(10) 5YP1 (10),YP4( 10), A14( 10) A4( 10),A12( 10) )XYS (10),SST (45) DIMENSION A(2565,AD.),B(2565,BD),C(2565,CD),D(2&6'5,DD) VECTOR VALUES AD=2, 1, 45 VECTOR VALUES BD=2 1,. 45 VECTOR VALUES CD=2, 1, 45 VECTOR VALUES DD=2, 1, 45 READ AND PRINT DATA EXECUTE SETRKD. (3, Y(l1), F(1), Q,- X, STEP T=1 ZET=.O ZET=ZET+1 WHENEVER ((-1 ) P ZET)*L 0, YX=YDP+ZET-1 WHENEVER ((-1).P.ZET).G.0 YX=YDP-ZET+1 R(O ) =15 R(.1)=i N=1 Q=CF*(R(N)/R(N-1)-1) WHENEVER Q*.G87.20, Q=87*20 WHENEVER Q.L.(-87*20). Q=-87.20 DX=-DY*U(2.35040/(EXP.(Q)-EXP.(-Q)'))+1) DY=DX N=N+1 YX=YX+DY WHENEVER T.L.TMAX Y(3)=YX OTHERWISE Y(2)=YX END OF CONDITIONAL J=O X=O WHENEVER T*.E1i Y(1)=CONST WHENEVER T.G.1, Y(1)=0 WHENEVER T*L.TMAX~ Y(2)0= J=J+1 P(J)=X Y1 (J)=YU1) Y2(JJ)=Y(2) WHENEVER T.L.TMAX Y3(J)=Y.(3) F3(J)=FU3) OTHERW ISE F3(J)=F(2) END OF CONDITIONAL F(1)=Y(2) 78

WHENEVER T.L.TMAX, t i2; =Y(3) WHENEVER T'E.1 F(3).-(Y(2)*P.2)-Y(1)*Y(3)-1 WHENEVER T.E.2, F(3)=4*b(1J)*Y(2)-3*C(1 J) *Y(1)-A(1 J) *Y(3)11 WHENEVER T.E.3, F(3)=b(,1J)*Y(2)-A( 1J)*Y(3)-1 WHENEVER T*E.4. F 3)=3*B 1.J) *Y 2-2*C (1 J )*Y 1 ).-A (1 J) *Y 3)11 WHENEVER T.E.5. F(3)=3*B(1-J)*Y(2)-2*C(1,J)*Y(1).-A.(1,J)*Y(3)11+4*(B(29J)*B(3,J)-A(2,J)*C(3,J)) WHENEVER T.E.69 F(3.)=B(1.J)*Y(2)-A( 1J)*Y(3)+B( 3J)-1 WHENEVER T.E.79 F(3)=3*B(1,J)*Y(2)-2*C(6,J)*Y(1)-A(1,J)*Y(3)+.B(4.J)-0.333 WHENEVER T.E.89 F(3)=3*8( 1J)*Y(2)-2*C(1iJ)*Y(1)-A(1iJ)*Y(3)+ 1B(5,J)+4.0*(B(2,J).*B(6,J)-A(2.J)*C(6,J)) WHENEVER T.E*9, F(3)=B(1,J)*Y(2)-A( i,J)*Y(3)+B(6,J) WHENEVER T.E.10.. F(3)=3*B (1J )*Y(2)-2*C (1J)*Y(1)-A(1J)*Y (3) 1+B(7,J) WHENEVER TE.llit F(3)=3*8(1,J)*Yi2)-2~*C(1,J)*Y(1)-A( tJ)*Y(3) 1+B(8,J)+4*(B(2,J)*B('9,J)-A(2,J)*C(9,J)) WHENEVER T.E.129 F.(3)=2.0*B(l J)*Y(2)+2.0*b(3 J)*b (4J)-A (J1 1)*Y(3)+2.0*C ( 3J)*A(4,J)-C(1 J)*Y(1)-0.666 WHENEVER T.E.13. F(3)=2*0*8( 1J)*Y(2)+2.0*B(39J)*B(59J)-A(1,J 1)*Y(3)+2.0*C(3,J)*A(5,J)-C(1,J)*Y(1) WHENEVER T.E.14. F(3)=2Q0*B(1iJ)*Y(2)+2.0* B( 12 J )+2,0*B(3 J)* 1B(7,J)-A(1,J)*Y(3)-2.0*C(6,J)*A(4,J)-C(1,J)*Y(l) WHENEVER T.-E. 15 F ( 3 ) =2- 0*6 (13 J ) +2.0*8 (1 J )*Y ( 2 )+2 0* ( 3,J ) * 1B(8,J)-A(1,J)*Y(3)-2.0*C(6,J)*9J*A(5,J)-C(1J)*Y(1) _.~WHENEVER- T~E. 16i F(3 ) =2.0**B. I'1J)*Y(2)-A( 1 J) *Y( 3 -C(1 J ) *Y (1) 1+B(33J)MB(10,J)+B(9,J)*B(4,J)-B(6tJ)*B(7,J)-C(9,J)*A(4,J)-C(3 2i J*L AlL..).. +C ( 6, J)*A ( 7, J) WHENEVER TeE.17, F(3)=2.0*B(1,J)*Y(Z)-A(1 J)*Y(3)-C(1 J)*Y(1) L.'3-J~B4..11W ) +B(9 J )*B( 5,J )-b (6,J) *B I J)-C(9, J)*A( 5J)-C(3 2,J)*A(11,J)+C(6,J)*A(8,J).lHEEV.ER T.E.-18 F(3) =2 U*B(1,J)*Y (2 )-A (1 J)*Y(3) -C ( 1 J)*Y (.) 1+8(3iJ)MB( 10,J)+B(9,J)*B(4,J)+B(6,J)*B(7,J)-C(9,J)*A(4J)-C(3 2. JJ *AI1O.J)-C ( 6-J )*A ( 7,J )'+B (14. J ) WHENEVER T.E.19s F(3)=2.0*B( 1J)*Y(2)-A( 1 J)*Y(3)-C(1 J)*Y(1) 1+B.(34.JJ,*B.( 11 J )+B ( 9 J )* ( 5 J)+B (6 iJ ) * (8,J)-C ( 9 J)*A (5 J)-C( 3 2~J)*A' (11 i J)- C(6, J)'*A(8,J)+8( 15J) WHE.NEVER.. _T E 20 F (3 )=6*B ( 1,J ) *Y (2)-5 *C ( 1,J ) *Y ( 1 ) -A ( 1.J) *Y (3 1)-1. WHENEVER TEA.21, F(3)=6,*b( 1,'J)*Y(2) +8.* (2,J)*b (2,J)-b5*C(1,J 1)*Y(l)- **A(2,J)*C(2,J)-A( 1J)*Y(3)-0.5 WHENEV.ER T.*E.22.F 3 )=5* t 1,J )*Y (2) -A ( 1,J )*Y (3 )-4e*C (1 J ).Y ( 1)-1. WHENEVER Z - TE23, F (3) =5.*B ( 1 J)*Y(2.)+12. *B (2 J)*B(4 J)-A(1 J) 1*Y(3)-7C2*A(2,J)*C(4,J)-4**C(1,J)*Y(1)-4'.8*C(2vJ)*A(4,J)-1l WHENEVER..T_=E 2.4. F(3)=5**B(lJ )*Y(2)+6..*B.(3,J)*t (20,J)-A(1,J)* 1Y(3 )-4.MC( 1J )*Y (1) - WHENEVER T..eE~25 F(3)=5*B( 1 J)*Y(2)+12.*B( 2 J )B(5 J) +6.*B(3 1J)*B(21TJ)-A(1, J)*Y(3)-7.2*A(2tJ)*C (5~J)-4.*C(l~J)*Y(l)-4.8 2*C(2 J..*A 5,J) WHENEVER T.Er26,F(3)=4 *B( 1J)*Y(2)+3 3333*B(3,J)*B(22,J)-A( 1.1 J)*Y(3) -33333*C(3,J)*A(22 J)-3**C (,J)*Y(1)-0,6667 WHENEVER T.E.27,F(3)4. =*B( 1,J)*Y(2) +3.*B(4,J)*b (4,J)-A(1,J)* 1Y(3)-3**A(49J)*C(4,J)-3,*C(1,J)*Y(1 -0.3333 WHENEVER T*E.28 F(3)=8.*b(2,J)*B(12,J)+4.*b( 1 J)*Y(2)+3*3333* 1B(3 J)MB.(23,J)+6*B (4,J)*B(5,J)-A(1,J)*Y(3)-6.*A(2,J)*C(12,J) 2-3* A(4TJ)*C(5 J)-3*C(3 J)*A( 5J )-33333*C(3 J)*A(23 J)-2**C 79

3(2,J)*AU12,J)-3.*C(1J).Y(1 ).-W.E:ER.-T-E 2F3. =^.a L *2.J-.).*e(13.3J ) +4**B ( 1,J ) *Y ( 2 ) +3.3333* - 1B(3,J)*B(259J)+3,*B(5SJ)*B( 5,J)-A(1 J)*Y(3)-6*A(29J)*C(13,J) 2-3-~*C 15f ~J}~) ~A5JL-3 * 3333 C (3 ~J-)PA (25.J ) -2.*C (2 J *A( 13 J) -3 * * 3C( 1,J)*Y(1) WLLENEVER T.EEa QtLf(3.).=4.*B( 1,J ).*Y(2) +33333* ( 3J) *B (24,J)-A(1 1 J)*Y(3D-3,3333*C(3J')*A(24+J)-3*0*C(llJ)*Y(1) WHENlEVERT _3L-^.-4 L2- -PRA -.1,J) *Y( 2) WHENEVER TeE.329 F'2)=PR*(2.0*B( 1J)*.Y(1)-3O0*A(2,J)*B(31 J)1. A(L1., 44Y-2rl-L. WHENEVER T*E*339 F2.)=PR*(B(C1J)*Y(l)-A(1,J)*Y(2)-3.0*A(4,J)* 1B_(31.J.. WHENEVER T'E.e34+ F (2) PR*(B( 1J)*Y( 1 )+40*B( 3J )*A(32 J)-A (1 LJ.) *Y.L2 )-3 0*A (5.J) *('31 J ) ) WHENEVER T*E.35. F(2.)PR*(B(1,J)*Y(1)-A(1,J)*Y(2)-3.0*A(7~J)* 1B ( 31.I1J. ~ WHENEVER T*E*369 F(2)=PR*(B( 1J)*Y(1)+40*BC(6J)*A(C32,J)-A(Cl lJ.*Yt.2.) -3ai 0*A( 8 J ) *B (31 J ) WHENEVER T'*E37~ F(2) PR*(B(1J)*Y( 1)-A( 1J)*Y(2)-3O0*A(1UeJ) WHENEVER TEE*38~ F(2)=PR*(B( 1J)*Y( 1)+4.0*b(9,J)*A(32,J)-AC(1 1JL*Y12 -3L0*At11,J ) *B ( 31,J ) WHENEVER T.E.39. F(2)=PR*(O.666*B(3,J)*A(33,J).-A( 1J)*Y(2)-A( 112 )*.B ( 31. J)..). WHENEVER T.Ee40 F(2)=PR*(0.666*B(3,J)*A(34~J)-A(lJ)*Y(2)-A( J13,4J.I*B (31.J4.) WHENEVER TeEe41~ F(2)=PR*(0.666*B (6J)*A(33 tJ)+0*666*b(3 J)*A l.l35s JJ+2.O0*A 39) J. )-A (J) *Y (2)-A(14 A JJ) *B( 31 ) ) WHENEVER TeE.42, F(2)=PR*(0.666*b(6,J)*A(34J)+0666*B(3,J)*A.1136L.Jl +2. Q*A(.40 J ) -A ( 1J)*Y ( 2)-A ( 15J)*b ( 31J) ) WHENEVER TeE.43v F(2)=PR*(b(9*J)*A(33,J)+bC(3J)*A(37,J)-b(tbJ 11....AL35. -J... -A.(1 J) *Y (2) -30*A(16J ) *B ( 31J ) ) WHENEVER T*E*44+ F(2)=PR*(B(9,J)*A(34,J)+8(3~J)*A(38,J)-bI(6J lltA.(.36, J.I-A.I isJ 1)*Y (2) -3.0*A (17,J)*B(31 J)) WHENEVER T.E+45,F(2)=PR*(B(9,J)*A(33,J)+B(3.J)*A(37~J)+B(6,J) 1*AI35J..J -A(.,Al.JLjY..I-12)-3*A ( 16 J ) *B (31 J )+3*A ( 41 J ) ) WHENEVER T*E*46tF (2) PR*( B (9J)*A(34 J )+B(3J )*A( 38 J)+B(6 J).1*AL-36.J) -A.1J.).*Y 12 ) -3.*A (17. J ) *B ( 31 tJ )+3 *A ( 42,J ) ) WHENEVER T.E.47,F(2)=PR*(-A(1J)*Y(2)+4.*b(1~J)*Y(1)-5.*A(20O *J.B-( 31.aJ. WHENEVER T.E.48F (2)=PR*(-A ( 1J)*Y(2)+4.* (1J)*Y(l)-5.*A (21 l..i. L3s J.1~Za66.7*( 2.*B( 2 J ) *A.( 3ZJ ) -3 ~ A (,J ) *B ( 32,J ) )) WHENEVER T*Ee49.F(2) PR*(3**B( 1J)*Y ( 1)-A(1 J)*Y(2)-5*B(31 J 1)*Al22sJ. ). WHENEVER T.Ee50F (2) =PR* (2*B(2 J.) *A (33J )+3e*B (1 J) *Y (1)+6.* 1A( 32jJD*.B-l.SJ ).-6 *A( 2 J ) *B.133 J) -A (1 J)*Y ( 2 )-6*A ( 4 J ) *B ( 32 J 2)-5.*A(23#J)*B(31,J)) WHENEVER.....IE.51 F(2 ) PR* ( 2.*.B (2 J)*A ( 34J)+3 O*B(1,J ) *Y(1) +6 1*B(3,J)MA(48,J)+6t*B(5,J)*A(32,J)-6**A(2,J)*B(349J)-A(1,J)*Y(.22)-.6b*sB32J.. *A (5.J) -5.*A( 25 J1*B..31 J ) WHENEVER T.E.52 F(2) =PR* (3*B( 1,J)*Y( 1)+6.*B ( 3J)*A(47.,J)-A( 1.sJ..*Y.12J. -5 *_Bl3-)s4L.4*A (.24.J).L)WHENEVER T.E.53.F(2)=PR*(2.*B( 1J)*Y(1)+B(5,J)*A(33,J)+B(4,J).lAtLs.JL+4_a.*LB12z J)* A 32 J L.-.6.*Aj.2J L*B 39 J ) -A (1 J )*Y ( 2 ) -2* 2A(5,J)*B(33,J)-2.*A(12,J)*B(32~J)-2.*A(4,J)*b(34,J)+2.*b(3,J) 23*Ai.LaJ l.-3oAL2a.J *B (31.J ) I WHENEVER T.E.54F (2) PR*(2*B( 1 J)*Y 1)+B (4J)*A(33,J)-A (1J) *Y1LZWHENEVER T.EZ-*A 5 )B33 R*2J) *-3 B ~*A 27 J)*B Y1 (31.)J. ) WHENEVER T EE55 tF(2 ) =PR *( 2~*B( 1 J )*Y (1 )+B (5 J )*A( 34 ~J)++4 *B ( 1 8o

13J )*A( 32J )-6*A ( 2 J )* (40J)-A( 1 J )*Y(2 )-2*A 5J)B(34,J )22.*A(13,J)* ( 32J ) +2.* ( 3,J )*A (51 J )-3.*A (29J)*.B(31 J) ) WHENEVER T.E.56F (2)=PR*(2.*b( 1J)*Y( l)-A ('J)*Y(2)+2.*B(3,J) 1*A(49.J)-3.*A(269J)*B(319J)) WHENEVER T. E579F(2)=PR*(2*B( 1J)*Y ( 1)-A (1J)*Y(2)+2.*B(3 J) 1*A(52,J)-3,*A(30,J)*B(31 J)) CALC S=RKDEQ.(O) WHENEVER S.E1l.0, TRANSFER TO SW1 WHENEVER J.E.GRSIZ, TRANSFER TO SW3 TRANSFER TO START SW3 WHENEVER T.Eo.19 TRANSFER TO SW8 YHENEVER T.E.2, TRANSFER TO SW7 WHENEVER T.E.39 TRANSFER TO SW8 WHENEVER,T*E.4 TRANSFER TO SW16 WHENEVER T.G.4 *AND.T.L.20.TRANSFER TO SW11 WHENEVER T.E.20,TRANSFER TO SW17 WHENEVER T.E.21.TRANSFER TO SW11 WHENEVER T-E.,22' TRANSFER TO SW18 WHENEVER T.G.22.AND.T.L.TMAX.TRANSFER TO SW11 WHENEVER T.E.TMAXTRANSFER TO SW12 WHENEVER TG6.TMAX.TRANSFER TO SW13 SW7 WHENEVER.ABS(Y (2)-.25 ).GO.001.AND. N.L.NMAX R(N)= (YU2)-.25 ) TRANSFER TO SW2 OTHERWISE TRANSFER TO SW9 END OF CONDITIONAL SW8 WHENEVER.ABS. (Y(2)-1).G.0.001.AND. N.L.NMAX R(N)=Y(2)-i TRANSFER TO SW2 OTHERWISE TRANSFER TO SW9 END OF CONDITIONAL SW11 WHENEVER.ABS. Y(2).G.O.001.AND. N.L.NMAX R(N)=Y(2) TRANSFER TO SW2 OTHERWISE TRANSFER TO SW9 END OF CONDITIONAL SW16 WHENEVER.ABS.(Y(2)-0.333).G.O.OO0.AND. N.L.NMAX R(N)=Y(2)-0.333 TRANSFER TO SW2 OTHERWISE. TRANSFER TO SW9 END OF CONDITIONAL SW12 WHENEVER.ABS. (Y(1)-l )*G..0001.AND.NeL.NMAX R(N)=Y(1)-1 TRANSFER TO SW2 OTHERWISE TRANSFER TO SW15 END OF CONDITIONAL SW13 WHENEVER.ABS.Y(1).G.0.001 *AND.N.L.NMAX R(N)=Y(1) TRANSFER TO SW2 OTHERWISE TRANSFER TO SW15 END OF CONDITIONAL SW17 WHENEVER.ABS.(Y(2)-0 16667) ~ G..Q.1 ANDN*^.LL.MA R(N)=Y( 2 )-0.16667 81

TRANSFER TO SW2 OTIHERW ISE TRANSFER TO SW9 END OF CONDIT.IONAL SW18 WHENEVER *ABS.(Y(2)-0.2)G.O.O001 *AND*N*L.NMAX RI.N)=Y(2)-0.2 TRANSFER TO SW2 ~OHERW.ISE TRANSFER TO SW9 END OF CONDITIONAL SW15 WHENEVER *ABS4 Y(2)9GO0.001 *AND. ZET.L.ZMAX, TRANSFER TO 1SW10 SW9 WHENEVER *ABS. Y(3).G.0O001 *AND. ZET.L.ZMAX9 TRANSFER TO SW 110 PRINT RESULTS T PRINT RESULTS ZET WHENEVER ZET.E*ZMAX* PRINT COMMENT $EU. CAN NOT bE MADE TO SA 1TISFY BOUNDARY CONDITONS $ WHENEVER N.E.NMAXtTRANSFER TO SW14 PRINT COMMENT $ R(M) $ THROUGH SW5. FOR M=29 1, M*GeN SW5 PRINT FORMAT EPS, R(M) THROUGH SW4, FOR I=1 1i I.G.GRSIZ A(TI)=Y1('I) B(T.I)=Y2(I) WHENEVER T.L.TMAX C(TJI)=Y3(I) D1.T*I )=F3( I) OTHERWISE C(T.)=F3( I) END OF CONDITIONAL WHENEVER T.L.TMAX PRINT FORMAT DEL, P(I), Y1(I), Y2(I), Y3(I), F3(I) OTHERWISE PRINT FORMAT ROH, P(I), Y1(I), Y2(I), F3(I) SW4. END OF CONDITIONAL WHENEVER T.L.TMAX PRJJiUT FORMAT DEL. X*Y(1), Y(2). Y(3)' F(3) OTHERWISE PRINT FORMAT ROH~ X, Y(1), Y(2)9 F(2) END OF CONDITIONAL Z= RSIZ A(T Z)=Y (.1) B(T Z)=Y(2) WHENEVER ToLeTMAX C( T.Z)= Y (3) D(T Z)=F(3) OTHERWISE C(TtZ)=F(2) END OF CONDIT.IONAL T=T+1 WHENEVER. TI.et.A..T.EXE.Q.T.E...E XU TR.D. E. (.2 Y (1 ),F ( 1 ),,X STEP) WHENEVER T.LE.TENDTRANSFER TO SW6 TH RUGH. S.W21fEQ R.Z _ILZ. G..iZMA THROUGH SW21.FOR M=1 1,M G MMAX THROUGH SIW21,EOR J1, IJ *eG.JMAX THROUGH SW21.FOR KzliK *GKMAX IHIRUGH.. SW2 QtEOR_.IL..t 1, G GRS I Z P(I)=STEP*(.I-1) 82

XYS(J)=XY(J)*XY(J) YP1(J)=3**A2(M)*XYS(J) YP2(J) =3.*AO*A3*XY5(J.)/Al US(I)=AO*B(3,I)+YP1(J)*B(4,1)+YP2(J)*bi(5, ) UT(I)=AO*B(6,I)+YP1(J)*b(7I )+YP2(J)*b(b,I) UU( I )=AO*B(9I )+YP1(J)*(O, 1)+YP2(J)*b(li ) WA(K)=W(K)*W(K)/(Al*Al) Q1(I)=WA(K)*UU(I)-US(I) Q1S(I)=Ql(I)*Q1(I) Q2(I)=W(K)*UT(I)/Al Q2S(I)=-2(I)*Q2(1) AMQS()=QlS(I) +Q2S(I) AMQ(I)SQRT.(AMQS(I)) ANQ(I) Q2(I)/Q1(I) PHAQ(I)3ATAN.(ANQ(I)) AlS=1.5*AO/(SQRT.(Al)) A25=AO*A3/A1 A05=AO*A5/A3 A14(Z)=Al*A4(Z)/A3 A12(M)=Al*A2(M)*A2(M)/(AO*A3) SSR(I)=2.*AlS*XY(J)*(A2(M)*A(12,I)+A25*A(13,I)+2.*A3*(A2(M)*A 1(28,I)+A14(Z)*A(269I)+A12(M)*A(27,I)+A25*A(29,I)+A05*A(30,I)) 2*XYS(J)/A1) YP3(J)=2.*A2(M)*XY(J)/A1 YP4(J) =2*AO*A3*XY(J) /(A*Al) TS(I)=YP3(J)*A(33,I)+YP4(J)*A(34,I) TT(I)=YP3(J)*A(35,I)+YP4(J)*A(361I) TU(I) =YP3(J)*A(37,I)+YP4(J)*A(38,1) S1(I)=WA(K)*TU(I)-TS(I) S1S(I)=S1( I )* 1( I) S2(I)=WUK)*TT(I)/A1 525(I)=S2(I)*S2(I) AMSS(I)=S1S(I)+S2S(I) AMS(I)=SQRT.(AMSS(I)) ANS(I)=52(I)/S1(I) PHAS(I =ATAN.(ANS(I)) SST(I)=3.*AO*(A2(M)*A(39,I)+A2S*A(40,1)+2.*A3*(A2(M)*A 53,1) l+A12(M)*A(549I)+A25*A(55I 1)+Al4(Z)*A(56,I)+AO*A(57, 1) )*XY5(J 2)/Al)/(Al*Al) SW20 PRINT FORMAT BETP(I),US()TI)UT(I),UU(I)~AMU(I),PHAU(I),5SR(I) 1lTS(I) TT(I) TU(I) AMS(I) PHAS(I),SST(I) NTAU(K)=Al**XY(J)*(A2(M#*C(12,1)+A25*C (131)-WA(K)*(A2(M)*C( 11691)+A'2S*C(17,1))+2.*A3*(A2(M)*C(28,1)+A14(Z)*C(26,1)+Ali (M) 2*C.(27,1)+A2S*C(29,1)+A05*C( 30,) )*XY6(J)/Ai) NNUS(K)3AlS*(A2(M)*B(39,1) +A2S*B(40,1)-WA(K)*(A2(M)*B(431 )+ lA2S*B(44,1))+2.*A3*(A2(M)*B(53,1)+A12(M)*B(54,1)+A25*B (D51)+ 2A14(Z)*B(56,1)+A05*B(57,1))*XYS(J))/A1 SW21 PRINT FORMAT ALPXY(J),W(K),NTAU(K)9NNUS(K) TRANSFER TO BEGIN SW14 END OF PROGRAM $DATA ZMAX=6,TMAX=3lTEND=57,CF=IlNMAX=50,TEP=O. 1,RSIZ=45bPR=1 i DYO.OO iYDP=l.OOOTCONST=O,Al=4,A3=-2.666b tXY ( 1)-0XY(2) =005 XY(3)=Ol,XY(4)=OC2,XY(5)=03, XY(6)=04,XY(7)=0O5,XY( )=0=Ot XY(9)=l1, W(l)sOKMAX=2,AO=lC.MMAX=lJMAX=99A5=0.53333,A4(i)O=,ZMA-iA(1))-0, W(2)=0.2 * 83

C. THE COMPUTER PROGRAM FOR ROTATIONAL OSCILLATION OF CYLINDER SURFACE $COMP I LE BEG.I N SW6 SW10 SW2 START MAD, EXECUTE, PRINT OBJECT, UUMP, PUNCH OBJECT PRINT COMMENT $ THE COMPUTtR PROGNAM F-O ROTATIONAL USCILLAiI 1ON $ DIMENSION Y(3), F(3), Q(3). INTEGER J, I, N. M. K, Z, T, ZET, PRI VECTOR VALUES DEL=$1H,F6.494F15.8*$ VECTOR VALUES ROH=$1H, F6.4, 3F15.8*$ VECTOR VALUES ALP=$1H,F6*4, 3F15.8*$ VECTOR VALUES BET=$1H,F6.4,12F10.6*$ VECTOR VALUES EPS=$1H. 1F20*4*$ DIMENSION P(45)9 Y1(45), Y2(4.5), F3(45), R(55), Y3(45), 1US(45) UT(45) (45)UU,45Q (45) QlS(45),Q(45) Q25(45),AMQS(45), 2AMQ(45DANQ(45 ) PHAQ(45),TS(45),TT( 45) TU(45),S1(45) S1S(45) 3S2(45),S2S(45),AMSS(45),AMS(45),ANS'(45) PHAS(45) YP3(10), 4XY(10)TWA('10) YP2( 10) NTAU(10),W(10) NNUS(10) SSR(45) A2(10), 5YP1(10)TYP4(10),A14(10),A4(10),A12(10) XYS(10) SST (45) DIMENSION A(2565,AD),b(2565,BD), C(2565,CD),D(2565,DD) VECTOR VALUES AD=2. 1. 45 VECTOR VALUES BD=2, 1. 45 VECTOR VALUES CD=2, 1. 45 VECTOR VALUES DD=2, 1, 45 READ AND PRINT DATA EXECUTE SETRKD. (3, Y(1), F(1), Q, X, STEP ) T=1.. ZET=O ZET=ZET+.1 WHENEVER ((-1).PeZET).L.O, YX=YDP+ZET-1 WHENEVEI. ( (.-1 ).P.ZET) G.O YX=YDP-ZET+1 R(O)=15 R(1)=l N=1 QaCF*(RUN)/R(N-1)-L ) WHENEVER Q.G.87.20, Q0=7.20 WHENEVER Q.L.(-87.20) Q0-87.20 DX=-DY*((2.35040/(EXP*(Q)-EXP.(-Q)))+1).DY=DX N=N+1 YX=YX+DY. WHENEVER T.L.TMAX Y(3)=YX OTHERWISE Y (2)=YX END OF CONDITIONAL J=0 X=O. WHENEVER T*E.1. Y(1)=CONST WHENEVER T;.G.'1 Y('1)=0 WHENEVER T.E.1.Y(2)=0 WHENEVER T.E,2,Y(2)=0 WHENEVER. T.E3,Y(2)=1.0 WHENEVER T.E.4,Y(2)=0.3333 WHENEVER T.*G4 *AND.T*L.22.Y(2)=0 WHENEVER T.E.22 Y'(2)=0.2 WHENEVER..T..G.22 *AND. TL. TMAXY (2) = J=J+1 P (J)=.X Y1(J)sY(.1).Y2 (J) ='Y 2 WHENEVER T.L.TMAX 84

Y3(J ) Y( 3) F3(J) =F (3 ) OTHERWISE F3(J)=F(2) END OF CONDITIONAL SW1 F(1)=Y(2) WHENEVER T.L.TMAX, F(2)=Y(3) WHENEVER T.E.1. F(3)=(Y(2).P.2)-Y(1)*Y(3)-i WHENEVER T. E.2 F (3 )4*B ( 1J) Y (2 )'3*C (1,J ) *Y(1 )-A(1,J ) *Y(3)1 WHENEVER T.E.3* F(3)=B(1,J)*Y(2)-A( 1J)*Y(3) WHENEVER T.E.4, F(3)=3*d( 1,J)*Y(2)-2*C(1,J)*Y(1)-A(1,J)*Y(3) WHENEVER T.E.5o F(3)=3*b(1,.J)*Y(2)-2*C(1,J)*Y(1l-A(1,J)*Y(3)+ 14.*(B(2TJ)*B(3,J)'A(29J )*C(3,J)) WHENEVER T.E.69 F(3)=B(1,J)*Y(2)-A(i1J)*Y(3)+B(3,J) WHENEVER T.E.7, F(3)=3*b(1 J)*Y(2)-2*C(6,J)*Y)(1)-A(1 J'*Y(3)+ 1B(4,J) WHENEVER T.E.89 F(3)=3*B(1iJ)*Y(2)-2*C(1~J)*Y(i)-A(1J)*Y(3)+ 1B(5,J)+4.0*(B(2,J)*B(6,J)-A(2,J)*C(6,J)) WHENEVER T.E.9~ F(3)=b(1,J)*Y(2)-A(1,J)*Y(3)+b(6,J) WHENEVER T.EelO, F(3)=3*B( 1J)*Y(2)-2*C(1,J)*Y( 1)-A( 1J)*Y(3) 1+B(7,J) WHENEVER T.E.ll F(3)=3*b(19J)*Y(2)-2*C(1,J)*Y(i)-A(1,J)*Y(3) 1+B(8,J)+4*(B(2,J)*B(9,J)-A(2,J)*C(9,J)) WHENEVER T*E.12, F(3 ) 2.0*B(1,J)*Y(2)+2o0*B(39J)*B(4,J)-A(iJ 1)*Y(3)+2,0*C(3,J)*A(4,J)-C(1,J)*Y(1) WHENEVER T.Eo13, F(3)=20O*B(1 J)*Y( 2 )+2O*B (3J ) *B (5 J)-A(1 J 1)*Y(3)+2.0*C(3,J)*A(5,J)-C(1,J)*Y(1) WHENEVER T.E.14, F(3)-=2.0*B(1 J)*Y(2)+2 0*(12 J )+2.0*b(3 J)* 1B(79J)-A(1iJ)*Y(3)-2oO*C(69J)*A(4,J)-C(1(J)*Y(1) WHENEVER T.E.15, F(3)-2.0*B(13,J)+2O0*B(1,J)*Y(2)+2.0*B(3,J)* 1B(8tJ)-A(1, J)*Y(3)-2.0*C(6,J)*A(5,J)-C(1,J)*Y(1) WHENEVER ToE.16, F(3)=2.0*B(1,J)*Y(2)-A(1,J)*Y(3)-C(1,J)*Y(1) 1+b(3,J)MB(109J)+B(9,J)*b(4+,J)-B(6,J)*b(7,J)-C(9J)*A(4,J)-C(3 29J)*A(10,J)+C(6vJ.)*A(79,J) WHENEVER TeE.17, F(3)=2.O*B(1,J)*Y(2)-A(1,J)*Y(3)-C(1,J)*Y(1) 1+8 ( 3, J)*B( 11 J)+B ( 9, J) *b(5J )-8 (6,J) b(69J)-C(9,J)*A(5,J)-C(3 2,J)*A(11 J)+C(6,J)*A(8,J) WHENEVER T.E.189 F(3)-2.0*8(1,J)*Y(2)-A(1,J)*Y(3)-C(1,J)*Y(1) 1+B(3 J)MB(10, J)+B(9J)B(4,J)+B(6J ) *b( 7J )-C (9.J)*A(4J)-C (3 2,J)*A(10,J)-C(6,J)*A(7,J)+b(14,J) WHENEVER T.E*199 F(3)=2.0*B(1,J)*Y(2)-A(1iJ)*Y(3)-C(1,J)*Y(1) I+8(3,JD*B(11,J)+B(9,J)*B(59J)+B(6,J)*6(dJ)-C(99J)*A(5,J)-C(3 2.J)*A(11,J)-C(6)J)*A(89J)+ ( 15,J) WHENEVER T*Eo20OF(3)=6.*b(1J)*Y(2)-5.*C(1,J)*Y(1)-A(1iJ)*Y(3 1) —1. WHENEVER T E. 21 F ( 3) 6.*b ( 1 J)*Y(2)+ -*bj ( J )*(2 J)-5.*C( 1,J 1)*Y(1)- **A(2,J)*C(2~J)-A(1,J)*Y(3)-0.5 WHENEVER T.E.22F(3) 5.*B( 1,J)*Y(2)-A(1 J)*Y(3)-4 *C( 1J)*Y(1 1) WHENEVER T.E,23,F(3)=5.*B( 1J)*Y(2)+12*b8(2 J)*b(4,J)-A(1 J) 1*Y(3)-7C2*A(2,J)*C(4,J)-4.*C(1J)*Y(1)-4+.diC(2~J).*A(4J) WHENEVER T Eo24 F(3)=5.*b( 1J)*Y(2)+6,*b(3,J)*B(20,J)-A(1 J)* 1Y (3)-4qMC(1,J)*Y (1) WHENEVER TE.25~,F(3)=5.*b(1,J)*Y(2)+12e-*b(2,J)*b(5,J)+6'*B(3, 1J)*B(21TJ)-A(1,J)*Y(3)-7.2*A(2 J)*C(5,J)-4*'C(1~J)*Y(1)-4+8 2*C(2,J)*A(5,J) WHENEVER T.E.26,F(3)=4.*b( 1,J)*Y(2)+33333* (3,J)* (22,J)-A(1 1,J)*Y(3)-3o3333*C(3,J)*A(J*A(22J)-3*C(1J)*Y() 85

WHENEVER TeE.27,F(3)4z*B( 1 J)*Y(2)+3.*B(4+J)*b(4,J)-A (1 J)* 1Y (.3) -3.MA'(4j ) *C (4 J)-3**C( 1.J )*Y (1 WHENEVER T.E.28 F( 3) =8.*B( 2J)*B( 12 J) +4.*B( 1,J)*Y(2 )+3i3333* 18(3.J)*B(23,J)+6b*8(49J)*B(5,J)-A(ilJ)*Y(3)-6,*A(2mJ )*C(12.J) 2-33*A(4,J)*C(5,J))-3*C(3.J)*A(5,J)-3.3333*C(3,J)*A(23J))-2.*C 3(2,J)*AU12.J)-3.*C(1,J)*Y(1) WHENEVER'T.E.29,F(3)=8*B(2,J)*B(13,J)+4.*B(1,J)*Y(2)+3.3333* lB(3 J)*B(25.J)+3*B ( 5J )*b( 5 J).-A(.J) *Y(3 ).-.6 *A(2 J )*C( 13,J) 2-3.*C(5,J)*A(5 J)-3.3333*C(3 J)*A(25,J)-2.*C (2J)*A(13 J)-3.* 3C( 1J)*Y(1) WHENEVER T. E30 F (3) 4 *b( 1,J)*Y(2)+3.3333*B( 3,J)*B (24 J)-A ( 1,J)*Y(3D-3.3333*C(3,J)*A(24,J)-3eQ*C( lJ)*Y(l) WHENEVER T.E.31, F(2)=-PR*A(1IJ)*Y(2) WHENEVER T.E*32. FL2)=PR*(2.0*B( 1 J)*Y(1)-3.0*A(2,J)*b(31,J)1A(1,J)*Y(2)) WHENEVER T.E.33, F(2)=PR*(B(1*J)*Y(1)-A(1,J)*Y(2)-3.0*A(4,J)* 1B(31,J)) WHENEVER T.E.34, F(2)=PR*(B(1,J)*Y(1)+4e0*B(3,J)*A(32,J)-A(l, 1J)*Y(2)-3o0*A(5,J)*B(31,J)) WHENEVER T.E.35.. F(2) PR*(.(1,J)*Y(1)-A(1 J) *Y(2)-3.0*A( 7J)* 1B(31,J)) WHENEVER TE. 36. F(2)=PR*(B(1.J)*Y(1)+4.0*b(6 J)*A(32,J).-A(1i 1J)*Y(2)-3.0*A(8,J)*8(31,J)) WHENEVER T.E.37, F.(2)=PR*(B( 1J)*Y-(1 )-A(1,J)*Y(2)-3.0*A(1OJ) 1*B(31,J)) WHENEVER T*E~38. F(2)}PR*(B(1lJ)*Y 1)+4O0*bI(9,J)*A(32,J)-A(1, 1J)*Y'(2)-30 *A ( 11J )* (31-,J)) WHENEVER T.E.39, F(2)-PR*(0.666*B(3,J)*A(33,J))-A(1,J)*Y(2)-A( 112,J)*B(31,J)) WHENEVER T.E.40. F(2)=PR*(0.666*B(39J)*A(34,J)-A(1,J)*Y(2)-A( 113,J)*B(31,J)) WHENEVER T.E.41 F(2)=PR* (O666*B(6,J)*A(33,J )+0O666*B( 3 J) *A 1(35,J)+2.0*A(39,J)-A(1,J)*Y(2)-A(14,J)*B(31.J)) WHENEVER T.E.42, F(2)=PR*(0.666*B(6,J)*A(34,J)+0.666*B(3,J)*A 1(36,J)+2.0*A(40.J)-A(1,J)*Y(2)-A(15 J)*B(31,J)) WHENEVER T.E.43, F(2)=PR*(B(9 J)*A (33,J)+ (3,J)*A(37,J)-B(6,J 1)*A(35,J)-A(loJ*Y(2)-3.0*A(16,J)*B(31 J)) WHENEVER T.E.44, F(2)=PR*(B(9,J)*A(34,J)+B(3,J)*A(36,J)-b(6,J 1)*A(36,J)-A(1,J)*Y(2)-3.0*A(17,J)'*B(31tJ)) WHENEVER TEE.45 F(2)=PR*(B(9,J)*A(33,J)+B(3,J)*A(37,J)+B(6,J) 1*A(35,J)-A(1,J)*Y(2)-3.*A(16,J)*b(31,J)+3.*A(41,J)) WHENEVER T*.E46,F(2)=PR*(b(9,J)*A(34,J)+B(3,J)*A(38,J)+B(6,J) 1*A(36J )-A( 1,J)*Y(2)-3.*A( 17 J)*B (3 1 -J )+3 **A(42,J)) WHENEVER T* E47,.F(2) PR*(-A ( 1J)*Y ( 2 )+4.*b ( 1,J) *Y( 1 )-5.*A (20 1J)*B(31,J)) WHENEVER T.E.48 F (2) =PR* (-A (1 J)*Y ( 2 ) +4.* ( 1J )*Y( 1)-5.*A(21, 1J)*B(319J)+2.6667*(2,*B(2,J)*A(32,J)-3.*A(2,J)*b(32,J))) WHENEVER T.E.49,F(2)=PR*(3.*b( 1,J)*Y(1) -A(1,J)*Y(2)-.*b(31,J 1)*A(22,J)) WHENEVER T.E.50 F (2) PR*(2*B (2.J)*A(33,J)+3.i (1 J ) *Y (1 )+6.* 1A(32,JD*B(4,J)-6e*A(29J)*b(33,J)-A(1,J)*Y(2)-6.*A(4,J)*b(32,J 2)-5.*A(23,J)*B(31,J)) WHENEVER T.E.51,F(2)=PR*(2.*B(2,J)*A(34,J)+30*B(1,J)*Y( 1)+6. 1*B(3,J )MA(48,J)+6**b(5 J)*A(32 J)-6 *A(2,J)*b(34,J)-A( 1J)*Y( 22)-6.*B(32,J)*A(5,J)-5.*A(25,J)*b(31,J)) WHENEVER T.E.52,F(2)=PR*(3*b( 1J)*Y( 1 )+6.*b(3,J)*A(47,J)-A(1 1,J)*Y(2)-5*B(31 J)*A.(24,J) ) WHENEVER T.E.53,F(2)=PR*(2e*B(1,J)*Y( 1)+B(,J)*A(33,J)+b(4,J) 1*A(34,J )+4.*B(12,J)*A(32,J)-6.*A(2 J)*6(39,J)-A( J)*Y(2)-2* 86

2A( 5 J)M43 ( 33 J)-2*A ( 1,J ) *b( 32 J )-2 *A(49J ) (,J). - o 3j ) 3*A ( 50,JD-3.*A( 28, ) *b( 31J) ) WHENEVER T.E.54,F (2)=PR*(2e.*(1,J)*Y( I)+t(4,J)-A('.3,J)-A(1j) 1*Y (2 )-2C*A (4,J )B (33 3J)-3.*A (2.7,J )* ( 31,J) ) WHENEVER T.E.55,F(2) PR*( 2.*b( 1 J)*Y ( )+b ( J)*A( j4,J)+4. ( 139J )*A( 32,J ) -6.*A( 29 J) *6 (40, J )-A ( i J )'*Y (2) -2.'*A (. J ) o(.,4j)22.*A ( 13,J )*B ( 32,J )+2*' b( 3,J ) *A ( 51,J ) -3 - A ( 2,J ) - b( 31,J) ) WHENEVER T.E. 56, F (2 ) =PR* ( 2. b ( 1,J ) -Y ( 1 ) -A ( 1 J ) ) 'Y ( Z ) +2 * * ( 3,J ) 1*A(49,J)-3.*A(26,J)*B(31,J)) WHENEVER T E 57, F ( 2 ) =PR* (2. - * b( 1,J ) A Y ( 1) -A ( 1,J ) - Y ( ) +2.-o *( 3,J 1*A(52,J)-3,*A (30,J)* (31J) ) CALC S=RKDEQC(O) WHENEVER S.E.1.0, TRANSFER TO SWi WHENEVER JoE.GRSIZ, TRANtorER TO o3 TRANSFER TO START SW3 WHENEVER T.E.1 TRANSFER 10 SW8 WHENEVER T.E.2, TRANSFER TO SW7 WHENEVER T.G.2 *AND.T*.L.20,TRANSFER [O SWll WHENEVER T*E.209TRANSFER TO SW18 WHENEVER T*G.20.AND.I.L.TMAXTRANSFErI TO 'Wl11 WHENEVER T.E.TMAX,TRANSFER TO SW12 WHENEVER T.*GTMAXTRANSFER TO S.W13 SW7 WHENEVER *ABS.(Y(2)-.25 )u.0.01.ANU. N.L.NMAX R(N)=(Y(2) - 25) TRANSFER TO SW2 OTHERWISE 1RANSFER TO SW9 END OF CONDITIONAL SW8 WHENEVER AB3S. (Y(2)-1 )..0001.AND. N.L.NMAX R(N)=Y(2)-1 TRANSFER TO SW2 OTHERWISE TRANSFER TO SW9. END OF CONDITIONAL SWll WHENEVER.ABS Y(2).G.*O.01 *AND, N.L.NMAX R(N) Y(2) TRANSFER TO SW2 OTHERWISE TRANSFER TO SW9 END OF CONDITIONAL SW12 WHENEVER *ABS. (Y(i)-l ).*U.0001 *AN'D.N.L.sNAA R(N)=Y(1)-1 TRANSFER TO SW2 OTHERWISE TRANSFER TO SW15 END OF CONDI'IONAL 5W13 WHENEVER.ABS.Y(1) o.O.0 01 *ANL).NeLNivAX R(N) =Y( 1) TRANSFER TO SW2 OTHERWISE TRANSFER TO SW15 END OF CONDITIONAL SW18 WHENEVER *ABS.(Y(2)-O.166b6) *G.O.vOl.ANDL.NL.NMAX R (N ) =Y ( 2 )-0 1666 7 TRANSFER TO SW2 OTHERW I S-E TRANSFER TO SW9 END OF CONDITIONAL SW15 WHENEVER *ABS. Y( 2 ).oO.UOi *ANu. Lit 1.L. 'vlMAX, AiA A K, -FE lu

I1SW 10 SW9 WHENEVER ABS. Y.(3)oG*0.001 *AND. ZET*L.ZMAX. TRANF-ER [0( SW 110 PRINT RESULTS T PRINT RESULTS ZET WHENEVER..ZET.E.ZMAX, PRINT COMMENT $EO. CAN NOT BE MADE.TO SA 1TISFY BOUNDARY CONDITONS $ WHENEVER N.E.NMAX.TRANSFER TO SWl4 PRINT COMMENT $ R(M) $ THROUGH SW5. FOR M=2 1, M.G.N SW5 PRINT FORMAT EPS. R(M) THROUGH SW4. FOR I=1, 1, I.G.GRSI.Z A(TII)=Y1(I) B(T.I )=Y2( I ) WHENEVER TeL.TMAX C(T~I )Y3( I) D(T I )=F3(I ) OTHERWISE C(TTI)=F3( I END OF CONDITIONAL WHENEVER T.L.TMAX PRINT FORMAT DELt P(I), Y1(I), Y2(I). Y3(I), F3(I) OTHERWISE PRINT FORMAT ROH, P(I), Y1(I). Y2(I), F3(I) SW4 END OF CONDITIONAL WHENEVER T.L.TMAX PRINT FORMAT DEL, XtY(1)9 Y(2), Y(3)9 F(3) OTHERWISE PRINT FORMAT ROH. X~ Y(1) Y(2), F(2) END OF CONDITIONAL Z= RSIZ A(T Z) —Y ( 1 ) B(TtZ)=Y(2) WHENEVER T.L.TMAX C(TZ)=Y(3) D(TZ)=F(3) OTHERWISE C.(TZ)=F(2) END OF CONDITIONAL T=T+1 WHENEVER T.E.TMAX. EXECUTE SETRKD.(2.Y(1).F(I)QUXSTEP) WHENEVER T.LEETENDTRANSFER TO SW6 THROUGH SW21iFOR M-= 11M.GGMMAX THROUGH SW21OFOR Z=ll,,Z.G.ZMA THROUGH SW21tFOR J=9llJ.oGJMAX THROUGH SW21,FOR K=11K.oG.KMAX THROUGH SW20,FOR I=1'1,I.GeGRSIZ P(I)=STEP*(I-1) XYS(J)=XY(J)*XY(J) YP1(J)=3.*A2(M)*XYS(J) YP2(J)=3**AO*A3*XYS(J)/A1 US(I)=AO*B(3,I)+YP1(J)*bC(4,1)+YP2(J)*b(5I ) UT(I)=AO*B(6,I)+YP1.(J)*3C(7,1)+YP2(J)*b3(6,I) UU(I)=AO*B(9,I)+YP1(J)*b(iOI)+YP2(J)*b(11,I) WA(K)=W(K)*W(K)/(A1*Al) Q1(I)=WA(K)*UU(I)-US(I) Q1S(I)=Q1( I )*Q (I) 2( I)=W(K)*UT(1)/Al Q2S(I)=Q2(I)*Q2(I) 88

AMQS I )=Q15( )+Q2S(I) AMQ(I) SQ-RT*(AAMQS( I)) ANQ(I)=Q2(I)/Q1(I) PHAQ(I)3ATAN (ANQ(I ) A1S=1-5*AO/(SQRT.(A1)) A2S=A0*A3 A... A05=AO*A5/.A3 A12 (M) =Al*A2 (M.*A2 UM) / ( AO*A3.) A14(Z)=A1*A4(Z)/A3 SSR(I)=2.*A1S*XY(J)*(Ai(M).*A(L121t)+A2S*A(13,I)+2.*A3*(A2(M)*A 1(28.1 )+A14(Z)*A(26I )+A12(M)*A(27,I )+A2.*A(29,1 )+A05*A(301) ) 2*XYS(J)/A1)/A. YP3(J) =2,*A2 (M)*XY (J)/A1 YP4(J )=2*AO*A3*XY (J) / (A*A1) TS( I )=YP3(J)*A(33 I ) +YP4(J)*A(3491 ) TT(I )=YP3(J)*A(35.I)+YP4(J)*A(36 I ) TU( I)=YP3(J)*A(37,I )+YP4(J)*A(38, ) S1( I )=WA(K)*TU( I )-TS(I) S1S( I )=S1( I )*S1( I ) 52(1 )=WUK*TT( )/A1 52S()S2( I 2(I)*S2(I) AMSS(I)-S1S(I )+S25(I) AMS(I )=SQRT.(AMSS( I)) ANS(.) S2 2 I.) /.S1 ( I) PHAS(I)=ATAN(ANS( I)).SST(I.)=3.*AO* (A2(M)*A(391 )+A2SA( 401 )+2.*A3*(A2(M)*A (53, ) 1+A12(MD*A(54,I)+A2S*A( 55,I )+A14(Z)*A(56,I )+A05*A(57,I ) )*XAY(J 2)/A1 )/ (A1*A.1. SW20 PRINT FORMAT BETP(I),US(I) UT(I),UU(I ) AMQ(I),PHAQ(I ).SR(I) 1,TS( I ).TT( I) TU(I ),AMS( I ) PHAS( I),SST( I) NTAU(KD=A1S*XY(J)*(A2(M)*C( 12,1)+A2*C (131 )-WA' K) * (A2(M)*C( 11691)+A2S*C(17,1) ) +2 *A3*(A2(M)*C (28 1) +A14(Z).*C(26,1)+A12(M) 2*C(27,1) *C(291)2S*C(29 +A5*C(301))*XYS(J)/A1) NNUS(K)3A1S*(A2(M)*b( 39,1) +A25*b (40,1)-WA (K )*(A2(M) *b(431 )+ lA2S*B(44,1))+2.*A3*(A2(M)*B(53,1)+A12(M)*B( 541,)+A25*B(5b,1)+ 2A14(Z)*B(56,1)+AQ5*(.57,1) )*XY5(J) )/Al SW21 PRINT FORMAT ALPXY(J),W(K),NTAU(K) NNUS(K) TRANSFER TO BEGIN SW14 END OF PROGRAM S DATA ZMAX=6,TMAX=31'TEND=57,CF=lNMAX=50,STEP=0*lGRSIZ=45,PR=i-, DY.-Q-_-.O.Ql.-.YDP= 00 T CONST=O,Al=4,A3=-2.6666,XY(1)=OXY(2)=0.05, XY(3)=0elXY(4)=0C2,XY(5)=0*3,XY(6)=0.4,XY(7)=0.5,XY(8)=0O8,XY(9)=1., W(ll.).=0OKMAX31..AO=1C*MMAX=l1JMAX=9,A5=0.53333,A4 (1)=1. ZMA=IA2 (1)=1 * ZMAX=6,TMAX=31,TEND=57,CF=1,NMAX=509,TEP=Ol..GRSIZ=45,PR=i.O0 D.Y.=-0OQlY.YDP=1-OOOTCONST=O.1.,Al=4.,A3=-2 6666,XY(1) =0 XY i2)-=0.Ot. XY(3)=0.1XY(l)=0C2,XY(5)=0.3,XY(O6)C2=034,XY(X(7)=-0.5,XY(8)=o08XY(9)= W..(.-.l)..,KMAX=.1.AO=lCMMAX=lJMAX=9,A5=0b53333,A4(1)=0,ZMA-1 A2(1)-0 * 89

UNIVERSITY OF MICHIGAN III5111 11111111 9015 03527 $745