THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Physics Technical Report CONTRIBUTIONS TO THE THEORY OF HYDRODYNAMIC STABILITY I. V. Schensted UMRI Project 05114 under contract with: DEPARTMENT OF THE NAVY OFFICE OF NAVAL RESEARCH NAVY THEORETICAL PHYSICS CONTRACT NO. Nonr 1224(15) WASHINGTON, D. C. administered by: THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR June 1960

This report has also been submitted as a dissertation in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1960.

TABLE OF CONTENTS Page ABSTRACT v INTRODUCT ION 1 CHAPTER I. FORMULATION AND GENERAL NATURE OF THE BOUNDARY VALUE PROBLEM 4 1. Fundamental Equations 2. The Boundary Value Problem for the Plane Parallel Flows 6 3. The Boundary Value Problem for the Flow Through a Circular Pipe 17 The Fundamental Equations in Cylindrical Coordinates 17 Axially Symmetric Disturbances 19 CHAPTER II. THE EXPANSION THEOREM FOR PLANE PARALLEL FLOWS 26 1. Introduction 26 2. Preliminary Results 27 3. Explanation of the Method of Investigation 30 4. The Main Theorem 37 CHAPTER III. EXPANSION THEOREM FOR THE FLOW THROUGH A CIRCULAR PIPE 44 1. Introduction 44 2. The Adjoint Boundary Value Problem 45 3. Explanation of the Method of Investigation 46 4. The Main Theorem 52 CHAPTER IV. APPLICATIONS OF THE EXPANSION THEOREMS 62 1. Introduction 62 2. The Initial Value Problem 63 3. The Solution to the Forced Problem 67 4. The Non-Linear Problem 69 CHAPTER V. APPROXIMATE LOCATION OF EIGENVALUES FOR LARGE aR 74 1. Introduction 74 2. Approximate Solutions to the Orr-Sommerfeld Equation for Large aR 74 3. Approximate Location of Eigenvalues for the Poiseuille Flow Case 81 4. Discussion 86 iii

TABLE OF CONTENTS (Concluded) Page APPENDIX I. THE VALIDITY OF THE ASYMPTOTIC EXPANSIONS 88 APPENDIX II. THE REGULAR SOLUTION TO THE DIFFERENTIAL EQUATION FOR THE FLOW THROUGH A CIRCULAR PIPE 100 APPENDIX III. THE CASE THAT x IS A NON-SIMPLE ZERO OF THE CHARACTERISTIC EQUATION 104 APPENDIX IV. SIMPLE EXAMPLE OF AN EXPANSION THEOREM SIMILAR TO THE THEOREM OF CHAPTER II 110 REFERENCES 113 iv

ABSTRACT This work deals primarily with the proof of expansion theorems relating to the expansion of arbitrary functions in terms of the eigenfunctions of the plane parallel flow, stability problem and those of the stability problem for the flow through a circular pipe. The investigations herein contained show that such expansions are valid for functions which satisfy certain boundary conditions and regularity conditions which are specified in the text. Application of these expansion theorems to the solution of the initial value problem, to the solution of the forced oscillation problem, and to the non-linear problem are given. The approximate location of some of the eigenvalues of the parallel flow stability problem and of the stability problem for the flow through a circular pipe are also given. v

INTRODUCTION In this thesis we shall discuss topics relating to the study of the stability of viscous incompressible fluid flows. The problem has a long history dating back to the beginning of this century. The general approach has been to examine whether or not small disturbances superimposed upon a steady state flow would tend to be amplified in time or not. This study of the behavior of a small disturbance leads to what is called the linear theory of stability. The system of linear equations which results when one neglects terms quadratic in the disturbance components has been studied intensively by physicists and mathematicians during the last half century. Even in the cases of the simplest steady state flows such as the plane parallel Poiseuille and Couette flows, the problem has proved to be difficult. Of greatest physical interest is the nature of the solution for large Reynolds numbers since it is in this domain that turbulence is observed to occur. However, it is precisely here that it becomes impractical to solve the mathematical equations by power series techniques and as a result asymptotic expansions have been used to make specific calculations. The problem of defining the validity of these asymptotic techniques has engaged a long line of mathematicians and it is only recently that rigorous mathematical justification for some of the calculations has been given. The particular stability problems with which we shall deal here 1

2 relate to the plane parallel Poiseuille and Couette flows as well as to the Poiseuille flow through a circular pipe. All these problems have long histories. The plane Couette flow problem was studied by Hopf6* 14a early in this century and more recently by Wasow and Corcos and Sel2 lars. All these investigators came to the conclusion that the plane Couette flow is stable, i.e., that all small disturbances ultimately tend to die out. This fact however has not been rigorously proved. The plane Poiseuille flow problem was studied by Heisenberg5 who concluded that the flow was unstable. Later workers however found many of the points in Heisenberg's work to be obscure, C. C. Lin8 later clarified many of these points in Heisenbergts analysis. He too concluded in favor of instability. The controversy however did not subside until 1953 when L. H. Thomas12 confirmed Lin's results by numerical calculations on a high speed digital computer. More recent confirmation has come from the work of Dolph and Lewis.3 Lin's calculations are now considered to be one of the triumphs of the linear theory of stability. In connection with the stability problem for the flow through a circular pipe only the axially symmetric disturbances have been intensively studied. Sexl in 1927 investigated this problem and concluded the flow was stable with respect to such disturbances. Pekeris9a and Corcos and Sellars2 have written more recent papers. Although these later authors have criticized the validity of Sexl s calculations, they too have concluded that the flow is stable with respect to axially *The raised numbers refer to the bibliography at the end of this work.

5 symmetric disturbances. In this thesis we shall deal with certain areas of the stability problem in which relatively little research has been done. The usual manner of treating the problem is to substitute into the linearized equations a disturbance which is periodic in the direction of the basic flow and which has an exponential time dependence. A boundary value problem results to which one can find solutions only for certain values of the time constants which are called the eigenvalues of the problem. In Chapter I we shall introduce this boundary value problem and shall go on to show the general nature of the spectrum of eigenvalues. The problem of whether the eigenfunctions corresponding to these eigenvalues form a complete set in the sense that arbitrary functions may be expanded in terms of them has received little attention. In 1912 Haupt4 investigated this problem for the plane Couette flow. His expansion theorm however is not general enough to include the plane Poiseuille flow. In Chapter II we shall prove an expansion theorm for plane parallel flows which is sufficiently general to include both types. In Chapter III we shall prove the corresponding expansion theorem for the axially symmetric eigenfunctions of the circular pipe stability problem. In this case, the discussion will help dispel some of the misconceptions which have appeared in the recent literature. In Chapter IV we shall discuss some of the applications of the expansion theorems. In Chapter V we shall outline the asymptotic techniques of Lin and Heisenberg and shall use these to present approximate formulae for the eigenvalues to these stability problems for large Reynolds numbers, Much of the material in this chapter will be a recapitulation of the work of others.

CHAPTER I FORMULATION AND GENERAL NATURE OF THE BOUNDARY VALUE PROBLEM 1. FUNDAMENTAL EQUATIONS The equations which form the basis for all investigations relating to the stability of viscous incompressible fluid flows are the NavierStokes equations and the equation of continuity. As is customary we shall write these equations in a dimensionless form by measuring all lengths and velocities in terms of a characteristic length L* and a characteristic velocity V*. The unit of time is taken correspondingly as Lt \ and of pressure as g Vf where is the density of the fluid which will always be considered to be constant. The basic equations can then be written in the form: atu + (LC>xa >aW=- a^p^L Dt ZT (u,^^ c)^ ^ + U-dX A 1 U = - tP + /R <I F (1.1.2) Dt r +(UL. +va+t^^w - P+ l/R +A where (uv,w) are the three components of the velocity, p is the pressure and R = V/ /AA is the Reynolds number which is a measure of the ratio of the inertial to viscous forces in the fluid; )UL is the coefficient of viscosity; i is the operator ( A - o To these equations we must add the boundary conditions which are that the fluid in the neighborhood of a solid boundary must be stationary with respect to it. In particular at a stationary boundary we have 4

the conditions: UC(S) _ ty = ir ()= D (1=153) where s represents any point on the bounding surface. The study of the stability of a given steady state basic flow according to the linearized theory proceeds according to the following pattern. One assumes a small disturbance velocity superimposed upon the basic flow velocity. One then obtains a set of equations linear in the disturbance components by substituting this form for the velocity distribution into (1.11o) and (1.12) and neglecting all terms of second order in the disturbance components. If the solution of the resulting equations indicates that the disturbance tends to increase after a long time the corresponding basic flow is termed unstable. If all small disturbances tend to decrease after a long time the steady motion is termed stable. Let, U,VW denote the components of the basic flow velocity of the fluid and let u,v,w be the components of the disturbance velocity. Let P and p be the pressures associated with the basic flow and the disturbance respectively. Then if we follow the procedure outlined above and drop terms of second order in u, v,w and furthermore take into account that U,V,W and P satisfy (1.lol) and (1.1,2) and the boundary conditions we obtain the following equations for the disturbance components: t' -(ul^, Va. (.a +,') = -(+, Up+U - w+ 4v+ (UN4+ VU~* 44),t (U.x*YA ) =u~ - VP+ \/t V (1.1.4) Asrc(Uast^^4Wde)ar W)g t o(as t)w= - a v + Yq WAS

6 3,@ S^ Af^t at w C(1.1.5) The boundary conditions are: U(S - (5)) =(6)= (1.1.6) In the remaining sections of this chapter we shall consider the specialization of these equations to the cases of plane parallel flow and the flow through a circular pipe. 2. THE BOUNDARY VALUE PROBLEM FOR THE PLANE PARALLEL FLOWS In this section we shall consider the specialization of the boundary value problem stated above to the cases of the plane Couette and Poiseuille flows. Let the bounding planes be parallel to the xz plane of our coordinate system and let them be situated at y = +1. For the case of the Couette flow we imagine the lower bounding plane to move in the x direction with the constant speed 2 (2V* in the actual physical system). Corresponding to these boundary conditions we have the following steady state solution to the equations of motion: U0 -y \ 0v ~ - t (1.201) For the plane Poiseuille flow both bounding planes are stationary and we have the following steady state solution: U= -4, \ 0 I,W O (1o2.2) We shall assume a 2 dimensional disturbance of the form:* *This is not a specialization since the more general 3 dimensional disturbance A = Cl^) (LP^I^V-C ^-i(t)$ Vsy- 945 LptiX* xt-XCtA)3 Ur'S- l~(cti (dx.>-_CtCcan always be reduced to the form (1.2.3) by choosing the x axis in the direction of propagation of the wave. See C. C. Lin (Refo 8b, po 27).

7 1 wCk -Ant) I= o,p = Bpy ei(tHere cZ is any real number and c is a constant in general complex, which must be determined by solving the boundary value problem. A positive value of the imaginary part of aCc will correspond to instability. Upon substituting (1.2.3) into (1.1.4) and (1.1.5) and making use of the fact that for the Couette and Poiseuille flows V = W = Owe obtain the following system of equations: -'_d c, I U it - ld, +/ - D ot) -t A A\A ) ^^ 1 (1.2.4) -LisCC +Uiir =-W+VP ui- ) LAdz. = A 0 (1.2.5) The boundary conditions are:, (1.2.6) (j) = ~(-0- o Further simplification may be achieved by the introduction of a stream function' (tV*) = ~(W)k,(Y —Cat)} which is related to the velocity components by the equations: u Ati -F (1.2.7) Equation (1.2.5), the continuity equation will be automatically satisfied. If we substitute (1.2.7) into (1.2.4) and then eliminate the pressure by differentiating the first equation of (1.2.4) with respect to y

8 and multiplying the second equation of (1.2.4) by ia and then subtracting one equation from the other, we obtain the following ordinary differential equation for 0 which is called the Orr-Sommerfeld equation. D nt;$w);- t47t-) ) ( _ o (1,2.8) where D The boundary conditions become: 40()= D+(l) =(>C-!)- = -)= o ( 1.29) Equation (1.2.8) has four linearly independent solutions (, C), = l 1,3.4 ~ Since U for the Poiseuille and Couette flows is an analytic function of y, it can be shown that the functions TV will be analytic in y on the finite portion of the complex y plane and in addition may be chosen to be analytic in c on the finite portion of the complex c plane.* The function () may be expressed as a linear combination of the i in the form: +(i)_ 8 A~ff (1.2,10) *The theorem which applies here may be stated as follows: If we have a linear differential equation of order n of the form [( ~ + t, P. _,,'"''F v. ) - o where the coefficients P,(,C) are analytic in y on OX L r and analytic in c for c in the region S of the complex c plane, then a set of n linearly independent solutions nn,C) can be found which are analytic in y on o_ L \ L and in c for c in S. A proof of this theorem appears in Ince (Ref. 7, pp. 72-75).

9 where the AZ are constants. Substitution of Eq. (1.2.10) into the boundaryconditions (1.2.9) yields the determinantal equation: |f,(l,) LMi45 F3(l) fsi0,)| fit) r( ),-) - (1,2.11) gD^(-r,)) D k%-i4 c) O 3(-l <) C4->)), For given values of a2 and CIR the permissible values of c may be determined from (1.2.11). Since the determinant is an analytic function of c in the finitec plane its zero's, if any, will form a discrete set. Corresponding to each member c, of this set there will be at least one eigenfunction (Y' R/C4) which satisfies all the conditions of the boundary value problem stated above. We shall now show that Eq. (1.2.11) possesses an infinite number of eigenvalues ct and that corresponding to these eigenvalues there are an infinite number of eigenfunctions 0. We shall in the following discussion exclude a = O. This case represents no special difficulty and will be treated separately at the end of this section. to prove that there are an infinite number of eigenfunctions we shall need explicit forms for the solutions to (1.2.8) for large values of Ic\. Let us introduce the parameter x which we define to be equal to QiR,. In addition let us define the quantitykas follows: K-Cixr e1^(12.2

10 where )C is the argument of K. In what follows W will be restricted so that o_ tL. L Tt and consequently the argument of k will be restricted to the range t/~ Jo~t 3$!T/l. If we substitute in the Orr-Sommerfeld equation (1.2.8) for () a series of the form: <(S-)= e* 9 ( f (1.2.13) e=o then, by equating successive powers of k, the functions Q and G.(i can be determined. Equating the coefficients of k one gets: uT0 - Q'^ ) 0= (.2.14) This equation has four solutions for which one can take: Qi(y)s-(ot ql); =i +i') Q1- Q4- (1.2.15) These solutions are all arbitrary to within additive constants which we have chosen in a manner which will be convenient for the work in Chapter II. For the cases where Q(~ O we obtain from the coefficient of the k% term: J C)3^ X- X =O (1,2e16) This equation is consistent with (1.2.15) only if /o Hence we can choose CO e Co j,. Proceeding further we can obtain arbitrarily many of the functions ( and. For the cases where Q O we obtain the following relationships for the coefficients ~. and C. (D1-~ _ _ o (1.2.17)

(D-)6- U)d-4R E(L -)- U3) j (12. 18) From Eq. (1.2.17) we see that we may choose ( e G- ) - (1.2.19) Succeeding terms may be obtained by the use of (1.2.18). Hence one obtains the four formal solutions: f_ =+e r_ 0 T. g- = e t tw et (1.2.20) t~ - ~ —tfc, T - - t t Z_ The functions ^e are analytic in y on "l_ i C and do not contain k. For the Couette flow e are exact solutions so in this case the series a and {, reduce to one term. Of course the formal series solutions (1.2o20) arenot necessarily convergent. The relation of such formal series solutions to the true solutions of a differential equation containing a large parameter has been investigated in detail in the mathematical literature, especially by Trjitzinsky.13 For this work we can conclude that there exists a set of true solutions fg to (1.2.8) of which the f of (1.2.20) are the asymptotic expansions in I/R. Or more precisely the true solutions,, are of the form:

12,Qi:) r tA ^ (a) E (*; =,e X it @'- 1" Z i" Je where the Q, and R ) are as in (1.2.20); the value of M is arbitrary and the remainder terms E.i(,3 ) are analytic in y on -~ i t \ and bounded in k provided lk[ is sufficiently large and in the sector ~i/~ ~> t 3Q1V. In addition one can prove that the first three derivatives of the T~ are similarly related to the series obtained from r by differentiating the corresponding number of times. An outline of the proof of these statements is given in Appendix I. In what follows we shall use the following notation. Let the quantity al) designate a finite series of the form (i + Ca -', -+ V' ); where the functions 0,, CL,.,., C are bounded functions of y on - 1. f \ l and bounded with respect to k as Ikl -- oo. The manipulation of such quantities is simple. We have for example that CG[l r C[<u3 and t4 _ 3L providing that -, i O. Using this notation we may write the solutions to (1.2.8) and their first three derivatives in the form: T- e ii3l 4- e%6 + 13 * v-\e-en D t- (1.2.21) -r 3 A L U 3

13 F3 ~. e CYt1)3 fy -ie-e^l it -^ Ltet>%tl) O $- M3 ((1.2.21) Of3= CeOY3 P tl)+ D3 t- - 3e-t'4*"01 Using these formulae, asymptotic expressions for the eigenvalues in the limit of large |c may be obtained easily- Substituting the above expressions into (1.2.11), we obtain the following equation! e'-21\, e21'tn3 [ed Le-4 i 3 U13 [e-"3 e" 3 -ilK h-I ideV 2 0 (1.2.22) |- aln ^tUCi1 I_ e-3 L- tae7 Multiplying out the determinant while making use of the appropriate rules for the manipulation of the quantities in brackets we find that it may be factored into two factors and equating each of these separately to zero yields the two equations: et ^U -t e 3(1.2.25)..k U 3"3 = Locc-td(3 <.\n- ^ -<'n (1.2t24)

14 To obtain the roots of (1.2.23) we expand the functions on the left hand side about the points inc where n is any large integer. We then get for the roots of (1.2.23): NW _ i-? t O(k (1.2.25).. To obtain the roots of (1.2.24) we expand the left hand side about the points ti t+tl)tr. We get the following formulae for these roots: K _ (I.TL C' 0g) X *=i (tt)s oet \ (12.2.6) at.i_ -+ 0O (0) We observe that for n sufficiently large the imaginary part of the quantity ac will always be negative so that the corresponding modes will be stable. Equations (1.2.23) and (1.2.24) may also be arrived at in the following manner. One observes that it is possible to construct from the functions of (1.2.21) two linear combinations, the leading terms of which are even in y, and two which are odd in their leading terms. Since the boundary conditions (1.2.9) are the same at y = +1, it follows that, for IkI large, one may obtain approximate eigenfunctions by considering separately linear combinations of the two functions which are even in their leading terms and of the two which are odd. The "even" eigenfunctions will have leading terms of

15 the form A = ACo K%~~ o bCk). Applying the boundary conditions at y = 1 to this function we get: (K ay 1 = OL ga0d s which is essentially the same as (1.2.25). Making use of (1.2.25) we see that to terms of OC/r)nthe function Oe is given by the formula: c- c C Ld ) (1.2.27) The "odd" eigenfunctions will have leading terms of the form: )= ASL(qt 4)+- P S(AOJ ). Applying the boundary conditions we get (1.2.24). Making use of (1.2.26) we obtain that to terms of OI/v O^, = 5 (?cy) Ot (-,Z,4oL' 3QIL + (1.2.28 We should remark here that these limiting eigenvalues and eigenfunctions are the same for all viscous flows between two parallel planes. From the equations above we see that the nature of the function U(y) does not affect the leading terms of c, or 0,. In the next chapter we shall consider the conditions under which arbitrary continuous functions f(y) may be expanded in terms of the eigenfunctions of plane parallel flows. The problem of the completeness of the eigenfunctions of the Couette flow has been considered by Haupt.4 The proof given by Haupt depends on the fact that for the Couette flow 0 /= 0 and hence (1.2.8) may be factored into two simple equations of the second order. The proof which we shall present in the next chapter will apply to viscous flows for which ll0O as well

16 I \t as to those for which i 0=. All the remarks we have made above apply to the case oL O as well as to.70. Actually once one knows the eigenvalues C (?) and eigenfunctions ) (o*) corresponding to a given a those corresponding to -c may be obtained from the relationships: CHj-io) = C^<} i> (-4) rr ~c (1.2.29) Relationships (1.2.29) may be easily derived from (1.2.28) by setting a to -a in that equation. The stability characteristics of the modes are unchanged by this transformation since the imaginary part of the quantity -_L(-o) is equal to the imaginary part of the quantity c(&(o}) by (1.2.29). We conclude this section with a treatment of the a = 0 case. For this case there is no x dependence of the disturbance and therefore the y component of the velocity vahishes. To treat this case it is simplest to go back to (1.2.4). Let us set oC = Ct. We then obtain from (1.2.4) the equation: - Aga= -'/t act The boundary conditions are: A -^ The complete velocity function is U-L- U(t). The solutions to this boundary value problem are simple. We obtain a set of even and A a set of odd modes for which Wt and 67 are given by the following:

17 A h a:, ~ ~~~ 4r=~ k IW | =(12.2 30) L^u So4(VKY) n *Y.,!\',.atTh,,GO (1.2.531) a= IULPt( R = R i rl 1 O. 1)2,),1,. It is well known that the functions of (1.2.30) and (1.2.31) form a complete set. In Chapter II we shall consider the completeness problem when oBLO. 3. THE BOUNDARY VALUE PROBLEM FOR THE FLOW THROUGH A CIRCULAR PIPE The Fundamental Equations in Cylindrical Coordinates To treat the flow through a circular pipe it is most convenient to introduce a cylindrical coordinate system. Let the Z axis of this system coincide with the axis of the pipe. Let r represent the radial variable and t9 the azimuth angle measured from some fixed radial line. We shall set the characteristic length L* equal to the radius of the pipe. The characteristic velocity V* is the basic flow at the center of the pipe. The quantities u, v, w throughout this section will represent the r,9,z components of the disturbance velocity. The basic flow components are \=0; Uo;V^-/= \- Tr. Using this notation the linearized equations for the disturbance components are as follows:

18 t (- + ) 0 - I-) 1-^'^ [L T (r - 1 3 - ~1yvt Stc w - 0 (1.3.2) + r where L- L~ +r- + - The boundary conditions are U=, —' V V 0 at r = 1 and in addition u,v, and w must be finite at r = 0. If into (1.3.1) and (1.3.2) we substitute disturbance components and a disturbance pressure of the form: L= W(^ tQ ILe^ ^ (15 (5) P-P), (f+Fti^,i) where n is any integer and 3 is any real number, then we obtain the following set of equations: i:R (cw-), + l/ rP'-~+~ - (- T. W + L Y1t 4 Af +Af-' = (1.5.5) I t. _rl

19 The boundary conditions are A. /f tUaL3- must be finite for t - < It does not seem possible to simplify the problem further as in the case for plane parallel flows. Only for axially symmetric disturbances,:z- ~) can the problem be reduced to a simple ordinary differential equation. We shall in what follows consider only the axially symmetric case. Axially Symmetric Disturbances There are two types of disturbances for n = 0, the shearing and the radial modes. The equations for the shearing modes are obtained by setting iJ.I/"' p- 0. Equation (1.3.5) and the first and third equations of (1.3.4) are automatically satisfied. The remaining equation and boundary conditions involve only q; it~g(Vsl-ti-85t~w )- ts@L_ Edt try+ $ i (1,3.7) Ad Vis finite. Pekeris9a has treated this problem in some detail and has found all the modes associated with this type of disturbance to be stable. The problem of the stability of the radial modes is more interesting. For these modes. is set equal to zero. The second equation of (1.5.4) is satisfied identically. The study of the remaining equations is simplified by a stream function c r )'/( i^ - I t) which is

20 related to the velocity components as follows: * - r &dCtY' ); G,-cB (r) W= la.r(1) * c=l. (Wk(V()) (1.3.8) The continuity equation (1.3.5) is satisfied identically and by eliminating the pressure we obtain the following equation for 0: C[(L_- pR_ ie - R ( \ )_ L-pa ) +(t) o (1.3.9) where L- - + -- - The boundary conditions are: ) ) ^ ^r 0'1) (1.5.10) )A^ Vr o 0 is finite Equation (1.3.9) possesses four solutions ftP (,C) ) L- 1)2)3,t ~ Of these, two will be regular at the origin and the other two will be singular there. The eigenfunction must be a linear combination of the two regular solutions because of the regularity condition for 0 at r = 0. The boundary conditions at r = 1 then lead to the eigenvalue equation: cd ^J } dS1__ (1.5.11) It is easy to see that equation (1..39) possesses as exact solutions the functions, J[,(qt) and.yTr). ( J [t) is the Bessel function of order one and Y(lt) throughout this work will be taken to mean a Neumann function of order one.) These are called the inviscid solutions because they do not depend upon the Reynolds number. The other two solutions are called the viscous solutions. Of the two inviscid solutions

21 only Jl (Lt) is regular r = 0 and can therefore be taken for {- so that (1.3.11) becomes: ^i ( yv.t = d, it 1) (1.3.12) T, (LIP) This boundary value problem was first treated by Sexl. More recent calculations have been carried out by Pekeris9a and by Corcos and Sellars.2 All these investigators have concluded that only stable modes exist. We shall, in what follows, show that this problem possesses an infinite number of eigenfunctions and eigenvalues. Corcos and Sellars2 have suggested that only a finite number of eigenfunctions and eigenvalues actually exist so that arbitrary disturbances may not be expanded in terms of the eigenfunctions of this problem. They go on to suggest that it is necessary to investigate the response of the fluid to disturbances of a more general nature than that given by (1.3.3) (with n set equal to zero) before it can be concluded that the flow is stable with respect to axially symmetric disturbances. Below we shall show that there are in fact an infinite number of eigenfunctions to the boundary value problem stated above. In Chapter III we shall go on to discuss the conditions under which arbitrary functions f(v) may be expanded in terms of these eigenfunctions. All these eigenfunctions correspond to stable modes and since, as we shall see in Chapter III, any function, f, can be expanded in these eigenfunctions one can conclude that for axially symmetric disturbances the Poiseuille flow is stable.

22 Let us consider the nature of the solution fi for large eigenvalues. Let us define - C 6 RC. In addition let us define the quantity k as follows: - -—, X1 L U1. where CO is the argument of \. In what follows 0) will be restricted so that o _ L _ I TT and consequently the argument of k will lie in the interval TT/L_ 0 j - L_ 1 /. As in the case of the plane parallel flows considered in the last section we may obtain formal series solutions to (1.3.9) by substituting for f. (i = 1,2,3,4) expansions of the form:'F 2pW4KY (] I )/K t'-o One finds that two of these formal series solutions, f, and i say, have the form: _ e -L\ + 7 \ - ) _ her~ t~w +;a (uios r^ (1. 3.14) The other two formal solutions reduce to the two exact solutions J\-(t andY, (ift) which have already been mentioned. If we bound the variable r away from zero, we may apply the results of Trjitzinsky and state that for, 06t i_ t1, Eq. (1.3.9) possesses two true solutions p, and f, which are asymptoticallyequal to arbitrarily many terms to i and f. The function fA(r) which is regular at r = 0 must be expressible as a linear combination of f and i$. To find out what the appropriate combination of FO, and H is we proceed heuristically as follows. Let us consider what happens, if in (1.359), we neglect the quantity

23 1 - r2 with respect to c and a2 with respect to \. We shall then obtain a differential equation which may be:.factored in either of the following ways: (L+x%-)( t~o) o (153.15) or(L - e)(L t\) 0 The four solutions to (1.3.15) are'(i ~),Yi )~~r) and yC(t] Of these T} (X'Tf) is the one of interest to us since it is regular at r = 0. If we compare the asymptotic formulae:*.t e (T &r _) I O_ o. It L ZT J, t) C (rTr/^y"-) (. tt) ( )' with equations (1.3.14) we see that if we put: iT= T Vf,) e f (t)' =.: -3/ ~t:X cat ~ J b 3T4, (1.5.16) s = 3s/ ~, 3~IL8/ 4' - ^ *See for example Watson (Ref. 15), p. 202.

24 then F,() will behave asymptotically like A(k) J(X V)and therefore can be expected to be the appropriate linear combination of i and. which is regular at the origin. A rigorous proof that this is the case is given in Appendix II. Substituting (1.3.16) with -3f/ into (1.3.12) we obtains K ( e in - e ) = X Ci) ) -- (1.3-17) tK-'i w (:L.3 ZK (d) Since k increases without bound while the right hand side of (1.5.17) does not depend upon k at all, the solutions of (1.5.17) must lie close to the zeros of the quantity e'.e.. These zeros lie close to the points k = (C 2/4)T/f. Expanding about these points we obtain the following formulae for k and c:,w. = ( 3/C ) T r + 0( Cr) (1.3.18).C - "(M+3/4 1YT - O (l) where n is any sufficiently large positive integer. Corresponding to these eigenvalues the approximate eigenfunctions are:,j.. j-, (T.3.zg) <,^,,-) = J, -) - JC^) T,[( m ^ r] ^(1-3*19) This completes the proof of the fact that there are infinitely many eigenvalues to the problem for the case 3 O Let us now consider the case L = 0. Going back to (1.2.4) and (1.2.6) we see that if we set G- 3C we obtain the following equation: _1^ A t 3 1-'1 t-ur t T.. 0_ (1.5.20) dl^c r dr

25 The boundary conditions are: t(i)-O t, CO ) is finite The solution of (1.3.20) which is regular at r = 0 is Jo(CG-F R ). The eigenvalues ( are the roots of the equation: o (( cGR'L')- O (1.5.21) Since o (i) possesses an infinite number of zeros on the real axis there are an infinite number of eigenvalues G all of which correspond to stable modes. It is well known that the corresponding eigenfunctions form a complete set. In Chapter III we shall discuss the completeness of the eigenfunctions when * H 0 o

CHAPTER II THE EXPANSION THEOREM FOR PLANE PARALLEL FLOWS 1. INTRODUCTION In this chapter we shall consider the conditions under which arbitrary functions f(y) may be expanded in terms of the eigenfunctions to the stability problem for plane parallel flows. That is to say we shall investigate the validity of the expansion: where the functions pj are the solutions to the boundary value problem for plane parallel flows introduced in Chapter I. This exapnsion is of interest in connection with the solution of the initial value problem for the disturbance which we shall discuss in Chapter IV. The conditions which,j must satisfy are as follows: L ('$-R -to R iU-c( Y-)- U%]4~ ~~ i O (2.1.2) (A) (1) - DO) =0 o (-i) = D l-' ) o (2..j3) where D~ -. For the discussion in the chapter it suffices for us to know that a and R are fixed real non-zero numbers; U(y) is analytic in y on -I L- L |, and c is the eigenvalue parameter. Throughout what follows we shall use the designation, (A), to mean the entire system of equations given above including both the differential equation, (2.1.2), and the boundary conditions, (2.1.3). 26

27 In Chapter I we showed that the boundary value problem possesses an infinite number of discrete eigenvalues, cj, for each C and each R, and that corresponding to these eigenvalues there is an infinite number of eigenfunctions, 0j. The discussion below of the validity of (2.1.1) will not depend on the specific form of the analytic function U(y). 2. PRELIMINARY RESULTS Let us introduce the system of equations adjoint to (A). C (o^- )-all O S(D-9)(-UC) -X= o (2.2.1) (A') l)^ tX) tal ) 0O (2.2.2) (-l)= Dx(-l) =o The set of eigenvalues of (A') will be shown to be the same as that of (A). Furthermore it will be shown that the eigenfunctions of (A) and those of (A') form a biorthogonal set, with an orthogonality relationship of the form:* J ^ (^ ( 0 d) i( dy = J ^() (02t o) Xt ( y = C) -1 -& ((2.2-5) To prove that the eigenvalues of the two problems are the same, we select an arbitrary eigenvalue of (A), ci, let us say. The corre*Throughout this discussion we shall assume for the sake of simplicity that the eigenvalues cj are not degenerate and correspond therefore to only one 0j. Whenever this is not the case some of the results derived below will have to be modified. The nature of the necessary modifications will however be clear from the procedures used in Sections 2 and 3 of this chapter to obtain the expansion theorem.

28 spending eigenfunction of (A) we designate as 0i. Since (2.2.1) is a differential equation of the fourth order and therefore possesses four linearly independent, solutions, we can find for this fixed value of c, a function %X which satisfies any three of the four boundary conditions (2.1.3). One then shows that the function 3i, must necessarily also satisfy the fourth condition and hence is an eigenfunction of (A') corresponding to the eigenvalue ci. To demonstrate this let us assume we have constructed a solution' of (2.2.1) which satisfies the three boundary conditions 0(I)- = (-I) = i(1) - O. We now prove that DX%(-I) must be zero. To do this we multiply (2.1.2) by' and (2.2.1) by 0i and integrate over -\ L LI \ Then we subtract one of the resulting equations from the other. Because S S U(P1-^ Y = X1 U(DQ >)()) U $ as shown by partial integration using the boundary conditions on 0i, one obtains S t ( D4 ) - fD -), X = 0 (2.2.4) Again by integrating each term by parts twice and using the four boundary conditions on 0i and the three on i, one gets: b')Xe$(-. D^) -" b 0 (2.2.5) If D I/(-) 0 then we have b X(-I)O so that ci is an eigenvalue of (At). If DC (-I)-O then we construct X so that it satisfies instead of the three conditions assumed above, the three conditions: i(CI5 X(l) =m)= DXi(-I)-O. Now we must prove that X%(-) =O. Equation (2.2.4) still applies but upon integrating by parts one now gets:

29 X.;(-, bID (k-f)- 0 (2.2.6) Both D H-, and D3C(-) cannot be zero for then would be identically zero. It follows that X-i) -O and ci is an eigenfunction of (A'). The orthogonality relationships (2.2.3) may be proved in a similar manner. We assume:j(y) is an eigenfunction of (A') corresponding to the eigenvalue cj and that 0 is an eigenfunction of (A) corresponding to the eigenvalue ci. We substitute j and cj into (2.2.1) and we substitute ( and ct into (2.1.2). Then we multiply (2.2.1) by e and (2.1.2) by Xj and integrate over -1 _ I 1. Upon subtracting one of the two resulting equations from the other we obtain the relationship: frch the oit rCDa e)liti = O (2.2.7) from which the orthogonality relationships follow. Let us normalize -j(y) and 0s(y) so that* S ~r(-')~ ~ - i. (2.2.8) If we multiply both sides of (2.1.1) by (DZ- 2)(1j and integrate over- i - 1 L, we obtain as a necessary condition for the validity of the expansion (2.1.1) where D('I " - j-( ) (_ ) 7 where DI- da. Using the notation (f,'-)( we have if the set is complete: "T )..,1 (2.2.9) *We shall show later that the integral,; )(DI-d )..y does not vanish so that the normalization procedure may actually be carried out.

30 3. EXPLANATION OF THE METHOD OF INVESTIGATION The technique we shall use to investigate the conditions under which (2.2.9)is valid is analogous to the one used by Birkhoffl in his investigation of the possibility of expanding an arbitrary function f(y) on the interval X _ e _'r in terms of the eigenfunctions Uj(y) to the following boundary value problem: where Pi(Y) ae a naly ti on _.e where pi(y) are analytic on oQ L, The eigenvalue parameter is \. In addition the functions Uj must satisfy n homogeneous linear boundary conditions at a and b. The essential difference between the boundary value problem (A) and the problem treated by Birkhoff lies in the presence of the operator t)- (-. multiplying c.. Because of this difference we must modify the Birkhoff proof considerably to suit our purposes but the broad outlines remain the same. Let X.- /i c. We shall begin by constructing a function aL4,/,), which is analytic in \ in the entire A plane except at the discrete set of points X = \j where it has simple poles, the residues of which are (0~- (~) C (). If we multiply I/2 i ( a. X )) by f( ) and integrate over the interval -i _ $ 1 in the 7 plane and over a large circle on the \ plane which does not go through any of the singularities of the function g(y,, x) we obtain, I (2.5.1) _,

31 where the sum on the right is taken over all the poles contained within the circle r. If we construct a sequence of circles il which do not pass through any of the singularities of g(Y)i, ) and are such that the sequence of radii KA approaches infinity as A approaches infinity, then the sum on the right will approach that of expansion (2.2.9). Hence the problem of evaluating the expansion (2.2.9) is in this manner converted to the process of evaluating L ip 06 i 1'(,s,) F1dx (2.5.2) and showing that this limit is f(y). The advantage of this process lies mainly in that (2.532) will depend only on the character of the solutions to the differential equation for large values of \ where as we have seen in Chapter I, they are quite simple in nature. We have thus far not defined explicitly how to construct the function g(i. )X ). We can show that aols8Q) = (Dt - i>-^)&c^;o G t (2.3.3) where G('A \) ) is the Green's function for system (A) defined uniquely by the following relationships: (i) G(~4j ) satisfies the differential equation of (A) for all points y on-1L - yl except at - f where (2.3.4) (ii) G(, \)X ) satisfies the boundary conditions of (A). (iii) G( 4,)X ), G(,)% A,), and: G(B, XI ) are all continuous at y =

32 (Iv) G,, 1_ Similarly one defines the Green's function H(%,$) o ) of the adjoint system (A'). It can be shown that* bH(^Yf,)- G(ck, (2-3.5) The function G( C),X ) may be written explicitly in the form:** G&4,)= z - NL (2.53.6) X(kX where the quantities A(\) and N(%, )) are defined as follows:, (-i) 0 E&) F3 i 7( i) - I Df-) Di-ji) o,0(-)V 0,),-) (2.3.7) |1,) W oy() W3 tco) f,(eb tV^A f+3i ) (^ ^(Y8v4 ) f, c-0) &C-~- f3-) c -^t + 7 (-t>S, k5 N(4'i>9- | c(ll) ft(l) 8Cl() ti(1 7t )8> | (2.3.8) Dl((-0) D0^-l) OfM ) D4q(-l) 07 (->, ^ A) D,( l Df{v() Ot(15 bi(lA DOc 7tP^ ^,& where D refers always to a differentiation with respect to y, and for instance the notation O (-\ t) ) means 7 ( l ) for y =-1. The functions, fi(y), i = 1,2,3,4, which appear here are the four *A proof of this is given in most standard texts on ordinary differential equations. See for example Ince (Ref. 7, pp.255-256). **The formulae which follow are also to be found in Ince (Ref. 7, p. 259).

33 linearly independent solutions of the differential equation (2.1.2)o They will, of course, depend upon \. The function 7 C * > ) is defined as follows: 1 / I) ) O,) @|1(t5 f_(0) ^f) i$c) 2. 1 (I) itS7, III (24.39) 1ill.-II, The plus sign applies when 7; the minus sign applies when 3 ~ The primes designate differentiation with respect to the argument. These expressions for G(i) },X ), satisfy conditions (i)-(iv) for all x ~ \j. Clearly G(,,X )- is a linear combination of the four functions fi(y), so that it satisfies (i). To see that the boundary conditions are satisfied, we evaluate G and DG at y = +1. For these cases the first row in the determinant expression for N bccomes identical with one of the other rows and hence N vanishes. To see that conditions (iii) are satisfied we note that the only discontinuity in G or its derivatives must enter through the term N(Yl)). By expanding the determinant N with respect to the elements in the first row, one sees that

54 G = E (M)X)t 7+ ( ~x where E is analytic for -1 3 L. If we consider both right hand and left hand limits of the quantities I, D, and D. as Y- f, we see that in each of these quantities the top row in the determinant in the numerator of the expression for 7 becomes identical with one of the other rows of this determinant; hence these quantities vanish. Therefore G, DG and D2G are continuous at y =,. However for the quantitiy D3G we get plus or minus 1 depending on how the point 5 is ^^ ^ - I/.I Hence D - =. approached; D%1 = /7~ r 3: —/; Hence7.)? =i -. We see therefore that (iv) is also satisfied. We shall now show that the function g(jd), ) defined by Eq. (2.3.5) has the properties mentioned at the beginning of this section. For x = \j, G(.,A ) will have a pole of finite order due to the presence of the factor A(\) in its denominator. We shall assume in what follows that the poles are simple. This actually appears to be true for the hydrodynamical cases considered and corresponds to the assumption that the eigenvalues cj are not degenerate.* At a simple pole X = Xj, the residue Rj of G(t, A)\ ) will be of the form: (c, fxi At % = \j, the function, N( i )~ ), considered as a function of y alone satisfies the differential equation at all points of the interval-i_ ~1i. *hIn Appendix III some of the possible complications will be discussed in the case that the poles are not simple.

35 This is due to the fact that the coefficient of I(NCd)r) is equal to zero at \ = \j. Hence N is analytic at y = ~, and since it is a linear combination of the functions fi it satisfies the differential equation (2.1.2). Furthermore, since N satisfies all the boundary conditions it followw that N considered as a function of y is an eigenfunction of (A). We see therefore that Rj may be written in the form: R,- ea(Oa Y Where 0j(y) is an eigenfunction of (A) corresponding to the eigenvalue \j. To'find the form of the function, e w[l), we apply a similar line of reasoning to the determination of the residues of the function H(,7,N ) which is the Green's function of the adjoint system (A'). The residues of H(,); ) at A = %j, must be of the form t( ([^)C. Applying the fact that H(i ^, ) = G($ )'X ) we obtain the result that e(S) must be of the form b. X(a) and mc( u) must be of the form b I (4) where bj is a constant. Hence we have: IRaf} b )X.() (2.5.4) The explicit value of bj can be determined as follows: Consider the inhomogeneous system of equations corresponding to (A) and write it in the form: [ (o D;;X+XfDi;)' ^ + (yD)]g += te),-=DP&'-I~~~ )~ - o (2.5.11) (Y, r) s -aRiltUr od-m "3 where r(y) is arbitrary. From the definition of the Green's function

36 the solution to this system of equations can be wirtten as: )', J- S (I,),.(rJ)d (2.3.12) The function (y) satisfies the equations: 3C C( DL + x(D-d ) t (., D )} ( =CX-XjD)-d ),(Y ) Hence making use of (2.3.12),.(Y) can be written in the form: Q,(a) = (t-; *'% J lA)a u\^D-~< )^(t i) d^ (2.3.14) Taking the limit of (2.3.14) as X- -.. and noting that l^ pec (X-^-a &e(W?,) = R(,$) = bgX^^)^) we obtain the following: b "- -... -i. (g g -;)(i $ c f (2. 53.15) From this equation one may conclude that the integral cannot be zero, and hence can be put equal to one by the proper normalization of X. and?4. The residue of G will then be %($)(Y) and therefore the residue of g(w l) A ) will be (D-o) Z()'Q( at \ = kj, which was the property used to derive Eq. (2.3.1). We now turn to the evaluation of the integral (2.3.2), for which the explicit forms of the solutions Cf,() for large X are needed. We have already in Chapter I made use of such forms in order to show that

57 there are an infinite number of eigenvalues. For convenience of reference we rewrite the following expressions for the solutions to the differential equation of (A) and their first three derivatives. (2.5.16) DS - Ci ea-s, ine ^Yf" series of the form (,,/+ where a al. a 2, etc., are bounded in y on- i ~ _ 1 and in k as jk/ approaches infinity in the sector O/ A K 4, K -_V1 ~ We shall make use of the above expressions for the functions fi(y) for the explicit evaluation of (2..2) to be carried out in the next section. 4. TE MAIN THEOPREM Having completed the discussion of these preliminary results we shall now prove the following theorem. Let f(y) be a function which possesses a continuous second derivative on-.a_ tLL %. Furthermore let f(l) = f(-1) = o. f(l) = f(-l) = 0.

58 Under these conditions, zi^ S' (J,)X(s), ~ \f.- f(' & (2.4.1) for-1 L y L i ~ Proof: From the definition (2.5.9) it is clear that 7 (i,)() is a linear combination of the functions fi(y)o We will write: 7( yS,)^ =~ + (Y) hj() (2.4.2) where the plus sign applies when > l and the minus sign when 4 & One can write therefore. f~( | t -c) p( 4 () -~ (1 iV) G= 1- i(-i) ^ ^-i) v-^-~^ ^ (2.4-3) ^ Dfi) D() DV(in b +K) -+ Dk(-td DR(-1) DR(-1} DR(-i)-Z Df(-l^h(t) Since G contains the variable only in the last column of this determinant, the effect of applying the operator (;-K ) in order to construct g(m J X ) is the same as applying it to the last column alone on hi( o). Consider the denominator of 7 ( y ) in (25..9). This becomes when k is large enough: -i k3f-0tO uIz Lo^^(S 41 r 3e~t)rz+1)L L-~-e 1^ 2, | ii" e- C" e~(t ) ud e^(qt^l5 CL^ _te-o') | In e_ [i (Sztl) t ] ke19(5W L e4't+')l [c e- t''

59 The value of this determinant is -8 (C (' K-( K-^ ( >tWo -8 ock S(2.4. 4) Now consider the cofactors of f (y) in the numerator of 7 (, ) ) Using again (2.3.16) one obtains for the numerator: t e e dak' tl 4z- t~e e mS )Ol (2.4.5) +.K(1 4) Le_ Ol - J( -o e )'[ Therefore: ^hls =- L K3 4 h = Kh3( Le-^*"'J 4^ 1^ h; 0=(2.4.6) Applying the operator 7.-.i we obtain: (D;-^); v^g) = - 4 K t1n; ( a)^ = ) hK 3 For the purpose of evaluating: ZITr L ^ ys)( (2.4.8) it is convenient to consider separately the two intervals -1 L - and ~ L f a 1.* First consider the interval1 L ~ C 4. The plus sign applies *In what follows we need only consider y to be an interior point of the interval. This is because we have demanded that f(y) = 0 at y = +1. Since it follows from the boundary conditions in G that (+4, )- O it is obvious that (2o4.1) is true at y = +1o

4o in (2.4.5). In order that all the elements of the determinant in the numerator of g(', ) X ) be bounded for large k when the real part of k is less than or equal to zero (this corresponds to the condition Y/>k ik 1,o we multiply the first column by minus c(iD;- hiC and the second column by plus ( - ^ ) hC($ and add these columns to the last column. In addition we factor out of the first column of the numerator and denominator the factor P, and out of the last two rows of the numerator and denominator the factor k. The resulting expression for g( J.1 ) can then be written in the form: ~_.,,.,. __ 2'LK whe re.I] [11 a21 2L 1(k(~S')_\ 3 (.204.1o) KKle and whr ~h \e cK o\fo and where the ai are functions of y and k which are regular for-l.-LyL and remain bounded for Ik| — oo. By our restriction on the contours rQ,9,(K) will be bounded and will never be zero on Ir, and all ex

41 ponentials (always for -1 L'j, -1 l I1. ) will be either small or unimodular on n. We now show that the only important term in (2.4.9) arises from the first element in the last column. For the proof we shall make repeated use of the following: Lemma: For Irl/ 6_ C~ t' 3 / and for X 7 0 if A(k) is bounded on the circles. The proof of this is straightforward and will be omitted. Let us now expand the numerator of (2.4.9) in terms of the minors of the last column, and consider first the contribution, say, of the third term to the integral (2.4.8). Clearly for i- < the contribution of the term proportional to (/t3 will go to zero because it is multiplied by a bounded function in k and the range of integration (the circle P ) is proportional to k2. Also the contribution of the term, proportional to, will go to zero if f(l) = f(-l) = 0. This is seen as follows: The term has the form: x, 5 i" at[ BCY)e^R(,)FC 4 Xi -1 A K t where B is bounded in k on. Integrating by parts on the f variable twice one obtains: S/he ded 04aaz IAmt us~ -#e J(e te/ms ) (2.4.11) The term O(iK4 hnas a zero limit just like the terms ). If f(-l) = O the second term in (2.4.11) is zero. Since y is an interior

42 point* the limit of the first term in (2.4.11) is zero according to the lemma. By a completely similar argument one shows that the contributions of the second, fourth, and fifth terms in the development of the determinant (2.4.9) in the minors of the last column vanish for ->o. There remains the contribution of the first term, which is: t <~ 2 (-t t1+e + o(s> (2.4.12) where f(y_) denotes the limit of f( ) as -_ y from below. Turning now to the calculation of (2.4.8) for the interval "LtZi, now the minus sign applies in (2.4.5), so that if one makes the same transformation of the determinant in g(' ~ )'' ) as before one will again obtain exponentials which are small or uniniodular on r. Assuming again f(l) = f(-l) = 0, one can then prove in exactly the same way as before that only the first term in the development of the determinant in the numerator of g(%,,$ X ) according to the minors of the last column will give a contribution in the limit as -. This contribution is: (k _1 ) (XK (2.4.13) This completes the proof of the main theorem. An interesting point about this proof is that in order to ensure convergence at an interior point of the interval we have had to assume that the function *See footnote on p. 39.

43 f(y) satisfies the boundary conditions at y = +1. In this our result differs from that of Birkhoff. In the case treated by Birkhoff, as in the case of ordinary Fourier series, convergence of the series to f(y) at an interior point of the interval is not influenced by whether or not the function f(y) satisfies the boundary conditions. This property is a peculiarity of series expansion in which the scalar product involves a differential operator. We shall give a simple example of this in Appendix IV. For the case when f(y) does not vanish at either or both of the boundaries y = +1, it is not difficult to prove the following: S nr4 o((-1) a-)a _ r s( nA(pendix (2.4.14) A proof of (2 4) appears in Appendix IV. A proof of (2.4.14) appears in Appendix IV.

CHAPTER III EXPANSION THEOREM FOR THE FLOW THROUGH A CIRCULAR PIPE 1. INTRODUCTION We shall, in this chapter, consider the possibility of the expansion of arbitraryfunctions f(r) in terms of the eigenfunctions 0j, of the following boundary value problem: [ (L-) -- (l-t-t)L")} (r) o (3.1.1) (B) 4(I)=o Odor f(1) =~ (5.1.2) CA h ~-y 0o __ remains finite r where L -L h - Ii and 0 and R are real non-zero numders. In what follows we shall use the letter notation (B) to stand for the entire system of equations consisting of both (3.1.1) and (3.1.2). This boundary value problem, as we have seen in Chapter I, arises in connection with the stability problem for axially symmetric disturbances superimposed upon the steady state viscous flow through a circular pipe. In Chapter I we showed that this boundary value problem has an infinite number of discrete eigenvalues, cj, j = 1,2,..., and an infinite number of eigenfunctions, 0j, corresponding to these eigenvalues. The following discussion of the expansion of an arbitrary function in terms of the functions Xj will be analogous to the corresponding 44

45 discussion in Chapter II. For this reason we shall only sketch in some of the details. 2. THE ADJOINT BOUNDARY VALUE PROBLEM We introduce the boundary value problem adjoint to (B) which we shall designate as (B'). { (L-p)t-F3R (L-(a)(i0-WC)^ ^ XCb=0 (3.2.1) (B') tt - A O t }= o ~ (3.2.2) lim __ remains finite The set of eigenvalues, c-j, to (B') can be shown to be identical with the set for (B). Furthermore the eigenfunctions G% of (B') and the functions 0j satisfy the following orthogonality relationships:* 3 Jt!'I Aft(L-8X) XQcv) dr - tt72(t)(L-k)'~), o8 (3.2.3) o 0 The derivation of these results is similar to the derivation of the corresponding results in Chapter II and for this reason will be omitted here. We shall, in what follows, assume that the functions and 0j are normalized so that 0 As in Chapter II, it will follow from the proof of the expansion *We shall assume here for the sake of simplicity that the eigenValuca are all simple roots of the characteristic equation and are therefore non-degenerate, each corresponding to a single eigenfunction. When this is not the case some modifications are necessary, however the nature of these modifications will be clear from what follows.

46 theorem that this integral is not zero, so that the normalization will always be possible. If we consider the expansion of an arbitrary function f(r) in terms of the functions 0j, then, applying (3.2.3), we see that a necessary condition on aj is: Ad = S rim (L-pe antre Let us use the notation: (9,)n- 5rf(Lo- ^)q ro We shall in the remaining sections of this chapter investigate the validity of the expansion: +f - 2- ( dj)n AT (I) (3.2.4) 3. EXPLANATION OF THE:.METHOD OF INVESTIGATION To investigate the validity of (5.2.4), we shall construct a function g(,( )X ), where X t. (,ri,, which as a function of \ possesses simple poles at the points j - i R C, the residues of g( f, X ) at these poles being 4 (L -P ) "-d )^ A ) If we multiply g(( ))X ) by fj and integrate over the interval 0 8 4L1 in the. plane and over a large circle 1 in the x plane which does not go through any of the singularities \j, we obtain: A I S i r,^ W ^f, t d _ Z.,t) (') (3.35.1) p aL

47 The sum on the right is extended over all the residues contained in the circle f. If we construct a sequence of circles f7 of radii RL which do not pass through any of the points \j and which are such that RB approaches infinity as 1 approaches infinity then we have:, apprache in int!a y? as,'3.3.y2s ^-t-^, ^(1~-, 4, xl Fct~~ $,5 d''~ z V(3.3.2) The process of examining the validity of (3.2.3) is converted by (5.3.2) to the process of seeing whether the integral on the left in (5.3.2) approaches f(r) as, approaches infinity. The advantage of this conversion lies again in the fact that the solutions to Eq. (3.1.1) for large values of \ are of a simple nature. The desired function g(, 74 ) may be constructed in a similar way as the corresponding function in Chapter II. Put: cult,; I F(L eerl) @(rcn (5.5.5) where G(\r,$, ) is the Green's function to (B) defined as follows: (i) G as a function of r satisfies the differential equation (3.1.1) for all points r on o _ T -A\ except at r =; (ii) G and its first two r-derivatives are continuous at r = $; (iii) )r 6i Aei}X)t > t - 3r &( a,,)X b s_ \/4 (iv) G satisfies the boundary conditions (3.1.2). These conditions uniquely define the function G(r) 4,/ ). Let fl' f2, f3, and f4 be four linearly independent solutions of (3.1.1). Furthermore let us designate as fl the "inviscid" solution of (3.1.1) which is regular at r = O. Let f2 be the regular "viscous" solution, Let f3 be the irregular "inviscid" solution, and fl, the irregular "vis

48 cous" solution. Then since G must satisfy the regularity conditions at r = O, we must have: G(ra,>%= -- c'- +S42jr, ro (353.4) where < and n do not depend upon r but may depend upon t. For r 7 1? G is a linear combination of all four solutions, G (t, fi, ) = S3 Fr +6 t + r3(r) + (5.5.5) where the functions,..., so may depend upon 4 but not upon r. The conditions (ii), (iii) and (iv) imply the following set of six equations in the six coefficients ^, i = 1,2,34,5, and 6. i ^ -to, 4, o~),~ 6 —i &Z'-i The solution of this set of equations leads to the following expressions for the functions 6(i i = 1,., = 6. ^, (F,+; id ('an -f, f ) 2 = i )S (3..76) so^ _ (> )( F.wcr (ftF) (F )\.

49 ignate the Wronskian determinant ofs the functions contained within the Using these expressions for i = 12..., 6 the explicit form for g(r)$)X ) may be written down: (~f). =.. ( -f7if4(-L, (3. 3.7) 3' > (f\lf^f5)(a) In these expressions we have used the symbol (f, f, * ~ ^-\ f,\7 to designate the Wronskian determinant of the functions contained within the brackets evaluated at the point S, i.e.,; "* * 2 ) | l' /^c - (,-~ e \('" ( ( b t(,A$ t 0) U L< > $ 3*3

50 To see that g( r) \ ) constructed in this way does have the required residues (4L-g2) as i cr,) at kj = l R a, we note that since the Wronskian (flf2f3f4)(4) of four linearly independent solutions of (351.1) has no zeros as a function of X, the function G( T) ) X ) has poles at the points \j = C R' where (flf2)() = ~. We saw in Chapter I that this equation determined the eigenvalues of (B). Therefore the poles of G(V, X ) lie at the appropriate points. The residues at these poles, assuming that they are simple,* must b f te of the form,(. To see this we note from (53.57) that S and 0) are regular at T = Xj. The residue of G(r) 4,) X) at \ =.j contains therefore only the regular functions fl(r) f2(r). Furthermore one sees from (3.3.7) that at X = j\ the residues of al and A3, and of A) and 4 are equal to each other so that the residue of G(f), X ) and all of its derivatives with respect to r are continuous at r = ~. Furthermore the residue satisfies the boundary conditions at r = 1. We can see this directly from expressions (3.3.7) if we make use of the condition (flf2)(1) = 0 at \ = Xj. It can also be seen from more general principles that the boundary conditions must be satisfied. Since G(ti0, ) = 0 and X G(+t1$, ) = 0 for all \, it follows that each coefficient in the expansion of these functions about the point X = Xj in powers of \ - Xj must also be zero. Since the residue satisfies the differential equation (35.1.1) as well as *Some of the complications which may arise when the poles are not simple are treated in Appendix III.

51 the boundary conditions (3.1.2), it follows that it must be proportional to the eigenfunction Oj(r), and therefore we may write it as ($)d (r). By making use of the fact that the Green's function, H( )X > ), which is the Green's function for the adjoint problem (B'), is equal to G(,, r)X ) we can show - $ ) = b' ~6(C) where bj is a constant independent of both r and 4. To evaluate bj we write Eq. (3.1.1) for the eigenfunction 0j in the form: { (L-0Y) -6 R(O-t)( -~ ) X(L-L' H ) (t)- (X-\^KL-B ) 4(t) Using the well known properties of the Green's function we have: 8) (oi)) bi - C (,lAx (LY-, )~)IL Taking the limit of this equation as X- X j and making use of the fact that the residue of G( T), ) at. % = Xj is equal to bj(C) (r we obtain the following equation for bj: <t6r)= J ib f X( W L-p )^ (3,,39) I _ or ba = _) = ^ Equation (3.3.9) constitutes proof of the fact that (0jXj) does not vanish and hence that the eigenfunctions 0j and j may be normalized so that (0j, Y j) = 1. The residue of G( r, %,X ) is then [% Fr). We have now shown that the residue of G( r,A ) at its simple poles are of the form X~~)~(Cr). It follows that those of g(rl ) are of the form (L%-p1) ~(& ) ().

52 4. THE MAIN THEOREM Having completed the discussion of these preliminary results we shall now prove the following theorem. Given a function f(r) which possesses a continuous second derivative on O0 X _ 1 and, in addition, f(O) = 0 and f(l) = O, then for each point of the interval 0O _ _1 meto dt ~ 5 aX,$ dr - ber (35.4.1) Proof: To evaluate the integral in (3.4.1) we shall find useful the following expressions for the functions f2 and f4.*'[,)j t \31 L3T (3.4.2) o r' xL1, tr/ ^A_ K T/L^ These expressions were introduced in Chapter I and their validity is discussed in Appendix I and Appendix II. The proof of the theorem at r = 0 and at r = i is trivial. Because of the boundary conditions on G( T) 4) ), the function g( r, I ) is zero at r = 0 and r = 1. Hence since f(O) = 0 and f(l)= 0 the validity of (3.4.1) at these points can be immediately deduced. In what follows, therefore, we shall consider r to be an interior point of the interval (0,1). *The fact that the coefficient of the "small" exponential e x in f2 changes as we pass from one half of the range of integration in k to the other half will not at all affect our results. These would be the same even if the coefficient of elx/~r were any bounded function of k on Tlzz G^ kJ lc 3Tr/2..

53 We shall now develop relationships which will be useful in proving (3.4.1). First we note that the functions fl and f3 satisfy the equation: (L-P13),FI3 = (3.4.3) We shall also make use of the fact that f2 and f4 satisfy a differential equation of the second order. Given any two functions j l(r) ~ 2(r) the differential equation of the second order which they satisfy may be written in the form: 4 1x` - |kI \ L'r~1^ \ due, it ~ (3.4.4) a'^ ^ Iji'z t^\ Using the asymptotic forms for f2 and f4 we obtain the following: | i' f _ i =,C]; l Kl = II4 if L (3.4.5) stittig ( it Substituting (3.4.5) into (5.4.4) we obtain that the differential equation for f2 and f4 may be written as follows: (L+)x) F, = {, aLr 9 }L (3.4.6) where a and b are bounded functions of k as ikl - oO. Throughout we have used the brakcet notation as it was used in Chapter I and Chapter II. By the use of (5.4.3) and (3.4.6) we can greatly simplify the Wronskian expressions which appear in the expressions for the functions

54 ($i). As an example we shall carry out explicitly the calculations of (flf2f3) ( )' (l(%) Lit fC'f) - -S itC fcL 13.4.8 _:\, - t-),, ( -':i3) a(f>f^f~c) t fAc) "3 -(?J (5.4.8) -f- fL I(4B' (:l -ff ^ If in (3.4.8) we use (3.4.3) to express fl/C ) and f3'l(t) as linear combinations of fl (} and fl (, ), and of f3 L() and f3 (7) respectively, we find that all the two by two determinants of (3.4.8) are proportional to the Wronskian (flf3)( ). Upon collecting terms we obtain the following: (Cf1 +XJ3)i-( = (-^ Wt d-B$ (L ) t(f) (3.4.9) All of the relationships which we list below for easy reference may be developed in a similar manner by making use of (3.4.3) and (3.4.6). (fl if,< -(Lest (ik)^(s) 2abF o Uh e (X4eil ( Cf) c = -((| -) A )[- )f _ + ] Anothe set o(fr elationships whiche shal (,.4.10) (fAnthe (se of reiC-nsh - -we shinde l Another set of relationships which we shall find useful are:

55 tS'f~fer$ - 1 s i'i i (f 1 = ('f) (3.4.11) (fzfr)(,0 = mKn\1 These may all be obtained by the use of the asymptotic expressions for f2 and f4 and also of the fact that fl and f3 are Bessel functions. It is convenient for the evaluation of (3.4.1) to break the region of integration into two parts, one part corresponding to Z _ r, and the other to;? r. Consider first:.,k w I $H t(r,,} (3.4.12) The expression for g( Y X,\ ) which is appropriate to this range is given by the first equation of (3 3.8). To see which are the significant terms in (3.4.12), it is useful to remember that functions of the form ekx where x 7 0 will be small except near the end points of the range of integration in the k variable. On the other hand, the function, e kx, where x ) 0 will be large except near the end points of the k range. Thus functions of the form (flf2)(1) contain the "large" exponential e-k, and the function (flf4)(1) contains the "small" exponential ekx On the other hand the function (f2f4)(1) has neither a "large" nor a "small" exponential as these cancel one another in it,

56 If addition it will be useful to remember that the functions f1 and f3 do not contain k at all and that they are anihilated by the operator L - 2. By use of these facts we can show that all the terms in g(rI )X ) which contain the Wronskians (f1f2f4)Q() ) and (f3f2f4)(/) are of order 1/k3 and will not therefore contribute to (3.4.12) in the limit of,- o0. Consider a typical term of this form: ( Ls - #X)(l () i t3;,4(sbYl^ (3.4.13) By (3.4.10) and (3.4.11) we obtain: #( L T ) oCS Q(K43\) (3.4.14) Now let us consider the term: + (L - 0 )(\_^\ (fw (,, f, ff3, \t ) i'L (Fll(fn( FrfAj )Js (3*.415) By the use of (3.4.10) and (3.4.11) we obtain that (3.4.15) may be written in the form: -T\4F&)^ _ A(~ e^~ ~(35.4.16) where A is bounded as \~k\ - O on r. Multiplying by f( ) arid integration over r L_ 1, and over the circle A,, we have upon integrating by parts on the, variable a term of the form: 8(K;r) (F(', E (,r) e O (, L " n" 1<

57 where both B and E are bounded as Ikl| - O. The first term in the brackets vanishes by virtue of the boundary condition f(l) = 0. The second term can be shown to go to zero as A —> o> by application of the lemma stated in Chapter II. The quantity l-r is positive since r is an interior point of the interval (0,1). Hence the lemma applies here. Hence all the terms in (3.4.17) will go to zero as -Q 0. We can also show that the following term in g(r, iX ) will integrate to zero. SQ(L>v- ) _L_(_, ff)(, 3?)^) ( K This term is of the form: AV, ) et(24i) where A(k, ) is bounded in k on P. Upon integration by parts on the f variable it can be shown to be of the form which will integrate to zero by virtue of the lemma. Thus we are left only with the term: - ^ - (Lr- n y F,@;QI { SA j*')l (3.4.18) By making use of (3.4.10) and (3.4.11) this term may be expressed in the form: >,2 eas ( e(i-A) (3.4.19) Upon multiplying by f(f ) and integrating over r - -. and over n we obtain:

58 Y'f%^ O 2 t)51'i(e'i! -e-L I)/ Kd (3.4.20) Upon integration by parts on the i variable we obtain: Qr,'r1~~~~~~ (5.4.21) t ->,a t/sni + i-Q~e)+ (\<, where B is bounded as 1kl — > and t-V~' O. Applying the lemma we obtain: oL^t+i.^A n l/gWL 2. 9(t7i,) V^a) = for (35.4.22) This completes one half of the main result. We turn now to a consideration of the integral r __t %rn F yX z)i%) ct$ 68) (3.4.23) For the range of integration of 0 - C ~, the second equation of (3.3.8) applies. Again due to the fact fl and f3 satisfy (3.4.3), we can show that all terms in g( r, 4)\ ) which involve the Wronskians (flf2f4)( ) and (f3f2f4)( ) are of order 1/k3 and therefore will not contribute in the limit as lkl — oo. Furthermore the terms: C L - n i(f~f4_ic _ C__F_) 4L \t) V^fF\, (:,^W^,1) 1s

59 and and > ( Lt- f2)(\ (^^ _Ulf t1 %LA) (4,)\ 4 can be shown upon integration once by parts on the < variable to be of the form which will go to zero as o- oe by virtue of the lemma. We are therefore finally left only with the term: ^t -^ (Ljy- go ) ^ (cuff jee%_ (3.4.24) By the use of (3.4.10) and (3.4.11) this may be written in the form: / ( La Lk- f ) a (3.4.25) 4 t thform ). We cannot now, with rigour, use the asymptotic form (3.4.2) for f (2) as we are dealing with 4 on the range 0 4 r. However we may use on this range for f2( ) the series, developed in Appendix II, of Bessel functions. Application of the operator (La- )'- to each of these yields a term, the dominant member of which is proportional to k2 times the corresponding Bessel function. Now upon integration by parts on the variable the term at the lower limit will be of the form /() where A(k) is bounded on. This lower limit term will integrate to zero by the lemma. The term due to the upper limit can be evaluated by the use of the asymptotic expression for f2( ). The result turns out to be exactly the same as if we had used the asymptotic formula (3.4.2) for f.( A) throughout the range. With this justification in mind we shall use the asymptotic formula for f2(P) as if it were valid in the entire range o L _ rf. Expression (3.4.25) then becomes:

6o (3.4.26) 2. K Multiplying by /,Ti f( ) and integrating over 0L i and over r, we get upon integration by parts on the ~ variable: (3.4.27) fhn r This completes the second half of the proof of the main theorem. We should note that in this part of the proof we had to assume only that the function f( ) was regular at r = 0 in order to ensure the convergence of the integrals involved and of the validity of the integration by parts process. We have not applied the condition f(Q) = 0 except to ensure the validity of the theorem at r = 0. For the application of the theorem at an interior point the regularity condition at the origin is enough. At r = 1, we saw in the first half of the proof that it was necessary to require the condition f(l) = 0,* in order to have (3.4.1) be true at an interior point. The situation is analogous to the situation we found in Chapter II where it was found necessary that the function f(y)vanish at the boundary points in order to ensure the validity of its series expansion at an interior point. For a function f(r) which does not necessarily satisfy the condition f(l) = 0 but is regular at r = 0 and is twice differentiable on O I-'_, we can write the following result which is analogous to *See Eq. (5.4.17).

(2.4.14) of Chapter II T ~ TrtdXC-)py-) (354.27) o.t.i The proof of this result is analogous to the proof given in Appendix IV of Eq. (2.4.14).

CHAPTER IV APPLICATIONS OF THE EXPANSION THEOREMS 1. INTRODUCTION In this chapter we shall consider some applications of the expansion theorems which were proved in Chapters II and III. For the sake of brevity only problems relating to the flow between parallel plates will be discussed in some detail, since the analogous problems for the axially symmetric flow through a circular pipe can be treated along completely similar lines. Specifically we shall consider the following problems: a) The initial value problem; given the disturbance of the flow at t = 0, how will it develop in the course of time? b) The forced oscillation problem; what will be the effect on the flow of an outside force which varies in time like e? c) The non-linear problem; in which way do the various characteristic modes of a disturbance interact when the non-linear terms in the hydrodynamical equations are taken into account? We shall always consider first the case when the zeros of the characteristic equation (1.2.10) are simple. It is likely that this is actually the case, although it seems difficult to prove. We shall there62

65 fore also consider briefly the modifications which are necessary when some eigenvalues are not simple. 2. THE INITIAL VALUE PROBLEM The Fourier transform ^ (3,t) -^ s e - ~x of the stream function ( (x, y, t) of a two dimensional disturbance of a plane parallel flow fulfills the equation: ((D=-y); see Eq. (2.1.2)) i ( )Fi Rt ( U 2-) -Iu")1 P= D-d ) a (4.2.1) D)t where U(y) is the basic flow. In addition ca must fulfill the boundary conditions: Zo (, t) =D ( ( t )=o Let f(x,y) be the initial value of JQ(x,y,t) and let fc(y) be the Fourier transform of f(x,y). It also has to fulfill the boundary conditions fj(tl)= DfC((+l) = 0. Assuming for the present that all the roots of Eq. (1.2.1) are simple, one can therefore according to our expansion theorem, expand fc((y) in the eigenfunctions 0 ce(Y) of problem (A) of Chapter II: fott^ = Q Qp^Q^}1) and for the expansion coefficients ape one obtains: art = (I,fa = f f>(az(4 825 (a(l _ | i fJ.^7 (4.2.2) It is clear therefore, that the solution of (42.1) which for t = reIt is clear therefore, that the solution of (4.2.1) which for t = 0 re

64 duces to fO(y) is given by: <Llr~ - ~~~ 21 %"dJ) (4.2.3) where a, is given by (4.2.2). The stream function'(x,y,t) for t ) 0 is then given by: 0O At (X g t)- = d0 H dX D (Y t Xt) It is interesting to note the changes which must be made in these formulae in the case of the eigenvalues X, = iOCFR are not simple. From the development in Chapter II, it follows that in general:. JL _\t/R W -^ ^ ^ ^ ^ =;e^,, dfrL'4 ~, c, -1 ) (4.2.4) Suppose now that there is a point x = 2j which is a non-simple zero of the characteristic determinant (2.3.7) which determines the poles of the integrand in (4.2.4). As shown in Appendix III only one of -the following two cases can occur: (a) the order of the zero is equal to the number of linearly independent eigenfunctions corresponding to A = Xj, (b) the order of the zero exceeds the number of eigenfunctions. Both these cases are discussed in Appendix III. The main results derived there are as follows. For case (a) the Green's function has only a simple pole with a residue which may be written as i i^'v^ where to are the k-fold degenerate eigenfunctions of the adjoint problem (A') corresponding to % = Xj and 0 are the k-fold degenerate eigenfunc

65 tions of system (A). They obey orthonormality relationships of the form: (I- k q, (4.2.5) The contribution to a)/(yt) due to this residue will be R,, L( cv.,, f^ CeDs\'YL -Al Ija, t h'-I g o (4.2. 6) This is an obvious generalization of (4.2.5) which is to be expected since the case of a simple zero is a special case of case (a). Case (b) although perhaps not a probable case for these hydrodynamical problems is mathematically more interesting. For this case the Green's function possesses a pole of an order which exceeds unity and which can be at most m - k + 1 where m is order of the zero of the characteristic determinant and k is the number of eigenfunctions.. The residue of the product A)rt G(8Cs) at x = A: will be of the form of a polynomial in the variable t i*~ Ot. Ifc times the exponential e. It is interesting to discuss in detail the case m = 2, k = 1. For this case in the neighborhood of X = Xj it is shown in Appendix III that the Green's function may be expanded as follows: G = 40w) r+' + ( )c Xadi b X, };;) 0~a\ t L>-^^>) (4.2.7) + 1-E&,+X) where 0 jz and a ja are eigenfunctions of (A) and (A') respectively. E(y,,\) is analytic at X = \j, and the functions a3 %jCa and` t jac are defined by:

66 ax ~- -,N~iAu x _( ibx )_a (4.2.8) Finally b is a fixed constant which is determined by expanding the Green's function about the point X = Aj in the manner shown in Appendix III. As shown in Appendix III the function i %Oja must be added to the OCe's in order to complete the set in this case. As a result of Eq. (4.2.7) we can write the contribution Rj, to j (y,t) from the pole at x = %j as follows: Rs* =- lt - i t>> ) X A(t) C-(K +, +jl ~ (y. ( arX*^tlF,_ fit 9(4.2.9 Let us check to see if this function Rjc((y,t) satisfies the differential equation. First we note that by differentiating the Orr-Sommerfeld equation with respect to X, we find that;%kjcj satisfies the equation: [ (D0^,){-ioR (U0(IX) -U")-X(P -D c^ - (4.2.10) Now substituting (4.2.9) into (4.2.1) and making use of (4.2.10) we see that Rjc(y,t) does indeed satisfy (4.2.1). We should remark here that while a degeneracy such as the one we have discussed above may occur at a particular value of Ca ao let us say, at adjacent values of a this double pole will be resolved onto two simple poles. The sum of the residues of these two simple poles will approach (4.2.7) in the limit as ot —>o. The main result of this section has been to show that the stability or instability of any initial disturbance satisfying suitable boundary

67 conditions at y = + 1 and at x\ = ~ can be characterized by the eigenvalues ~ot of problem (A) thereby justifying the historical approach to the problem. The single possible exception which can occur is when Im. cj = 0 and cj satisfies the condition of case (b). Because of the term proportional to t in (4.2.9) such a mode should not be considered to satisfy the condition of neutral equilibrium, but should be considered unstable. However no examples of the conditions of case (b) are known. 3. THE SOLUTION TO THE FORCED PROBLEM In this section we shall consider a small outside force F(x,y,t) which is "causing" the disturbance to a steady state flow characterized by the velocity profile U(y). Only the curl of F will affect the development of the disturbance. That portion of the force field which is derivable from a scalar potential will serve only to alter the pressure distribution. Let f(x,y,t) be the z component of the curl of the force field, then the disturbance stream function will satisfy the equation: [ xS-^ (< -2 X t 0 ( )- } - +,yt) (4..1). We shall be especially interested in the response of the flow to an alternating outside force of fixed frequency.j, and therefore we set: - (X t) = fe (L, 2 Let us assume that all the \'s are simple zeros of the characteristic equation so that for each a - the functions 0 aQ form a complete set. Let us write:

68 rl)l^)- r %e ),t) (43.2) Substituting Eq. (4.3.2) into (4.3.1) we obtain: <20 ai d, eats C( ib)^ (D~j-d) ) cL.d =d(,) (4.5.3) If we let foC(Y ) be the Fourier transform of f) (x,y) then (4.3.3) becomes: C L (~L + ( ~C*^X ) /() f u () By making use of the orthogonality relationships we obtain: ao _ - (4.3.4) As is to be expected, in the forced oscillation each of the characteristic modes will get excited to an amount which depends mainly on the resonance denominator CO)t+- + TOo. It is again of interest to see how this has to be modified if one has degenerate eigenvalues. Let us assume for example that at C = 0o we have an eigenvalue X = X: which has a degeneracy of the type of case (b) with m = 2 and k = 1. Let t1o (y,t) be the Fourier transform of (,(x,y,t) at a = %a. Then if we substitute for Vao (y,t) a series of the form Je.. / Qai., Go( + a 3t (4.3.5) v K-^ "~~~~~~~~~~

69 we have making use of (4.2.10) and of the properties of the functions. If we make use of the following orthogonality relationships given in Appendix III ex a ( %ti6 > Qo05 e i t tit^t < (XJoU^^Or r Hi (c a%$>br ) ~) ); (+X^ o^/A^ ck (4.3.7) ( "X udd +T ^i o.A~o O -O we obtain for the non-degenerate terms expressions which are identical with (4.3.4). In addition we obtain the following expressions for a1 and a2.. Q. I d,r) ii^;) o 5 7iI*t?(' d~~~g + 4 h( 1*.10 )Xlao4 40X< a7 ( a) UL> i~4 B v ( ) ti ot h o>) 4. THE NON-LINEAR PROBLEM As a final application we shall consider briefly the expansion of the solution to the non-linearized stability problem in terms of the eigenfunctions of the linearized problem. Again we shall assume that the zeros of the characteristic equation are simple. The exact non-linear equation which the stream function b(x,y,t) for a two dimensional disturbance must satisfy is' A- (as at ( t0 = (^at (t4.41l)

70 We shall assume the disturbance periodic in the x direction with a periodicity of 2ic/a. For this case we may write Y & u) )e(4.4.2) where 0kn is the kth eigenfunction of system (A) corresponding to the value a -- nc. The sum over n goes'from n = - O to n = + 0. We shall for each value of n label the modes in k proceeding from the least stable according to the linearized theory to the most stable, k = 1 corresponding to the least stable. Let ~'(LQ.1tL/Cr. Substituting Eq. (4.4.2) into Eq. (4.4.1) and multiplying by X tW )e and integrating over the area 0 x x 2jT/c,-l e y e 1 we obtain the following set of equations for the coefficients On' (L'+ ^W1^ G. - t I B,' At1, ^ (4.435) where the coefficients e,,, are given by: | rK^( irvn, C^^ ^ (1. i') Diwl,- % \) (4.4.4) where m = n - ml, and ( rNm;l = 0 for m z n - m The equations (4.4.3) show that if initially say only one of the modes 0kn was present after a while also all the other modes will become excited. The non-linear terms of (4.4.1) produce an interaction between the modes. The results of this interaction are in general so complex that

71 it is difficult to make predictions as to the form of the disturbance after a finite length of time. One of the many possibilities that may take place is that the excited modes may approach after a. time a new steady state equilibrium. Such a state is called a secondary flow. We shall illustrate this possibility by truncating the system above and considering only the interaction between the modes corresponding to the coefficients a1O, a_11 and all. The coefficient a10 corresponds to a mode for which a = O. Such modes were discussed in detail in Chapter I. They are always stable. The coefficients all and a,_1 correspond to modes which are the complex conjugates of one another (see Eq. (1.2.29)). For a real disturbance all = al. We shall take a10 to correspond to the least stable odd mode for n = 0 and we shall take all and al_1 to correspond to even modes so that the constants P corresponding to the interaction of these modes do not vanish. We obtain the following equations: O + o({ I\ ID 0\ ago ri, +Kr,1 a- = t'O0lo (4.4.5) dLo t ITo, _o 3 \=, \^ where \ 5- L j >o,u1'n t >,o', t\) ^t;. t~l? o'l-l X \0-U By multiplying the first equation by al-1 and the second equation by all and adding and making use of the fact that wll = - G -l we obtain the following equations: bs,=-,bi+ bs,; b6-R B'K (4.4.6)^

72 where b-(^t^ 3 b e (Ito V Lo For any K1 - K2 pair we can by means of Eq. (4.4.6) for any initial values of b1 and b2 trace the system in time and we can see for which initial values the system tends to go to the origin of the b1 - b2 plane (stability) and for which initial values it does not go to the origin. We should note that since the quantity |a1L12 must be positive only one or the other of the two half planes b1 L 0 and b1 7 0 is permissible physically. Just which half plane is permitted depends upon the sign of the quantity (A\t)3 If we set bl = b2 = 0 in (4.4.6) we can obtain the point in the b! - b2 plane corresponding to a secondary flow assuming this point lies in the permissible half plane. We have: b)L= - At; b, -' -9; (4.4.7) We can go further and investigate the stability of this secondary flow by assuming that b1 and b2 are close to the values given in (4.4.7) and then seeing whether or not the disturbance tends to move away from the point (4.4.7) or not. All these remarks are meant to be merely suggestive as the truncation of the system (4.4.3) to (4.4.6) will be valid only so long as the modes all, al_1 and a10 are the only ones which are appreciably exicted. As a conclusion to this section we remark that the utility of expanding the solution to the non-linear problem in terms of the eigenfunctions to the linearized problem depends largely upon the ease with which the coefficients f defined by (4.4.4) may be computed as well as

73 the sensitivity of the system (4.4.3) to accuracy in these coefficients. It is quite possible that the behavior of the system will be qualitatively the same for large ranges of the coefficients P so that rough evaluations of these constants suffice to give us the qualitative behavior of the system. For example in the two mode case discussed above knowledge of the sign of the quantities ( ~lt ) and Y3 tells us a great deal about the behavior of the system.

CHAPTER V APPROXIMATE LOCATION OF EIGENVALUES FOR LARGE CR 1. INTRODUCTION In this chapter we shall derive approximate formulae for some of the eigenvalues of the stability problem for the plane Poiseuille flow for large c \ We shall use a method which is similar to the method used by Heisenberg and which amounts to using as approximate solutions of the OrrSommerfeld equation the first terms in the formal expansions of the form: oo which are similar to those discussed in Chapter I. In contrast to the results in the previous chapters a rigorous justification for the approximate formulae which we shall obtain is difficult to give. Our aim is not so much to give an accurate calculation for any particular eigenvalue but to obtain a qualitative picture of where in the complex c-plane the eigenvalues are to be found for large aR. 2. APPROXIMATE SOLUTIONS TO THE ORR-SOMMERFELD EQUATION FOR LARGE aR If we substitute in the Orr-Sommerfeld equation (1.2.7) a solution of the form (5.1.1) and equate to zero the coefficients of successive powers of (aR)1/2 we obtain for Q(y) the equation: (Q) -C ( Q)2. (5.2.1) The solutions are Q = C (u-c) /% and Q' = 0 where yo is an arbitrary point in the complex y plane. Corresponding to Q = L-) the leading 74

75 coefficient 6Co(y) must satisfy the equation: 5Q it + Q a Q -O' I = (5.2.2) This yields (O(y) = A/(U-c)5/4 where A is a constant. We have therefore two formal solutions with leading terms which are given as follows: f, = A (-c<V)` 4 - jdP)t J. CU 6-)"o 4 = b (5.2.3) where A and B are arbitrary constants. The functions fl and f2 are rapidly varying functions whenever, as is assumed here, the parameter aR is large. It is clear they are not single valued functions. They possess branch points at those points y = yi where (U-c) = 0 and also become infinite there. These points are often referred to as critical points. We know therefore that these functions cannot represent true solutions to the Orr-Sommerfeld equation in all sectors of the complex y plane and in particular they cannot be good approximations near the critical points. Before going on to consider the nature of the two other formal solutions to (1.2.7) which correspond to Q' = 0 we shall show how to construct out of the two functions f1 and f2 two functions which are "quasi-single valued" near y = yi. The techniques we shall use are equivalent to what is usually called the W.K.B. method. Consider first the properties of the double valued function H(y) = f. R (V-") zA. Emanating from the branchpoint y = yi are three curves along which H(y) is real. When y is near yi we get: H ") 2/(i.i^"^ ^ (y-^)yY;))/ (,.2.4)

76 Hence if we set 1/2 = eir/4 we obtain <xn. He%) =- t/-I S-l % ) 1 2- ) (5.2.5) for y in the vicinity of Yi. For the purposes of illustration we choose U'(yi) to have a real part which is negative and an imaginary part which is relatively small so that we may consider the argument of U'(yi) to be approximately -Trr This is actually typical of some of the cases studied below. For this case we can see from (5.2.5) that in the neighborhood of y = yi the three lines along which H(y) is real will be separated by an angle of 2Tr/3 and that one of these goes off in a direction which is approximately parallel to the negative imaginary axis. We shall label this line S2 snd proceeding counterclockwise we shall label the other two S3 and Si. A schematic diagram of these lines appears below. Bisecting these lines are the lines Pi 3 i = 1,2,3, alongwhichH(y) is purely imaginary. We have also included these in the diagram. PL" For the values of the argument of U'(yi) which are not close to -T we need to rotate the lines Si and Pi through the appropriate angle. The sign of H(y) on the lines Si depends upon the argument chosen for (y - Yi). The function eH(y) will be very large if H(y) is positive and very small if H(y) is negative. We shall label the region between S2 and S3 as I or I' depending upon the argument of (y - yj) there. In I the argu

77 ment of (y - yi) is zero when (y - yi) is real and positive. In I' it will be 2j. We are tacitly assuming here that the argument of U'(yi) is such that the sectors I and I' contain the line through yj which is parallel to the real axis. This is true for all the cases which we shall consider in the next section. The region between S3 and S1 will be labeled II or III depending on whether one enters it from I or I'. Similarly the region between S1 and S2 will be labeled III or III'. The sequence proceeding counterclockwise from I will be I, II, III, I', II', III. We construct a quasi-single valued function of the form ( A-c5,ry A t i I tu dR 2 K/ (} as follows. The numbers A and B are to be constant within any given sector but we allow them to change as we enter a new sector by crossing a line Si. To ensure that the discontinuity thus introduced is small we allow only the coefficient associated with the term which-is exponentially small to jump as we cross the line Si. If we start out in region-:I with a function of the form: f - {L AI e i PI5 } (o-c) q Then upon altering the coefficients according to this rule and upon demanding that upon entering I' the function f be the same as in I, we obtain the result that the necessary alteration of the coefficients A and B is uniquely determined. In the following we shall make frequent use of the following connection formulae which relate AI and BI to AII and BII

78 in II, and to AIII and BIII in III, N = d-V L e,') i- "L ^A (5.2.6) At-As, i Y f The fact that we are able to construct such a quasi-single valued function out of the functions f1 and f2 does not justify their use. One must in addition show that there are two true solutions to the Orr-Sommerfeld (1.2.7) which are asymptotically equal to fl and f2 within each of the three regions I, II, and III, provided that the variable y is bounded away from the point y. By an extension of the work of Trjitzinsky, 14b Wasow has shown this to be the case for an annular region about the point yi with an outer boundary which extends out to the nearest other zero of (U-c). Specifically Wasow has shown that in any two P sectors the solutions to (1.2.7) are represented asymptotically by fl and f2. A P sector is an annular region contained between two adjacent P curves. Since any two such sectors contain one complete S sector we are able to use his results to justify the use of fl and f2 within I, II, and III. For c - 1 for the Poiseuille flow the critical points which are at y = + (l-c)l/2 both approach the origin so that a circle which extends from one critical point only out to the other would not include either of the boundary points y = + 1. However it seems plausible to assume that the functions fl and f2 if continued analytically toward that boundary which is away from the other critical point would remain asymptotic representationsof the true solutions to (1.2.7). We are fortified in this belief by the fact that the results we obtain by using this assumption agree

79 closely with the results obtained by Pekeris9b using a completely different approach to which such objections do not apply. We shall also use fl and f2 (in their quasi-single valued form) in calculating eigenvalues for which one or the other of the boundaries y = + 1 approach the critical point y = Yi. To make computations for such cases Heisenberg5 expressed the solutions to the Orr-Sommerfeld equation around yi in terms of the variable 7=- d). Then by taking the leading terms in 4 3 HR) he obtained four solutions of the form: a, -d ^d;5 &1 ^ 7-, ^ bo,(, 7, 1)2 where o = U'(yi) and H1,3 designate Hankel functions of the first and second kind of order 1/3. By using the asymptotic formulae for the Hankel functions Heisenberg showed that gl and g2 correspond to fl and f2. The connection formulae of (5.2.6) may be derived by using the appropriate asymptotic representations of the single valued functions gl and g2. We have preferred not to make use of the properties of gl and g2 in deriving (5.2.6). We shall make no explicit use of the functions gl and g2 even for calculations where y r- y. By so doing we cannot expect any great accuracy in these calculations. However, results of similar calculations done in connection with other problems indicate that the results we obtain will be qualitatively correct. The solutions to (1.2.7) which correspond to fl and f2 are generally called the viscous integrals. Their rapidly varying nature plays a most

8o important role in the determination of the eigenvalue spectrum when c is not large. We now turn to a discussion of the other two formal solutions to the Orr-Sommerfeld equation which correspond to Q' = 0. We shall call these f3 and f4 and they can be shown to satisfy the second order differential equation: D'- - U'"/(c)? 4 Q= ~ (5.2.8) Equation (5.2.8) is usually called the inviscid equation. When U'I = 0 +cy as is the case for the Couette flow f3 and f4 are the functions eWhen U" i 0 the solutions are more complicated. The differential equation has singular points at the points y = yi and the solutions will be general 14b be multivalued. Wasow4 showed that within any two P sectors the inviscid solutions f3 and f4 represent asymptotically in the limit of large oR true solutions of (1.2.7). Heisenberg expanded the solutions to (5.2.8) in powers of a2 He obtained the convergent expansions: 43AM - 2u- 1C-1, ) d we yisnria poin Of greatest significance touswill, b e(5.2.9) where yo is an arbitrary point. Of greatest significance to us will be the slowly varying character of these solutions when a2 is not large. We shall in what follows assume that it is always possible to choose the branch of f3 and f4 so that they represent true solutions to (1.2.7) at the boundaries of interest to us.

81 3. APPROXIMATE LOCATION OF EIGENVALUES FOR THE POISEUILLE FLOW CASE For the case of the Poiseuille flow it is possible to simplify the eigenvalue equation considerably by considering even and odd eigenfunctions separately. We shall show that the two viscous integrals can be combined to yield an even and odd function and one may also show that it is possible to form an even and an odd function from the inviscid functions (5.2.9). We shall label the corresponding vicsous functions as fe and fo and the inviscid functions as f and f. It is now possible to restrict the variable y to the interval 0 ey l. For the even eigenfunctions we set = Af + Substituting into the boundary conditions 0, Do = 0 at y = 1 we obtain: [^ -- 8j —-- (5.5.1) For odd eigenfunctions 0 = Afv + Bfo we obtain from the boundary conditions: \ i77o / Y- (5.3.2) v v At the outset in constructing fe and fo we are beset with the difficulty that the appropriate expressions for these functions in terms of fl and f2 depends upon the position of the unknown eigenvalue c. We must use the trial and error method. We first assume that c lies within a certain region. Then we calculate fo upon this assumption. Using e, 0 these functions we calculate c from (5.3.1) and (5.532) and then we check to see if c lies in the region assumed. In what follows we shall report

82 only the result of successful trials. For all of these it has turned out that t i is consistent to take y = 0 in III and y = 1 in I. We construct an even viscous solution as follows: Consider The functions e+H(y) may be written in the form: e%5 = eCO) (pt 0.d12^- R ^(ll ii e*.= e- a. (,-C"sV^ at Ct) C,"1 de Hence for y in the neighborhood of the origin we have: H e 14(e~po te.R8I-)CBiid To make fv an even function we apply the condition f'(0) = 0 and we get: J\V = e- e B s where q = 2H(0) (5.3.3) An explicit formula for q is (5.5.4) Making use of the connection formulae (5.2.6) and of (5-3.5) we obtain: Ax - e *' Ar.-Lt Asr Substituting (5.535) into (5.5.1) we obtain after some manipulation: wehre K ~ 2H(1). By a similar line of reasoning we obtain for the odd eigenfunctions the eigenvalue equation:

83 -----— L —----- (5.3.7) t~t ^)t -V~ tb 1 - 5/ Let us first seek solutions to these equations in which c is in the neighborhood of c = 1. The critical point yj approaches the origin and the point y = 1 will lie between P1 and S3 where the quantity K has a real part which is large and positive so that eK is enormous, The right hand side of (5.3.6) and (535.7) are of order unity for c near 1. It follows therefore that the eigenvalues are given to a good approximation by the equations: (e ~ +) O for even modes. (5.3.8) \(( A /~)" Ofor odd modes. Equations (5.5.8) furnish the information that the point y = 0 must lie upon the line P3 where H(y) is negative imaginary. From this fact in conjunction with equation (5.3.8) and (5-3.4) we obtain: 4ymtl -Jl t = 1 -- I+ ot) e for even modes (5.3.9) Ct (^ S for odd modes (5.5310) In these formulae n is any integer which:is sufficiently small so that the assumptions we have made are valid. Hence n should be less than (caR) /2 The eigenvalues (5.3.9) for the even modes were first discovered by Pekeris9b using a different approach. Their existence, however, was denied by Tatsumi. Although Tatsumi's criticisms of other portions of Pekeris's work seem to be well founded his arguments against the existence

84 of the eigenvalues given by (5.3.9) are not valid. Here we reassert the existence of these eigenvalues and introduce as well as those of (5.3510). These latter were not considered by Pekeris. Let us now consider eigenvalues, c, near c = 0. Now the critical point iY,-it. For this reason the use of the exponential functions fl and f2 rather than the functions gl and g2 of (5.2.7) is extremely doubtful. However calculations of a similar nature indicate that we may expect to obtain a picture which is qualitatively correct through the use of the functions f1 and f2. When c — 0, the point y = 0 now lies between P3 and S1 and hence eq is very small. For this case we obtain the equation eY/t R c-V4 t' - e 1 C) - 96 "V) - 5/,( There are three cases to be considered which may be enumerated as follows: 1) The point y = 1 lies between S3 and Pi (eK is very large); 2) The point y = 1 lies betwen Pi and S2 (eK is very small); 3) The point y = 1 lies on Pi (eK is of order unity). In case I we must have that c is near a zero of the denominator of the right hand side of (5.3.11) and we have the equations:

Equation (5.5.12) -defines a single eigenvalue which corresponds to the unstable mode studied by Heisenberg5 and afterwards by Lin. Lin showed how this equation may be used to trace qualitatively the curve of neutral stability which separates that region of the a-R plane for which Im. c " 0 from that for which Im. c 4 O.* Lin also carried out a more accurate tracing of this curve using the functions gl and g2 of (5.2.7). On the other hand there appears to be no solution of (5.3.15) which does not violate the assumptions upon which (5.3.13) is based. K. Considering case 2 when e is very small we obtain the equation: i e~i it <Rt"t t + />e SO(5-3-14) From the slowly varying nature of the functions f in the neighborhood e,o -. of c = 0 we expect that this equation will define at most two eigenvalues one even and one odd. Both of these must be stable as according to our assumption the point y = 1 lies betweeen P1 and S2 and therefore the point yi must have an imaginary part which is greater than zero. This means that Im. c must be less than zero and hence the solutions to (5.3.14), if any exist, must be stable. Finally let us consider case 3 where e is of order unity. Here we obtain a whole family of modes. Due to the complicated nature of the functions fe and fo we can give no simple formula for those members of the family which are closest to c = 0. However if we restrict the integer n *We are considering here a to be greater than zero so that Im. c 7 0 corresponds to instability and Im. c L 0 corresponds to stability.

86 as follows 1iL n L4(aR)1/3 we can write the following formula for the leading term for both the even and odd eigenvalues for the members of the family which are sufficiently removed from c = 0. 14 r (3 MaR R)ll (5.5-15) This finishes our discussion of the approximate positions of eigenvalues for the Poiseuille flow case. There remain large gaps in the picture of the distribution of eigenvalues. We have not examined at all the region intermediate between c = 0 and c = 1. We have also not shown anything of what happens as the eigenvalues grow larger and approach those of the limiting case considered in Chapter I. 4. DISCUSSION To summarize, the results we have obtained indicate that for the Poiseuille flow the eigenvalues may be divided up into three groups. In one group c lies near the lowest value of the main flow velocity and in the second group c lies near the highest value of the main flow velocity and finally the third group is that studied in Chapter I in which tcl -— 0. These results agree with results of the investigations of other stability problems. For example Corcos and Sellars2 in their investigation of the axially symmetric disturbance of the flow through a circular pipe find a set of modes which approach c = 1 and which are given by the formula:

87 c -= I+' (5.4.1).AR)'IIn addition they find there exists a set of modes near c = 0. A more accurate calculation than the one we used to obtain (5.5.15) was carried out by these authors to yield a set of eigenvalues which, while not lying on the ray arg. c = -j/6 come in pairs which are near reflections in this line. They found no mode in this family corresponding to the unstable mode of Lin in the parallel plane case. Corcos and Sellars also investigated the case of the plane Couette flow as did Wasow4 earler. All these investigators found two famillies of modes for which the value of c approaches the velocity of the main flow at one or the other of the two walls. For the case in which one wall is at rest and the other is moving with a speed of 1 the two'families approach c = 0 and c = 1 respectively. In addition for both the circular flow case and the Couette flow case there are the modes for tc\ -- oO which were discussed in Chapter I. If for large aR we were to plot the positions of the eigenvalues in the complex c-plane we see, therefore, that for all the cases discussed here the typical configuration would have the shape of a Y the tips of which approach the main flow velocity at the boundaries and the tail of which trails off to infinity in the stable portion of the complex c-plane.

APPENDIX I THE VALIDITY OF THE ASYMPTOTIC EXPANSIONS In Chapters I, II, and III, we have made use of asymptotic expressions for solutions to the differential equations (2.1.2) and (3.1.1). The justification for the use of these expansions is to be found in Trjitzinsky's work13 which, however, is more general and complicated than is necessary for our purposes. The approach we shall use below is most similar to that used by Birkhoffla for a similar class of problems. We shall begin by discussing the nature of the solutions to the following differential equation of the second order for large jk| {^ 9- Age, 43 =O ~(I.1) where q(y) is analytic on a L y. b. The parameter k will be confined to the sector S of the complex k plane where rt/2 L arg. k L 35t/2 throughout the following discussion. Our purposes in discussing this differential equation are twofold. First it will illustrate in a simple manner the same techniques which we shall use in discussing the fourth order Orr-Sommerfeld equation. Secondly it will be of direct use to us in verifying the validity of the asymptotic forms for the circular flow case. If we substitute into (I.1) formal series such as those used in Chapter I of the form o _ e [) < c 88

89 we find that we may choose Q(y) to be either +(y-a) or -(y-a) and that corresponding to both of these choices r7 may be taken to be 1 so that we obtain two formal series the leading terms of which are ek(y-a) and e k(ya) We shall now show that on the interval a 4 y _ b there exist two solutions to (I.1), fl and f2, which, together with their first derivatives, are given as follows:. e-K (t -"ti + E cl*> F; t )K [}V - -e1 t -1 +^ Ke v a lel E t1 (1I.2) where E(y,k) is used as a generic term for a function which is analytic in y on a 4 y L b and bounded uniformly in k for k in S and for kl 7/ ko where ko is some fixed positive number. From this proof of the validity of the leading terms of the formal seriesit will be clear how we must proceed in order to prove the validity of the series to arbitrarily many terms. We begin by writing (I.1) in the form: (AL >8+)f = W(y4 (I.5) where ({L)E CbQ Q Since we know the solutions of the homogeneous equation corresponding to (1I.), we can, by the method of variation of parameters, write the general solution to (I.5), and hence of (I.1), in the form: ~ eK- je. eK-( ) YIP f - f) ^ (I.4)

90 We shall now seek the special solution fl which is of the form: Fl e — K(b) { + ( e o./To do this we set cl = 1 and c2 = 0 in (I.4). In addition we set The resulting integral equation for Zl(y) may be written as: t- e,K 1+ CA X (1.5) We note that since y >/ the quantity i | - 4i for k in S. Equation (I.5) is of the Volterra type and it can be shown to have a unique solution Z1(y) which is analytic on a _ y L b. Let us designate the maximum value of Zl(y,k)l on a y z b as ZM(k). We shall now show that ZM(k) is uniformly bounded for tk| 7/ ko for k in S where ko is some fixed positive number. Let B be the maximum of q(y) on a' y a b, then if we apply (1.5) at the point yt where |Zl(y,k) assumes its maximum value ZM, we obtain: ^ _______-g&~~~~~ (I.6) We see therefore that for Ik 7/ ko = B lb-oiC + where S 7 0, ZM will be uniformly bounded. Hence we have shown that there exists a solution of (I.1) of the form: -F, = e'lyQc7& { io, (SX ( (I.7) From the fact that Zl and \e \are bounded appropriately we see that this function has the form required in (1.2). By differentiating

91 (I.7) we can establish that the function Df2 also has the required form. Now we turn to the problem of constructing the function f2 of (I.2). To do this one may not simply set cl = 0, c2 = 1 in (I.4) for then the remainder term would be exponentially large for k on S. Instead we set: X KI'- ) Using (I.8), Eq. (1.4) takes the form: c _ /~ (57 GA C (~-c)) GI&U (I.9) If we set f2 = e t,,,.)in (1.9) we obtain the following equation for Z2o ":~l.(.% t2,'\'= ~ 9'.....1 }tj;(Io10) + aKC^),': This may be written in the form of the Fredholm equation: (^)-whe (re.) where ^C< (. 12) lt (D f) - % 8 L y We see that the kernal K(y, ) is continuous on a:L b' a y ~ b, and that K(y, t ) is uniformly bounded for k in S. We may apply the results of the Fredholm theory to show that the inhomogeneous equation (I.11) possesses a unique solution if the corresponding homogeneous equation has no non-trivial solution. We shall show that this is in fact the case if

92 k is sufficiently large because if there were such a solution then its maximum value ZM would have to satisfy the relationship: I M Z rD |t.- 6 C ^ \ -\ from which we obtain i b e -r\~ (I.15) It is clear that for ik\ k 7 B\b-a\ Eq. (1.13) cannot be satisfied, and therefore the homogeneous equation has no non-trivial solution. It then follows that (I.11) has a unique solution Z2(y,k). Let us designate the M maximum value of l Z2(yk)[ on a L y b as Z2. We now show that for k sufficiently large Z2(k) is bounded uniformly for k in S. To do this we apply (I.11) at the point y' where I Z(y,k)| = Zr(k). We get: iL. J:___ (I. 14) - I -Bi X- We have therefore that for k 7/ ko = B ltr-+\ where 0 7 O Z2 is uniformly bounded for k in S. We have therefore shown that there exists a solution f2 to (I.1) which is of the form: ia = e'^ t4+I kir($, l^4~) (I.15) Making use of the bounds on Z2 and K(y,,) we see that this solution is of the form required in (1.2). By differentiating (I.15) we can establish that Df2 also has the required properties. We note that essential to this proof.has been the fact that through

93 out the sector S the real part of k is negative, so that for all y on a y aL b, Re. k(y-a) L Re - k(y-a). It was due to this fact that it was possible to construct the two solutions f1 and f2 with bounded reaminders. We see that as soon as k leaves the sector S the remainder terms in (I.7) and (I.15) become exponeitially large as fkl -- 0. This does not mean that we cannot construct solutions of the form (I.2) in the sector S1 for which -jr/2 ^ arg. k Z it/2. On the contrary, it is obvious from the methods we have used in the proof that on S1 we can also construct two such functions with appropriate remainder terms. The proof would be the same except the roles of the two functions e k(ya) and ek(y-a) would be interchanged for on S1 the function ek(y-a) is the dominant exponential. However the solutions f1 and f2 which we have constructed in S do not necessarily have as their respective analytic continuations in Si the functions of the form (I.2). This is due to the fact that the corresponding remainder terms are not necessarily the same. Similar considerations apply in connection with the asymptotic solutions to the Orr-Sommerfeld equation. With the aid of these results it is easy to develop asymptotic solutions to the differential equation for the flow through a circular pipe. Consider the following equation which is identical with (3.1.1) { L-w-herRe\- < -'19 3 -P ") 4t =O (1.16) where L -'+ ~ - - _.~i~i — ~

94 Two solutions of this are - J ain) and. q - Jl( ). Two other solutions gl and g2, let us say, are the solutions of the inhomogeneous equations: (L-,), t, ='' t -f),?- S (I. 17) here hi and h2 are two linearly independent solutions to the equation: A(} -A -;~R(\-u')-l I) h =0~ (1.18) Since we know the two solutions of the homogeneous equation (L - P2)f = 0 we may by the method of variation of parameters write two solutions and g2 to (1.17) as follows: |% I (a ( tt)Y, )- 1 ) amp) hl(Shda (1.19) g: I^rsnyc^-Tia^Ci''~,Ylrh,*^c (1.20) Let us obtain explicit forms for the solutions hi and h2 of (I.18). First we note that by means of the substitution h = 1 /yl/2 in (I.18), we obtain the following equation for. (D;- % (e) -k2)7= o (I.21) where C6 ) { i (I ) } This is essentially of the same form as (I.1) so that for 0 4L y 1 where the function q(y) is analytic and bounded we can immediately write down two solutions of (I.18) in the form: here e \ c 4hoo s (1.22) where we choose N 7 2.

95 Substituting (I.22) into (I.19) and (1.20) we. obtain: (1.23) By integrating by parts twice on the variable we can establish that gl and g2 are of the following form: 5e' t - + W+3 VIC l i / " *c "? (1.24) 9L cC SC )~x)j) I1(4AJer {LS By subtracting off from these solutions appropriate linear combinations of the functions J1(L, ) and YJ (igf) we can construct two solutions ga and gb of (I.17) which have the form: 0& =e I,<t {AT4 ar() I ", (I.25) These are of the form of the functions we have called fa and fb in Chapter I, In the construction of the Green's function in Chapter III it is necessary to use the viscous solution f2 which is regular at y = 0. This solution must be expressible for k in S and 0 L E L y c 1 as a linear combination of fa and fb in the form: The completeness proof given in Chapter III does not depend at all upon the specific form of the function B(k) as long as it is bounded in SO

96 We now turn to the problem of showing that the Orr-Sommerfeld equation possesses four solutions which with their first three derivatives are of the form: c^t~, ^^ {^ ^} ^ [( f (o ( { + Ek} =C- ri d E (I.26) We first note that the leading terms in these expressions satisfy the differential equation: The Orr-Sommerfeld equation may be written in the form: -(d ) ^t\ f )^ = (1.27) (5,).) (I.28) where r(y) - { + i jandq(y) - (LoRU +; ) and q2(y) - i ~ R ( ~<)- V A- d Using the method of variation of parameters we can immediately write for f(y) and its first three derivatives the following system of equations: We shall now go through the details of proving that there exists a solution to (.28) which with its first three derivatives may be expressed in the form: fLL) & tQ; Ee+)

97 We note that the functions eddy do not depend upon k and therefore we treat them as functions of the form ekQ(Y( e+jY)with Q - 0. Of importance to us is the following relationship which holds form k in S and y on the interval -1 < y 4 1. ^R Wh Myl-) o s RQ- uK^ (I.29) Because of (I.29) we can apply with success the following transformation which is analogous to (I.8): C[ -I KW(Wi) I' = t * = _W e + )- o (I-30) If in addition we set: c -U.) _ e UO'.U^) ('"- - ^ (I-32) We obtain the following system of equations for the functions Z2i. a -I+ K a x ( _'yl) {vS Oiakb4 t?o )} ce i-"1 -+ +^ ad bv1 )l^) >St ) I. (I.33) es n ich al the eorne als o h e resttin ( 1a absolt We note that all the exponentials which appear in (I33) have absolute values which are less than or equal to 1 on the corresponding ranges of integration for k in S. The system (5533) is equivalent to a Fredholm system of equations with bounded continuous kernels. From the Fredholm

98 theory one can show that if there is no non-zero solution to the corresponding homogeneous system there will be a unique solution to (I.33). We shall now show that for k sufficiently large there is no solution to the homogeneous system. First we carry out an integration by parts on the last integral in (I.33) making use of the fact that according to (I.2): eY 4 We obtain therefore: + LT = + 1,i )i? ~(1:L /)} V, Ko l -() ~gCca, vN) t o(()ld' where kje). _ - Let us designate the maximum of all the quantities ZZ2i(yk)|I i = 0,1,2,3 M on -1 z f' 1 as ZM. Furthermore let us choose the positive number B to be greater than 1 and also to be greater than any of the quantities: \,f^ \), \% k\L'(, \%1lv\,It~c( l I, _ _ l Also let fkl be greater than both I2Ca and 1, so that the quantities ai/ki, i = 0,1,2,3, l/k and | are less than 2. Furthermore let b be the greater of the two quantities 1 and l/a. If we assume there is a non-trivial solution to the homogeneous equation corresponding to (5533) then we obtain the following relationship for ZM'

99 M L 3 I35) or 1 A. l b Clearly for Ik I sufficiently large (I.35) cannot be true. Therefore we have the result that for Jkl sufficiently large (I.33) possesses a unique solution. It is easy to show the maximum ZM(k). Of all the functions Z2i on 1 Z y L 1 is uniformly bounded on S for Ikl sufficiently large. Hence it follows that Eq. (1.27) possesses a solution f2(Y) which together with its first three derivativds is given as follows: CL- k ct t,, where the remainder terms E(y,k) all have the necessary properties for k in S. We shall not go through the details of proving that there exist solutions of the other three forms given in (I.28). The details for these cases are completely analogous to what we have done above. The only difference lies in that instead of choosing the constants cl, c2, c3, and c4 as in (I.30) we must choose them as follows: To construct the function fl tCi _ 1C o ) CV-OC -QO To construct the functions f3 and f4: ~,- ^1< O; aq t =^ 0 ^,'e_.,,+ o t

APPENDIX II THE REGULAR SOLUTION TO THE DIFFERENTIAL EQUATION FOR THE FLOW THROUGH A CIRCULAR PIPE In Appendix I we showed that there exist two true solutions to the circular flow differential equation which for large K- tl X are asymptotic to the formal series solutions designated as fa and fb in Chapter I. Here we shall show which combination of these one must take in order to construct the regular solution designated as f2 in Chapter I and in Chapter III. From the work in Appendix I we see that f2 may be written in the form:'C (,). (tz 5 at {yt(X8rb [,(XrDi^( i~S h(P (II.1) whete the function h(r) is the regular solution of the differential equation: [ L-K p —' -QK(l ^2 ) (II.2) where L - and -k2 PRC. For the sake of brevity we shall designate the above integral operator as (L - 2)-1 By making the substitution g = rh and in addition the change of variables s = r2 we get the following differential equation for g(s): 4S "C) X S(t 9 ~o (11.5) where I\_ - (W+t'-2 (k1) 100

101 Equation (11.3) is linear in the independent variable and may be solved by the use of a Laplace transform. One obtains for the regular solution of (II.5) the following contour integral: where - ((iPR). The contour P is one which encircles the two points 1 = +1 and does not cross the branch cut which we imagine to extend along the real axis from? = +1 to 7 = -1. Since r and can be chosen so that everyhwere on L p, the integrand above can be expanded in a series in 1/, which is absolutely and uniformly convergent for 7 on r for any r on the interval 0 z r K 1. This series may be expressed in a product of two series in the form: S s @/ (I+''' ~\{'1\- +C 6 - 17 X A Due to the uniform convergence on the path of integration term by term integration is permissible and yields a convergent expansion representing an analytic function of s - r2. If we make use of the formula: "/ 5 e(Et1 /a_\- ) " which is derived from the well known relationship then we obtain the following expansion for h(r):

102 +A3A= Q -- tt B k t) 7 Ad; 1 3( t 4 e as i (IIe5) The coefficient of the general term in (II.5) is difficult to obtain in a simple form. The important property for us is that all the terms after the first involve only the functions - Jy ( tl)multiplied by coefficients of the form Qn/)l- where j 7/ 1. It is evident therefore from the general nature of Bessel functions that if the quantity is bounded away from the zeros of Jl(Z) then the ratio of the first term in (II.5) to successive terms can be made arbitrarily large as Po\ c- O so that the dominant term in this expansion is TJ ( 2r). Now applying the operator (L- p ) to (11.5) we obtain: iF() = L- 1-' I " E' (L- " D<,... t (11.6) Explicit integration using the formula:* 2 e ~(Kt) a(trd _ i {iP') Ge&I) {4 > e where A and A> are any two Bessel functions of orderA4 yields the result: (L-(^-'St(^)=r - _ ^ i,('t (II.7) Thus we have evalulated the first term in (11.6). We shall not evaluate the other terms explicitly but shall show that the first term in (II.6) maintains the same type of dominance over the remaining term in the limit as ~T- 7o as that maintained by the first term in (I.5) ~*See Watson (Ref. 15, p..134).

103 over the remainder of that series. To obtain an estimate of the relative sizes of the succeeding terms in (11.6) we carry out successive integrations by parts using the formula: A Nr' &71) d 7 = - C, Thus we can establish that for n 7/ 2 a- - )r,-' {u-, ( ( Making use of this expression we see that the first term in (11.6) is indeed the dominant term when ~ is bounded away from the zeros of Jl(Z). By making use of the asymptotic expressions for J i~'t-i) in conjunction with the fact that,lt may be expressed as K11] we can establish the validity of Eq. (1.3.16) for the regular function f2. In conclusion we note that the completeness proof in Chapter III does not depend upon which combination of fa and fb one must use to construct the regular function f^. However for the purpose of locating the eigenvalues in Chapter I one needs to know the appropriate combination.

APPENDIX III THE CASE THAT X- IS A NON-SIMPLE ZERO OF THE CHARACTERISTIC EQUATION In Chapter II we considered in detail the nature of the residues of the Green's function at a simple zero X = ki of the characteristic equation (1.2.11). Here we shall consider the form of the residues of the Green's function when Xi is not a simple zero of (1.2.11). In what follows we shall designate the characteristic determinant as A(%). There are two types of cases which may occur and we shall treat them separately: (a) The determinant A(x) possesses at. = ki a zero of an order which is equal to the number of linearly independent eigenfunctions corresponding to 2 = %i. (b) The order of the zero of A at ki exceeds the number of eigenfunctions corresponding to x = ki. It is easy to see that the case where the order of the zero at 2 = ki is less than the number of linearly independent eigenfunctions cannot occur. Let k equal the number of linearly independent eigenfunctions. For system (A), k must be less than or equal to four. The rank of the determinant A(X) at X = ki is then (4 - k) which is to say that the determinant and all its minors up to and including the (k - l)St minors vanish at 2 = ki. Sincec g is a linear sum of its (k - 1)St minors it follows that the (k - 1)t derivative of A vanishes at X = X- so that A approaches zero at? = Xi at least as rapidly as (2 - ki)k Case a. Let us consider the case when there are k eigenfunctions of system (A) corresponding to X = Xi, which we shall designate as 0J 104

105 j = 1,2..., k, and let the order of the zero of A(\) at X = ki be equal to k. If there are k eigenfunctions for X = \i to the system (A) then there will also be k eigenfunctions to the adjoint problem. Let us denote these as'ya, j = 1,2,..., k. This fact follows from the theorem that the index of compatibility of an nth order homogeneous differential system with n boundary conditions is equal to the index of the adjoint system. (See for example Ince (Ref. 7, p. 215)). Although the determinant A(\) vanishes as (2 - _i)k at x = Xi nevertheless it is easy to see that the Green's function (see Chapter II for the notation) sion of the determinant (.) 13) in the minors of the first row is A()X) and hence will not contribute tothe residue. It is then easy to see X (( 1,) that the residue must be an eigenfuncton) a c(IIe 1) still has a simple pole at s = in. To determine the residue at this pole, first observe that the coefficient of 7 ( to) in the expansion of the determinant (III.1) in the minors of the first row is A(x) and hence will not contribute to the residue. It is then easy to see that the residue must be an eigenfunction of problem (A) and hence may be expressed as a linear combination of the functions 0J(y). By applying a similar line of reasoning we can show that the residue of the Green's function to the adjoint system H(y, j,A) is a linear combination of the functions x i'. Hence since G('^ ~,.) = H(,y,7) we can deduce:

106 where the CQ, are constants which cannot all be zero. From the fundamental properties of the Green's function and the linear independence of the functions!J one then deduces: 4 Q S t sl (D7 i b -^) =, (III.2) Hence by using instead of the the functions: X = 2 Q ^ (III.3) M -A one obtains, using the scalar product notation of Chapter II: (h /i ) c a c i (III.4) which is clearly a generalization of the normalization condition (2~2.8) of Chapter II. The residue of the Green's function at X = \i is then: j~i The case of a simple zero of A(X) is the special case corresponding to k = 1. Case b. In this case the order of the zero of the determinant exceeds the number of eigenfunctions. Let m designate the order of the zeros of A(\) at x = ki. Let k designate the number of eigenfunctions. Then the rank of the determinant A(\) will be at X = ki (4 - k) and it is not difficult to deduce from (IIIol) that the Green's function G(y,,,) will have a pole of an order which is at most m - k + 1, When the pole is of order

107 m - k + 1 then the denominator and the numerator of G(y,,X) for x near hi can be expanded as follows: A( Q) = (X-a)( {,Co+ I( X-X4J v\ + } N (%,Xv) ~~ 4 ~~O= %-^^^^^^W^^ { h(2,s,>h + t92 ) X(III.5) Thus the form of the Green's function in the neighborhood of X = Xwill be: a<io; hE~,(n->^ Io ^-rl)< (III.6) Each succeeding term contains higher derivatives of the function h(y, \)o The general nature of function h(y,, ) may easily be determined. In the expansion of N in terms of the minors of the first row we see that since 1 (y,,k) is multiplied by A(x) it will not contribute at all to the series of negative powers of (A - Xi). Hence we can say that h(y, 7,i) is a linear sum only of solutions to the differential equation and furthermore it satisfies the boundary conditions so that it can be written as Z_ [j)% (i). As an illustration let us specialize these results to the case of m = 2 and k = 1. In this special case the expansion of G(y, >,X) takes the form: +(Y),tii X_ aJ^da ix Afl~tf GM (III 7) _cr~s~xi _______(kil) ("'.r7) where E is analytic at x = i. The functions ~i, 2 i are eigenfunctions for X = Xi of the systems (A) and (At) of Chapter II respectively, and b _ -a1/ao. We see further that the residue of G at x = \i for this case may be written in the form:

108 Qs G&- = 14.ax)(9> {^ X t i(+(7), t- Lx(sE ib) (InI.8) Therefore in the expansion of an arbitrary function f(y) we must have the term: (X,f ) x( Y)- (Cax,; ) Cj) (III. 9) The function (hk)\i\= is not a solution to system (A). It satisfies the boundary conditions but not the differential equation. The significance of this fact comes out when one considers the initial value problem as is done in Chapter IV. Here we merely note that by substituting (0. into the Orr-Sommerfeld and differentiating with respect to \ we obtain the result that () XA~_) satisfies the differential equation: [ ( D -2)2 idR(g (y-c)(-d )- () 33(aA, -_ - (III.10) It what follows we shall designate (aXk) Xk.u as \<, and (Gk) ) as'$X(. The function ab must be added to the set of the eigenfunctions On of problem (A) to complete the set in the event of the occurrence of the situation described. For the more general case it may be necessary to add higher derivatives of the functions 0A with respect to 2 to complete the set of functions ne. Although straightforward we shall not further elaborate this and we conclude by giving a short list of the properties of c)t0 and AXeJ when m = 2 and k = 1. All of these properties may be obtained without difficulty from (IIIo7) and (III.9). (III 11) (3x/4X^ )^(^oX ~ Q

109 The first of these equations if interesting in that it shows that for the Hermitean self-adjoint case where the complex conjugate of system (Av) is identical with (A), case b cannot occur. The foregoing results all have their analogue in the theory of matrices. Any finite matrix whose latent roots are all distinct may be diagonalized and its eigenvectors form a complete seto When however two or more latent roots concide the number of eigenvectors corresponding to these may be less than the number of roots. In this latter case additional vectors must be added to complete the set. In our case finding the residues of the Green's function prescribes for us the "vectors" which must be added to complete the set of functions 0n which are the ".eigenvectors" of (A). Thus for the case m = 2, k = 1, we must add to the set of 0 functions the function ak~ and to the ( functions the function Xi *4+.)X( Equations (III.1l) tell us that 3X is orthogonal to all the 9''s except'6 and is, in addition, orthogonal to 4X~i -i Ad We can think of Xt as being the function in the ~ set which corres ponds to 34/, and tpe function Xi we can think of as corresponding to bXRi-At~X. When the 0Is and 1 Is are supplemented in this way we can see immediately by the use of (III.11) that a necessary condition on the coefficients in the expansion of an arbitrary function f is that the coefficient of bKB should be ( if) and that of 0i should be ( 3X frX,,f) which is exactly the result obtained from (III.8).

APPENDIX IV SIMPLE EXAMPLE OF AN EXPANSION THEOREM SIMILAR TO THE THEOREM OF CHAPTER II Here we shall give a simple example of an expansion theorem for eigenfunctions in which the scalar product involves a differential operator just as the scalar product ($ At ) in the hydrodynamical stability problem does. Consider the boundary value problem: (IV. ) The solutions to this boundary value problem are: ~(^Q) -- m — a'i1 ^. aft (IV.2) The eigenfunetions are: 4M. I= i w (IV.5) We note that n = 0 is not a member of the above set for this would correspond to 00" 0 on 0 o y L 1. The boundary value problem which is adjoint to (IV.1) is: (IV.4) The solution to this boundary value problem is a^C(t) = l\ -^~^ ^ _,+~a,9 (IV.5) The eigenvalues are the same as (IV.3). It is easy to see that the I n and the 0n satisfy an orthogonality relationship of the form: 110

1ll S ( 1o ^ g - f&&iS^m L (IV.6) If we attempt to expand an arbitrary function f(y) in terms of the ~n, we see from (IV.6) that the expansion must have the form: r | L1 )}t,>5) >$$) 4a (O-_ e^m6%) (IV.7) where the summation is taken over all the positive and negative integers and n = 0 is excluded. Equation (IV.7) may also be written more explicitly as: luT L A, ZTL:^e-)~I \&a~ W @)d(g_( ) (IV.8) on —0 In Eq. (IV.8) we may include the n = 0 term as we see that it will not contribute anything to the sum. From our knowledge of ordinary Fourier series we may evaluate the terms which appear in (IV.8). We have: 00 I -~w^,..yn^^ r, ~ e e(IV.9) Also i0 u o (IV.10) Making use of (IV.9) and (IV.10) we obtain: $ e a F9s- ( IV.11) 4 -e_

112 We see that unless f(l) = f(0) = 0, the series representation of f(y) will not converge to f(y). Instead it converges to f(y) plus a term which can be described as a linear combination of f(l) and f(0) times a function (namely a constant) which is anihilated by the differential operator D which appears in the scalar product. The form of (2.4.14) is analogous to this. There we have the series t ($ T)t) converging to f(y) plus terms which are linear combinations of f(l) and f(-l) times the functions +ay.2 2 e- y which are anihilated by the differential operator D - a which appears in the scalar product defined in Chapter II. We shall now prove Eq. (2.4.14). To do this we construct a function f*(y) which is related to the function f(y) as follows: J'CY) -- JCm - Si^^dC\-Y) f - Si^<W -foQ T-) _in 5-1) _s - - ------ (I. 12) s$t h a9 Sluhaoh We see that f* defined in this manner satisfies all the conditions of the main theorem of Chapter II including the boundary conditions, f*(l) = f*(-1) = 0. Hence applying that theorem we have: r - (IV.13) By integration by parts on the P variable applying the boundary conditions on G($ l ) as well as the fact that the functions 5seh i() and Sh~ i L~ —) are anihilated by the operator (D2 a2), we can show that: -I -o and hence one obtains Eq. (2.4.14).

REFERENCES la. Birkhoff, G. Do, Trans. Amer. Math. Soc., 9, 373 (1908). b. Birkhoff, G. D., Trans. Amer. MathO Soc., 9, 219 (1908)o 2. Corcos, G. M. and Sellars, J. R., J. Fluid Mech., 5, 97 (1959). 3. Dolph, C. L., and Lewis, D. C., Quart. Appl. Math., 16, 97 (1958). 4. Haupt, 0., S. B. Bayer. AkadWiss., pp. 289-301 (1912). 5. Heisenberg, W., Ann. Phys. Lpz., 74, 577 (1924). 6, Hopf, L., Ann. Phys. Lpz., 44, 1 (1914). 7. Ince, E. L., Ordinary Differential Equations, Dover Publications, First American Edition. 8a. Lin, C. C., Quart. Appl. Math., 3, 117, 218, 277 (1945). b. Lin, C. C., The Theory of Hydrodynamic Stability, Cambridge University Press (1955). 9a. Pekeris, C. L., Proc. Nat. Acad. Sci., 34, 285 (1948). b. Pekeris, C. L., Phys. Rev., 74, 191 (1948). 10, Sexl, T., Ann. Phys. Lpz., 83, 835 (1927). 11. Tatsumi, T., J, Phys. Soc. Japan, 1, 619 (1952). 12. Thomas, L. H., Phys. Rev., 91, 780 (1953). 13. Trjitzinsky,W. J., Acta Math., 67, 1 (1936). 14a. Wasow, W., J. Res. Nat, Bur. Std., 51, 195 (1953). b. Wasow, W., Ann. Math., 52, 350 (1948). 15. Watson, G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press (1958). 113

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