the effect of external forcing on the stability of plane poiseuille flow

The Effect of External Forcing on the Stability of Plane Poiseuille FlowAuthor(s): P. HallSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 359, No. 1699 (Mar. 17, 1978), pp. 453-478Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/79653 .

Accessed: 07/05/2014 14:12

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of theRoyal Society of London. Series A, Mathematical and Physical Sciences.

http://www.jstor.org

This content downloaded from 169.229.32.136 on Wed, 7 May 2014 14:12:13 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=rsl

http://www.jstor.org/stable/79653?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Proc. R. Soc. Lond. A. 359, 453-478 (1978)

Printed in Great Britain

The effect of external forcing on the stability of plane Poiseuille flow

BY P. HALL

Mathematics Department and Physiological Flow Studies Unit, Imperial College, London SW7, U.K.

(Communicated by J. T. Stuart, F.R.S. - Received 20 May 1977)

The stability of plane Poiseuille flow in a channel forced by a wavelike motion on one of the channel walls is investigated. The amplitude e of this forcing is taken to be small. The most dangerous modes of forcing are identi- fied and it is found in general the critical Reynolds number is changed by 0(6)2. However, we identify two particular modes of forcing which give rise to decrements of order eI and e in the critical Reynolds number. Some types of forcing are found to generate suberitical stable finite amplitude perturbations to plane Poiseuille flow. This contrasts with the unforced case where the only stable solution is the zero amplitude solution. The forcing also deforms the unstable suberitical limit cycle solution from -its usual circular shape into a more complicated shape. This has an effect on the threshold amplitude ideas suggested by, for example, Meksyn & Stuart (I95I). It is found that the phase of disturbances must also be considered when finding the amplitude dependent critical Reynolds numbers.

1. INTRODUCTION

The linear and weakly nonlinear stability characteristics of Poiseuille flow between rigid boundaries are now well known. Linear stability theory predicts that the flow becomes unstable to infinitesimal disturbances when the Reynolds number reaches a certain critical value. Experimentally it is found that the flow becomes turbulent at significantly lower values of the Reynolds number. This is in contrast to the case in centrifugal instabilities (Taylor vortices) and Benard convection where the theoretical values of the critical Taylor or Rayleigh agree well with experimental observations.

For the Taylor problem weakly nonlinear stability theory can also describe successfully the development of the flow after the linear critical Taylor number has been reached. (See for example the review article by Stuart (I 97 ).) The essential physical difference between the Taylor vortex and plane Poiseuille flow instability is that, in the neighbourhood of critical linear stability parameter, nonlinear effects are stabilizing in the former case and destabilizing in the latter. (For a discussion of nonlinear effects in plane Poiseuille flow see, for example, Meksyn & Stuart

[ 453 ]



454 P. Hall

(I95I), Stuart (I960), Watson (I960) and Pekeris & Shkoller (I967, I969) and Reynolds & Potter (i967).)

Linear stability theory has also been used to study the stability of the Blasius boundary layer on a flat plate. This problem is complicated by the fact that the basic flow depends on the downstream variable. However, local stability analyses have been performed by ignoring this fact and taking the flow to be parallel (see, for example, Jordinson 1970). More recent work (Boutheir 1972, 1973; Gaster 1974) has taken this variation into account by an approach of the W.K.B. type. This is justifiable in the limit of the Reynolds number tending to infinity. It is found that taking such a variation into account improves the agreement between linear stability theory and experimental observations. The above authors all assume that any disturbance to the flow is confined to the boundary layer. However, experimentally it has been shown by Schubauer & Skramsted (I947) that the presence of free stream disturbances in the form of turbulence can significantly decrease the Reynolds number at which transition occurs. The transition Reynolds number decreases as a characteristic measure of the amplitude of the turbulence increases. It has been suggested that this is due to some kind of resonance due to the disturbances outside the boundary layer. For example, Criminale (I 971 ) has discussed the effects of travelling waves and random disturbance outside the boundary layer whilst Rogler & Reschotko (I975) have discussed the effect of an array of vortices. However, it appears that the configurations discussed by the above authors did not lead to resonance. The reason for this is clear and is that such a response will only occur if the motion outside the boundary is forcing a natural solution of the homogeneous Orr-Sommerfeld equation at some value of the Reynolds number. This situation was not considered by the above authors.

It has recently been shown by Kelly & Pal (I976) and Hall & Walton (977) that such a resonant situation can occur in Benard convection. Kelly & Pal showed that this can be achieved by allowing the temperature of one of the planes bounding the fluid to have a small amplitude periodic spatial variation with the same wavelength as the critical wave of linear stability theory. Moreover, Kelly & Pal showed how this resonance could be balanced by nonlinear effects. Hall & Walton (I977) have shown that a similar situation occurs in Benard convection in a rectangular channel when realistic boundary conditions are applied. The main result of these papers is that, rather than a sudden bifurcation to a convective regime, there is a so-called 'smooth bifurcation'. These effects are well known in elastic stability theory (see Budiansky I974). However, it is well known that nonlinear effects in Benard convection are stabilizing, whereas in parallel flows they are usually found to be destabilizing. In elastic stability theory it is known that when nonlinear effects are destabilizing the smooth solution becomes unstable and 'snap buckling' occurs at some value of the appropriate stability parameter. The latter result suggests that a similar effect might occur in parallel flows if they are forced in a suitable manner in the neighbourhood of the critical Reynolds number. In view of the incompleteness of our own knowledge of nonlinear effects in the Blasius



Stabtility of forced Poiseuille flow 455

boundary layer we shall in this paper consider the simpler problem of the stability of forced plane Poiseuille flow. In particular, we consider the case when one rigid boundary is replaced by a boundary defined by

y= l+2ecosk(-h h-2)) (1.1)

where x is the downstream coordinate and t is the time. Alternatively, we could impose boundary conditions of the form

U = eAcosk ( UO )) v = eBsink (- h t)

at a rigid boundary. The effect of the latter and former conditions is essentially identical so we restrict our attention to the former case.

It is perhaps useful at this point to discuss in more detail the known results concerning the stability of unforced plane Poiseuille flow. In a region close to the critical Reynolds number Re, it can be shown that the amplitude A of the most unstable wave of linear stability theory satisfies an equation of the form

dA/dr = (Re-Rec) kA + hA2A, (1.2)

where k and h are complex constants having positive real parts. We can show from (1.2) that the modulus of the amplitude A is determined by

d(1A12)/dr = (Re-Rec) krIAI12?+ hrIA14,

where the subscript r denotes the real part. It follows from above that if Re is less than Re, there exists a stable solution with A = 0. In addition, there exists an unstable time dependent solution corresponding to a circular limit cycle in the amplitude phase plane with radius { - [Re - Re,] kr h,- }I and centre the origin. When Re reaches the value Re, this limit cycle disappears and the previously stable singularity at the origin becomes unstable.

In this paper we shall investigate the effect of forcing on this phase plane be- haviour. In particular we isolate three particular cases:

(i) k = c, UO = c

(ii) k= 2a, UO= c

(iii) k=3a, UO = c

where ac and c are the wavenumber and wavespeed of the critical point on the neutral curve for plane Poiseuille flow. We shall see that the presence of the forcing generates extra terms in (1.2) which are proportional to A, A, A2 or are independent of A. The procedure adopted in the rest of the paper is as follows. In ? 2 we formulate the partial differential system determining the flow in the channel while in ? 3 we determine the generalizations of (1.2), corresponding to conditions (i), (ii) and (iii). In ? 4 we describe the numerical work required to determine the coefficients in the amplitude equations obtained in ? 3. The solutions of the amplitude equations are



456 P. Hall

discussed in ? 5 while in ? 6 we explain how more general forcing conditions can be investigated. Finally, in ? 7, we discuss our results and extend the analysis to allow for slow spatial modulations of the type discussed by Stewartson & Stuart (I97I).

2. FORMULATION OF THE PROBLEM

We consider viscous incompressible flow between the walls which are defined in terms of Cartesian coordinates (x', y', z) by

y =-h,

y = h{1 + 2e cos k[x'/h- UO vh-2t']}, (2.1 a, b)

where h is the mean depth of the channel and P and t' are the kinematic viscosity and time respectively. We assume throughout that e, the dimensionless amplitude of the wall motion, is small. Furthermore, we assume that k and UO the dimensionless wavenumber and wavespeed of the motion are of order unity. In the absence of a steady pressure gradient down the channel the motion of the wall induces a motion of the fluid. It is of interest to note that the oscillatory motion of the boundaries can generate a steady flow in the channel at order 62. However, we assume in this paper that there exists a steady pressure gradient down the channel which generates a mass flow 'FJ at any position along the channel. We define a typical velocity in terms of 'F by writing VO = 3WO/2h, (2.2)

and then a typical Reynolds number Re by writing

Re = 3V1o/2v. (2.3)

We define dimensionless variables x and y by writing

x = xh-', y = yh-', t = t'vh-2. (2.4)

If we assume that the motion in the channel is two dimensional then we find that the stream function Vi'(x, y, t) associated with the motion satisfies

a (V2~f+8i) 2f 1 25 at a(x y) R-e (2.5)

where V2 = 02/ax2 + 02/ay2 (2.6)

We now impose the no-slip condition at the boundaries so that

Rfx-fy =o, y = - t

3bz- (2ekIRe)Uosink(x-Uot), 3by = 0, y - 1+2ecosk(x-Uot). (2.7a,b)

Neglecting terms of order e we see that the motion is parallel to the planes y = ? 1 with velocity field given by 2=

1 (2.8)

which just represents the familiar Poiseuille flow in a rigid channel. If we now write 2f in the form y - -fr + X y, t), wI--s3g (2 . { 9)G



Stability of forced Poiseuille flow 457

then we can show from (2.5), (2.7), and (2.9) that the correction to Poiseuille flow, namely J(x, y, t), is determined by the differential system.

R-e ?Jt a ax 8(xx,y) (2.10)

It is well known that the flow given by (2.8) is unstable to small disturbances when the Reynolds number reaches the value 5774 = Re,. Clearly any disturbance to Poiseuille flow satisfies the differential system (2.10) with e = 0. The most dangerous disturbances have !f of the form

f = {V'o(y) exp ix{x - ct} + complex conjugate}

where a, c and the Reynolds number Re satisfy the eigenrelation

Y(cx, c, Re) = 0. (2.11)

If the Reynolds number Re = Re, and k, U0 satisfy the eigenrelation (2.11) then resonance will occur when Re approaches Re, from above or below. Following Hall & Walton (I977) it is an easy matter to show that near Re,

!1 % e/(Re-Rec). (2.12)

Thus in the next section we shall consider the above case in detail. We shall also consider two other wall motions which have significant effects, though resonance does not in fact occur. These motions are defined by

(a) k = 2a, UO = c, (b) k=3a, UO = c, (2.13a,b)

where ac, c satisfy the eigenrelation (2.1 1) at the critical Reynolds number Re = Re,.

3. THE STABILITY OF FORCED PLANE POISEUILLE FLOW

(a) Forcing of the fundamental

We first restrict our attention to the case when the wavelength and wavespeed of the travelling wave on the upper boundary are equal to the corresponding quantities at the critical point on the neutral curve. If we denote the latter quantities by a and c then we see that the boundary conditions to be satisfied are

vf =V =0? y =-1,

VJ,= -{1-(1+eE+eE)2}, -RI(eic/Re){EE}, y= 1+eE+eE, (3.1)

where for convenience we define E by

E = exp io{x-ct}. (3.2)

If the Reynolds number Re is less than its critical value Re, then the forcing rep- resented by (3.1) leads to a flow of order e. However, in the neighbourhood of Re, resonance occurs and the forced motion becomes of order e{Re -Rec}-'. In order



458 P. Hall

to remove this singularity we balance this term with a typical amplitude of an instability to the basic flow in this region. Thus, we require that

elRe-Re,1-1 -|Re-Reell. (3.3)

This suggests that we confine our attention to a region of thickness di about the critical Reynolds number. Thus we write

Re=Rec+Re2ell+, (3.4)

which, for any given values of e and the Reynolds number Re, determines the constant Re2. We now expand [ in the form

qu = el TO(y, r) E + e-w QWO(y,T) E + 6*'FM(y, T) = ieo(y,T)E2+c46['(y,T)E?+eiY(y,T)E

+ CJT2(y, T) E2 + el ['2(y' T.) ER2 + CI['OO(y' T) E

+eV00(y, T) E + eF3(y, T) E3 + F3(y, ) E3 + 0(), (3.5)

where a bar denotes complex conjugate, M denotes mean and T is a slow time scale defined by

T = ext, (3.6)

which is the appropriate time scale for the growth of Tollmien Schlichting waves. If we substitute for I[ from (3.5) into (3.1) and expand in powers of e we can

show that the appropriate boundary conditions for the functions appearing in (3.5) are given by

o[%( ? 1) = V%( ? 1) ~I'2( ? 1) = ~( ? 1) = ~~( ?1) = '( + -? 1) = I'F? ? 1) = 0,

[OO%(- 1) _ IOO(- 1) = 0, IO(1) = 2, FOO((1) = cRe-1, (3.7)

where a dash denotes a derivative with respect to y. If we substitute for IF from (3.5) into (2. 10) and equate terms of order el we see that '[o is determined by

{-C( <2Q+Uu+,2 -02) /o = 0, /FOo(? 1) = T;j(? 1)=0 (3.8)

Thus we can write Yf in the form

To = A () fo(y, r), (3.9)

where A(T) is an amplitude function to be determined later and ifi satisfies

Y/3;fo - O f,0o(? 1) = o(? 1)=0. (3.10)

Here the operator k is defined by




At order ed we find that F2 and t'M can be written in the form

IF2= A2 f2W),

'M= IA12M(y), (3.12) where the function V/2 is determined by

f2(? +l) -- f2(?+ 1) = O. (3.13)

Here the operator Nk is defined by

N= d2dy2 -k2a2

and Vrmis given by X = iaJ (#oo - fo)dy+;((-y2). (3.14)

The undetermined constant A can be fixed by stipulating that there should be no net flow down the channel associated with (3.14). Thus we choose A such that

2A = -ia dy J fo do (3.15)

Alternatively we could choose A equal to zero so that the pressure gradient down the channel were fixed. However, as pointed out by Stewartson & Stuart (1971), the latter is not possible when spatial modulation of the waves is considered. Since we shall in ? 7 briefly discuss the generalization of this section to take spatial modulation of the flow into account we choose A as defined by (3.15). At order e we find that the fundamental term Vfo is determined by the following differential system.

{7u-c} N1 loo -'Yo + (i/aRec) N2 V[oo = 7-i- NV 3 Re2 A AIA12S(y) 100 ic d-r icxRe,

= l'-1)-Fo(-t) = 0 W[(+ 1) = 2, q0oo(1) = c/Rec. (3.16)

where the function S(y) is given by

S(y) = {#M fo0 - N, fo + 2V2 Nl io' N2 V2 + Vf'Nl WO O N2 V2} (3.17)

In general the differential system (3.16) will not have a solution. However, it is easily shown that the orthogonality condition which ensures a solution of (3.16) exists is

dA/dr = (Re2/Rec) d1 A + el A2A +f1, (3.18)

which then determines the amplitude funcltion A(T). The constants dl, el and fg appearing in (3.18) are given by

d = NVro Ody R/Rec, iVroOdy)

el = ij SO dyj N1 ro 0dy,

fi = [(c/Rec) O"'(1) - 20'(1)/2Recj N1 Vfot0 6dy. (3.19a, b, c)



460 P. Hall

Here 0(y) is the solution of the differential system adjoint to (3.10). If we putf- 0 in (3.18) we recover the amplitude equation governing the stability of plane Poiseuille flow between two rigid boundaries.

We postpone a discussion of the solution of (3.18) until ? 5.

(b) Forcing of the first harmonic

We suppose next that the wavelength and wavespeed of the travelling wave on the upper boundary are given by

k=2cx, UO=c,

where oz and c are again the critical values of linear stability theory. In this case the forced motion will not resonate in the neighbourhood of the critical Reynolds number. Since the linear homogeneous Orr-Sommerfeld equation has no eigenvalues in such a neighbourhood for the above values of cx and c. Indeed, it is not clear if such an eigenvalue exists at any value of the Reynolds number. However, the forced motion is important in the neighbourhood of Re, in that it can interact with the fundamental to reproduce the fundamental at higher order.

Suppose then that the Reynolds number Re is given by

Re = Re,+ 1Re2 +

where a is small. In the absence of any forcing a disturbance with order of magnitude 4i can exist in this neighbourhood. Such a disturbance interacts with itself to generate a first harmonic term of order d. Thus, in order to balance the effect of the forcing with that of the usual nonlinear interactions we choose a to be of order e. Thus we write Re = Re,+eRe2+ (3.20)

and redefine the slow time scale T by writing

T = et. (3.21) We then expand T[ in the form

W = e Tio E + ife 70 + 2E2 + 6F2E2 +61FM(V)

+?J0oo E?ei T.oE +?elVI3E-?+eJY3E3+ 0(t)2. (3.22)

If we substitute the above expansion into (3.1) and expand in powers of e we see that the required boundary conditions for the functions apart from VI3 appearing in (3.22) are

Wo( ? 1) = Wo( ? 1) = NCr( ? 1) = V'2(- 1) Voo( ? 1) = 1) ?( 1) = 0

?2(1) c/Rec, ?oo(1) = -'(1), ?2(1) = 2. (3.23) If we next substitute from (3.22) into (2.10) and equate like powers of el and e we see that Eoo = A (T) o,

,= JAI2bM,

2 = A2V2 + ?22 (3.24)




where 3bo 'M, If2 are as determined by (3.10), (3.14) and (3.13) respectively and f22 satisfies the following differential system

2 ?f2 2= 0, if22(- 1) = ?i22(- 1) 0, fr22(1) = ciRec, *f22(1) = 2. (3.25)

We can now equate terms of order el proportional to E in (2.10) to show that

?? ia d i/re N~?R 1 o+A2AS(y) +AT(y),

O'( ? 1 ) = 0O(-1 ) = 0, Poo(1) =--A3fo0(1), (3.26)

where S(y) is as defined by (3.17) and T(y) is obtained from (3.17) by replacing i/2

by /22 and putting VfM = 0. If the above system is to have a solution we require that A satisfies the equation

dA/dT = (Re2/Rec)d1A + [gq +hl]jA +elA2A. (3.27)

Here d1, e1 are as defined by (3.19a, b) and gq by (3.19b) with S replaced by T. The constant h1 is given by

0 1 O) )"(1M) (3.28)

2Recf ON1, bo dy

The extra terms proportional to A in (3.27) arise from the interaction of the fundamental and the forced component of the first harmonic. We note that (3.27) determines the growth or decay of a perturbation to Poiseuille flow in a channel having a travelling wave on the upper boundary.

It is important to realize at this stage that we have only considered the stability of the flow to disturbances having a time dependence in phase with the forcing. In order to determine the possible effects of such a phase difference we now take the upper boundary to be defined by

y = 1 +eeiAE2+ee-iAE2, (3.29)

where the real constant A can take any value in [0, 2Xn]. The expansion (3.22) is left unchanged and the ,procedure described above is repeated. We find that the resulting amplitude equation becomes

dA/dr = (Re2!Rec) d1 A + [g1 + h1]A eiA + e1 A2A. (3.30)

The effect of the phase difference is easily removed by writing A = BelA!2 in which case B satisfies the amplitude equation (3.27). It is of interest to note that if we had allowed for a similar phase difference in ? 3 a then the resulting amplitude equation would have been

dA/dr = (Re2/Rec) d1 A + e1 A2A ?+f el'.



462 P. Hall

However, we see that the effect of the phase difference is again easily removed by writing B = e iAA

in which case B is determined by

dB/d-r = (Re2IRec) d1 B + el B2B +fj, so that B is independent of A.

We again postpone a discussion of the amplitude equation (3.27) until ? 5.

(c) Forcing of the second harmonic

Finally in this section we consider the case when the wave travelling on the upper boundary is the second harmonic of the wave associated with the critical point on the neutral stability curve. Thus, the boundary conditions for the function J

now become = -3icccRec[E3 -E3], Y?Y = 2eE3 + 2eE3 + 62[E6 +E6 + 2],

y= 1+eE3+eE3, '(- 1)= V(-1) 0. (3.31)

In this case we expand Re in the form

Re - Re + e2Re2+.... (3.32)

so that the second harmonic forcing term can interact with the first harmonic of the instability wave of the unforced motion to reproduce the latter. Thus, we now expand V[ in the form

VI = 0 E + T OE + [ 3OE3 +! [ E+33 +E+2[m + Me!22E2 + 622E2

? 62WI' E6 + 62q B6 + 62VI4 E4 + e2!F4E4 + e,T3' E + e63'ooE

+e3{terms proportional to Em, m : ? 1}. (3.33)

The boundary conditions for the functions appearing above can be obtained by substituting for T from above into (3.3) and equating like powers of e. We find that

V' ( ? 1) ( 1) = V( + 1) = (-3 1) = -_)2( ? 1) _ F2(- 1) = V"6( - 1) 0,

TU4 ( ? ) fo = IO() = V4(- 1) TJO(- 1), V13'(1) 2, VJ2'( 1 --fO'( Y4(1)-- o"(1), M(1) 2 - [f3"(1) + T3'(1) W3(1) = cR;;'-,

T'oo(l) = Wo'() o()-"/l-U(1))-W0,0. (3.34) We now define a slow time scale by writing

r = e2t. (3.35)

The functions appearing in (3.33) are obtained by substituting for 1 into (2.10) and equating like powers of e to find that, apart from T6, the functions up to order 62are given by = A2%, [f3

WM = IA I2/M + V/MM,

2= A2b2 + ?Ai/2,




Here A(r) is again an amplitude function to be determined while #k, "M, and k2

are determined by (3.10), (3.13) and (3.14) respectively. The functions VL3, PfMM,

?b22 and h are given by the solutions of

2 f2 =W{3 N,'R0-31F0N3 2,3 - f O N3Vf3 + 3Vf3 N1 Vf,222( ? 1) =-?, Y2 Vr222(?1)-0,~

p22( - 1) =0, pf22(1) = - ol

J { = T-23 Njf0-3? N3?#3 + ?O N3 ?K+ 3?3 N,RO'} #4(? 1) = 0,

V14 (-1) = 0, 0 f(1) = -O

Re' d4VMM/dy4 = 3ia{BN3 N3 #- 3'N 1 = 0,

"(1) = -[ '(1)+fr(1)]+2. (3.37a,b,c,d)

The differential system (3.37 d) which determines the mean flow correction is again solved such that there is no net flow down the channel associated with WfMM. The amplitude function A is determined by proceeding to order 63 and obtaining the solvability condition on the function Vfoo. We find that

dA/dr = (Re2IRe0) d1 A + k1 A + 11A2 + el A2A (3.38)

The constants d,, and el are as defined earlier while k1 and 11 are defined by

1 =(i J 0[ - 22 NI W~o + 2~o' N2 f22223 N2 2~2 + 3 W2' N3 2f3-2 N, W'O 11 = (i~~~~~~~~~f1 ~~1M

+ 2#2N3 ?Lr3 + f0oN2 L42-323NX2W2] dy+ (2Rec)- '`(1) a (1)/f ON,1 f dy,

k,t=(i I~~ [ N*fo + 3W2'2 N3 Vf3-2 N -23 N2 W22 + 4W'? N4 *f4

-V'MM fo-3#3 N2 22 + 2222 N3 3- 44 N3 #31+ 3f3 N4Vf4] dy

+ (2Rec)-' [O"(1) {~fr2(1) + f4t(1) +fO'(1)} + O'(1) (N)]}/J' Nflt dy. (3.39)

Finally in this section we again consider the effect of a phase difference between the forcing and the disturbance. If the upper boundary is taken to be defined by

y = I + e{E3 eIA + E3 e-iA},

then it is easily shown that (3.38) is replaced by

dA/dr = (Re2/Rec) d1 A + k, A + 1 e+IAA2 + eA2A.

However, as was found to be the case in ? 3 a and ? 3 b the effect of the phase difference can be removed by writing

in which case we see that B satisfies the differential equation (3.38).



464 P. Hall

4. THE NUMERICAL WORK

In order to discuss the solutions of the amplitude equations appearing in ? 3, the complex constants in these equations must be evaluated numerically. The determination of these constants requires the solution of several forced Orr- Sommerfeld problems of the type

Sk -H(y),

0q(- ) - 0'(-1 ) = O,

OM(t)-o, 0q'(1) =f, (4.1)

where a, fi, H(y) are given. In addition, the homogeneous differential systems for the fundamental and its adjoint must be solved. All these systems were solved by the finite difference method used for the homogeneous problem by Thomas (I953) and later for inhomogeneous problems by Pekeris & Shkoller (I967, I969).

The first step in this method is to introduce an auxilliary function g by writing

g 0 - w2qW" + 90w40V

where w is the step size which we shall use. The purpose of the transformation is that the resulting difference equations for the auxillary function are more accurate. The function 0 is split up into an even and an odd part so that it is enough to consider only the interval 0 < y < 1. The step length w was taken to be 0.01 so that, after satisfying the boundary conditions, we had to solve 101 equations for 101 unknowns. These equations are in general independent except in the homogeneous cases where one equation becomes redundant. The eigensolution 3bo and its adjoint were normalized so that

Vro(1) = 0(1) = 1.

We further note that in contrast to Pekeris & Shkoller the mean flow equations were always solved such that there was no mass flow through the channel. Some checks were available since the constants d1 and el, defined by (3. 19 a, b) respectively, have been computed elsewhere. We found that

di = 0.009816 + 0.046195i,

e= 30.76 -173.3i. (4.2 a, b)

which agree well with the results quoted by Stewartson & Stuart (1971). The remaining constants were found to be given by

f= -0.01708+0.0199i, h= 0.01474 + 0.76i,

1= - 1.321 - 0.6773i,

11 = 58.74 + 63.57i,

k, = -277.9 -1402.i. (4.3a, b, c, d, e)




5. SOLUTION OF THE AMPLITUDE EQUATIONS

We first turn our attention to equation (3.18) which determines the amplitude of the motion forced by a wave on the upper boundary having wavenumber and wavespeed satisfying the eigenrelation of plane Poiseuille flow. We first note that any singular points of (3.18) in the phase plane satisfy

0 = (Re2/Rec) dl AE + e1 A2 AE +f1, (5X1)

where AE is the value of A at the singular points. We can show from (5.1) that

JAEI2 is determined by the cubic equation

If 12= IAE12 I(Re2lRec)d +elIAE1212. (5.2)

This equation can be solved for JAEI2 and then substitution into (5.1) determines AE. The equation (5.2) has only one real positive root for negative values of R2 so that for Re2 < 0 there is only one singular point. In order to determine the nature of this singular point we write

A = AE+B,

and substitute into (3.18) to show that if B is small then

dB/drT= {(Re2/Rec) d, + 2e, IAE 12} B + el AE B. (5.3)

If we now look for a solution of the form B - eAT then we find that the exponent A is determined by

A = (Re2/Rec) dir + 2 IAE 12elr ? V[1el A2 12 - {(Re2lRec) dl + 2 IAE I2e i}2], (5.4)

where subscripts r and i denote the real and imaginary parts, respectively. The quantity inside the square root is always negative for the values of el, d1 and fi given in ? 4, so that the singularity is, in fact, a focus. The focus becomes unstable when Re2 reaches the critical value Re* given by

Re*/Rec -9.23. (5.5)

When Re2 is less than the above critical value an unstable limit cycle encloses the singular point. However, when Re2 approaches Re* the limit cycle shrinks into the singularity which itself changes into an unstable focus. Thus we can recognize the value Re* as being the correction to the critical Reynolds number at which infinitesimal perturbations to the forced motion become unstable. Hence the critical Reynolds number

Re = 5774. {1-9.23i. . .} (5.6)

gives for a given value of c the Reynolds number at which the forced motion becomes unstable. Such a phenomenon has long been known under the name 'snap buckling' in elasticity (see, for example, Budinasky 1974). It is of interest to note that the amplitude dependent critical Reynolds number obtained by Meksyn & Stuart



466 P. Hall

(95I) has {Re-Re,} proportional to the square of the amplitude in the neighbourhood of Re,. This critical Reynolds number is defined to be such that the solutions of the amplitude equation

dIA 2/dr = (Re2/Rec)d,rIA12+e,rlA 14

increase indefinitely if a disturbance with amplitude greater than (Re2dirlRe/ er)

is imposed on the flow. We further note that a related nonlinear critical Reynolds number can be defined in our case.

We first note that if Re2 is less than Re* then (3.18) has a stable point AE in the phase plane surrounded by an unstable limit cycle. In contrast to the unforced case the limit cycle is non-circular. Suppose we define rmin and rmax to be the radii of the circles centred on AE where rmin and rmax are the minimum and maximum distances of any point lying on the limit cycle from the singular point. If at this given value of Re2 we perturb the flow from AE by a disturbance with modulus greater than rmax then the disturbance will grow indefinitely whereas if the modulus is less than rmin the flow will return to its equilibrium state. If the modulus is less than rmax but greater than rmin then the disturbance will be either damped out or will grow depending upon its phase. The quantities rmin and rmax will be functions of Re2. Thus, for a given value of e and hence Re two amplitude dependent Reynolds numbers could be defined in terms of rmin and rmax. If a disturbance imposed on the equilibrium flow has modulus greater than rmin instability is possible whereas if the modulus is greater than rmax instability is assured. We note that if ,8 is a measure of the modulus of such disturbances then the amplitude dependent Reynolds numbers would both be of the form

Re =Re-0()2

in the neighbourhood of Re. It is of interest to note that Re, the critical Reynolds number of linear stability

theory, given by (5.6), can only be obtained by the consideration of nonlinear theory. Clearly this apparent inconsistency is because the basic equilibrium flow is non-zero in the forced problem.

In figure 1 we have plotted the critical Reynolds number given by (5.6) as a function of e. In figure 2 we have shown the limit cycle solution of (3.18) for a value of Re2/Re, -20.0. As stated above, the limit cycle is not present when Re2 > Re* If the limit cycle solution is disturbed such that the total amplitude lies inside the limit cycle, the solution spirals in towards the stable focus. On the other hand, if the disturbance is such that the total amplitude lies outside the limit cycle then the solution spirals to infinity.

When Re2 is positive, we find that there is again only one singular point except when Re2/Re, lies in the range (19.0, 20.95). For values of Re2/Rec inside this range, (5.2) has three real positive roots so that three singular points exist. In figure 3 we have shown the dependence of the real and imaginary parts of the singular points on Re2/Rec. We can show by standard phase plane methods that in the regions where there is just one singular point the singularity is either a stable or unstable



Stability of forced Poiseujille flow 467

Re

5600

5200 -

4800 -

4400 -

4000 L I I I I 0 0.0008 Q0016 0.0024

FIGURE 1. The critical Reynolds numbers associated with the forcing of the fundamental (lower curve) and the first harmonic.

AE

/ ~~~~-0.04\ / ~~~~~~stable focus \

-0.08 -0.04 0 0.04 0.08 A

\-0.04- - unstable limit/

\ ~~~~~~cycle /

FIGURE 2. The phase plane description of (3.18) for Re2 =-20. Rec.

focus depending on whether Re2/Rec is less or greater than -9.23. In the region where three singular points exist, there is always one saddle point and an unstable focus. In figures 4, 5 we have shown the dependence of the singular points of (3.18) on Re2/Re, in the region (19.0, 20.95) in more detail. We further note that, as shown in figure 3, the amplitude of the singularity behaves like IRe2 -' for large values of



468 P. Hall

-0.08

AlJ/ (A )r

40 80

-80 -40 0 =0Re,/Re,

(Av,)

-0.04-

FIGURE 3. The singular points of (3.18) for different values of Re2IRec.

(AE),

0.06

0.04 -

0.02_

0

-0.070 . I II - - s1

200 22 Re

FIGURE 4. The real part of AE in the region (19.0, 20.95).

(A8,)i

-0.03 -

-0.05 - s

-0.071III 18 20 22 Re

FIGURE 5. The imaginary part of AE, in the region (19.0, 20.95).




JRe2 . We have shown some phase plane trajectories of (3.18) for Re2/Rec = 19.5 in figure 6. We see that at this Reynolds number there exist two unstable foci and one saddle point. There are no limit cycles at this value of Re2/IRe so that any disturbance to the equilibrium flow associated with any of these three singularities ultimately leads to an infinitely large amplitude motion. The precise path which the amplitude takes to infinity will clearly depend on the phase of the disturbance to any of the equilibrium solutions. For large values of IA I the trajectories appear to correspond

Al

0.4

0.04 0.08

-0.08 -0.04 0

FIGURE 6. Some typical phase plane trajectories of (3.18) for Re2fRe0 = 19.5.

to a single unstable focus at the origin. When Re2/Rec decreases the singular points lying in the third quadrant approach each other and annihilate each other when Re2/Re, = 19.0. Before this happens the unstable focus first changes into an unstable node. On the other hand, when Re2/Re, increases, the saddle point and the focus in the second quadrant approach each other before annihilating each other when Re2/Re, = 20.95. The unstable focus again changes into an unstable node before this occurs.

Next we consider the amplitude equation (3.27) obtained when the wave travelling on the upper boundary is the first harmonic of the critical wave of linear stability theory. The singular points AE of (3.27) are found to be determined by the equations

Jh1+ g 12 jAE12 - AE2 I (Re2IRec) dl + e IAEI212,

AE= 2 - jAE12(h + gl) (5.7a,b) ( Re2 fRec) d1 + el IA, 12 ' (.,6



470 P. Hall

We see that the origin is a critical point of the amplitude equation. The nature of the singular point can be investigated by standard phase plane methods. If we define the constants Res ReT and Re, by

Res = h1d 1 = 27.93, ReT - jell 74.38

Re, = _hjglI - 28.56, (5.8a, b, c) dl

then we find that

(a) If Re2 Re`j1 <- RI, the singularity is a stable focus. (b) If - Re, < Re2 Re-' < - Res, the singularity is a stable node. (c) If -Res < Re2 Re- < Res, the singularity is a saddle point. (d) If Res < Re2 Re`j < Re, the singularity is an unstable node. (e) If Re, < Re2 Re- 1, the singularity is an unstable focus.

/ ~~~~0.04--

f-0.08 -0.04 0.04 0.08/ A, stable focus /_

-0.041 - unstable /lmit cycle

\ ~~~-0.08- /

FIGURE 7. The unstable limit cycle of (3.27) for Re2 -30. Re,.

Furthermore, in the region where the singularity is stable (i.e. Re2Recj <-Re5) an unstable non-circular limit cycle surrounds the singular point. Such a limit cycle is shown in figure 7 for Re2 =- 30. Re,. We note that a disturbance to this solution giving a total amplitude inside the limit cycle gives a solution spiralling into the origin whilst otherwise the amplitude spirals away to infinity. When Re2 > - Res Rec the singular point at the origin becomes an unstable saddle point. However, the limit cycle solution is still present.

Apart from the singular point at the origin we can show that when Re2 =- Res Rec two further singular points, having the same value of IAEI and differing in phase by X arise at the origin and move outwards as Re2 increases further. These singularities are initially stable foci. When Re2= Res Re, two further unstable singular



Stabtlity of forced Poiseutlle flow 471

points move away from the origin and annihilate the other two singularities when Re2 = ReT ReC.

Increasing Re2 from -Res Rec we find that when Re2= 17.69Re, the two foci change into unstable foci and the limit cycle disappears. We note that for Re2 in the range Rec(- 27.93, - 17.69) two possible stable non-zero steady solutions exist. In contrast to the unforced case we see that the zero equilibrium solution is unstable for Re2/Rec in this range. A disturbance to the zero solution, provided it does not lie outside the limit cycle solution, will tend to either one of the foci depending upon its amplitude and the phase. The situation is shown clearly in figure 8 where we have shown some trajectories of (3.27) for Re2/Rec -24. We note that two of

FIGURE 8. The phase plane description of (3.27) for Re2/Rec = -24.

the lines leaving the saddle point spiral into the foci whereas the two lines entering the saddle point spiral out towards the limit cycle solution whenT -? - oo. We can see that an infinitesimal disturbance to the zero solution of the form A = Cei' spirals into the singularity in the first quadrant if 0 in the range (6, X + d) where 6 is the angle shown in figure 8. A disturbance to the limit cycle solution can lead to two possible stable solutions. The two trajectories shown in figure 8 spiral towards the limit cycle solution as r -? - oo. Thus, there will be an infinite number of adjacent regions lying inside the limit cycle which are such that a trajectory beginning in such a region will eventually cross the Ai axis between the points p and q or between -p and - q where p and q are as shown in figure 8. These regions become infinitely thin as they approach the limit cycle. Moreover, of the trajectories beginning in adjacent regions near the limit cycles, one crosses the Ai axis in (p, q) and the other in (- p, - q). Trajectories crossing in the former and latter intervals spiral towards the singularities in the third and first quadrants respectively. However, any



472 P. Hall

disturbance away from the limit cycle spirals to infinity. As stated earlier the limit cycle solution disappears and the stable foci become unstable when

Re2IRe - -17.69.

Thus, we can define the critical Reynolds number

Be = 5774. {1-17.69c...} (5.9)

0.06 -

(A}<)i I - 0. 04 - /

0.02 - -17.82 \74.38

-20 0 20 40 60 80 0,02 -

Re2lRe,

-0.04 _-,

FIGURE 9. The dependence of the imaginary part of the singular points of (3.27). (Note that the axis (AE)i = 0 is always a solution.)

to be that Reynolds number at which an infinitesimal disturbance to the basic flow will grow indefinitely. However, the flow is unstable when Re is greater than the value 5774 . {1 - 27.93e. ..} but any disturbance in this case tends to either of the stable non-zero steady solutions if Re is less than the critical value given by (5.9). In figure 1 we have shown the dependence of Re with C. In the range of validity of (5.6) and (5.9) the most dangerous mode of forcing is that of the fundamental. In figures 9 and 10 we have shown the dependence of the singular points of (3.30) on Re2IRe,. We note that since if AR is a singular point of (3.30) then so is -A E, the reflection of the curves in the horizontal axis are also solutions. The only stable singular points are those corresponding to the zero solution to the left of

Re2IRec - - 17.69 and the region X Y of the bifurcating solution.

When Re2/Rec reaches the value 27.93 two further singular points arise from the saddle point at the origin. When this bifurcation takes place, the saddle point changes into an unstable node. These further singular points are unstable and eventually approach the other two singular points away from the origin. Eventually the two pairs of singularities annihilate each other when Re2/Re, = 74.38 and then only the singularity at the origin remains.

Finally, we turn our attention to the amplitude equation (3.38). The critical points of this equation are found to be given by the solutions of the equations

A -= hIAEI4/{(Re2/Rec) d] + k, + e, IAE1j2},

11121AEI4 = IAEEI2 I(Re2IRec)d, +kl+el IARI22. (5.10a,b)



Stabbility of forced Poiseuille flow 473

We see from (5.10) that the origin is again a singular point. In addition, there are a further six unstable singular points when Re2/Rel lies in the range (Re,, Re2) where Rel and Re2 are the solutions of the quadratic equation

{2[xd, i, +kiei]r-1112}2 = 41el121Xdl+k112. (5.11) If we substitute for d,, el, k1, 11 from ? 4 into the above equation, we find that

Rel = 30430,

Re2 = 33980. (5.12a, b)

(AE)r

0.12-

0.08 -

0.04 -

:-17.82 7438

o X -40 0 40 Re2/Re. 80

FIGuRE 10. The dependence of the real part of the singular points of (3.27). (Note that the axis (AE)r = 0 is always a solution.)

The singular points arise with non-zero magnitude and initially all lie on a circle in the phase plane. When Re2/Re, increases from the value Re,, the singular points split into two groups of three, each group lying on a circle in the phase plane. Initially there are four saddle points and two unstable nodes. The nodes first become unstable foci and then unstable nodes again. The circles coalesce when Re2/Re = -Re2 and the singularities disappear. The singularity at the origin is a focus and is stable or unstable depending on whether Re2/Rec is less or greater than the critical value Re2/Ree=-k rldrI 28310. (5.13)

In addition for Re2/Rec less than this critical value an unstable non-circular limit cycle encloses the origin. Thus, the basic flow becomes unstable to small disturbances when Re is greater than the critical value

Re = 5774{1 + 28310e2 ..}. (5.14)

so that the forcing is stabilizing in this case. Indeed, the correction term is relatively large compared to the corresponding

terms in (5.6), (5.9) even for quite small values of e. In figure 12 we have shown



474 P. Hall

Re, given by (5.14) as a function of e. In figure 1 1 we ha-ve plotted the solutions of (5. 10b) which have IAE 2 real and positive. Each of the non-zero solutions leads to three singular points when they are substituted into (5.10a). However, as stated above, all these non-zero solutions are unstable, the only stable flow being the zero flow for Re < Re.

IAEI 0.8 /

/ -

0

30400 31600 32800 34000 Re2 /Re,

FIGURE 11. The solutions of (5. 10a). Each point on the curve corresponds to three singular points in the phase plane.

Re-

7000-

6000 -

5000 0 0.001 0.002 0.004 e

FIGuRE 12. The critical Reynolds number given by (5.14).

6. EXTENSION TO MORE GENERAL FORCING CONDITIONS

In ? 3 we have isolated three particular forcing conditions and investigated the response of the fluid to this forcing in a neighbourhood of the neutral point of linear stability theory. For the sake of completeness, we now consider the case when the upper boundary is defined by

y = 1 + e exp iA(x-jut) + complex conjugate. (6.1)




We stipulate only that A and It are not given by any of the three cases discussed in ? 3. In such a situation we can expand the Reynolds number Re in the form

Re = Re, + c2Re2 + ...

and then V[ in the form

V = AcE/fo + A6LjNO + cE1 21 + 6E_ 0[1O + 0(6)2 (6.2)

where E is as defined in ? 3 and E1 = exp iA(x - jtt). The order 62 terms include all possible terms due to the interaction of the order c terms with each other. It suffices for our purpose to note that the amplitude function determined at order 63 satisfies an equation of the form (6.3). The term proportional to t1 in (6.3) arises from such interactions as that of the fundamental with the correction to the mean motion produced by the forcing.

dA/dr = (Re2fRet) d1 A + e1 A2A + t1 A. (6.3)

Here di and e1 are as defined in ? 3 and t1 is a constant determined by the order 6 and 62 terms in (6.2). Thus, in this case, the critical Reynolds number for the flow is decreased by the amount 62tir Re, dirl. Moreover, the limit cycle associated with (6.3) for Re2 less than - tir dj-' is circular as is the case in the unforced problem and the only singular point of (6.3) is at the origin. Thus, we see that it is only when the forcing motion is as given in ?? 3 a, b that the critical Reynolds number can be decreased by an amount greater than order 62.

We now investigate the case when the boundary conditions contain more than one wave. We note that the basic flow set up by such a motion can be found by expanding T in the form 00

V- I enVfl(x, y,t) n=1

If the forcing motion does not generate (if at all) the fundamental until order cm with m > 3 then it can be shown that the Reynolds number is simply shifted by an amount of order 62. However, if the fundamental is generated at order 62 then the corresponding change is of order c'. In the latter case the phase plane description of the amplitude of the forced motion of fundamental is similar to that of (3.18).

We conclude this section by making the following statements: (a) If the motion on the upper boundary has the fundamental component of

linear stability theory in its decomposition at order c, then the critical Reynolds number is decreased by an amount of order el. For values of the Reynolds number less than the critical one there exist a non-zero stable equilibrium flow proportional to the fundamental. In addition, there is a periodic solution corresponding to a limit cycle solution. This solution is unstable and disappears when the stable solution disappears. There also exists a finite range of Reynolds numbers after the limit cycle has disappeared for which two further steady unstable equilibrium points exist.

i6 Vol. 359. A.



476 P. Hall

(b) ff the motion of the upper boundary contains at order c the first harmonic of the fundamental component but not the fundamental itself, then the critical Reynolds number is decreased by an amount of order e. The only stable suberitical flow proportional to the fundamental in this case has zero amplitude up to a certain value of the Reynolds number at which the latter flow becomes unstable and two possible non-zero stable steady equilibrium flows then exist. When the Reynolds number increases further all the possible equilibrium flows are unstable and the unstable limit cycle enclosing the origin disappears.

(c) If the fluid motion induced by the boundary motion has no first harmonic term of order e and no fundamental term up to order e2 then the critical Reynolds number is either increased or decreased by an amount of order 62. The only stable subcritical flow proportional to the fundamental is the one with zero amplitude. We further note that the unstable suberitical motion which also exists is associated with a circular limit cycle in the amplitude phase plane only if the second harmonic is not present in the boundary conditions.

(d) If the motion of the boundary generates the fundamental at order 62 in the fluid motion then we have a situation identical to (a) above except that the decre- ment ei is replaced by et.

7. DiscussioN OF RESULTS

In figures 1-12 we have shown the critical Reynolds numbers given by (5.6), (5.9), and (5.14). We see that the effect of the forcing destabilizes the flow when either the fundamental on first harmonic is forced. It is of interest to compare the orders of magnitude of the changes in the critical Reynolds numbers from the unforced value with corresponding quantities in similar problems. In particular we discuss the results of Hall (I 975 b) in connection with the stability of plane Poiseuille flow modulated by allowing one of the boundaries to oscillate with amplitude y say. For small values of y it was found that the critical Reynolds number was decreased by an amount of order y2. Indeed, in several related problems in which some basic flow is modulated by some small velocity or temperature fluctuation of order y the change in the critical values of the appropriate stability parameter is found to be of order y2. (See, for example, Venezian (i969), Rosenblat & Herbert (I970) in connection with Benard convection, and Hall (I975a) and Seminara & Hall (I 975) in connection with centrifugal instabilities.) The reason for the relatively weak response in these problems is that the forced motions investigated do not lead to a resonant response of the type discussed by Kelly & Pal (I976) and Hall & Walton (I 977).

However, it is interesting to note that in the absence of the fundamental, the most dangerous mode of forcing is the first harmonic which leads to a change in the critical Reynolds number of the same order of magnitude as the forcing. This is surprising since this mode of forcing does not lead to resonance so that it might be expected to lead to a change of order of the square of the forcing amplitude.




Finally, we close by discussing how the analysis of ? 3 can be generalized to take into account spatial modulation of the disturbance in the manner of Stewartson & Stuart (I97I), Hocking, Stewartson & Stuart (I 972), Davey, Hocking & Stewartson (I974). We restrict our attention only to the case of spatial modulation in the x direction and follow the approach of Stewartson & Stuart (I97I). The latter authors have shown that if a slow variable in the x direction is defined by

= IRe Recjidlr(x + art) (7.1)

where d1 is as defined by (3.19a) and a,r, the group velocity, is a real constant defined i terms of an integral condition involving the eigensolution V/o and its adjoint 0 (3.18) is generalized to give an equation of the form

aA + 2A Re2 d, A +e,,A2(7 ~+b~=~% - ei A(7.2)

where b is a constant again defined by an integral condition. For certain values of b, dl, el it can be shown that the solution of the above equation develops a singularity at finite values of T and 6 when some localized disturbance is imposed on the flow at - = 0. The analysis of Stewartson & Stuart can be extended to take into account the forcing conditions discussed in ? 3. The appropriate generalizations of (3.18), (3.27), (3.38) which take into account a slow spatial variation as defined by (7.1) are

A A 2A R2 2A +fi,

- + b- = Re2 d, A +elA2A + f1, + h, A) aT a2 Re, +

aA a2A Re22 2 T+b = Re d A+e1A2A+11A2+k,A. (7.3a,b,c)

The determination of the properties of (7.3) represents a non-trivial computational problem buLt it would be surprising if the extra terms present greatly alter the conclusions reached by Hocking & Stewartson (1972) for the unforced problem. This is because when the solution begins to 'burst' the essential balance is between the terms on the left hand side of (7.3) and the cubic term. Since the extra terms are proportional to the amplitude or its square, we would not expect them to be important where and when the solution 'bursts'. For the values of b, dl, el appropriate to plane Poiseuille flow Hocking & Stewartson (I 972) found that no 'bursting' solutions develop. Thus it would be of interest to know if the extra terms in (7.3) can alter the latter result. Clearly, this question cannot be answered without extensive numerical calculations.

The amplitude equations (7.3) have been derived assuming that the wave on the upper boundary is unmodulated in time and space. We now let the upper wall be defined by

y = +eS(6, T) exp inx(x - et) +CO8MPLTEX n = 12, 3. i6-2



478 P. Hall

Here S(6, T) is some given function of the slow variables 6 and r and allows for a slow modulation in time and space of the wave on the upper boundary. The appropriate generalizations of (7.3) are

Oa + b ag2A Re2 d, A + el A2 + 8l (6, -r) A+71S()A OA 62A -e

+b Be2d A +eA2A + , (6,-r)A+h 86)A

67 6J62 Re, I +

OA a2A __ 22T)1 - +b- = dd A + e A2A + I1 S(, T) A2 + kIS(, T)2A. (7.4 a, b, c) 6fJ -r 62 Re-c

I F

We note that the function S(6, T) is a given function. The determination of the properties of (7.4) again requires a great deal of numerical work. However, it is interesting to speculate whether a 'bursting' solution for A can be induced by letting S(6, T) become singular at some finite values of 6 and T. If this were the case then we would have a possible mechanism for the generation of 'bursting' motions by external forcing. This idea might have some relevance to boundary layers since it suggests that a bursting type motion in the free stream might induce a similar motion in the boundary layer.

The author acknowledges the help of Dr P. K. Sen in connection with the com- putations described in ? 4 and some useful discussions with Professor J. T. Stuart.

REFERENCES Bouthier, M. 1972 J. Mec. 11, 599. Bouthier, M. 1973 J. Mec. 12, 75. Budiansky, B. I974 Adv. appl. Mech. 14, 1. Criminale, W. 0. 1971 I.U.T.A.M. symposium on unsteady boundary layers, p. 940. Davey, A., Hocking, L. M. & Stewartson, K. 1974 J. Fluid Mech. 63, 529. Gaster, M. 1974 J. Fluid Mech. 66, 465. Hall, P. 1975 a J. Fluid Mech. 67, 29. Hall, P. I975 b Proc. R. Soc. Lond. A 344, 453. Hall, P. & Walton, I. C. 1977 Proc. R. Soc. Lond. A 358, 199-221. Hocking, L. M. & Stewartson, K. 1972 Proc. R. Soc. Lond. A 326, 289. Hocking, L. M., Stewartson, K. & Stuart, J. T. 1972 J. Fluid Mech. 51, 705. Jordinson, R. 1970 J. Fluid Mech. 43, 801. Kelly, R. E. & Pal, D. 1976 Proc. 1976 Heat and Mass Transfer Institute. Meksyn, D. & Stuart, J. T. 1951 Proc. R. Soc. Lond. A 208, 517. Pekeris, C. C. & Schkoller, B. I967 J. FluidMech. 29, 31. Pekeris, C. C. & Schkoller, B. I969 J. Fluid Mech. 39, 611. Reynolds, W. C. & Potter, M. C. I967 J. Fluid Mech. 27, 465. Rogler, H. L. & Reschotko, E. 1975 SIAM J. appl. Math. 28, 431. Rosenblat, S. & Herbert, D. M. 1970 J. Fluid Mech. 43, 385. Schubauer, G. B. & Skramsted, H. K. 1947 NACA Report no. 909. Seminara, G. & Hall, P. 1975 Proc. R. Soc. Lond. A 346, 279. Stewartson, K. & Stuart, J. T. 1971 J. Fluid Mech. 48, 529. Stuart, J. T. I960 J. Fluid Mech. 9, 353. Stuart, J. T. I97I A. Rev. Fluid Mech. 3, 347. Thomas, R. L. H. 1953 Phys. Rev. (2) 91, 780. Venezian, G. I969 J. Fluid Mech. 35, 243. Watson, J. i96o J. Fluid Mech. 9, 371,



the effect of external forcing on the stability of plane poiseuille flow

Documents