THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING INSTABILITY OF A LAYER OF LIQUID FLOWING DOWN AN INCLINED PLANE AT LARGE REYNOLDS NUMBERS Sung P. LTn A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Engineering Mechanics 1965 April, 1965 IP-703

Doctoral Committee: Professor Chia-Shun Yih, Chairman Associate Professor Walter R. Debler Professor Arnold M. Kuethe Professor Vi-Cheng Liu

ACKNOWLEDGMENTS The writer wishes to express his appreciation to Professor Chia-Shun Yih who suggested the subject and also directed the study of the problem. The writer is also grateful for the interest and time of all other members of his committee. The work has been partially supported by National Science Foundation and Army Research Office (Durham). The writer has also received supports from Rackham Graduate School and the International Center of the University, in the form of scholarships, and the indispensable assistance from the Computing Center of the University of Michigan, which made its facilities available to the writer, Thanks are also due to the Industry Program of the College of Engineering, for its aid in the final preparation of the manuscript. ii

TABLE OF CONTENTS Page ACKNOWLTEDGMENTS o o o o o o Q o o o o o o o o o o d o o o o o. o. o o. 3.O r ii LIST OF TABLES4. a -. 0 4 0 6 aa, O O O O O.o 5.a a. o a a iv LIST OF FIGURES eoooao owoee o eoeo o o *vo oo ooooo v I. INTRODUCTION.......................... 1 II. FORMULATION OF THE PROBLEM...................................* 4 A. Two Dimensional and Three Dimensional Disturbances....... 4 B. The Governing Differential Equation...................... 4 C. Boundary Conditions...................................... 11 D. Hydrodynamic Stability Problem as an Eigen-Value Problem..O~vvv**vevO~o~ovvvo0~o~vo**6***@@@ 13 III. METHODS OF COMPUTATION AND THE RESULTS....................... 31 A. C. C. Lin's Method............................... 31 B. Method to be Used for Problems in Which ce is Larger than Unity.............................................. 38 IV. DISCUSSION ON THE RESULTS AND ON THE METHOD OF COMPUTATION... 56 V. CONCLUSIONS............. AND THEIR NUMERICAL VAS U.E 58 VI. APPENDICX A SOME FUNCTIONS AND THEIR NUMERICAL VALUES USED IN C. C. LIN'S METHOD......................... 61 VII. APPENDIX B - COEFFICIENTS OF THE POWER SERIES SOLUTION OF THE INVTISCID EQUATION............................................ 65 BIBLIOGR APHY.............................................. 68 iii

LIST OF TABLES Table Page Io Eigen-Values Obtained by Co Co Lin's Method for S = 0.0 and 5 - 1 Min.................a..... o.....o.......a.. 39 IIo The Functions F(Z) and((Z).... e..,....., o.... 49 III. Eigen-Values Obtained by a Modified Method for S = 0.0 and =-in- 1 Min......,...........,..,.*.......,....,..... 51 IV. Eigen-Values for S = 0.0 and p = 1~..................... 52 V. Eigen-Values for S = 0.2, p = 1~...................... 52 VI. Eigen-Values for S - 0.59 = 1~............................53 VII. Critical Reynolds Numbers................................... 58 iv

LIST OF FIGURES Figure Page 1, Definition Sketch 0 a 0 0 & o............. 6 2. Two Neutral-Stability Curves Obtained with Two Different Methods for a Given Angle of Inclination and a Given Surface Tension... e o o o o o, o0 0 o a a 40 3. Neutral-Stability Curves Showing That Surface Tension is a Stabilizing Factor.....,..............0...0.. 54 4, Neutral-Stability Curves Showing That the Reduction of Angle of Inclination is a Stabilizing Factor,........ 00aa...... 55 5. General Features of Neutral-Stability Curves.................0 60 v

I. INTRODUCTION The problem of instability of a layer of liquid flowing down an inclined plane was first studied by Kapitza(4) and others. The first correct formulation was given by Yih, (11) who showed that the problem can be solved by a method of regular perturbation. The governing differential equation is the well known Orr-Sommerfeld equation. The boundary conditions are the non-slip condition at the bottom plane and the stress condition at the free surface. The Orr-Sommerfeld equation and the boundary conditions constitute an eigen-value problem. Yih(12) and Benjamin (2) have obtained solutions to the problem with different methods. It was found that for the case of large angle of inclination, surface waves govern the instability of the problem, which occurs at small Reynolds numbers. It was also found for the case of long waves that the imaginary part of the complex wave speed is zero at a(wave number) = O. And this imaginary part of the complex wave speed will increase or decrease when ca increases from zero, according as R >( cotO)/6 or R <(5 cot(3)/6 where R is the Reynolds number and B is the angle of inclination, That is to say the neutral stability curve has a bifircation point at a = O, R = (5 cot V)/6. The above relation indicates that the neutral stability curve will be shifted to the right in the a - R plane if a is decreased. However the above relation was found under the assumption that caR remains small compared with unity. The question arises -1

-2naturally as to whether the instability does occur at small aR for a given small A. The answer depends on whether the critical Reynolds number (for a given f and S = 0), obtained by assuming CeR larger than unity in the problem is greater or smaller than (5 cot $)/6. This thesis aims to give a definite answer to the above question, by studying the stability of a mode of disturbances for which aoR is much larger than unity. A neutral stability curve for f = 1 min. and zero surface tension is obtained by use of a method which closely follows the method used by C. C. Lin(8) for the problem of plane Poiseuille flow. A slightly different method is used to check the above neutral stability curve. This slightly different method consists of obtaining asymptotic series solutions of the Orr-Sommerfeld equation by solving the inviscid equation and a related differential equation by use of Frobenius method. The neutral stability curves obtained by these two methods check closely with each other for small wave numbers and differ slightly for large wave numbers, as expected. By use of the second method several' neutral stability curves for different angles of inclination and zero surface tension have been obtained. A set of neutral curves for different surface tension and f = 10 are also obtained. These numerical results indicate that the surface tension as well as the reduction of the angle of inclination are all stabilizing factors. The main difficulty in the computation for this problem arises from the fact that the flow parameters as well as eigen-values appear in the boundary conditions and the imaginary part of the secular equation is not a function of the wave speed c alone even for small values of a.

-3By comparing the results obtained by Yih(12) and Benjamin(2) and the results obtained in this thesis, it is shown that a layer of liquid flowing down an inclined plane does occur at small aR for any B, where R is given by R = (5 cot p)/6. That is to say, the "soft" waves which grow or decay more slowly than the shear waves do govern the instability of the problem for all values of P.

II. FORMULATION OF THE PROBLEM A. Two Dimensional and Three Dimensional Disturbances Squire(9) has shown that the solution of hydrodynamic stability with respect to three-dimensional disturbances in undirectional flows between rigid boundaries is related to the solution for two-dimensional disturbances by a simple transformation. Yih(13) has extended this result to flows with free surfaces, interfaces, or density stratification. Therefore, it is sufficient to study the case of two-dimensional disturbances only. B. The Governing Differential Equation Consider a layer of liquid flowing down an inclined plane, under the action of gravity. The plane is of infinite length and the flow is assumed to be parallel to the plate, so that the pressure gradient in the X-direction is zero, and the velocity component parallel to the X-axis -does not change along this axis. For the primary flow, the Navier-Stokes equations are simply (c.f. Figure 1)

-52L and - a My = e9 COS # (2) in which p is the constant density, g the gravitational acceleration, t the viscosity, and p is the pressure in this primary flow. All other quantities in Equations (1) and (2) are defined in Figure 1. The boundary conditions are:=o at Y-a -d, and cday)=o a Y' (3) Equation (1) can be integrated with the boundary conditions in Equation (3). The result is 21-= (atslnP /zv)(dI —>) or in which being the maximum velocity of the primary flow, i.e., um being the maximum velocity of the primary flow, i.e., ", = 92 siSin (4)

-6Y Fgr1DentoSec Figure 1. Definition Sketch.

-7The disturbed flow is not steady. The Navier-Stokes equations and the continuity equation for this unsteady flow are a — + uZ + V —? = —- -i + V'/ + a U ~ Dv CV v ladp Zu _ =/V o oX by in which t is time, p the pressure and A is the Laplacian. Making the following substitutions,P,- _l(/m v, C, Z -t UV, / /d v one can write the equations of motion and continuity in the following dimensionless form: AI 1 + V +r (6) Z v. ac, ~, a F: os + V (6)

-8au,+ aV, (7) where R is Reynolds number and F is Froude number, i.e., (8) Combining Equation (8) and Equation (4), one obtains the relation between R and F 2 F= t Kin 3p Assume that the velocity and the pressure of the unsteady flow consist of two parts in the following manner: u,=U1'U )U C= - (9) in which accented quantities are velocity and pressure perturbations, and U and P are velocity and pressure in the primary flow. Then, Equations (5), (6), and (7) can be written as u1LT;; V'v -,U (10) UT 4 IJ U + UV J4 Qb~ i

-9U' t V/ (12) in which high-order terms in perturbation quantities are neglected. Subscrpts in the above equations denote partial differentiations. Equation (12) is a necessary and sufficient condition for the existence of a stream function A in terms of which u' and v' can be expressed as In terms of stream function, Equations (10) and (11) become J~it$ U t, -IX<=-X t_ R aa (13)'t I U; t(14) Since the flow extends to infinity, one can assume that all disturbances propagate to infinity. Furthermore, since any disturbances can be constructed from sinusoidal disturbances by use of Fourier transformation, it is sufficient to study the case of sinusoidal disturbances. Thusy it is natural to assume

-10 4' - )ef'[ic)((A- ct)J, (15)'pk c((t e~p i t(tr CTI/3]A (i6) where o<(wvave numbers 2Trd/A A - iove lept8 i C O rcl4e wave speed- CrtC It can be seen from (15) and (16) that cr is the wave velocity and cia is the rate of amplification if positive, or the rate of dampitg if negativeo Substitution of (15) and (16) into (13) and (14) gives -i owheilccho te ieot if reniato wi' r epc (17) C -DZLC r/+ - 44'- ) ) ) (18) in which the primes denote differenitiation with respect to y.

-11Elimination of f between (17) and (18) by cross-differentiation gives a vorticity equation which is known as the Orr-Sommerfeld equation. + i=R 1(U - C)(C, t_ (19) Equation (19) is the governing differential equation of hydrodynamic stability of parallel flows. C. Boundary Conditions The boundary conditions as formulated by Yih(ll) consists of the non-slip conditions,U_=O= (i) and at the bottom, and the stress boundary conditions Xy = O (iii) and YTtt +T K o (iv) at the free surface. In (iii) and (iv) T is the shearing stress, Tyy the normal stress, T the surface tension and K is the curvature of the free surface. Condition (iii) is true because the viscosity of of the free surface. Condition (iii) is true because the viscosity of

-12the air is very small compared with that of the liquid. Condition (iv) must be satisfied, otherwise the fluid particles at the free surface will experience an infinite acceleration because of the finite jump in T at free surface. More explicitly (iii) and (iv) can be written as 9Xl t J> = a Q (iii) z ~Z O 4= fT (iv) in which nd is the displacement of the free surface from its mean positions. Using the Taylor series expansions of U and P up to the first order term in i, one can write To= > / and the free surface conditions can be written as if terms of higr or tn 02 a n (iii) e+qol a t eat y t= t 0 (iv) if terms of higher order than 0(n2) are neglected. In the above two equations, all quantities except r are to be evaluated at y = 0.

Noticing P(O) = 0 and Py(O) = cos a/F, one can simplify (iV) to the form C'0 /)05 S -S X -(iv) The dimensionless displacement r is related to r by the kinematic condition at free surface, i.e., V,=' = tr t ( rA t ), = o Since Uyly=O = O, ry = 0 and u'Tx is very small, the solution of the above equation is readily obtained by p 17= onst. e ctx - C ) Thus, 4 - (O e) f ri 2 (at-p c ) (20) where -e'= c - 1 Do Hrdrodynamic Stability Problem as an Eigen-Value Problem Equation (19) and the boundary conditions (i), (ii), (iii), (iv) constitute an eigen-value problem. A non-trivial solution exists if there exists a relation between R, F, c, S and x. In this problem, R and F are related by (9) for given values of P. Therefore the task is to obtain relation of the form c_ CR, d ) for given values of B and S, so that a nontrivial solution exists.

-14In general, c is complex i.e., c = cr - ici, in which cI,"Cr( R, o) (for given S and B) As can be seen from (15) and (16), the flow is stable if ci < 0, and unstable if ci > 0. If ci= O, there is a sustained oscillation. The relation gives the neutral-stability curve in the R - a plane. The wave velocity c on the curve of neutral-stability is then a function of a only. Now, the complete solution of the Orr-Sommerfeld equation can be written as q- c, Q,, c, Ab t c,', t C, 4, where 01, 02, 3 and 04 are f6ur independent particular solutions of the equation. Substituting this into the boundary conditions, one has a system of simultaneous equations in C1, C2, C3 and C4. Non-trivial Bolution of this system exists if

-15 - 4~4 a/ Ct)' P4r l (21) // " r/ in3L which;t - (d R. c., Z. 3, 4j d-, Z) l - Z C/ P f > S R ) Equation (21) is the so-called secular equation. In order to solve it, one has to solve the Orr-Sommerfeld equation first. Langer's(6) method readily yields four particular asymptotic solutions which are, in fact, the samne asymptotic solutions Heisenberg obtained with a less systematic method in 1924.(5) In order to use Langer'smethod, one writes the Orr-Sommerfeld equation in the following form L+-*e~tA'" ~ "A t2''A go + +;+h (22)

where X.is a large parameter ( o R) /2 and M- o loz —'~tC / 7. Ass-ume a solution of the form t A( A)d 2,, 9), (23) where Xo are determined by z (24) This equation is actually obtained by substituting (23) into (22) and then letting X approaches infinity. (A. in (23) being kept constant~ ) Solving (i24)9 one has 3 ie

It follows from (23) that A3 (~,A)e A,)e where Ai(y X), (i - 2 3,4) are to be so chosen that the Orr-Sommerfeld equation is asymptotically satisfied, Following Langer, let u,,/,= 27A (j 132.934) Then substituting ~01 02 into (22)j one has sJ,' * AsJ s. + s -— =o c Iz) 3 (25) in which Sjo o = (p P-o<)cZ R ~ U /,, ) - 97 (26) I, -.. - ( D-~ )o,, I T C J S"j,, o = ete O

where D - in (26). dy Letting the first two equations in (26) equal to zeros one obtains two inviscid equations of the same form, This is drue to the fact that (24) has so-called simple multiplicities, They are called inviscid equations because they are of the same differential eicuation as those which are obtained from the Orr-Sommerfeld eqruation'by letting R go to infinityo Obviously and ch, A ) = 51+ c, A ( ) (27) terms smaller than 0(+) in (25) being neglected One will lose no information by letting the constant in (27) be equal to ze ro as far as the eigen-value problem is concerned. since the factor (1 + const,/X) in the expression of j(j J= 192) can be factored out of the secular determinant (21). To solve the inviscid equation for aj, one can expand the s ition in a uniformly convergent series of a29 sine the solution must be analytic in Ca2 as can be seen from the inviseid equationo Substituting ad, ( -C ) L 0D (L i C, U)f ) / C I C, )t

G19into the inviscid equation and equating all coefficients of a2n(n 0 1i 2, o ) to zero, one has P9( U(c)-Y t U'JO =F C D( tr- C)f(, -(U- c, U W - The solution of (29) is readily obtained where U' r 2 hra-s been used. Thus, two independent solutions are obtained by properly assigning values to Cl and C02 Choose C and C2 so that f or c)l C frC2 sc, o I h

-20Then I, (r..,t....u-all~b~ Cz Cz___ _ _ _ _ _ IC =A,) -Cu(Tj-Cz C, +0 ( 1( )!?(30) The above solutions are good asymptotic solutions even near y where J c if the path of integration is properly chosetn, Since Y is not a singular point of the Or-roSomme.feld equatlon although it is a singiiar point of the inviscid equation~ C. C. Lin(8) showed in a heuristic manner that this proper path of integration must be such that it passes throuigh Yc from below, Lin's conclusion was later confirmed by the rigorous mathematical analyses of Wasow (10) To obtain 03 and 4 one sub stitute (j 3,4) into (22) and thus has L Axj c = _ [Xi O ] p,,4 J. -~ 4 6, ( 31 -,(u-c)dz(. pAj. + jAj pXj )j 4 *... (31)

-21 in which X, (j - 3 4) are the other two roots of (24). Letting the coefficient of X3 to zero. one has 4 7j (U-c) D joj, +6i cTj-c)j, D Pj - ( u-c)(z ~1/Dj, o -ft 6?jo X1j ) o Thus 2ja) +J' o (32) if U6 ~i~ C (33) The solution of (32) is Hence A~, 44 =.onit /(u-~c + 0 (b — ) terms smaller than O(x2) being neglected in (31), in comparison with terms of O(X3). Therefore

A 1X c)-dq iq 4~ 4=e i2 U (t7-c) + C () ) (34) e I C-c) tO (U )) Condition (33) indicates that the above solutions are not valid at y = yco They are also not good asymptotic solutions near yco Since at those points U - c is nearly zero and the coefficient of X3 in l >)(Z gg Ds V 5 aJ A ) may be so small that the X3-term is even smaller than the X2-term in (31)o In such cases the error terms involved in (34) is larger than 0( R )~ In the secular Equation (34), ~3, 04 and their derivatives are evaluated at y= -i1 which is close to Yc +Y: " + iYci in this prob lem~ (Since c are found to be much smaller than unity in this problem as well as in the problem of plane Poiseullie flow, and c = -- i y ~ ) yonse2uently, it is necessary to find solutions which are more accurate for small y - yeo To do this, one introduces a change of independent variables i2_-~ _a,~~~ gm~~~ t 7)(35)

in which _<K 2 Taylor's series expansions of U and D2U around yC are (UJ- c)=(U c) t IT ( )t~ -Z X ( ~') - -- and'~t~ //U."+ in whSch J / U BU U / etc,, and Substitution of (3"t) into Equation (19) gives 44- Z 4 d 1)4 2'17 =Sub ~ stitutn of { k in)to Eq ua2 tion - () give se L t Tie.~- Z-.,-~~~~~~~~ e~~~~~i~~~~r~~~~

Now, n in (35) is to be so chosen that both sides of (37) are of the same order of magnitudeo Obviously, the proper choice is n - 3" Thus, neglecting terms which are smaller than 0(;-), one has from (37) 4j = U; Lc d (38) To solve this equation, one puts 3~'& t - C./3/z Thus, (38) is transformed to the well-known Bessel equation ___" IC- d 3 (40) Two particular solutions of (40) are f-z) Hence ~,..Sd'70fC Stl /kut l S (A,((41)

-25where Ot ( U) (43) The above solutions were first used by C. C. Lin in his computation for the problem of plane Poiseuille flow. Substituting asymptotic expansions of Hankel functions into the above solutions and then compare the result thus obtained with solutions (34), C. C. Lin also determined the proper path of integration for obtaining 01 and 02 to be below the critical point y = ye. Having obtained four particular solutions 01, 02 and P3, mk, one is in a position to use them to solve the secular equation. Before doing so, one can simplify (21) by order of magnitude analysis. To start with, one divides the third column by $31 and the fourth coltrnn by 042. Thus (21) is rewritten as (hi iR)1~~~~~~ (+Ztl~z ((44) 4 ab-2t ~C 2z3)

The last row in (45) has been divided through by m o Each element in the first two columns are of order unity if j > m, then j should be factored out instead of m9 and the same is true), Therefore it is legitimate to simplify the determinant by comparison of order of magnitude of each element in the last two colmns before expanding the determinant. The order of magnitude of relevant quantities are listed below. I. I, /Y i /( gZ *5 "S) ) /, ) /I )' B,~~~~~~ s4i ( (3- - - -'L~4))

4-Z Cx> (t- BC + + where (46) B > ) | >4 (g 1, 2) (47) For very smnall c real part of P is positive. Therefore for large value of aR, e-P is much smaller than unity. Thus. neglecting quantities of order e~P one can reduce (4)3 to 6,, %21 -t ~ i (48) IZX lz 1( t 4-U j ( t

iP'~~~~~~~~~~~~~~~~~~~~~1 i<;:;ic; P+4 1 2 0 m' N -- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _-S -'I W c~~~~~~~~~~~~~~~~~~~t~~~~~~~~~~~~~~~~~~~a,S- k,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_-A — n -e $- -,,9 hi,__~~~~~~~~~~~______________ c- - N CD -4- c - AP- D Xf l s ~ \c-. __ t~18 ~- - C (I' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1~~~~~~~~~~~~~~~~~~~~~~~~~~~1 Ac —~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~S KU-F~;tt 9\

The first term of the above equation is of the order ( aR)1/ /2 whereas the second term is of order (aR)3/ Hence, for aR so large that (gaR) can be negIected compared with unityj one can appr,x:imate (49)'by Cll -0SZ~~ 41/l5O) E;s'( 4 C+'~ L.,.4? dz. rr> 0 7or the case Dr in-nitje surface tensionr, ie S -r one has m i'.L. l, c an b, fu: rther redu:ced to _ 0 (51-) #QF4: ~ ~ ~ ~ =!5'}4o cpl, 4,6I O

-30 This is the secular equation for the stability of plane Poiseulle flow, with respect to symmetric disturbances (I odd in y). Thus the vestige of the shear waves is completely traced in the course of the order of magnitude analysis of each element in the secular equationo It should be pointed out that termwise differentiation of the asymptotic series expansion of a function does not give the derivative of the function, in general~ It can be shown easily that termwise differentiation is legitimate if the derivatives of the function also possess asymptotic series expansions. Fortunately, this is the case in this problem. Since it is known that the exact solution of the Orr-Sommerfeld equation must be analytic in vr; ioef, derivatives of the solutions of the Orr-Sommerfeld equation possess asymptotic series expansionso There fore the termwise differentiation in the above analysis is legitimate,

III. METHODS OF COMPUTATION AND THE RESULTS A. C. C. Lin's Method It follows from (50) that the secular equation to be solved for this problem is /~~5~~ -- ----- ~~~~~~(52) | I t7M t-e p (m4z AL t42)l

-32If one makes use of the explicit solutions of 01 in (30) to reduce the right-hand side of the above equation, one has, after some manipulation _ (-c) rru c 4 (jm Q(L j J' ) _ ____t_) U_'( 4, -K~j") —(J-.c~j~1 T _t# _ _ _ 7 z'1 % ({ ~ Q, Zf - f ~ Yn t- To simplify the computation, the above equation can be rewritten as I i5)L + 3 f t (53) -' t _i In the above expressions., = (,) -' ) o Y n c 54 Z, -( 9 ta, t i ~ 4 — - - -.' (l )!.:Le i

.-I ~T~r 2r 0t~/ Z a. n~ Z(56) where real in the above euation.o S2ubstituting (40) into the left hand side'', ( ) 1- ) t ( -) T- 5) J ETof (53) for ~ one has L+' C - 1- ( t + L ) FC -' ),

where dS 2~f;;~~I 7 ( U (58) g_ g -(Sf)lTC ) 4 Thus the secular equation c53)9 can be rewritten as (I +~ —l ~- -~ I where s the rg(h>,,h<ard side':53): one has from (59) (61) 7,~~~

Fr z) or Fi(z) LS the real or the imaginary part of F(z ) respectivelyo They are tabulated in Table II, By use of the relations in (4) 60) cane tn a U t * V L'- (62) To bring out the dominant terms in'the functions dl2 l 22 and 22,' the following transformation is sedo NIdi.K FH,. K nzz Si1nce is H\ < n n. Sine 1Hn < H:n i.... < H1 and Kn < K!.. < K1.9 as can be seen from \ 57) and t64)) the above transformat'ion does serve the purpose of briingng out, the dominant ter s in the functions 012~'2~, 22 and — sing the above transformation) one has from (54) ct L =( )(CIi-2 2)(( " n-i) Xzz Z Iz, s hs | ) ( 63) - _I; where H1l K3, etc.@ and K29 K49 et to are defined in (57)j Other fuacetions are defined as follows

3, 1(64) K3 =(t'T-C) ( (i-C) etc o The above functions in algebraic forms and their numerical values for different values of c are listed in Appendix Ao Substituting (63) into (62) one has u+ i v'- I-+ /c K< t,( — C)' ( _Z o'Z."lw, )t (P::-ix -.. /viZ/lt,, ),5 (65) in which Substituting these expressions for q and m in (65), one has U t/ tTC K, + J,c (ID-I )Z Y~~~~~~~~~

in which If C and S are given in Equation (66), u and v are ffunctions of. - and Z only, Hence one can solve (61) for a and c for various values of Z,, Then corresponding R are obtained from For actual computation, the explicit expressions of u and v are needed. After a straightforward manipulation they are fouand to be UK I+t 6 t z L X 7 t Y )I 4,) t Y (r (68) irn whilch C~^ -c j) a Sz ( tI- [_ (I - ) /.'- i~,,. f.~ ~J...... ( 7( P C 4 Z Nk jY7 t, )S4 i//.7 4 7 1 E Ct ) >1 7 D_ A

=38= Note that only the first two terms in the infinite series in (66) are retained. In the above expressions the subscripts i or r indicate the imaginary or the real part of the function they subscribe, To solve (61), the so-called Newton s method(3) is used, For given values of Z the corresponding values of c and a are obtained from the following iteration procedure. ( ) l (aC ) C,'tt =(QCfc, ( )r' C+ (69) where an and en are the values of a and c at the n-th iteration, and (Q)cn etco are the values of the functions evaluated for c ac n given by cn etco The numerical results thus obtained for the case of S 0- O and. 5 - 1 mino are given in Table Io The results are plotted in Figure 2. B3 Method to be Used for Problems in Which a is Larger than'Unity After a change of the independent variable where a -z- C

-39TABLE I EIGEN-VALUES OBTAINED BY C. C. LIN'S METHOD FOR S = 0.0 AND B = 1 MIN. Z c a R 2.414759.080000.341187 3.159721 x 105 2.594636.175000.265139 4.702408 x 104 2.788177.236000.219866 2.823893 x 104 3.051198.266000.272340 2.070277 x 104 3.369871.256000.452931 1.886246 x 104 3.577772.235000.615388 2.159633 x 104 3.767519.210000.751831 2.911494 x 104 3.975282.178500.838797 5.032895 x 104 4.182047.145000.834318 1.108617 x 105

I I I I I I I I I I I I 1.2 S- =0.0 P=1 mi'n.-.- OBTAINED BY C.C. LIN'S METHOD STABLE 1.0 OBTAINED BY A MODIFI ED METHOD I 0.8 II / X 0.6- /UNSTABLE 0.4_ 0.2 2 3 4 5 6 7 8 9 10 20 30 40 50 60 R(104) = Umd/v = Reynolds Number. Figure 2. Two Neutral-Stability Curves Obtained with Two Different Methods for a Given Angle of Inclination and a Given Surface Tension.

-41the inviscid equation can be rewritten as (ins )i Z O ->a (70) It is seen from (70) that z = 0 is a regular singular point, and the region of varidity of the power series solution E _2 6a " 5 (71) includes the end points zl = a - 1 and z2 = a; at these end points O1 and $2 are to be evaluated in the secular equation. Substituting (71) into (70) and equating all coefficients of zn+s to zero, one has n= o: (S- = )-c - O. YK =. SC S-l)a~ = -Z ( ) - O m a 3 A i [ k+5- Cnris-2)! z._,-z4 (nts)(n+5-t) a. < 4 3 ZYE> nr4-i5 - C

Using the exponent S - 1, one has oaI a,(l- -/za The second solution is of the form m;2 =, -+ t7it+3 (73) ) Substituting (73) into (70), one finds 2y)a 2 ) bn= 0 O'b on =- n Substitutingn~tn -73 iinZ t7-0z1) on f, inJ The proper branch for in z has already been determined in the previous method

-43Having obtained the recurrence relations for the coefficients of the series solutions 1 and 42, one can write = = E eI- i-tazi29> A -3 3 1(. t4+ q + (i,~K + A4)> t' 4f > )g t (nAbo i446 A<)~ (75) where A z4 - 1/72 Z o A44 - - zo, (76) A4r =({8e __~ - i )/6ca, tc. Similarly, 4 - iP, jhI~ +0 icd t~7+ f( giz SZ1 )!3 62z Z3 3 (77) where

644 /R 4 2 It follows, after a straightforward manipulation, (>, ~)=4-, t + ~ a c ] ~ (~~,/) C ~ -F 4@ cC1#(oh c)a (79) sI (,~,) -,; in 2 t qi3Tw'1~ (X C) —-2 N 2 2 I tA 514 34 (2 x r" ~~~~1

-45t,/=l +1 = C, t Z 5,1 c ~ /zz 0? ZV S, qtenQ f'- t+3Z f Z = I 1 &z. z -r 3 -z), = - E C R (b Z -)J,, -( h'i~~n Jwhich, wFith N - 1,23, et oo( nr'lI 2 c,._ = 2~,1,,, d h'=7.~

-46IKN _;2 (nt 1)1 (r-l) L I J a 2 Cj7 1~7 g 6( n- 2 The above coefficients ioe, QN, RN etc. for different values of c are listed in the Appendix. In evaluating 021l the proper branch of in z has been usedo Having obtained each term in the right side of the secular equation, one can separate both numerator and denominator of the right

side of (52) into real and imaginary parts. That is to say, one can rewrite (52) in the following form H1-;/ l= t (81) where Now, tbe left side of the4secular equationris / Now, the left side of the secular equation is ~C a. —_~) F~(~.Y~.),~ ~(83) where F(Z) is defined by (58)o To evaluate F(Z) for any values of Z, one can do the following. Letting z - (-i )1/3 and T = (-i) /3 d in (38), one has 2 ha s

-487 Tdo ~ %(84) It follows directly from the above change of variables and (38), (39) that the solutions of (84) are 2. (85) On the other hand, the solutions of (84) can also be expressed in terms of power series o They are T3 iS, ra~ = 0 3...~t - 2 )(86) in which 2 ) - [ - / 73fl (87) ___ (- ) an- - t (7) (31 )(3 va — ( j With the relations (85), (86) and (87) one can evaluate F(Z) and hence

The coefficients in the Maclaurin's series (84) are tabulated in Reference 1. The functions F(Z) as well as 3(Z) thus obtained are tabulated in the following table. TABLE II Z Fr(Z) F i(Z) J r(Z) i(Z) 1.5.692520 -.185059 2,387441 -1.436897 1.6.669260 - o159868 2.450892 -1,184677 1.7.649086 -.135825 2.478400 -.959293 1.8.631381 -.112566 2, 481432 -.757760 1.9.615631 -.089794 2.467026 -.576332 2.0.601395 -o 067264 2.439287 -.411628 2.1.588285 -o 044772 2.400476 -.261041 2.2.575945 -.022146 2.351772.122821 2.3 564042 o000755 2.293793.003973 2,4.552249.024045 2,226963.119594 2,5.540240.o47809 2,151780.223759 2.o 6.527679.072099 2,068994.315830 2, 7.514214 o.096928 1.979706.395009 2.8.499472.122261 1,885400.460535 2.9.483054.147997 1.787898.511861 3.0.464539.173955 1.689265.548791 3.1 o443490 o199843 1.591662.571568 3.2.419476.225235 1.497204.580894 3.3.392109.249535 1,407811.577896 3.4.361100.271957 1.325096.564045 3 5.326343.291518 1.250300.541053 3 o6.288022.307068 1,184254 510756 307.246722.317381 1.127395.475009 3.8.o203514.321309 1.079792.435597 3,9.159963.318012 1.o041204.394168 4,0.118013.307201 1.011136.352185 4.1.079634.289320.988900.310898 4.2 o046971.265583.973673. 271336 4o 3.021011.237808.964548.234300 4.4.002374.208107.960580.200379 4.5 -.o 009193 o 178515.960827.169960 4,6 -.014553.150703.964378.143250 4,7 -o 014926 o 125822.970380.120300 4.8 - 011638 o 104493.978061.101025 4.9 - 005933 o o86894.986739.085236 50 -o 001134 o 072882. 995833 o 072660

-50The subscripts r or i in the above table denotes the real or imaginary part of the function it subscribes. With the left side being given by (83) and the right side by (82), Equation (81) can be rewritten as (-I)F() U(a,C i,,, ( V 5) (88) where U= X r +YVU= -_ Y' - X VOr equivalently (89) A- (a-,) Ft )-Vh~bF (F-/Zr )- V" I For given values of P and S, (89) is a system of simultaneous equations in three unknowns o, c, and Z, Thus, for various values of c, one can solve (89) for o and Z. Then the corresponding R can be obtained from KG~ = - ((67) a ll ~ "'67

-51A plot of the relation between ac and R, thus obtained is a neutral curve. In solving (89) the following iteration scheme is used. (90) te bofan elesctronibd cmpuero The fresultis aeplobttadinFres With the above described process the following result is obtained by use of an electronic computer. The results are plotted in Figures 3 and 4. TABLE III EIGEN-VALUES OBTAINED BY A MODIFIED METHOD FOR S = 0.0 AND = 1 MIN. Z c a R 2.414964.0o80000.336732 3.202343 x 105 2.598376.175000.264869 4.727604 x 104 2.802864.235000.227307 2.811159 x 104 2.999574.258500.263662 2.218076 x 104 3.195362.261000.346283 1.982209 x 104 3.397784.250000.469903 2. 004248 x 104 3.599316.230000.608539 2.374924 x 104 3.795285.204500.722210 3.360005 x 104 4.003972.173000.783976 6.052522 x 104 4.200095.141500.773417 1.304778 x 105 4.386976.112000.712072 3.281373 x 105 4.598310.082000.610088 1.132477 x 106

-52TABLE IV EIGEN-VALUES FOR S = 0.0 AND X = 10 Z c R 2.400196.070600.371858 4.152194 x 105 2.500516.129500.539594 5.164133 x 104 2.591080.172500.656387 1.976407 x 104 2.799279.236700.849766 7327.3423 2.995686.261900.966309 5791.7020 3.212561.264000 1.039897 6476.6970 3.368392.255000 1.062555 8126.8064 3.552588.237000 1.060654 1.195319 x 104 3.800499.204400 1.014059 2.406453 x 104 3.992671.175000.946671 4.799069 x 104 4.191120.143000.852935 1.138520 x 105 4.400189.110000.736120 3.382574 x 105 4.602193.081500.618260 1.141233 x 106 TABLE V EIGEN-VALUES FOR S = 0.2, 5 = 1~ Z c o R 2.400041.070500.310212 4.997700 x l05 2.507280.130000.449408 5.765097 x 104 2.600485.176200.538139 2.284523 x 104 2.800388.236000.672941 9348.09155 2.996595.260500.760146 7491.3995 3.197549.260000.817137 8222.5165 3.599313.230700.847652 1.689200 x 104 3.800190.200000.823235 2.981317 x 104 4.000727.173500.773467 6.066319 x 104 4.201071.141300.701191 1.446378 x 105 4.403286.109500.611922 4.134361 x 105 4.594385.082500.521939 1.296325 x 106

-53TABLE VI EIGEN-VALUES FOR S = 0.5, = 1~ Z c C R 2.399266.070000.256467 6.170290 x 105 2.493119.125500.356033 8.531772 x 104 2.578898.167000.423036 3.336996 x 104 2.761200.227000.525043 1.293781 x 104 2.995937.259600.611403 9407.138916 3.202144.262000.660228 1.034078 x 104 3.399280.250600.686155 1.364330 x 104 3.599472.230200.691719 2.084060 x 104 3.798191.204000.675956 3.625166 x 104 3.999799.173500.638265 7.346216 x 104 4.171495.146000.590360 1.522784 x 105 4.399831.110000.510017 4.880966 x 105 4.662390.074000.409423 2.398347 x 1O6

II I I I' I I I I I I I I lI I I I I I I I I I I.I /3=l 0.9 0.7 ~~~~~~~7: S ~~~~~~~~~~~~~~~~~~~~~~~~~=.O ct3 I S=.2 Q5 ~ STABLE \1 0~ ~ ~ ~~~~~~~~~~~~~~~~~ ~s=.5-4 Od' UNSTABLE i S=.O Cj 0.3 S=.5 0.1 II I III I I I I 11 I I I I I I 11 I I I I I II I 104 105 106 R = Umd/v = Reynolds Number Figure 3. Neutral-Stability Curves Showing That Surface Tension is a Stabilizing Factor.

I.' 0.9 0.07 e 0.75 0.5 ~ STABLE 0.3 10 10 0.,~~~~~~~~~~~~~~~~~~~~1 R Umd/v R eynolds Number Figure 4. Neutral-Stability Curves Showing That the Reduction of Angle of Inclination is a Stabilizing Factor.

IV. DISCUSSION ON THE RESULTS AND ON THE METHOD OF COMPUTATION A study of the neutral curves for different values of S in Figure 3 shows that surface tension is a stabilizing factor. It is also observed in Figure 3 that the space between neutral curves for different values of S decreases as a decreases. Therefore the stabilizing effect of the surface tension decreases as the wave length becomes larger. It is easily seen from Figure 4 that the reduction of the angle of inclination is also a stabilizing factor. As a limiting case, one naturally expects that the neutral curve will be shifted to R - oo when P -- O. Since when 5 = 0 there will be no flow, the liquid layer lying flat will be always stable with respect to any infinitesimal disturbances, The two neutral curves shown in Figure 2 are obtained from the two different methods developed in the preceding chapter. These two neutral curves for the same P and S check with each other when a is small and differ slightly when a is large. This result is quite to be expected since only two terms are retained in all power series in the first method, whereas more than twenty terms are retained in all power series in the second method. For all values of a, the power series in a2 in the secular equation which appe rs in the first method converge faster than the series which appear in the second method. Unfortunately, in the first method the coefficients of the series, i.e., K1, N2, N3,....Ng, etc., cannot be obtained by the computer alone, This is due to the fact that -56

these constants are actually a series of multiple integrals, the path of integration of which do not lie entirely on the real axes, Consequently, one has to obtain a series of multiple integrals by hand, The evaluation of which requires considerable amount of algebraic work. The amount of work becomes formidable for the problems in which a are greater than one, Since for this value of a, the series converges very slowly. Although the series obtained from the Frobenius method converge even slower, their coefficients are obtainable from some recurrence relations, one can rely upon the computer to carry out the lengthy computation. Thus, it is very likely that the second method is more desirable for the problems in which c > 1. For the problems in which the primary flow is of higher order than two. one may have more than one singularity within -the domain of flowo If the first method is used for these problems, one has to decide which path to take around the singularities. As a result the evaluation of the multiple integrals mentioned above will'become even more complicated. On the other hand, if the second method is to be used) one has to obtain two different power series' one of which is to be used for the bottom boundary conditions and the other for the top boundary conditions. The diffiiculty of the computation in this problem arises from the following two facts, First of all, the imaginary part of the secular equation is not a function of c alone, therefore a fairly simple computational scheme used by CO C, Lin(8) in solving the problem of plane Poiseuille flow cannot be applied, Another fact is that the eigen-values as well as flow parameters appear in the boundary conditions, As a result the secular equation becomes considerably more complicated,

V. CONCLUSIONS For various values of 5 and S, several neutral stability curves have been obtained in Chapter III. These neutral curves, however, are neutral curves with respect to the disturbance of a mode in which aR is much larger than unity. The instability of the same problem with respect to the disturbances of another mode in which aR is much less than unity has been studied by Yih(l2) and Benjamin(2). In their study, R = (5 cot P)/6 is given as a critical Reynolds number for very long wave and zero surface tension. The critical Reynolds number (denote by R in Table VII) from this relation for three values of X and the corresponding critical Reynolds number (denoted by Rh in Table VII) obtained in Chapter III for S = 0 are listed in the following table for comparison. TABLE VII CRITICAL REYNOLDS NUMBERS P 30" 1' 10 90~ Rs 6,875-5 2,865 57.3 0 Rh ------- 19,600 5,790 --- Although the neutral curves for P smaller than 1 min. have not been obtained, the general trend of the neutral curves for different P shows that probably Rs < Rh also for P < 1 min. Thus, the results obtained in this thesis together with the results obtained by Yih(l2) and Benjamin(2), show that R5 < Rh for all P. The general features of S h -58

the neutral curves for the soft and hard modes are shown in the following figure. The two curves never intersect with each other since in their ranges of validity~ (ciR)s < (aR)h along the neutral-stability curves. The above mentioned two different modes represent two different type of disturbanceso The mode in which oR. is much larger than unity corresponds to hard. waves which damp or grow rapidly, The other mode in which aR is less than unity corresponds to soft waves which damp or grow less rapidly'than the former mod.e, This can be seen easily from the following argument, Given c i, (aRs < (aR)h for all a and Ro For a given flow condition one has the same R for both modes. Thus aK < ah - for given ci C onseIqently ac < ci)h. That is to say the rate of growth or decay of soift waves is smaller than that of hard waves, In conclusion, one can state that the soft- waves which arises from the presence of the free surface do govern the instability of a layer of liquid flowing down an inClined planeo

-60a( STABLE BOTH MODES NEUTRAL CURVE FOR HARD MODE: UNSTABLE, HARD MODE NEUTRAL CURVE FOR SOFT MODE UNSTABLE,SOFT MODE Rs Rh Figure 5. General Features of Neutral-Stability Curves.Rh Figure 5. General Features of Neutral-Stability Curves.

VIo APPENDIX A SOME FUNCTIONS AND THEIR NUMERICAL VALUES USED IN C. C. L!N'S METHOD HK,1 =t — 4~ 7 46S2t3 3D c IS 4~37 + tl,= t z 4'- _!7 f () I 4') Z /i_.n.:. - (, ( -- ) -_ + -- _ -_- - -! (a-/)5 _ (a-^) 1 4( (() 2 Oa 4~2jl) S- 9 F" ( Y<L

~ - p)o 4 - 0) t 94 ft1 (I I+ 7 4} ji V f t2 /r), J -tT- (Ii — I+ i - ((iL) ") n [s(wz )dg ) 7~V i)d 5~&r,4 11 (''2 -,- -,- T [ /( bZ g + (F~J 3 g /

-63where a- i-c.,=,t — / V ( 1 62L {GAGS 4 9- 7~~~~Z 0t0 - 16Hf

-64c Klr Kli H1 H2r H2i 005 -9.35007.84821.46917.20696.00002.10 -4.49066.91987.41000.19516.00155.15 -2.89951 1.00222.35583.18278 ~00573.20 -2.11623 1.09763 o30667.16982.01503.25 -1.65287 1.20920.26250.15626.03256.30 -1034799 1.34104.22333.14209.06284 035 -1 13287 1.49872.18917.12729 o11236.40 -.97338 1 68991 o 16000 11182.19058 045 -.85069 1.92551.1-3583 ~09566.31165.50 -.75355 2.22144.11667.07875.49729 c N N N N M M 2r 2i N3r 3i 3r 3i.05.20696.00002.20401 o 00051 05045.000000 o10.19516.00155.21521. 00220.04181.000000.15.18278.00573.22668 o00534.03439 o000000.20.16982 o.01503.23826.o01036.02811,000002 o25.15626.03256.24957 - o01782.02291 0000009.30 o14209.06284.25984 -o02851.01869 ~000030 035.12729.11236 o26761 -.04354.01535 0000oooo84 40 o.11182.19058.27016 - o06449.01278 o000215 o 45 o09566 o 31165.26243 - 09372 01083 oo00504.50 o 07875.49729.23474 - 13477.00935.001113 The subscript r or i denotes the real or imaginary part of the function they subscribe,

-65VII. APPENDIX B COEFFICIENTS OF THE POWER SERIES SOLUTION OF THE INVISCID EQUATION Qc 1 Q2 Q3 Q4 Q5 Q6.05 1.078 x 10-4 3.448 x 10-9 3.257 x 10-14 4.678 x 10-19 2.725 x 10-24 1.148 x 10-29.10 4.468 x 10-4 5.859 x 10-8 3.664 x 10-12 1.338 x 10-16 3.199 x 10-21 5.679 x 10-26.15 1.044 x 10-3 3.158 x 10-7 4.563 x 10-11 3.851 x 10-15 2.128 x 10-19 8.968 x 10-24.20 1.931 x 10-3 1.066 x 10-6 2.815 x 10-10 4.342 x 10-14 4.388 x 10-18 3.475 x 10-22.25 3.145 x 10-3 2.790 x 10-6 1.184 x 10-9 2.939 x 10-13 4.779 x 10-17 6.265 x 10-21.30 4.735 x 10-3 6.226 x 10-6 3.921 x 10-9 1.445 x 10-12 3.489 x 10-16 6.993 x 10-20.35 6.757 x 10-3 1.246 x 10-5 1.103 x 10-8 5.710 x 10-12 1.939 x 10-15 5.632 x 10-19.40 9.284 x 10-3 2.309 x 10-5 2.759 x 10-8 1.930 x 10-11 8.862 x 10-15 3.588 x 10-18.45.012409 4.041 x 10-5 6.328 x 10-8 5.809 x 10-11 3.500 x 10-14 1.922 x 10-17.50.016254 6.773 x 10io-5 1.360 x 10-7 1.601 x 10-10 1.238 x 10-13 9.020 x 10-17 c R1 R2 R3 R4 R5 R6.05 1.036 x 10-1 5.492 x 10-3 1.320 x 10-4 1.813 x 10-6 1.610 x 10-8 1.055 x 10-10.10 9.819 x 10-2 4.929 x 10-3 1.122 x 10-4 1.460 x 10-6 1.229 x 10-8 8.046 x o10-.15 9.273 x 10-2 4.397 x 10-3 9.452 x 10-5 1.162 x 10-6 9.235 x 10-9 6.033 x 10-ll.20 8.728 x 10-2 3.895 x 10-3 7.880 x 10-5 9.116 x 10-7 6.820 x 10-9 4.438 x 10-ll.25 8.182 x 10-2 3.423 x 10-3 6.492 x 10-5 7.042 x 10-7 4.939 x 10-9 3.196 x 1o-1l.30 7.637 x 10-2 2.982 x 10-3 5.278 x 10-5 5.343 x 10-7 3.498 x 10-9 2.246 x 10-11.35 7.091 x 10-2 2.571 x 10-3 4.226 x 10-5 3.973 x 10-7 2.415 x 10-9 1.535 x 10-11.40 6.546 x 10-2 2.191 x 10-3 3.323 x 10-5 2.884 x 10-7 1.618 x 10-9 1.016 x 10-11.45 6.000 x 10-2 1.841 x 10-3 2.559 x 10-5 2.036 x 10-7 1.047 x 10-9 6.472 x 10-12.50 5.455 x 10-2 1.521 x 10-3 1.922 x 10-5 1.391 x 10-7 6.504 x 10-10 3.941 x 10-12 c R12 R12 R13 14 R15 R16.05 2.534 x 10-1 2.527 x 10-2 8.815 x 10-4 1.584 x 10-5 1.735 x 10-7 1.356 x 10-9.10 2.401 x 10-1 2.268 x 10-2 7.494 x 10-4 1.276 x 10-5 1.324 x 10-7 1.038 x 10-9.15 2.268 x 10-1 2.023 x 10-2 6.313 x 10-4 1.015 x 10-5 9.949 x 10-8 7.807 x 10-10.20 2.134 x 10-1 1.792 x 10-2 5.262 x 10-4 7.963 x 10-6 7.347 x 10-8 5.764 x 10-10.25 2.001 x 10-1 1.575 x 10-2 4.335 x 10-4 6.151 x 10-6 5.321 x 10-8 4.164 x 10-10.30 1.867 x 10-1 1.372 x 10-2 3.524 x 10-4 4.668 x 10-6 3.768 x 10-8 2.936 x10-10.35 1.734 x 10-1 1.183 x 10-2 2.821 x 10-4 3.470 x 10-6 2.602 x 10-8 2.013 x10-1.40 1.601 x 10-1 1.008 x 10-2 2.218 x 10-4 2.519 x 10-6 1.743 x 10-8 1.337 x 10-1.45 1.467 x 10-1 8.471 x 10-3 1.708 x 10-4 1.779 x 10-6 1.12 8 x 10- 8.541 x 10-11.50 1.334 x 10-1 7.030 x 10-3 1.283 x 10-4 1.215 x - 7.o6 x10-9 5.217 x10c R21 R22 R23 R24 R25 R26.05.27581 8. 975 -2 063 x 10 -3 1.249 x 10-4 1.734 x i6 10 646 x lo-8.10.26846 8.276 x 10- 4.421 x 10-3 1.034 x 10-4 1.359 x 10-6 1.301 x 10-8.15.26089 7.596 x 10-2 3.832 x 10-3 8.463 x 10-5 1.051 x10-6 1:012 x 0-8.20.25310 6.936 x 10-2 3.292 x 10-3 6.844 x 10-5 8.001 x 10-7 7.731 x 10-9.25.24507 6.2956 x 10-2 2.800 x 10-3 5.460 x 10-5 5.984 x 10-7 5.794 x 10-9.30.23676 5.677 x 10-2 2.356 x 10-3 4.288 x 10-5 4.387 x 10-7 4.246 x 10-9.35.22814 5.079 x 10-2 1.956 x 10-3 3.308 x 10-5 3.143 x 3033 x 10-9.40.21919 4.505 x 10-2 1.601 x 10-3 2.500 x 10-5 2.192 x 10- 2.104 x 10-9.45.20986 3.954 x 10-2 1.287 x 10-3 1.843 x 10-5 1.482 x 10-7 1.410 x 10-9.50.20010 3.427 x 10-2 1.013 x 10-3 1.320 x 10-5 9.651 x 10-8 9.065 x10RcR R R c R31 R32 R33 R34 R35 R36.05 -.52302.20024 2.316 x 10-2 8.482 x 10-4 1.547 x 10-5 1.827 x 10-7.10 -.52302.18971 2.077 x 10-2 7.211 x 10-4 1.246 x 1-S 1.493 x 10-7.15 -.52302.17917 1.851 x ia-2 6.074 x 10-6 9915 x 10-.20 -.52302.16863 1.638 x 10-2 5.o63 x 10- 7.780 x 10-6 9 522 x 1i-8.25 0.52302.15809 1.438 x 10-2 4.171 x 10- 6.010 x -611.30 -.52302.14755 1.251 x l0-2 3.391 x iO- 4.560 x -6 61 x -8.35 -.52302.13701 1.077 x 10-2 2.714 x -4 3.390 x 106 4.211 x 8.40 -.52302.12647 9.165 x 10-3 2.135 x 10-4 2.461 x 10-6 3.055 x 1i-8.45 -.52302.11593 7.688 x 10-3 1.644 x 10-4 1.738 x 10-6 2.148 x 1-8.50 -.52302.10539 6.339 x 10-3 1.235 x 10- 1.187 x 10-6 1.456 x 10-8

-66c s1 S2 S3 S4 5 S6.05 -3.142 x 10-4 -1.673 x 10-8 -3.572 x 10-13 -4.087 x 10-18 -2.910 x 10-23 -1.413 x 10-28.10 -1.264 x 10-3 -2.754 x 10-7 -2.412 x 10-11 -1.133 x 10-15 -3.313 x 10-20 -6.606 x 10-25.15 -2.857 x 10-3 -1.436 x 10-6 -2.904 x o10-10 -3.153 x 10-14 -2.132 x 10-18 -9.830 x 10-23.20 -5.103 x 10-3 -4.674 x 10-6 -1.728 x o10-9 -3.430 x 10-13 -4.242 x 10-17 -3.579 x 10-21.25 -8.007 x 10-3 -1.177 x 10-5 -6.994 x lo-9 -2.235 x 1o-12 -4.449 x 10-16 -6.043 x o10-20.30 -1.157 x 10-2 -2.517 x 10-5 -2.221 x 10-8 -1.054 x 10o-11 -3.119 x 10-15 -6.293 x 10-19.35 -1.579 x 10-2 -4.815 x 10-5 -5.970 x 10-8 -3.985 x 10-11 -1.659 x 0-14 709 x.40 -2.066 x 10-2 -8.486 x 10-o5 -1.422 x 10-7 -1.283 x 10-10 -7.224 x 10-14 -2.774 x io-17.45 -2.614 x 10-2 -1.405 x 10-4 -3.o88 x 10-7 -3.661 x lo-10 -2.707 x 10-13 -1.347 x 1o-16.50 -3.220 x 10-2 -2.214 x 10-4 -6.245 x 10-7 -9.506 x loo10 -9.031 x 10-13 -5.847 x 10-16 CS11 S12 S13 S14 S15 S16.05 2.080 x 10-4 2.645 x 10-6 8.465 x 10-11 1.291 x 10-15 1.149 x 10o-20 6.698 x 10-26.10 8.288 x 10-4 2.149 x 10-5 2.821 x 10-9 1.766 x 10-13 6.455 x 10-18 1.545 x 10-22.15 1.852 x 10-3 7.369 x 10-5 2.233 x 10-8 3.232 x 10-12 2.731 x 10-16 1.511 x 10-20.20 3.260 x 10-3 1.775 x 10-4 9.825 x io-8 2.600 x 10-11 4.019 x o10-15 4.068 x 10-19.25 5.018 x 10-3 3.521 x 10-4 3.134 x 10-7 1.335 x 10-10 3.321 x 10-14 5.412 x 10-18.30 7.077 x 10-3 6.180 x 10-4 8.163 x 10-7 5.163 x 10-10 1.909 x 10-13 4.622 x 10-17.35 9.356 x 10-3 9.965 x 10-4 1.849 x 10-6 1.645 x 10-9 8.558 x 10-13 2.915 x 10-16.40 1.174 x 10-2 1.510 x 10-3 3.785 x 10-6 4.553 x 10-9 3.204 x 10-12 1.476 x 10-15.45 1.403 x 10-2 2.180 x 10-3 7.171 x 10-6 1.133 x 10-8 1.047 x 10-11 6.335 x 10-15.50 1.597 x 10-2 3.027 x 10-3 1.278 x 10-5 2.594 x io-8 3.080 x 10-11 2.393 x o-14 c T1 T2T T T4 T5 T6 T4 T6.05 -.287395 -2.951 x 10-2 -1.021 x 10-3 -1.8o8 x 10-5 -1.957 x 10-7 -1.432 x 10-9.10 -.265010 -2.578 x 10-2 -8.447 x 10-4 -1.418 x 10-5 -1.453 x 10-7 -1.008 x 10-9.15 -.243233 -2.234 x 10-2 -6.915 x 10-4 -1.096 x 10-5 -1.061 x 10-7 -6.957 x 10-10.20 -.222090 -1.920 x 10-2 -5.593 x 10-4 -8.343 x 10-6 -7.604 x 10-8 -4.694 x lo-10.25 -.201598 -1.634 x 10-2 -4.462 x 10-4 -6.240 x 10-6 -5.332 x 10-8 -3.088 x 100o-.30 -.181778 -1.375 x 10-2 -3.505 x 10-4 -4.575 x 10-6 -3.648 x 10-8 -1.974 x 10-10.35 -.162654 -1.143 x 10-2 -2.704 x 10-4 -3.277 x 10-6 -2.427 x 10-8 -1.221 x 10-10.40 -.144252 -9.354 x 10-3 -2.044 x 10-4 -2.286 x 10-6 -1.563 x 10-8 -7.262 x 10-11.45 -.126601 -7.525 x 10-3 -1.507 x 10-4 -1.545 x 10-6 -9.684 x 10-9 -4.130 x 10-11.50 -.109736 -5.930 x 10-3 -1.080 x 10-4 -1.01 x 10-6 -5.733 x 10-9 -2.226 x o10-11 Tc T12 T13 T14 T15 T16.05 -.342584 -.109817 -6.042 x 10-3 -1.454 x 10-4 -1.983 x 10-6 -1.749 x 10-8.10 -.324553 -.098561 -5.137 x 10-3 -1.171 x 10-4 -1.513 x 10-6 -1.265 x 10-8.15 -.306522 -.087914 -4.328 x 10-3 -9.318 x 10-5 -1.137 x 10-6 -8,985 x 10-9.20 -.288492 -.077876 -3.608 x 10-3 -7.312 x 10-5 -8.398 x 10-7 -6.251 x 10-9.25 -.270461 -.o68445 -2.973 x 10-3 -5.648 x 10-5 -6.082 x 10-7 -4.247 x 10-9.30 -.252430 -.059624 -2.417 x 10-3 -4.286 x 10-5 -4.308 x 10-7 -2.810 x 10-9.35 -.234400 -.051401 -1.935 x 10-3 -3.187 x 10-5 -2.974 x 10-7 -1.804 x o10-9.40 -.216369 -.043805 -1.522 x 10-3 -2.314 x 10-5 -1.993 x 10-7 -1.117 x 10-9.45 -.198338 -.036808 -1.172 x 10-3 -1.634 x 10-5 -1.290 x 10-7 -6.637 x o10-10.50 -.180307 -.030420 -8.809 x 10-4 -1.116 x 10-5 -8.039 x 10-8 -3.752 x lo10 T31 7 32 T33 34 3T 36.05 2.237432 -.227318 -.104981 -5.946 x 10-3 -1.443 x 10-4 -1.976 x 10-6.10 2.237432 -.215354 -.094219 -5.056 x 10-3 -1.162 x 10-4 -1.510 x 10-6.15 2.237432 -,.203390 -.o84039 -4.259 x 10-3 -9.256 x 10-5 -1.136 x 10-6.20 2.237432 -.191426 -.074441 -3.551 x 10-3 -7.255 x 10-5 -8.395 x 10-7.25 2.237432 -.179462 -.065424 -2.926 x 10-3 -5.605 x 10-5 -6.o88 x 10-7.30 2.237432 -.167498 -.056990go -2.379 x 10-3 -4.253 x 10-5 -4.317 x 10-7.35 2.237432 -.155533 -.049137 -1.905 x 10-3 -3.162 x 10-5 -2.985 x 10-7 40 2.237432 -.14357 -.041866 -1.498 -3 -2.296 x 10-3 -2.296004 x 10-5 -2.004 x 10-7.45 2.237432 -.131605 -.0351774 -1.154 x 10-3 -1.621 x 10-5 -1.299 x 10-7.50 2.237432 -,119641 -.029070 -8.669 x 10-4 -1.107 x 10-5 -8. o87 x 1i-8

c c1 C2 c1 C2 2 o05 - 999791 0339571 o999841 1o500000 -2o083289 o10 lo 998593 ~328789 998916 1,500000 -2,222175,15.-996628 o319525 0997362 1,500000 -2o352892 20 -~ 993589 ~309985 0994908 1o500000 -2.499947 ~25 -o989246 ~300142 0991320 1,500000 -2,6666510 ~30 -.983302 ~289964 o986292 1o500000 -2,857083 ~35 -o975367 o279417 o979414 1o500000 =3 076858 o40 -,964927 ~268455 ~970130 1o500000 -3,333263 ~45 -o951279 ~257026 ~957664 1o500000 -3~636287 ~50 -o933452 ~245065 ~940914 10500000 -3.999916

BIBLIOGRAPHY 1o Arnolds, H. A. and others, Tables of the Modified Hankel Functions of Order One-Third and of Their Derivatives, Harvard Univo Press, Cambridge, Massachusetts (1965). 2. Benjamin, T. B., "Wave Formation in Laminar Flow Down an Inclined Plane, " JO Fluid Mechanics 2, (1957) 554. 3. Hildebrand, F. B., Introduction to Numerical Analysis, McGraw-Hill Book Company Inc. (1956), 4, Kapitza, P. L,, Zh, Eksperim, i Teoro Fizo 18, 3 (1948).; 18, 20 (1948); 19, 105 (1949). 5. Heisenberg, W., "Uber Stabilitat fund Turbulenz von Flussigkeitsstromen," Ann, Phys. Lpz, 4, 74, 577-627 (1924). 6. Langer, R, E., On the Asymptotic Solutions of Ordinary Linear Differential Equations about a Turning Point, Maryland Univo Institute for Fluid Dynamics and Applied Mathematics Lecture, Series No. 29, 919540. 7. Lin, C, C,, The Theory of Hydrodynamic Stability, Cambridge Univo Press (1955). 8. Lin, C, C., "On the Stability of Two-Dimensional Parallel Flows," Part 1, 2, 3, Quart, Appl. Math 3, 117T42, 218-34, 277-301 (1945). 9. Squire' H. B,, "On the Stability of NTwo-Dimensional Disturbances of Viscous Fluid Flow Between Parallel Walls," Proco Royo Soco (London) A 142, (1933) 621-8, 10o Wasow, W., "The Complex Asymptotic Theory of a Fourth Order Differential Equation of Hydrodynamics,"' Ann. Math 2, 49, (194.8) 852-71. 1lo Yih, C. S,, Stability of Parallel Laminar Flow with a Free Surface, Proc. of the Second U SO National Congress of Applied Mechanics, (American Society of Mechanical Engineers, New York 1955) 623-28~ 12, Yih, C, So, "Stability of Liquid Flow Down an Inclined Plane," The Physics of Fluid9 69 No. 3, March 19630 13. Yih, C. So. "Stability of Two-Dimensional Parallel Flows for ThreeDimensional Disturbances," Quart1. Appl. Math, 12, (1955) 434. -.....