Derivation of amplitude equations and analysis of sideband instabilities
in two-layer flows
Michael Renardy and Yuriko Renardy
Department of Mathematics and ICAM, Virginia Polytechnic Institute and State University, Blacksburg,
Virginia 24061-0123
(Received 2 February 1993; accepted 24 June 1993)
Sideband instabilities following the onset of traveling interfacial waves in two-layer CouettePoiseuille flow are considered. The usual Ginzburg-Landau equation does not apply to this
problem due to the presence of a long-wave mode for which the decay rate tends to zero in the
limit of infinite wavelength. Instead of the Ginzburg-Landau equation, a coupled set of
equations for three amplitude factors is derived. The first corresponds to an amplitude of a
traveling wave, the second to a long-wave modulation of the interface height, and the third
results from the pressure. The criteria which determine the stability of the primary traveling
wave to sideband perturbations are presented. This scenario raises the possibility that as a result
of sideband instability of a primary traveling wave, the flow may eventually be dominated by a
long-wave mode. Experimental data on a gas-liquid flow are analyzed and models for air-water
waves are discussed. Finally, it is noted that the amplitude equations allow for possibilities other
than periodically modulated waves. In the concluding section, the presence of homoclinic and
heteroclinic orbits is investigated. These correspond to solutions which approach either a flat
interface or periodic waves at infinity.
1.INTRODUCTION
Many bifurcation problems in fluid mechanics involve
one or more spatially unbounded directions and a continuum of modes. A full description of the set of bifurcating
solutions in such a context is a rather formidable problem
that has not been solved. In the analysis of bifurcation, one
typically imposes some artificial periodicity on the problem
and then confines attention to those solutions satisfying
this given periodicity. The question then arises whether
such bifurcating periodic solutions are stable even under
perturbations which do not satisfy the periodicity requirement. Sometimes they may be unstable with respect to
slowly varying modulations.
This type of instability, known as sideband instability,
has been investigated by many authors. 1-7 To investigate
the issue, one derives an amplitude equation known as the
Ginzburg-Landau equation. This equation involves an amplitude factor for the critical mode which is allowed to vary
slowly as a function of rescaled space and time variables.
In the present paper, we are interested in the onset of
traveling waves on fluid interfaces in plane parallel shear
flow. Two immiscible liquids of different viscosities and
densities with surface tension at the interface lie between
parallel walls. The flow is driven by a combination of the
motion of the top wall and a pressure gradient in the direction of the flow. The arrangement with a flat interface is
a steady solution of the governing equations (Sec. II). The
linearized stability analysis of this solution 8 in terms of
eigenfunctions proportional to exp (iadx + ia2 y +At) yields
the eigenvalues A. At low speeds, the interfacial eigenvalue
determines the instabilities depending on the fluid properties. The interfacial eigenvalue is neutrally stable for the
wave-number pair (0,0). This study focuses on the situa2738
Phys. Fluids A 5 (11), November 1993
tion where the interfacial eigenvalue is also neutrally stable
at a critical wave-number pair (a,0), together with the pair
(- a,0), and is stable at other wave numbers. The weakly
nonlinear interaction of the wave at (a,0) with itself and
with the wave at (-a,0) determines whether the bifurcation is supercritical or subcritical. 9 If the bifurcation is
supercritical, then primary traveling waves are generated
at the interface as the bifurcation parameter, such as the
Reynolds number or the viscosity ratio, is raised past the
critical value. The present paper will address the stability
of the traveling wave solution to variations at large length
scales.
This situation does not fit into the usual framework of
the Ginzburg-Landau equation. The reason is that, in addition to the critical mode leading to traveling waves, there
is an additional neutral mode at zero wave number, which
corresponds to a shift of the interface. In a strictly periodic
situation, one can simply fix the amplitude of this neutral
mode to be zero, e.g. by requiring constant average interface height. If slow modulations are allowed, however, one
cannot ignore long wave modes. As we shall derive below,
one obtains instead of the Ginzburg-Landau equation a
coupled set of equations for the amplitudes. One of them,
denoted by A, is the amplitude of the primary traveling
wave mode and the other, denoted by B, is the amplitude
of a long wave. Here A is complex and B is real. In the
three-dimensional case, there is a third equation involving
a (real) pressure amplitude P. This pressure amplitude is
already present for shear flows of a single fluid.3 Coupled
amplitude equations involving long wave modes and modes
of finite wavelength have also arisen in convection problems, but the form of these amplitude equations is different
from ours.10 "' The forcing of a mean flow mode by slow
modulation of a traveling wave has also been noted by
0899-8213/93/5(11)/2738/25/$6.00
© 1993 American Institute of Physics
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Benney and Roskes 2 in their study of the stability of
water waves.
The amplitudes are scaled by a small parameter e.
Moreover, the space and time coordinates are rescaled as
t=e(x-cgt), 77=ey, r=et, where x is the streamwise coordinate, y is the unbounded coordinate perpendicular to
the streamwise direction, and cg is the group speed of traveling waves near the critical wave number. Finally, the
bifurcation parameter is scaled by c2. With these scalings,
amplitude equations of the following form will be shown to
result:
2
Ar~egt
A +5A B + epP¢A,
A.= EVAl~~+e A7+ eaA + I A
B= (cg-co)Be+e B+cWB,7,7+e2 (BP¢)¢
+2e (BP,)7+eI(B 2)+e(
1A I 2)g,
2
(1)
e(P9+P71)=ErO( 1A12)g+sOB£.
Here y, K, /J, 13, S and p are complex and the other
coefficients are real; co is the propagation speed for long
waves. We note that the set of equations (1) still contains
e. Indeed, there is no single scaling of the independent
variables under which e disappears from the equations. In
the study of eigenvalues for sideband stability in Sec. IV,
various scalings by additional factors of e will appear.
In the two-dimensional case, i.e. if there is no dependence on 77, we can integrate the last equation of (1) to
obtain ePg=erOJA12 +soB+eK, where K is a constant of
integration. We can then insert this result into the remaining equations and obtain a coupled system of just two
equations. They read as follows:
A,= eyAg;+E(a+ pK)A +e(,B+ pro) IA 12A
+ (S +pso)AB,
2
B= (cg-co) B¢+ej7Bg.+
E(/3+;so) (B )¢
+,ES'( IA12)¢(2)
We have neglected terms of order e2 in the second equation
of (2).
We note that if we ignore the second equation and the
term involving B in the first equation, then (2) reduces to
the usual Ginzburg-Landau equation. 9 If, on the other
hand, we ignore the first equation and the term involving A
in the second equation, then we have Burgers' equation,
which was derived by Hooper and Grimshaw 12 as an amplitude equation for long waves.
The problem of sideband instability for the GinzburgLandau equation has been studied extensively. If we ignore
the term AB in the first equation of (2) and scale time with
an additional factor e, then e scales out of the equation and
one finds a problem of the form
A,=,yg+oA +[JA 12A.
(3)
If Re y> 0, Re &> 0 and Re 13<0, this equation has traveling wave solutions of the form A =Ao exp (ipg+ior) as
long as M2 < Re /Re y (in terms of the original problem, a
nonzero ,u means that the wavelength of the traveling wave
is not exactly the critical wavelength, but differs from it by
2739
Phys. Fluids A, Vol. 5, No. 11, November 1993
order E). Eckhaus 4 assumes that Y,3 and 6 are real and he
shows that the traveling wave solution is unstable if
A2 > 6/( 3 ). If the coefficients are not real, then sideband
instability is possible even if p=0. This was first observed
by Benjamin and Feirl for an equation modeling water
waves. This system is conservative, and the coefficients in
the Ginzburg-Landau equation turn out purely imaginary.
The analog of the Benjamin-Feir criterion for dissipative
systems was derived by Lange and Newell. 5 Stuart and
DiPrima 7 study the problem for general coefficients and
general ti, thus achieving a unified treatment of the Eckhaus and Benjamin-Feir
instabilities. Davey, Hocking and
Stewartson 3 take account of the pressure mode in threedimensional shearing flows; their set of amplitude equations is equivalent to the first and third equation in (1)
(without the terms involving B). The recent work of
Cheng and Chang13 stresses the analogy in the derivation
of Ginzburg-Landau amplitude equations with center
manifold reduction. This analogy is also exploited in our
derivations below. Unfortunately, analysts have thus far
been unable to extend rigorous proofs of the center manifold theorem to situations like ours, which involve a continuum of modes. Cheng and Chang 13 report some discrepancies between their criteria for sideband stability and
earlier work. These discrepancies are due to errors which
can be traced back to the omission of the m =0 term in
their Eq. (22).14
Blennerhassett' 5 derives one amplitude equation for
traveling interfacial waves which allows for slow modulation. He does not show at any stage a coupled system for
two amplitudes such as (2). It turns out that his amplitude
equation can be derived as a special case from (2). To do
so, we scale B and
with an additional factor e. After
a/ar
doing so, e factors out of the first equation of (2), while the
leading order terms in the second equation reduce to
(cg-co)Bg+8(JA f2),=O. After integrating with respect
to the spatial variable ¢, this equation is satisfied by setting
(cg-co)B+±A12 =0. It is difficult to point out exactly
where this integration is performed in Ref. 15, since the
details of the lengthy calculation are omitted. However, it
is clear that somewhere in the derivation, there must have
been a spatial integration of the kinematic free surface condition [our Eq. (14) below]. Inserting this relationship between B and A into the first equation of (2) yields an
equation for A alone:
AS=yAg+aA+ (+pr 0 -
c3- C0
(8+Pso)
IAI 2A
The coefficients in this equation are discussed further in
Sec. V. The primary traveling wave solution of this equation is not the same as the one we shall investigate. Instead
of having a zero perturbation to the average interface
height, his involves a shift in the interface position which is
proportional to the square of the wave amplitude. While
this yields a mathematically valid solution, we find it difficult to interpret this procedure physically. This amplitude
equation is also incomplete; it would fail to predict sideband instability to long waves. In contrast, the approach of
M.Renardy and Y. Renardy
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our paper takes into account all modes which are close to
neutral stability and might therefore lead to sideband
instabilities.
Chen and Joseph' 6 (see also Ref. 17) simply set B=0.
This is a consistent assumption as long as one is interested
in periodic traveling waves, and indeed Ref. 16 is more or
less exclusively concerned with periodic waves. For studying sideband instabilities, however, it is not correct to set
B=0. This is because of the forcing term proportional to
IA | I, which only vanishes if there is no dependence of IA I
on
Recent experiments on pressure-driven gas-liquid systems described in Refs. 18 and 19 show the existence of the
sideband instability of a traveling wave, with the possibility
of subsequent feeding of energy into a mean flow mode.
Past analyses do not account for the growth of the mean
flow mode, and this motivated us to undertake this study.
We discuss the data of Ref. 18 in Sec. V.
Our paper is organized as follows. In Sec. II, we introduce the governing equations for the problem of parallel
Couette-Poiseuille flow of two fluids. The onset of traveling waves (without attention to sideband perturbations) in
this problem was studied in an earlier paper. 9 In Sec. III,
we show how (1) is derived and how the coefficients can be
calculated. In a formal sense, the derivation of the amplitude equations is analogous to center manifold reduction.
In Sec. IV, we derive criteria for sideband instability of
traveling waves of Eq. (1). We shall confine attention to
the case where the wavelength of the traveling wave equals
the critical wavelength. The general case of wavelengths
close to, but slightly different from the critical wavelength
is much more complicated and remains to be settled in
future investigations. Our numerical results are presented
in Sec. V for several models of gas-liquid flows. In Sec. VI,
we discuss some spatially nonperiodic solutions of (2).
These solutions asymptote to constant or periodic solutions
as ¢-=
oo. The leading order contribution for small e can
be given in explicit form. Solutions of the same nature have
been studied previously in the context of reaction-diffusion
equations.2022
II. EQUATIONS GOVERNING TWO-LAYER
COUETTE-POISEUILLE FLOW
Two fluids of densities pi (ii1,2), and viscosities jt,lie
in layers between infinite parallel plates located at z* =0,1*.
Asterisks are used for dimensional variables. The upper
plate moves with velocity (Up* ,0,0) and the bottom plate is
at rest. In the basic flow, fluid 1 occupies 0<z*(-1* and
fluid 2 occupies ltbz*<l*. The velocity of the interface in
the basic flow is (U*(1* ),0,0) and for brevity, we denote
U*( l*) by Ui. The velocity, distance, time and pressure
are made dimensionless with respect to Ui, 1*, I*/Ui, and
piUi. In Couette-Poiseuille flow, the basic flow has a pressure gradient - G* in the x-direction. Reynolds numbers
in each fluid are denoted by R,= UI**pI/1uL and
R 2 = Uil*P21A2. There are seven dimensionless parameters:
a Reynolds number, say R 1, the undisturbed depth 11 of
fluid 1, a surface tension parameter T7(surface tension
coefficient)/(Qi 2 U), a Froude number F given by
2740
Phys. Fluids A, Vol. 5, No. 11, November 1993
F 2 = Ui/gl* where g is the gravitational acceleration constant, a dimensionless pressure gradient G= G*l*/
(PIUt), the viscosity ratio m=1L1//' 2 , and a density ratio
r=p
1 /P2-
The dimensionless basic flow (U(z),0,0) is
I
GR;zA/2+cjz, O<z<11,
11 Z 1,
(4)
where
cl= (l+GRI12j12)11j,
12= 1-11,
(5)
c2 =m(-GRj+cj),
and the upper plate speed Up is
2
U~~=lym
1
Up +11
l - m1 2 GR, .
(6)
The basic pressure field P satisfies dP/dx =
dP= [- 1/,
Tz t- 1(rF2),
G and
-
0<z<l1,
(7)
11 <Z< 1.
Solutions that are perturbations of the above basic flow
are sought. The perturbations to the velocity, pressure and
interface position are denoted by (u,v,w), p and h, respectively. The Navier-Stokes equations in each fluid yield
au
au
at ax
- + U -+
av
-t+
at
aw
-t+
av
I
Ri
Uy---MuV+--=
Ax
pi ap
1
Wu,--
Ri
Au +-
Au
-
mu 7--V-_w
pi ax
Ax
pi ap
av
av
pi Y
u-UX-V
--
ax
aw
I
piap
Ax
Ri
pi z
UT--Aw+-y=
-U
Au
Ay
T~z
av
W
ay
aw
aw
ax
ay
-V
au
--
azZ-,
(8)
aw
wT.
az
Incompressibility reads
au av aw
(9)
AX-_+_+-=0.
y az
The boundary conditions are u=v=w=0 at z=0, 1. The
conditions at the interface are posed at the unknown position z=1j+h(xy,t). Since the unknown h(xy,t) will be
assumed to be small, the method of domain perturbation is
used, that is, the interfacial conditions are expanded as
Taylor series about z=l1 and truncated; our analysis requires terms up to cubic order. Continuity of velocity
yields
h2
au
F[t*a--2
1
h 2 a2u
-21a
rav 1 h2 a2 Vl
Go]
|2 L}
-V|d
Qjw1t=-h
(10)
IaWx
h2 i
2
wl
a2 ,
where Ix] denotes x(fluid l)-x(fluid 2). Continuity of
shear stress yields
M. Renardy and Y. Renardy
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j(Du + w\1
af/Du
a
wy I
Dh
aD a~l
2
aD2 U
Dh
~xI
h
Dw)
jDu+
ik za
Dx
a_\tDh(au _aw\ 1
Dx DyaixzDyz
~~
(D~~+vahD2w\
ptDwv aw
~_zXfZlT
. 1/Dv
(a + ap
L
(av
awl+
~t(a2~D2 v
D2V
2
D
U
DhL~t(
D2wa\
~~~
i~~TyZ
a hL
D2~~~\
V
aw
a
a 2
a-h +2hli
Dx
Dyh~1(\Dc' - -y-2
za(D w ~ Dh lq 4 D
(D
h'~2
-
ftf Dav
hajj
a
yx/V\D
Dw7
Daw
a2
U+
ay
x~
2
V
x~t-fl(1 1)
where ,i denotes Iif for fluid i. Here, [p U"]J =0 in the basic flow from the normal stress condition. The balance of normal
stress yields
[ 2j
dw
IR 1 Dz
1
T
mR,
adP]
d 2 w dp1 h2 f 2fL
hf 2A
Dz
dz
aRi 2 R, 1 3u
1
Ah 2
+ R
+
(ah\
2
h-y)
L
jT
Dh Ah2
/D2V D2 w \1
j
-l 2+azay)il
19(ua
2
yZRItuly axJ
ax dX
d2h (dh2 2
i
)
-
Dh
Dh
at+ U(l0aD- wW)
-h y- a Dzz-+ 2
Ah
_ 2
-Z2ax
avPT
(Dh 2
kay))
ah av(l)
(3
(13)
where the subscript 1 here refers to fluid 1. We remark that
there is an error in the statement of the normal stress
condition in Ref. 9. Instead of TU"], it should read
[(ut/tl) U"], which is actually zero. Unfortunately, this
error was also in the computer program. Most of the results listed in Ref. 9 are for plane Couette flow, however,
and hence unaffected by the error. The affected results are
discussed and corrected at the end of Sec. V.
In order to fit our equations into the abstract scheme of
the following section, we have to reformulate the kinematic
Phys. Fluids A, Vol. 5, No. 11, November 1993
X
ah ( (h\ 1
2
2mRIdx-,
&Z]
222 a
(Dh
2
5y)
_
11
-RylIdx
y7]
(ah\)
2
)
5x2{dA2(h2
dx
D2h
Ah Ah
2T
mRDxDyi x Dj
(12)
J
h2
u dz- hu()- 2 U'O)
h2 Au(l) h3 U,,
- 2
Dz - 6 U(1)I
r)
-u(1)T
V()y -h aDy D9z'
2741
1
7ax)
Ah a rI
at TX -U(li)h-
Dz
Ah h2
Ld
Ah 2 fPtd Avw
'ay R1 - -dz'y
free surface condition and the incompressibility condition.
First, we put the kinematic free surface condition in divergence form by using the incompressibility condition to replace the terms involving w in (13). This yields
The kinematic free surface condition yields
Ah Au(,) h2 D2w(l)
dh\2 2
D2w
x}
+ -vt+h(Vh)2rA<+ -
E{-
2mRl
;
{ah\ 2 pD/ 2u
tax) R1 lazax
dDw\9
a
+ Tx
ax
T
a2
a3W
a-f'
vdz-hv(,--_h DUI)
(14)
We note that the spatial average of the right-hand side in
(14) is zero if an average if such an average is meaningful,
e.g. in a spatially periodic situation. This property will be
important below.
During the course of the analysis, we shall need to
consider the equation of conservation of fluid volume, integrated over z. However, if (9) is integrated over z, the
term resulting from aw/Dz does not vanish, due to the
jump of w at the interface. We shall use (10) and the
incompressibility constraint to express tw] in divergence
M.Renardy and Y. Renardy
2741
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form. To this purpose, we multiply the first equati(ion in
(10) by ah/ax and the second equation by ah/ay and1then
add the two resulting equations. Ignoring fourth- order
terms, we find
ah
a (h 2\
Y-xt 2
ah
;X2+Y
a (h2 avi
au
'3|a--yt2 )3z]
( 15 )
%_az
-6 Uy
-2 )
We next use the incompressibility condition to replac nethde
terms involving aw/az on the right-hand side of the third
equation in (10). By adding (15) to the resulting equalation,
we find
a (h^
2
auv
h2\3
(16)
We now modify (9) by subtracting (16) from it:
III. DERIVATION OF THE AMPLITUDE EQUATIONS
In order to derive the amplitude equations to be analyzed, it will be convenient to write our equations in an
abstract schematic form. We shall use boldface letters to
denote the "abstract" variables. We let u denote the set of
unknowns, i.e. the quintuplet (u,v,wp,h). We shall regard
u as a function of the independent variables x, y and t,
taking values in a function space X, which incorporates the
z-dependence. An element of X consists of the four functions u(z), v(z), w(z), p(z) and the scalar A, where u, v
and w are required to vanish at z=0 and z= 1 and u, v, w
and p are smooth except for jump discontinuities at
z=1
1.Equations (8), (10), (11), (12), (14) and (17) can
be written in the form f= 0, where f takes values in a
function space Y. An element of the image space Y is
represented as f= (fi) 14 1. Here fl through f3 are associated with Eq. (8) for fluid 1, f 4 through f6 with Eq. (8)
for fluid 2, f7 with Eq. (17), f8 through flo with Eq. (10),
fl 1 andf 12 withEq. (11), f 13 withEq. (12) andf 14 with
Eq. ( 14).
We can represent our equations in the schematic form
au av aw
ax ay+ az -w
M(R)ut=L(RD,,,Dy)u+N2(RD,,,Dy;uu)
x(h[u]+ 2
U-z+-21U']+ -i U"|
+N3,(RD.,,Dy;uuu).
_a
(17)
(ht l +h' av)
We note that if we integrate with respect to z, there
n the
terms faw/laz dz and w]J cancel each other. All other
|
terms in (17) have either x- or y-derivatives on theym. In
particular, in the two-dimensional case, the integrrat of
(17) with respect to z yields
a+(
f
u(z) dz+h[uI±-2[±U]+u 6U-1)=i0.
(18)
(18)
The perturbation of the total flow rate is
11+h
j-
(1I
U(C)(Z)+U(l)(z) dz+
+U(2 )(z) dz-
f
U(2 )(Z)
U
U(l)(z) dz-
fi U( )(z)
2
dz
(19)
and Taylor expansion in h up to third order yield
isthe
quantity in brackets in (18). If we integrate (18)
with
respect to x and set the constant of integration to zero,
xtheIn
we are dealing with a situation of constant volume flu
Ix. In
studying two-dimensional flows, the situation of fixed
flow;
rate is often considered, and we have now made exillicitt
how it arises as a special case in our formulation.
Henceforth, we replace Eq. (13) by (14) and Eq I.(9)
by (17).
2742
(20)
Phys. Fluids A, Vol. 5, No. I1, November 1993
Here M and L are linear operators; the notation
L(RDX,,Dy) indicates that L is a differential operator involving derivatives with respect to x and y: See the Appendix for the explicit form of the operators. Moreover, N2
and N3 are symmetric bilinear and cubic terms, i.e.
N 2 (RDxDy;uv) is linear with respect to the arguments
u and v and N 2 (RDxDy;uv)=N2(RD.,Dy;vu); similarly N 3 (RDxDy;uvw) is linear in u, v and w and invariant under any perturbation of these arguments. We
need not consider nonlinearities of higher than third order.
Again, the listing of Dx and DV indicates the dependence
on derivatives of u as well as u itself. The number R is one
of the parameters on which the coefficients depend and
which will serve as a bifurcation parameter (in two-fluid
flows, R may be any of a number of things; it could be the
Reynolds number, but also, for instance, the viscosity ratio). All parameters other than R are considered fixed and
given, and we shall not carry them explicitly in the equations.
Our problem is translation invariant in the x- and
y-directions, i.e. the coefficients have no explicit x- or
y-dependence. In the linearized problem, we can then use
separation of variables: if we set u=v exp(ikx+ily+At),
where v does not depend on x, y or t, then the linearization
of (20) assumes the form
,tM(R)v= L(R~ikJi)v.
M. Renardy and Y. Renardy
(21)
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As usual, we call A an eigenvalue if (20) has a nontrivial
solution v. The quantity L (R,ik,il) is known as the symbol
of L; we can think of it as a function of three scalar variables. The derivatives of the symbol will be important later;
we denote them by subscripts, e.g. L 2 is the derivative with
respect to the second argument.
It can be shown that a Fredholm alternative holds for
(21), i.e. if A is not an eigenvalue, then the equation
AM(R)v-L(Riki1)v=f is uniquely solvable, and if A is
an eigenvalue, then there are some solvability conditions
whose number equals the dimension of the eigenspace.
A. Properties essential for the derivation of the
amplitude equations
In the following, we shall derive amplitude equations
for the situation where (21) has some neutral eigenvalues,
while the rest of the eigenvalues are stable. The amplitudes
appearing in the equation will correspond to factors multiplying the neutral eigenfunctions. We shall now state the
crucial properties which enter into the derivation of these
amplitude equations. The first three of these concern the
eigenvalues and eigenfunctions of (21). In general, we can
say the following:
(i) For k=l=0, there is an eigenvalue 0 that is independent of R. Corresponding to this zero eigenvalue, there
is a two-dimensional eigenspace spanned by two eigenvectors ao(R) and at, i.e. L(R,0,0)aO(R)=L(R,0,0)k=0.
Moreover, there are adjoint eigenvectors bo and bo such
that (bo,L(R,0,0)v) = (bO,L(R,0,0)v) =0 for every v. We
have (bO,M(R)ao(R))#70, while M(R)AD=O and
(bo,M(R)v) =0 for all v. We can normalize the eigenvectors ao(R) and b0 relative to each other so that
(bO,M(R)aO(R))=l. (As reflected in our notation, it
turns out that only ao depends on R, while k, bo and bo do
not.)
In the spatially periodic problem considered in Ref. 9,
we have set the average value of (bo,M(R)u) =h, or
f f h dx dy (the integration is over one period), equal to
zero. However, if spatial modulations are allowed, there is
in general no meaningful average.
The eigenvectors a( and aO, belonging to the eigenvalue
zero, are given as follows:
aO=(uO(z),OOpO(z),l), az=(0,0,0,l,0).
(22)
That is, ao corresponds to a shift in the interface position
and adjustment of the velocity and pressure, and k corresponds to a constant pressure with no change in the flow.
To determine uO and po, we have the equations
1
Tu
f
{
2
dP
1/F 2 ,
,
AHLI
l+m1
U0
~mU'I(l-z)
11 m12
Po=-Tz=
(25)
for l1 ~z(1,
,
for 0.~z<11,
2
1/(rF ),
for l1 ~z~l.
(26)
According to the Fredholm alternative, we have two
solvability conditions for the equations L(R,0,0)v=f.
From the right hand sides of (14) and (17), we find
that these solvability conditions are f14=0 and
ff 7 (z) dz=0. This yields the adjoint eigenvectors
ofo)
=fI4,
(b 0 ,f)
=
f A(z)
7
dz.
(27)
We are interested in situations where, apart from the
neutral modes for k=l=0, we have a neutrally stable
mode for some nonzero wave number. It can be shown
numerically 9 that this is the case for certain parameter
combinations. Specifically, we shall look at cases where the
following two conditions hold:
(ii) At R =R,, there is a simple imaginary eigenvalue
ico for k=ko and 1=0. Let a, and b, be the corresponding
eigenvector and adjoint eigenvector, i.e. icoGM(R,)a 1
= L(Rc,ikO,O)a1 and (b1 ,iJoM(Rc)v-L(Rc,ikO,O)v) =0
for every v. In a generic situation, (b 1 ,M(R,)a 1) will be
nonzero, and we assume this in the following. We can then
choose the normalization (b1 ,M(R,)a 1 ) = 1. Correspondingly, we get an eigenvalue - iwo at k = - ko and 1=0, with
the eigenvector i2f and adjoint eigenvector b1.
(iii) Except for the eigenvalues given by conditions (i)
and (ii), all other eigenvalues have negative real parts for
R =R,. The eigenvalue at icwo moves into the right half
plane for R > R, and into the left half-plane for R <R.
If the flow is periodic in x and y, then integration of the
right-hand side of ( 14) with respect to x and y yields zero.
Similarly, if we integrate ( 17) with respect to x, y and z, we
also get zero. Note now that Eqs. (14) and (17) are associated with the components f14 and 17, respectively. Using
(17) and (14), we obtain:
(iv) If u is periodic in x and y, say with periods Y and
Z, then
To
N 2,(R,Dx,Dy;u,u) dx dy)=0,
(b0 , fZ
~f0 N3 (R,DxDy;u,u1,u) dx dy) =0,
=0,
(23)
~
(28)
f
N2 (R,Dx,Dy;u,u) dxdy)=0,
(i~oJ'
with the boundary conditions uO=0 and interface conditions
lU a1=rIU'J,
u6 ' = 0,'
[Vol
I i
These equations are solved by
2743
for 0 <z<11,
Phys. Fluids A, Vol. 5, No. 11, November 1993
(24)
(
JO
r
{
N 3 (RDxDy;u~u~u) dxdy)= 0.
The significance of this property in the derivation below is
that it forces a number of terms to vanish which might
otherwise be there.
M. Renardy and Y. Renardy
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2743
Another property which will force some terms to vanish is symmetry. Since our equations originate from perturbing a two-dimensional base flow, we have a reflection
symmetry in the y-direction. This reflection consists of reversing the y-direction and at the same time reversing the
velocity field in that direction. We can express this as follows:
(v) There are linear operators II in X and HI' in Y
2
such that H2=1,
(H') 2
=1, and with the property that
Below, we shall frequently take the inner product
with bo and b0 . The form of the expressions
(boL(RDxDy)u) and (b0 , L(RDXDy)u) will therefore
be of interest. From (27) we see that applying b0 to
L (R, Dx,, Dy) u corresponds to integrating the linear part of
(17) with respect to z. We obtain the following: (vii) There
are elements w* and w* in the dual space of X such that
(boL(RDxDy)u) = (w* ,Du) + (w* ,Dyu).
Here, we have the representations:
[I'M(R) =M(R)rli, r1'L(RD,,,Dy) =L(R,D,,-Dy)I1,
N 2 (R,D,,-Dy;HuHu)=rH'N 2 (RDxDy;uu)
and
(w* ,u)= fu(z) dz,
N 3 (R,DX, -Dy;rlulu,rlu)=H'N3 (RDX, Dy;uuu).
The reflection operators II and HI' are given as follows:
flu= (u,-v,wp,h),
r'f=
(fl,
- A,
(w2*,u)= fv(z)
A3, f4, - A,
-f9, f1o, f
11 1-f
12,
A6,
dz.
7, A8,
113, f14)
(29)
That is, HIsimply reverses the velocity in the y-direction.
Condition (v) implies in particular that, if u(xyt) is a
solution of (20), then lIu(x,-yt) is also a solution. We
shall refer to uEXas even if fIu=u and odd if lIu=-u. A
corresponding terminology applies to elements of Y. Elements of the dual space (i.e. the space of linear mappings
from X or Y to the complex numbers) are called even if
they annihilate odd vectors, e.g. be Y* is called even if
(b,v) =0 for every odd ve Y. Similarly, elements of the
dual space are called odd if they annihilate even vectors.
We easily see from (22) and (27) that the eigenvectors
ao(R) and A) as well as the adjoint eigenvectors bo and
b0 are even. Moreover, a, corresponds to a twodimensional velocity field (v=0) and is hence also even.
Since the Fredholm alternative can be applied to even and
odd vectors separately, this also forces b, to be even. One
of the amplitudes in our equations will be that of a longwave pressure modulation. The rest of the properties we
shall list are related to this aspect of our problem, which is
responsible for much of the complexity of the derivation.
To derive the form of the pressure equation, it is essential
to observe exactly how the pressure appears in the equations. We observe that the pressure appears only in the
Navier-Stokes equations (8) and the normal stress balance
(12). Moreover, a pressure which is independent of z
does not affect (12). It appears only in (8) via the terms
ap/ax and play. We can state this as follows:
(vi) There exist elements wbw 2 eY such that
L(RDXDy) (P o)
0 = (DXP)wi+ (DyP)w 2 . In addition,
N 2 (RD.,Dy;u + Fao,u + Fa0)=N2 (RD.,Dy;uu) and
N 3 (RD., Dy;u + Piou + P5ou + PS0 ) = N 3 (RD.,
Dy;uuu). Moreover, w1 and w2 are in the range of
L(R,0,0).
In fact, we have the simple representation
w1= (- 1,0,0,-r,0,0,0,0,0,0,0,0,0,0),
w2 = (0, - 1,0,0, -r,0,0,0,0,0,0,0,0,0),
(30)
which we can directly read off from (8). The entries in
(30) are simply the coefficients of the pressure derivatives
in (8).
2744
(31)
Phys. Fluids A, Vol. 5, No. 11, November 1993
In an analogous fashion, we find from (14):
(viii) There are elements rear) and r*(R) in the dual
space of X such that (bo,L(R,Dx,Dy)u)=(r*(R),Du)
+ (r* (R),Dyu).
Explicitly, we have
(r*,u)=-U(1
1 )h- -
u(z) dz,
Jo
(32)
(r*,u)=-f
2
f~o,
v(z) dz.
Below, we shall need to solve problems of the form
Lu=g with given g. Of course, the solution, if it exists, is
only determined modulo an element of the null space of L.
To fix solutions uniquely, we require them to lie in a subspace X0 of X which is complementary to the eigenvectors
ao and ik. A convenient choice is
X
=fI(u~v~w~p~h) I h= f p(z) dz=0J.-
(33)
We note that X0 is invariant under the reflection 11. Also,
we have
(bo,M(R)v)=0
for
veX0 ,
(34)
because (bo,M(R)v) is the "h-component" of v and X0 is
a subspace of functions for which this component is zero.
In the context of deriving the pressure equation, it will
be necessary to consider the effect of a superimposed pressure gradient on the flow. Let now vi be the modification to
the parallel shear flow which results from superimposing a
unit pressure gradient in the ith coordinate direction
(without moving the interface). That is, vi satisfies the
problem L(R,0,O)vi= - wi. Taking note of the expression
for L in the appendix, we can find the explicit form
V1=(u1 (z),0,0,0,0), V2 = (0,u 1 (Z),0A00),
(35)
where ul (z) satisfies the equation [cf. (8)]
M. Renardy and Y. Renardy
2744
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1
Ri
(36)
Pi
h =e(usai
2(
0f0i 0)t
e
(41)
2q(uU2(Z)ati)o
where U2 satisfies the equation
zero boundary conditions and the interface conditions of
continuity of velocity and shear stress
(42)
(37)
fu]J=o.
[UiI=o,
Further, we find
This leads to the solution
'3 =(U3 (Z)+ U4(Z)A0W
RIz2
Rlz
+ml[ 2
2
2
U 1(Z)=
M 12'
Rlm(l-z)
2
ml 2
-2_
mR 1(l-z)
],
[m
l[
.1+m1 2
for 0<z<1
1,
1l+
X4= (0U4 (zA00,O),
±1J
_2
X6= (U3(Z)A00,W3(Z),P3(Z)).
(38)
We note that the following relationships hold:
(ix) We have
(wa ,vl (R) ) = (wn*v2d(R) ) =
2
The equations determining the components of X3 through
X6 are as follows:
-Ru4= Uu1 , u4 (0)=u4(1)=0, [u 4 l =O,
I-
u I (z) dz
W3 -W
(r* (R),vl (R) ) = (r2*(R),v2(R) ) = -J
I=-ui+
3
dz.
U(Z) C
If a and [3 are constant, then N 2 (R,Dx,D,;av1(R)
+,/v 2 (R),av1 (R)+,/3v 2 (R))=0. For arbitrary (i.e. not
necessarily constant) a and /3, we have (boN 2(RD.,
Dy;avi(R) +fv 2 (R),av 1(R) +/v 2 (R))) = (bO~,N
2 (RD.,
Dy;avt(R) +flv2 (R),av 1(R) +1v 2 (R))) =0. This is because the nonlinearities in all the equations except (8)
vanish if h=O, and the nonlinearities in (8) vanish for
parallel shear flows.
We define the following projection from Y onto the
range of L(R,0,0):
1rof=(Al A2, A I f4,I5A Ih
As fs t, ft 1s f12, fl 3 f°).
Let Xi(R) denote the solutions in X 0 of the following equations:
L(R,0,O)Xl (R) =iro(M(R)v2 (R)),
L (R,O,O)X2 (R) = ro(M(R)V2 (R) )
3 (R) =
L (R,0,0)X 4 (R)
=
Pi
Pt
3
,I It
-
21 _
UR1 u
(45)
=0,
3- 3
0
C1
dz=O,
(46)
1
-u3 t '=U'W
3 , U3 (0)=U3 (l=0,
1U
=0,
3]1
(47)
_Uu=0.
(W*,X3(R) ) = (W2*,X4(R) +X5(R) )
+ (w*',X 6 (R)), and (r*(R),X3(R))=(r*(R),X4 (R)
+Xs(R)) + (r*'(R),X 6(R)).
The properties (i)-(x) are the essential properties of
our equations which will be used in the subsequent derivation of the amplitude equations. The last two properties
(ix) and (x) will imply that certain coefficients in the
pressure amplitude equation are equal to each other and
hence the derivatives of the pressure amplitude appear only
in certain combinations.
~-iT0 (L 2(R,0,0)v 1 (R)),
- iro(L2 (R,0,0) v 2 (R) ),
(40)
L(R,0,0)X 5 (R) =-1ro(L3(R,0,0)v1(R)),
L(R,0,0)X 6 (R) = -7ro(L
3
(R,0,0)v 2 (R)).
By using the definitions of the various terms in (40), we
find that
2745
tw3hO0,
We can verify that the following holds:
() (wi ,(R)(2*,X2(R)),
(r*(R),X1(R))
fl
(39)
L(R,0,0)X
uuI(z) dz,
W3 (0)=W3 ()
=(r* (R),X2 (R) ),
foA(Z) dZ, Af,
u4] =0,
(44)
f~~~~~~o
and
_
~~~~~~~~
_. _n
1
for l1 z< 1.
1
3 (Z),P3(WM0,
12,
Phys. Fluids A, Vol. 5, No. 11, November 1993
B. The amplitude evolution equations
We assume that R is just above R,: R=R,+E2, and we
look for small amplitude traveling wave solutions with a
slow modulation: u = ev (kox+ cot,Ex,Ey,Et) = :Ev( g,7,Q,r).
Here ko and co are the critical wave number and frequency
as given by assumption (ii). For solutions of this form,
(20) takes the form
M.
Renardy and Y.Renardy
2745
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1
M(R,+e2) (coov~+ev1 .)
=
eL 3 (R,,kOD~-,O)v,1+-
L2L(R, ,k0 Dg,O)vg
L(R,+E2 ,koD~+eDg,ED,7)v
2 (R,+
+,EN
+E2N
3
+ e2L23(RkOD¢,)vb+
E2 ,kOD~+ ED.4,ED,~;v,v)
(48)
(R,+ e2,kOD 4 +EDg,ED,7,;v,v'v).
We now expand (48) in powers of e, retaining only terms
up to order e2. For the linear terms, this leads to (see the
Appendix for notation)
eIL 3 3 (RckoD¢,O)v7.
(50)
For the nonlinear term N2 , we get an expansion of the
form
N 2 (R,+ E2,kOD¢+eDg,eD,;v,v)
=N 2 (R,,koD¢,0;v,v) +eM 2 (D-;v,v¢)
2
M(R,+e2) =M(R,) + e M'(R,),
(49)
(51)
+eQ2(D¢;vv,7),
with M2 and Q2 denoting appropriate quadratic terms. In
N 3 , we only need the leading term given by
N3 (R~,koDDA;v'v'v).
We use these approximations and, in addition, expand
v'(g,77r)e'n¢. This
v in a Fourier series v(g,47,,-) = n=X
yields the infinite system
L(R,+e2,kOD¢+eD¢,eD,7)v
= L(R,,kOD,O)v+e 2L1(R,,kOD¢,O)v
+eL 2 (R,,kOD¢,0)va
incooM(R,)v'+e~inooMA'f(R,)v + eM(R,~)vn,
=L(R~,inkO
,O)Vn± E2L 1 (R ,ink 0 ,Q)Vn ± EL 2 (RC,JnkO,0)Vn + L 3 (R~,JnkO,0)v'+_ E2L22 (R~,inko,0)V"
±e2L23 (R~,Jnko,0)V- +-&2L
+
X
e 2M 2(D~-;vm exp(img),v-
00
+
33 (Rc,inko,0)V¾+
exp(i(n-n)g)) +
-
r=-w
-~~~
E2Q2(D~;vm~'exp(img),v-
m
exp(i(n-m)g))
7
e 2N 3(Rc,koD~,0;vm exp(img),vrexp(irg),Vnrn`exp(i(n-m-r)~)).
To satisfy these equations, we now assume, in analogy
to center manifold reduction, that the behavior of the solution is dominated by the neutral eigenfunctions ao, ko
and a,. This leads us to the expressions
vo= Bao(R,) +PaO+ep&vl(R,)+eP7V2(R,) +eg,
v'=Aal+ ef,
(53)
and all other Fourier components are O(e2). Here f and
g are such that (b1,M(R,)f)= 0, geXO, and A, B and P
are scalar factors. We note that in the third equation of
(1), there is a factor e in front of Pg-+Pr,7,, but no factor
e in front of B,. Consequently, we must allow derivatives
of P to be of order lIe relative to B. For this reason, we
have written out the terms involving Pg and P,7 in (53)
2746
X
m=
00
I
M=-M
m
EN 2(R,,koD~,0;vm exp(im~),v' -mexp(i(n-m)~))
Phys. Fluids A, Vol. 5, No. 11, November 1993
(52)
rather than incorporating them into g. In the following, we
shall also allow for the possibility that derivatives of P are
of order lIe rather than order 1. This is the reason for the
inclusion of terms which are seemingly of higher order in e.
We introduce the notation
rrlf=f - (blf)M(R,)a1.
(54)
That is, Tr, is a projection onto the range of
i6)GM(R,) -L(RJiko,O). We shall also use the projection
ia as given by (39).
We insert (53) into (52) and then consider the n = 2
component, the projection 1r, applied to the n =1 component and the projection ro applied to the n = 0 component.
This yields the following equations for f, g and h (at leading orders in e):
M. Renardy and Y. Renardy
2746
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2icooM(R,)h= L(Rc,2iko,O)h+A2N 2 (Rc,kOD¢,O;aI exp(ig);a1 expire)),
io)OM(R) f = L(Rc,7ikO) f +Ag1 (L 2(R ,ikOO)aj) +A r1
(55)
(IL3 (R ,iko,O) a1)
+2AB1TIN 2 (RckODDO;aI exp(ig),aO(Rc)).
(56)
iro(M(Rc)ao(Rc))B,+ecro(M(Rc)vl(R,))Pg,-ke~ro(M(Rc)V2(Rc))P,7,
= L(Rc,O,O)g+ B¢iro(L 2 (R,,O,O)ao(R,) ) + B,71ro(L 3 (RO,O)ao(R,) ))+EPgpro(L 2 (R ,O,°)vi(RC))
+ eP7grO(L3 (RC,0,0)v1 (Re)) +-EPgirO( L 2 (R ,AO,)v
+ B21roN2(R&,koD
+2E1rjON
2 (R,) ) +
,O;aO(RC),aO(RC)) +21 AI 2 ,romA2(RT
kODI
eP1711O( L 3 (Rc ,0,O)v 2 (Rc))
,O;aj expire) ,11 exp (-ig))
2 (RC,kOD0,O;PgvI(R,) +P~V2 (R,),BaO(R,)).
(57)
To simplify notation, we make the following definitions:
01= (2icoGM(R,) -L(Rc,2iko,0)
02= (i)oM(Rc)-L(Rc,iko0 ,)
) -N
2 (RrkoD;,0;a,exp(ig),aj
expire)),
) - lr'N2 (R,,koD¢,0;aI exp(i>),ao(R,)),
03= -(L(R,0,,o)) )-'roN2 (Rc,koD~,O;ao(Rc),ao(Rc)),
04= -(L(R,,O,O)
) -'vcON2(R^,kOD;,Oaj exp(i¢),iij exp(-ig) ),
V1J2= (i6o)M(R,) - L(Rc,ikoO) ) - 11r,( L3 (Riko,O)ai),
1/1=
- L(cik-o( 2 R,OO
'-rao(R~),kOa,
43= - (L(R,,O,0)) -'irO(L2(R,,O,O)ao(R,)),
-(iLG(R~,0)
4= - (L(RC,0,0)) -11 rO(L3(R,,O,0)aO(R,)),
ip5= (L(R,,OO))
iYO(ML(R,)aO(Rc))
)
(58)
XI= (L(Re,O,O) ) -'1rO(M(Rc)vj(&)),
X2 = (L roomO)
17ro(M(Rc)V2 (R,) )
X3 = - ((Rc,0,0) ) - 17ro (L2 (Rc,0,0) v I(R,))
X4 =
- ((R
,0,0) ) - 1iro (L2 (ROO)
V2 (R,) )
X5 = -(L(Rc,,0)
)- liro(L3(R,,OO) v 1(Rc))
X6 =- (L (RC,0,0) ) -i1To(L
3 (RC,0,0)v 2 (RC)),
X7= - (L(R,O,O) ) - 1 ro(N2 (RkoD¢,O;vI (R,),ao(R,)))
X8= -(L(Rc,0,0) ) -17ro(N
2 (Rc,koD¢,O;v 2 (R,),ao(Rc))).
We need to explain the meaning of (58), since, e.g. L(RC,O,O) has no inverse. However, the restriction of L(R,O,O) to
X 0 has an inverse, and similarly the restriction of ioGM(R) -L(RC,iko,O) to the subspace of all vectors f with
(b1 ,M(Rc)f) =0 has an inverse. The inverse operators occurring in (58) are to be interpreted in this fashion. With the
notations of (58), we can rewrite (55)-(57) in the form
h=A2
01,
f=2AB0
2
+A
1 I+Anb 2
,
g=B2 )3 +2 1A 12 q54+B) 3 + B#/ 4 + B143 + E(xlPr+x2Prr+x3P2+
X
(X4+X5)Pgq±+X6Pip77+2X7BPt+2X8BPrl)-
(59)
In the following, the symmetry condition (v) will make a number of terms vanish. We note that it follows from (v)
that 01, 02, 034), 04I
1,
03 XI, X3J X6, X7 and vl(R,) are even, while #2, 1/14, X2J X4J XS, X8 and v2(R,) are odd.
We now consider the n = 0 component of (52) and take the inner product with bo. We retain terms up to order e2,
except for terms involving the pressure P, which we shall need to retain up to order e3 . The conditions (iv) and (v) force
2747
Phys. Fluids A, Vol. 5, No. 11, November 1993
M. Renardy and Y. Renardy
2747
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many terms to be zero. By symmetry, (boq)
is zero whenever qb is odd. Also, such terms as, e.g.
(bo,N 2 (Rc,koD¢,0;ao(R,),ao(Rc))) must vanish by condition (iv). Taking this into account, we obtain, after division
by e,
B= (r* (Rc),ao(Rc)) Bg+e(r*(R,), 0 3) Bg+e(r4*(R,),04
4 ) B,7,7(r* (R,),b5) B
+e(r*(R,,),vl(R,)) (Pgg+P,7) +E(r* (R.
+E(boM
2 (D¢;ao(Rc),ao(R,)))BBg+e
2
03(2)g+er
(R)X
2)g
(boM 2 (D¢;ao(R,),vi(Re)))BPgg
+e2(bo,M 2 ((D4 ;v1 (Re),ao(Rc)))BpgP+e2(boQ 2 (D;;ao(R.),v 2 (R,)))BP,,,7
+e2(boQ2(Dg;;v2(R,),aO(Rc))) BvP7+,E(bO,M2(D;;;ajexp(ig),Yal exp(-i0) ))a~
+.E(bO,M2(D~;Yj exp(-ig),aj exp(i;))),AAg+e2(r*(R,),Xl)(P~gfP7
+e(i(RC),X3) (Pggg+P,7,7g) +2e2(r* (R,),X7) (BPg)g+2e2(r2*(R,),X8) (BP,7) 7.
We have used (ix) and (x) to simplify the expressions. The terms in (60) which involve AAg and AAg combine into
s,(IA j 2)g+isj(AA'g_-AA-), where
s: =(bO,M2( D;;aj expire),!, exp(-ig) ) ).
(61 )
2
If we set A =exp (ijt4), then we obtain sr( IA1 )g-+isi(AAg-AAg) =2,usi. This is compatible with assumption (iv) only if
si=O. Hence s is real. An analogous argument shows that the terms involving BP4t and BgPg must combine into a total
derivative, hence
(bOM
2 (Dc;ao(Rc),vi(Rc)))
= (bOM
2 (D4
;v1 (R,),ao(R)) )
(62)
and similarly,
(bo,Q 2 (D¢;aO(R.),v 2 (R,))) = (boQ 2 (D¢;v 2(R,),ao(R,))).
(63)
Next we take the n =I component of (52) and take the inner product with bl. Again we retain terms up to order
e2, except that we shall neglect terms such as e2ABg, eAgB and e2AB 2 relative to e AB. After dividing the equation by e,
the following is obtained:
A= -eiwO(bi,M'(R,)a1 )A +e(bi,Li(Rc,ikO,O)ai)A + (bi,L2(R ,ikO,O)a1 )Ag
1
2 -f(b ,L33(Rc,jko,°)aj)A777
+2e(b,,L22(R,,ikO0)a,)Agg+
e(b ,Lz(Rc ,iko0)+1 )Agg+2+e(bl ,L3(R,,iko,0)0,2)A,7,1+2(bl ,N2(R,,koD¢,O;aI exp(ig),aO(R,) ))AB
+2c(bl ,N2(R,,kOD¢,O;aj exp(ig),vl (R.) ))APg
+3e(bi,N3 (R,,kOD;,O;ai exp(ig),a1 exp(ig),Tj exp(-ig))) JA j2A
+4e(bi,N2 (R,,kOD¢,O;ai exp(ig),0 4 )) IA j2 A+2E(bi,N2 (R,,kODg,0;0j exp(2ig),TI exp(-ig))) JA12A.
(64)
Finally, we take the n =0 component in (52) and take the inner product with b0 . At leading order in e, this yields the
following equation
0=-E(w? ,v(R,) ) (Ptg+Prq)+ (w* ,ao(R,)) B±+e(w ,' 3)Bg+e(w'*,04) B7
+E(W* ,115)Bg +e(w ,03) (B 2 )g+2e(w*',0
4)
(IA 2l' )s+e(bo,M
2 (Dg;ao(Rc),ao(Rc))) BB9
+e 2 (b 0 ,M 2 ( D;;aO(R.),v1 (Re))) BPg+e 2(b 0 ,M2 ( D.;vl (Re) ,aO(R,) ) ) BgPg
+e2(bo,Q2(D¢;ao(R,),V2(Rc) ))BP,7,7+e2(bOQ2(D¢;V2(R,),ao(R,))) B't,7
+e(e
ig)))Ai~+e(bO,M2 (D);Tj exp(-ig),a1 expire)) )AAg
+
X2wl)(P~gg+P,7) + e(Wl*,X3)(Pgg+Pg77)+2e2(w*,X7)(BPg)g+2e (W2*X8) (BP7)17.
(65)
As above, condition (iv) implies that the coefficients of AAg and AAg in (65) are real (and hence equal), and that the
coefficients of BPa and BAPg and those of B 7 and B1 P1 7 are equal to each other.
In all order e3 terms in (65), we can replace Pgg+P,,7by theleading order term -(w ,ao(Rc))Bg/e(w ,v1 (RI)) We
now solve (65) for P±+P^,7, and insert the result into (60). We obtain a system of the form
2748
Phys. Fluids A, Vol. 5, No. 1 t, November 1993
M.
Renardy and Y.Renardy
2748
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A4= -cgA¢+eyrAg+ EKA,±+euA +Ef1A I2A+6AB+4EpPA,
B,=-cOBg+&7jB4e2>eBpe,(2ge(I
12)g+62p(Bpg)g+e4(Bpw,
E(Pg+Paw) =ero( IA 12)+soB,
(66)
where the last equation results from truncating (65) after the leading order terms. [The higher order terms in (65) had
to be carried along because they influence the first equation in (66), which results from inserting (65) into (60).] We can
transform (66) to (I) by using a coordinate system that moves with speed cg and replacing Bg7 by -coBr, which is
correct at leading order in e.
The coefficients in (1) are given as follows:
cg-
(biL
r=-2(b
1 ,L
2
22
(RcikoO)a,),
(Rc,ikO,O)aj) + (b 1 ,L 2 (Rc,iko,0)4' 1 ),
1
K=-(bl,L3 3 (Rc,iko,O)aj) + (biL 3 (Rciko,0)*2 ),
a= -io)O(b 1 ,M'(R,)a1 ) + (b 1 ,L
1
(RcikO,0)a1 ),
, = 3(bj,N3(Rc,kODC,O;ajexp(ig),ajexp(ig),Tal exp(-ig))) +4(bj,N2(R,,kOD¢,O;aj expU0g,04))
+ 2(bwN2(RckOD,
+
0;0jexp(2i g),Tjexp( -ig)
)),
5=2(brN2(R,,kODRO;a+ exp(ir),aO(R,) ) )
p=2(brN2(RckOD.[R;aj exp(iR)
so=
wv±(Rc)
)
r
(wrR,aO(Rc))1(w* ,vX(R))),
ro=-[2(w* ,04) + (60,M2(D;;aj exp(ig),Wal exp(-id) ) )]/(w* ,vj(Rc) ),
i'=iY - COi72,
co=--(r*(Rc),aO(Rc))(+R(r)(R,))v
(R,)(
,ao(Rc)) ) /(w*
+
',va(Rc)) J
Y7l= (r* (Rc),iP3) -[ (r* (R,),vl (Rc) ) (WA +) + (r* (Rc),X3) (W* ,ao(Rc) ) ]/(w* ,vl (Rc) )
+ (r*(R,),vl (Rc)) (w*,X3) (w
,ao(R,))/(w* ,vl(Rc) )2
y72= (r*(Rc),f5) - [(r* (Rc),vl (Rc)) (wl ,05) +(r* (Rc)Xl) (w* ,ao(Rc)) ]/(w*,vl (Rc))
+ (r
(Rc),vl (Rc)) (w*
',y) (w*
,ao(Rc))/(w
,vl (Rc) )2
i=(r2*(Rc),0~4)-(r* (Rc),vl (Rc) ) (w2* ,i4)/(w* ,vj(Rc) ),
8=2I(r*(Rc),0 4 ) + (bo,M 2(D.;a
1 exp(ig),ij exp(-ig)))
-(r
(R,),vi (Rc) [2(w* ,04) + (bO,M 2 (D¢;aj exp(ig),Ty exp -ig) ) ) /(w*,vl (Rc)).
We did not list the expressions for /3, p5 and k, because
these coefficients do not enter into the analysis of sideband
instability. To evaluate the coefficients numerically, we actually did not use the formulas above in all cases, but tried
to use our pre-existing programs, written for the calculation of the Landau constant in two-dimensional flows, as
2749
Phys. Fluids A, Vol. 5, No. 11, November 1993
much as possible in order to reduce both the programming
effort and the potential for error. The coefficients of the
linear terms in (66) can be related to the eigenvalue problem (21), since the eigenvalues for (66) must agree with
the eigenvalues of the original problem at leading orders.
Let A,(R,k,l) be the eigenvalue with AIj(RckoO)=iao.
M. Renardy and Y. Renardy
2749
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Then
aAI
-aR
all
Cg-1 7k
l a2.Z,
r=--2 ja
K
l a 2 A1
2-2w.
(67)
Let 2lO(R,k,l) denote the eigenvalue which assumes the
value A.O(R,0,0) =O. Then we have
aAO
.~
-
1 2A o
coEak '=-2
I 2AO
'K=-2
v*
(8
In two-layer shear flows co, f and K have been calculated
explicitly using an asymptotic expansion for long waves. 23
In our computations, we use a linear stability program and
difference approximations for the derivatives in (68). In a
similar fashion, we can identify 6 as the derivative of Al
with respect to a shift in the interface (for fixed pressure
gradient), and p as the derivative of Al with respect to the
prescribed pressure gradient. Moreover, the Landau constants for the cases of fixed pressure gradient (FPG) and
fixed volume flux (FVF) are /3 and 1+ pro, respectively,
and these constants have been computed in Ref. 9. Using
all this information, we have only 6 and so left to be determined, for which we use the formulas above. A more explicit description of the calculation of the coefficients is
given in Sec. V below.
As an aside, we mention that the Landau constants for
FPG and FVF can differ quite substantially in two-layer
flows; even the signs can be different. 9 Since the sign of the
real part of the Landau constant determines whether the
bifurcation is supercritical or subcritical, this means that it
may be supercritical in one case and subcritical in the
other. There is no contradiction in this. Suppose, to be
definitive, that we have pure Poiseuille flow and the bifurcation parameter is the Reynolds number (measured in
terms of the pressure gradient for FPG and in terms of the
flow rate for FVF). For the unperturbed parallel flow,
pressure gradient and mass flux are of course proportional
to each. Suppose instability occurs at some value of the
Reynolds number, where the pressure gradient is 20 and
the flow rate is 10. Suppose, moreover, that a bifurcated
solution has pressure gradient 21 and flow rate 9. Then this
bifurcated solution is supercritical for FPG and subcritical
for FVF. This kind of situation does not occur in Poiseuille
flow of a single fluid, but it can happen in two-layer flows.
IV. CRITERIA FOR SIDEBAND STABILITY OF
TRAVELING WAVES
B=0, P=~Po4±Pji,
(69)
where AO, PO and PI are constant. The analysis of sideband
stability for general [t and j! is quite involved, even for
much simpler equations7 . We shall in the following only
consider the case ju=jF=0, which turns out to be complicated enough. We also set the lateral pressure gradient
PI equal to zero. PO is arbitrary, but for the following we
are interested in two particular cases: PO = O and
2750
Phys. Fluids A, Vol. 5, No. 11, November 1993
2 =0, Co=oi+piPo0+f3jiAO2 . (70)
.r+PrPo+I3rIAoI
Here subscripts r and i denote real and imaginary parts.
Equations (70) can be solved uniquely for IAo 12 and co as
long as or and Or (for FPG) or fr+pro (for FVF) have
opposite signs. In the following, we make this assumption;
in addition, we assume or> 0, meaning that as the bifurcation parameter increases, the transition is from stable to
unstable. If the bifurcation is subcritical, then, as is well
known, the bifurcated solution is unstable even if no sidebands are involved. Since our equations are invariant under
a phase shift in A, we may assume without loss of generality that AO is real.
In
(1),
we now make
the
substitution
A= (AO+C)exp(iewi-), P=Po
+Q,
and
then
linearize at
0
C=O, B=O, Q=O. This yields the following set of linearized equations:
C,=eyrC4+ eKCP,+,Ef3(A 2C+A20C)
+6AOB+EpAOQg,
Bf= (Cg-Co+e2 PO) Bg+cfBg
+EKB,7,+e6Ao(Cg+±CE),
e(Qe+ Q,7,n) =er0 Ao0(C~+ 1~)+soBf.
(71)
Here we have used (70) to simplify the first equation.
Henceforth, we neglect the term E2pPo in the second equation of (71); we can regard this term simply as a small
perturbation to the coefficient cg-cO.
We now look for solutions of the form
C=-Cl exp(iv +i077+2r)
+C 2 exp -iv -iOq+A-r),
B=2 Re Bo exp(ivg+i077+AAr),
Q=2 Re QO exp(iv +iOq+Ar),
(72)
where the wave numbers v and 0 are allowed to take on
arbitrary real values. This leads to the set of equations
AC, =-eyv2C1 -eK
The basic solutions for which we shall study sideband
stability are what we shall refer to as traveling wave solutions of (1). By this we mean solutions of the form:
A =AOei(4+977eIE`7T
Po=rOIAO12. The first case corresponds to a situation of
"fixed pressure gradient" (FPG), and the second case corresponds to a situation of "fixed volume flux" (FVF). It is
easily checked that (for ju=4i'=0) (69) yields a solution of
(1) iff
2
CI+±e/(A2CI±+AC2 )
+±AoBo+ivpeAOQO,
AC 2 =- ev
2
C2 -eK0 2 C2 + eJ3(A2C 1
2AoC
2)
+ 3AOBO + ivpeAOQ0 ,
ABo= (cg-co)ivBo-Eyv'BO-EW 2 Bo
+ive6Ao(CI + C2 ),
_-E(V2 +02 ) QO=ro~oiv(CI+C 2 ) +soivBO.
(73)
We can eliminate QO from the last equation and insert into
the others. We thus find that A is an eigenvalue of the
matrix
M. Renardy and Y. Renardy
2750
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v2
(eI~
+13A 2]
K0
5A0
0
'V g O'
(74)
- 2iA
e~ivA0
Here we have set
The remaining eigenvalues of (8) are of order
e:2,3=EA2,3 . By inserting this ansatz into the characteristic polynomial of (74) and comparing leading order
terms we find
4=3 jorov 2
f3=I+2±0
(75)
2
s 0PV
I=5±
±01
8
A2+aA+b=O,
(80)
where
Sideband instabilities occur if (74) has eigenvalues of
positive real part for any v and 0. In discussing the eigenvalues of (74), we exploit the fact that e is small. We need
to distinguish various asymptotic limits. The simplest case
is when eV" and/or e6 2 are large or at least of order 1; in
this case the diagonal terms in (74) are dominant, and one
has stability if
a = 27rv 2 + 2Kr0
b = Iny
-
2,A
V _V2 (7 y+ z)A2+ IKI 204 ±v2 02 (KfY'rK)
2 4
_ (Ky+ - 02A2,
(81)
z: =3-b6/(cg-cO).
For later reference, we also introduce the notations
(76)
rŽ,O, Kr>O, iF>0, ,W>Q
z1 :=-t5
6
1/(cg-co),
2
It remains to study the case where eV" and e0 are
small. We shall first consider the situation where I 0 1 is at
most of the same order of magnitude as Ivl; the case
I01 > Iv I will require a separate investigation later.
Throughout, we assume of course that e is small. Suppose
now that Iv I is large relative to e. In this limit, one eigenvalue of (74) is of order v and the other two are of order
e. The first eigenvalue has the form iv(cg-co) +o(v), and
by inserting this ansatz into the characteristic equation of
(74), we find
Ail=iv(Cg-CO) +e(C-
v2702+
(82)
Z2 :=ZI +p(ro-SsO/(cg-co))-
For the eigenfunctions which correspond to the eigenvalues given by (80), B is of order e relative to A, hence these
modes are essentially associated with wave numbers close
to the critical wave number rather than with long waves.
Instability results if (80) has a root with positive real part
for any values of v and 0. We note that a and b are real,
and stability requires that a > 0 and b > 0. This is the case
for sufficiently small v and 0 if
Zr <0
C-C
)
and v 2 (r z+(K:Z)<0.
(83)
(77)
Here we have used the notation
at leading orders. The propagation speed as given by the
leading term in AI is that of long waves, and as far as the
eigenfunction is concerned, B is of the same order (in e) as
A4. We shall therefore refer to this mode as a long-wave
mode. We see that long-wave modes are stable (for sufficiently small v and 0) as long as
(84)
p:q 'Prq,+Pfiq
for complex numbers p and q. If (83) is to hold for all
ratios of v/0, we must have the following conditions:
(ZI)r<O, (Z2)r<O, K:Z 1 <O,
rYZ
2 <0,
(85)
and either
(78)
<0.
Cg- CO
If this is to hold for all ratios of v/0, then we conclude that
we must have
8(Sr+So r)
_____
<0,
CgCO
2751
<.o
Cg-CO
Phys. Fluids A, Vol. 5, No. 11, November 1993
(79)
K:Z 2 7'Y:Z1 <0
or (K:Z2,+rY:zi) 2 <4(K.Zi) (Y:Z2 ).
(86)
We note that the second and third conditions in (85) are
stability which arise in Blennerthe criteria for sideband
hassett's analysis. 15
The preceding analysis breaks down if Iv I is small of
the same order as e. In that case, we set v=6v in (74).
Since for the moment we are still assuming that 101 is at
M.Renardy and Y. Renardy
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2751
most of the same order of magnitude as Iv , we may also
set 0= Ee. We then multiply the third column of (74) by
... 2
_Ye2,p2_KE202+f3AO
-
e and divide the third row by e (this does not change the
eigenvalues), leading to the new matrix
i3Ao
2
SAO
flAO
C
SN1A0
RIMO
For the following, we set iA= el, and we expand the
characteristic polynomial for (87) in the form
-_3+ai2 -aj1+Ao=O,
(87)
i_ (Cg _CO) _E2j,2_ 2W622
(88)
lated. Hence there are no new instability criteria. The third
eigenvalue is of order c2 and its leading order term is given
by
I
where
(91)
2
a 2 = 2,AO +iv(cg-cO) +O(eki^),
ai=2zAis(cg-co)+O(e212),
(89)
a0 = -2i(cg-cO)AO e(fA7:z+ 2K:Z) + O(e4V,4 ).
which is negative if (83) holds.
If both v and e are small, then one of the roots of (88)
tends to 2/3,Ao. If (78) and (83) hold, this quantity is
negative. The other roots are of orders v2and e2v 2, respectively. If we set lIvl+v2lI+..., we find
We need to discuss the asymptotic behavior of eigenvalues
if e, and possibly v2,are small. If e is small, and v is of order
1, then two eigenvalues arise as solutions of the equation
4A2±
(2Jo+i12(Cg-Co))A-2,ohi(Cg-co) =0.
(90)
Equation (90) has a zero root if Zr=0 and a purely imaginary root if z,=,/,.; unstable eigenvalues occur if and only
if either Z, >0 or z, <13,, i.e. if either (78) or (83) is vio-
f3Ao
f#r
1
I-i(Cg-Co)LI
2P3AO
This also leads to stability if (78) and (83) hold. Finally,
the eigenvalue of order e2v is given by the same expression
as (91).
We still need to investigate the case where I 01> Iv I.
In this case, we find it convenient to set v = ev2.This yields
the matrix
6AJto
6i2A
.M
(92)
92
o
PAO
(93)
8i12Ao
Moreover, within terms of order e2, we have 13=1 3and
6=6. We shall now study stability of (93) for e=-0. A
perturbation analysis in e then also yields stability foir sufficiently small e.
For v2=O, we can find the eigenvalues explicitly; they
are _W2 and the eigenvalues of the matrix
fA2
23AA
(94)
If /3, is negative, then the trace of this matrix is negativ *e.Its
determinant is IKj 204 (K:13) 02, which must be positive
ve for
the eigenvalues to be stable. Hence we find that
2752
Z41=
(Cg-Co)
Phys. Fluids A, Vol. 5, No. I 1, November 1993
K:13 < 0
(95)
is a necessary condition for stability. This inequality does
not follow from our previous criteria. It turns out that even
if we add the condition (95), this is still not sufficient for
stability, and an additional condition is needed. This additional condition is quite complicated; we shall describe it in
the following.
Let A denote the eigenvalues of the matrix in (93). If
our earlier criteria (76), (79), (85), (86) and (95) are
satisfied, then the eigenvalues are stable for v2=O. If there is
instability, there must hence be imaginary eigenvalues for
some value of v. We therefore set A= iq, and we write out
M. Renardy and Y. Renardy
2752
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the characteristic polynomial for (93) (with E=0). This
yields the equation
(-_W
2
_iq)( _(q2-+2Kiq02 _2piqA42 -20
2
(K:/3)A_
TABLE I. Tests for sideband stability.
2
ev'2 and/or e& are large or 0(1): Eq. (76)
2
2
ev and e6 are small:
I 01
+ I sc 12d34)+
iv~g-Co)
+ IK1204)
-202 (K:Zl)A4
I VI:
jvIpjcI: Eqs. (76),(79),(85),(86)
_
_ q2+ 2i02
=0.
IvI -- je: Eqs. (79),(85),(86)
1l lvI: Eqs.
(96)
(76)2,(76)4,(79)1,(85)1, (85)3,
(95),(98)
We view (96) as a linear equation for v,, which we can
solve uniquely for given values of q and 0. The requirement
that v be real leads to the equation
q4 [ -KO +2(fr -(zi)r)A2]+q
2
,yr>0, f >O, 13,+P for<(z2), < , y:z
[2W'Kc2j6
-2W01(K:F,+K:Zj)Ao
-4iW
X
2
(K,0 2-pA
2
) (K,.02_ (z 1 ),A)
( I1I2e 4 202 (K:ZI)A2) -2(IK02
Like| 204-20 2 (K:13)A2)] _
;
+2(Kr02_pA 2g)
_ (Z ) 42)
( IKI2 0 4 -20 2AO(K':f3))
2
4-20 2A2(K:Zl))0.
X(|K 20
Equation (97) is of the form
2
a 2 q 4 +ajq +aO=0,
and if (76), (77), (85), (86) and (95) hold, then a 2 and
ao are both negative. In order to have real roots, it isi necessary and sufficient that a,>0 and a2>4a 0 a2 . Thusis our
final condition for stability is that
(98)
a,<0 or acl < aOa2
for all 0. The equation a = 0 reduces to a quadratic polynomial in 02, and by finding the roots of this equationtn, we
can determine the range in which a, > 0. We then hayve to
check whether we ever have a2f > 4a0 a2 within this r ^ange.
The equation a 2=4a0 a 2 leads to a fourth degree po lynomial in 02. We can either find its roots explicitly or use
Sturm's theorem (Ref. 24, p. 273) to decide whether iit has
roots within a given interval. Sturm's algorithm pro)ceeds
as follows. We want to know whether a real polynon nial p
has roots in an interval (a,b), presuming that p(a)) and
p(b) are nonzero. Set fo=p and f, =p' (the derivatiive of
p). Using long division of polynomials, we generatte the
sequence
fo=qjfj-f
2
,
fj=q2f2-f
3
,
,
(100)
For the FPG case, we need in addition ,ir< 0 to guarantee
supercriticality of bifurcation and thus stability to disturbances without modulation.
We also note that (79) and (85) imply that 13, and
,f,+ pro are both negative. That is, sideband stability in the
three-dimensional case holds only if the FPG and FVF
solutions are both supercritical. Moreover, if we consider
the complete list of our criteria for sideband stability, it
does not depend on whether we are dealing with the FPG
or FVF case. The amplitude Ao is the only thing that differs, and it appears only in (97). However, it can be eliminated by rescaling 0 and q.
V. NUMERICAL RESULTS AND APPLICATIONS
We summarize the methods used to compute the variables involved in our analysis. We first discuss
5=(AG(1j+e)-Aq(1 1 ))/e. More precisely, A1 (11+e) is
the eigenvalue at interface height 11 + e for the same values
of dimensional quantities such as Up*and G* and the same
nondimensionalization that was used for 11. This does not
simply amount to replacing 11 by 11 + e in the dimensionless
equations, because velocity was nondimensionalized relative to Uj and U1 changes. In order to calculate Al at interface height 11+e while holding dimensional variables
such as Up* and G* fixed at the same values as at interface
height 11, we need to transform the dimensionless variables
as follows. Let the tilde denote values for the situation at
height 11 and no tildes are used for the situation at height
11 + e. In both situations, the dimensional upper plate speed
Up* remains the same: Up*= Up*= U U1 . On the other hand,
Eq. (6) yields
fr-I=qrfr.
(99)
For instance, in the first expression, q1 denotes the quotient
and f2 isthe remainder. Let w(a) denote the number of
sign changes in the sequence fo(a),fl(a), --. ja),
with zeros disregarded. Then w (a) - w(b) is the number
of real roots of p which lie between a and b.
In summary, we find that we have sideband stability if
and only if all of the conditions (76), (79), (85), (86),
(95) and (98) hold. A summary of the preceding results is
shown in Table I.
It is also worthwhile to state the conditions which will
guarantee sideband stability to two-dimensional disturbances. These conditions are
2753
<0.
Phys. Fluids A, Vol. 5, No. 11, November 1993
up*=Upui=ui1+ M(12_0-
-M(1
-c)GRI)
(101)
Here
GIG*l* Uil*pl
GR1=U2 mA
GR1 Uj
Ui
(102)
Thus,
(103)
U, =aiU1 +bji1z
where
a,= 1I+m(1 2 -e)/(1j+e),bj= -m(1 2 -e)GR1j/2.
(104)
M. Renardy and Y. Renardy
2753
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Hence, aU+±b
and this given Ui in terms of
1 UEi= p~,
Ui. From this, we calculate
R1 =URI/UI, R 2 =U1 R 2/Ui, T=-T 1/U.,
F2 =F2 U2/Ui2, G=GU,2/U12.
(105)
The eigenvalues scale like 1/time under nondimensionalization; hence we must multiply the result by the corresponding factor: the dimensional eigenvalue is the nondimensional Al times U1 /l'*= ( U/Ui) ( U1 /l*). The
computed eigenvalue at interface l+ e is therefore multiplied by U1 /Ulto obtain AI(1 1+e).
An analogous method is used to calculate p. Again let
the tilde denote the unperturbed situation and no tilde the
perturbed situation. We perturb 0 to G-e (note that G is
a negative pressure gradient), but keep the unperturbed
nondimensionalization. This yields
G*= (Gdej)pU&2/l*,
(106)
and hence
G=l*G*/(p,Ui2)
=
(-e) Ui/U.
(107)
We can use Up*= UP as before to find U//U
1 . Once p is
known, we calculate ro from the Landau constants 6 for
FPG and,8+pro for FVF.
The quantity 6 is calculated as follows. The vector 04 is
given in Eq. (58). It is the perturbation to the mean flow
generated by the interaction of the primary mode with its
complex conjugate in the FPG case. We compute 04 numerically in a series of Chebyshev polynomials. This is
used in Eq. (32) to find (r* (R,),0 4 ), where the h and u(z)
on the right hand side belong to 04. The next term in the
definition of 8 above Eq. (67) is (bo, M 2(D~;
a, exp(i4),a exp(-i4))), where a, expire) denotes the
primary mode. By (27), (bo,f) =f14 ; moreover, the quadratic terms for M2 do not involve the did part of Eq.
(14), yielding
(bo,M 2 (4;f,g)) = -h(g)u({fe -h(N(9)-h(f)h(9)U)
=-2 Re (W(f)u(f)) )- Ih(f) 2 U)
(108)
for g=1 For the third term in 6, we have v1 defined in Eq.
(35) and
(r* (R,),vI(R,))
=- ul(z) dz
6
(4.M1
+
2)
m
(109)
Here ul(z) is given by (38). The fourth term in 8 is
(w* ,0 4 )=f'u(z)dz by Eq. (31), where u(z) belongs to
the perturbation of the mean flow 04. The fifth term in 8 is
(bo,M2(D¢;al exp(ig),N, exp(-ig)))
= 2 Re (h(f)[u](9))+ Jh(f)12[ U'],
2754
Phys. Fluids A, Vol. 5, No. 11, November 1993
( 110)
with f and g defined as above. The final term is
(w*,v 1(Rc))=fiu1(z) dzfrom (31) andthisisevaluated
using (38).
We have a code to compute the long-wave formula 23
for the interfacial eigenvalue AO. This is used to compute
c0
from
the
difference
ai./ak- (AO(k=e)
-AO(k=0))/e, where e is chosen sufficiently small. y is
calculated in a similar way. In order to calculate K of Eq.
(68), we have
a2AO
1
[AO(k=e',l=2e)-2AO(k=e, l=C)
±2LO(k=c',1=0)]1,
(111)
where n is chosen appropriately. The purpose of including
a small but nonzero value of k is to make Squire's transformation' 7 applicable. Squire's transformation is also used
to evaluate K of Eq. (67).
Finally, so is easily evaluated directly from its definition above Eq. (67); we have
so= -
uO(z) dz/ f uI(z) dz
and u0 and ul are given by (25) and (38).
As we have seen, some of the constants needed for
evaluating sideband instability are the coefficients describing the long-wave asymptotics of the interfacial eigenvalue.
It is of interest to note that at high Reynolds numbers the
range of validity of the long-wave asymptotics is rather
narrow, and our numerical results on the linear stability
problem show this. The reason can easily be understood.
The shallow-water theory for inviscid long waves yields the
wave speed A
, and hence eigenvalues on the order of
4i7 ki for small wave number k. On the other hand,
at wave number 0, the interfacial eigenvalue is zero and the
least stable of the one fluid eigenvalues is on the order of
- 10/R. A heuristic criterion for the transition from
shallow-water theory to long-wave asymptotics is therefore
that Vh7POk should be less than - 10/R. Our numerical
results agree with this.
Experiments1 8 1 9 have been performed on pressuredriven gas-liquid flows in a horizontal rectangular channel. We focus on Figs. 9(a)-9(c) of Ref. 18, where wave
spectra, surface tracings and bicoherence of the wave field
are shown to exhibit sideband interactions. In particular,
Fig. 9(a) shows the wave spectra as a function of fetch
(distance from the inlet): the plot upstream shows one
peak at the fundamental mode and smooth decay away
from it, while the downstream plots show the growth of
sideband modes accompanied by a growth of the mean
flow mode. The parameters are: liquid Reynolds number
R L= ULI*/vI = 5 (UUL is the average liquid velocity), liquid height 11=0.44 cm, liquid viscosity 1z,=2.38X 10-2
Nsec/m 2 , liquid density pl=1.18X 103 kg/M3 , gas Reynolds number RG= UG12*P2 /[t 2 = 6300, average gas velocity
Ug=4.5 m/sec, gas depth 12*=2. 1 cm, gas viscosity
IZ2 =1.85X10- 5 Nsec/m 2 , gas density P2= 1 . 1 5 kg/M3 .
This
yields
the
ratios
m=IZ,/1u2 =1081.0811,
r=pl/p2 =1026.1. For the surface tension constant, we
M. Renardy and Y. Renardy
2754
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used the value for air/water at 20 'C, which is 72.8 dyn/
cm. Moreover, we can determine UL from RL and v, to be
2.27 cm/sec.
In the experiments, the gas phase is turbulent. Hence
we would not expect the usual parabolic velocity profile to
prevail over the gas region. However, the gas flow may be
laminar in a thin boundary layer over the liquid. Our
model for the experiments will consist of a laminar liquid
layer with a thin laminar gas layer of thickness much less
than 12, above which we place a wall of speed U!, representing an average gas velocity. If the motion in the gas
phase far from the interface is not very important, then our
model may be applicable. We then seek a neutrally stable
situation occurring at a nonzero wave number, and then
apply the nonlinear analysis of Sec. IV, outlined in Table I.
In order to convert the parameters of Ref. 18 to ours,
we require the calculation of the dimensional interface
speed UJ from the upper wall speed UC and the average
liquid speed UL. From (6), we have
V;=Ut(1+ M+2
1
)mGR1
(113)
By eliminating G, we find
111 /
+
2112 Um12/4
Having found Uj, we can find RI and then calculate G
from (113). Our procedure is to fix the parameters for the
liquid to be those of Fig. 9 in Ref. 18, choose the gas
boundary layer thin enough to make the parallel flow stable to long wave disturbances and to vary Up until neutral
stability is attained. The value of 11must be close to 1 in
order to make long waves stable. For example, if we choose
11=0.9, then the first instability is due to long waves. We
choose 11=0.99.
This procedure has yielded UV=52.75 cm/sec,
G=0.43015E-2,
R =10.03,
cm/sec,
Uj=4.51
T=87 183, 1/F2 =21.379, ko=2.16 and ILI=-0.11E-3
- 13.523i. We note that the value of Up is much smaller
than the UG reported for the experiments. This is a general
difficulty in comparing with experiments. Usually experiments are not close to criticality because the waves on the
interface would then be too small to see. The Chebyshevtau method is used to numerically compute the eigenvalues. The growth rate is shown in Fig. 1. The Landau constants are 13=-45.99-217.53i for FPG and pro
=-41.87-63.90i for FVF. Our remaining constants in
the amplitude equations are computed as follows:
8= -18.6-1 12i, p=0.149+5.5 6 i, K=0.056+0.017i,
y=0.048-0.048i, cg= 6 . 2 1, f-=0.218, K'=0.020, co=6.19,
8=0.026, so= 12.8, ro=27.6. Working through our list of
criteria, we find that the only condition for sideband stability which is violated is (98).
2755
Phys. Fluids A, Vol. 5, No. 11, November 1993
'~~~~~~~~~~~~~~~~~~~~~~~~~~~
-0.01
-0.021
-0.03
Re A
-0.04[
-0.05-
0
0.5
1.5
1
2
3
2.5
k
FIG. 1. Growth rate Re A versus wave number k for the interfacial
mode. Rl=10.03032, G=0.430 15E-2, I1=0.99, m=1081.0811,
T=87 183.131, r= 1026.1, 1/FP=21.379 086, ko=2.16, U,*=52.75 cm/
sec, Uj=4.51 cm/sec.
(112)
and integration of (4) yields
UL= U2)GRII + 1 )
>
/
X
A second model was developed with 11=0.97, using
the same procedure as before to determine the other parameters. Hence m and r are as before. Criticality now
arises at U =41.47 cm/sec, yielding the dimensionless paT= 106 046,
G=0.1704,
R 1 =8.416,
rameters
1/F 2 =32.28. The critical wave number is ko=0.46 and the
eigenvalue is Al=-0.5E-5-2.885i. The outcome is
qualitatively different from the previous model. We find
that the following conditions are violated: the second inequality in (76), both inequalities in (79), the third inequality in (85) and (86), and (98). Hence there are sideband
instabilities to many different modes. The violation of
(76), i.e. K,<0, shows that the unperturbed base flow is
actually unstable to three-dimensional disturbances, although it is neutral for two-dimensional disturbances. This
does not contradict Squire's theorem1 , since Squire's
transformation also changes the values of the dimensionless parameters of the problem. In two-layer flows there is
more than one candidate for the bifurcation parameter,
and whether a transition is from stable to unstable or vice
versa depends crucially on how parameters are varied. In
our approach, we fixed the average liquid speed and the
physical properties of the fluids and then varied Up'. The
transition at U =41.47 cm/sec is from stable to unstable
as Up is increased. However, if we vary RI and keep the
other dimensionless parameters the same, then the transition is from unstable to stable as RI is increased. The violation of (79) shows a sideband instability to long waves.
The unperturbed base flow is stable to long waves at criticality; however, as U; is increased slightly beyond the
critical value we find that the band of unstable wave numbers spreads quickly. For instance, at U= 42.2 cm/sec,
the range of unstable wave numbers reaches all the way
down to zero.
Finally, we investigated a situation of Couette flow
with zero pressure gradient. Again, 11=0.97 and the physical properties of the fluids are as above. This led to neutral
M. Renardy and Y. Renardy
2755
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TABLE 11. Critical conditions for the BLI proffle.
G
R1
k6
0.1398E-2
0.6155E-3
0.7228E-3
0.3543E-3
1904.90
4331.86
3688.90
7525.06
2.8322
3.5002
3.9178
4.5
l/F
37963
33583
78664
77384
2
16.688
25.780
285.196
547.975
stability at R 1=2.815, G=O, T=317053, U/F2 =288.6.
The critical wave number is ko=0.45 and the eigenvalue is
A I= -0.8E-4-7.74i. In this case, we find that the only
condition for sideband stability which is violated is the
third inequality in (85).
The three models above yielded qualitatively different
results. The first and third cases discussed above have sideband instabilities only to three-dimensional modes. The
second case has sideband instability to two-dimensional
long-wave modes, in addition to a three-dimensional instability of the base flow. The onset of long-wave instabilities
in the base flow for slightly different values of the depths is
also intriguing in the light of the observed growth of the
mean-flow mode. In summary, the three cases above were
not very far from long-wave instability of the base flow and
were all subject to some sideband instability as well.
The remainder of this section concerns models for the
generation of water waves by wind used in Ref. 15. Tables
III-VI of Ref. 9 list the conditions for criticality in terms of
the dimensionless variables of this paper. No fit to experiments is attempted in Ref. 15, and the choices of some of
the parameters, especially the fluid depths, are rather arbitrary. Table II lists the critical conditions for the
boundary-layer profile named BLI in Ref. 15. This is a
correction of Table IV of Ref. 9. Table III lists the eigenvalues and Landau constants for BL1, BL2 and PPF and is
a correction to Tables VII, VIII and IX of Ref. 9, where
the normal stress condition contained an error.
The Landau constants for PCF are listed in Table X of
Ref. 9, where K(FPG) and K(FVF) represent ourt1 and
13+pro, respectively. We note that the first of the PCF
profiles in Table X,9 the first of the BL1 profiles and the
second and third of the PPF profiles in Table III have
opposing signs for the real parts of the two Landau constants and, by the remarks at the end of Sec. IV, they are
TABLE III. Linear eigenvalue and Landau constant for the BLI, BL2
and PPF profiles.
Flow
BLI I
BL12
BL13
BL14
BL21
BL22
BL23
PPF1
PPF2
PPF3
2756
't
0.302E-4-9.0722i
0.235E-4-12.245i
-0.268E-4-36.193i
-0.87E-5-52.945i
0.263E-5-7.74i
0.459E- 5-22.046i
-0.91E-5-28.077i
0.124E-4-8.41i
0.91E-5-19.394i
-0.23E-5-35.282i
3
(3+ pro
-0.139-3.942i
-0.547+8.901i
-0.596+2.282i
-0.272 +0.953i
-0.357 +0.782i
-0.23-9.055i
-0.075-12.88i
-4.3099-56.84i
-0.0428-6.513i
-0.346-18.989i
0.195-30.717i
-0.362-19.26i
-0.403-12.423i
-0.151-17.93i
-0.208-37.47i
-0.107- 34.72i
-0.0219-45.028i
-4.153-78.49i
0.0050-16.132i
0.2019-56.154i
Phys. Fluids A, Vol. 5, No. 11, November 1993
TABLE IV. Comparison with Blennerhassett's results. Here the flow
PPFn refers to the nth row in Table 2 of Ref. 15, BLIn is the nth row in
Table 3a, BL2n is the nth row in Table 4a, and PCFn is the nth row in
Table 5.
Flow
PCF1
PCF2
BLIl
BL12
BL13
BL14
BL21
BL22
BL23
PPFI
PPF2
PPF3
Im z2 /Re
z2
-59.5
-60.5
-92
-101
88
338
-94
-301
-1193
-8
1624
-374
Values found in Ref. 15
-59.5
-60.3
-92
-102
88
319
-93
-299
-1140
-8
1677
-372
-
necessarily unstable to sideband perturbations. We also
note that the Landau constant
reported
by
Blennerhassett' 5 corresponds to the quantity which we denoted by z 2 in Sec. IV (cf. the remarks in the Introduction). The absolute value of z 2 has no intrinsic meaning,
since it depends on the normalization of the eigenfunctions.
We shall therefore report the ratio of the imaginary to the
real part of z 2. Table IV compares our values with those
found by Blennerhassett. Our values for z2 reproduce the
Landau constants reported by Blennerhassett. The Landau
constant for PPF3 in Ref. 15 has a misplaced decimal
point; our table contains the corrected value. We note that
many of the coefficients in our amplitude equations go into
the computation of z2, and the agreement with Ref. 15
gives us some confidence in the correctness of our results.
Another independent check is the value of ro, which we
compute as the ratio of two complex values:
((13+ pro) -fl)/p. The value of ro must be real, and in all
our calculations the argument came out very small, on the
order of 10-5.
We tabulate the results of our investigation of sideband
instability for several of Blennerhassett's flows in Table V.
We see that all the flows violate several conditions. Thus
sideband instability seems to be rather ubiquitous. In addition, the base flow is unstable to long waves for the BL21
TABLE V. Stability criteria which are violated for the flows of Ref. 15.
Flow
Conditions which are violated
PCF1
PCF2
BLII
BL12
BL13
BL14
BL21
BL22
BL23
PPFM
PPF2
PPF3
(79)i,ii,(85)iii,(86),(98)
(79)i,ii,(85)iii,(86),(95),(98)
(79)iii,(85)iii,(86),(98)
(79)i,ii,(85)iii,(86),(95),(98)
(85)iv,(95),(98)
(85)iv,(95),(98)
(76)iii,(79)i,ii,(85)iii,(86),(95),(98)
(76)ii,(79)ii,(85)iii,(86),(98)
(76)ii,(79)ii,(85)i,iii,(86),(98)
(76)iii,(79)ii,(86),(98)
(79)i,ii,(86),(98)
(79)i,(85)ii,iv,(86),(98)
M. Renardy and Y. Renardy
2756
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and PPF1 profiles and to three-dimensional disturbances
for the BL22 and BL23 profiles. We note that our derivation of the amplitude equations assumed that the only neutrally stable modes were at wave numbers i ko and zero
and is therefore not valid for these profiles.
VI. HOMOCLINIC AND HETEROCLINIC SOLUTIONS
In this section, we shall consider only two-dimensional
solutions of our amplitude equations, i.e. solutions of (2).
a in
apr,
For convenience, we shall write 13 in place of +
place of o-+pK, 8 in place of 8+pso and /3in place of
13+ so. We first look for solutions which are of the form
A=exp(iecsr)A(t-ecuI), B=eB(t-ecr). Here (o and c
are a priori unknown constants which need to be found as
part of the solution. Recall that A is an amplitude factor
multiplying a traveling wave, which is proportional to
exp(i(kOx+o 0 t)). Hence e2c represents a frequency shift
of the traveling wave. Moreover, ,=e(x-cgt), so Ec is a
shift in the wave speed of the modulation relative to the
group speed.
Suppressing the hats on A and B, we obtain the following system of ODEs:
rA"+cA'+ (a-io)A +1,A 12 A+8AB=O,
(115)
2
2
EyB"+ (cg-co+ec)B'+e 3(B )'+(
IA
For E O, we evidently obtain
12,
B+e
erB'+ (C-Co+ec)
B ±+IAI =K,
For e =0 and c =0 Eq. ( 117) reduces to
yA"+
B' =- (cg-co+ec)B -E2,B
A=Csech (kg)exp(iEln cosh(kg)),
2 +1)-iu+_8K
k2 y(-2iE-E
7
2
|1A | +K.
For e=O, this system has a four-dimensional manifold of
by
the
equation
described
points
stationary
B=(K-81AI2 )/(CgCO),with arbitrary A and D. The
linearization at each of these stationary points yields a
fourfold zero eigenvalue and a simple eigenvalue
(co-cg)/y. If cosACg, then the center manifold theorem
can be used to show the existence of an invariant manifold
of the form B=O(eAD), which for the original equations
is to be interpreted as a "slow" manifold. Since the equatransformation
under
the
tions
are
invariant
A hA expirep, the slow manifold is also invariant, and we
can therefore put its equation in the form
B=4(e, IA 12, IA' 12,"AA). We may therefore investigate
the equation
rA" +cA' + (a-ico)A +1A 12A
2757
Phys. Fluids A, Vol. 5, No. 11, November 1993
(120)
&=O
(121)
k2 y(3iE+E 2-2)+Cd=0.
We multiply the last equation by k-2 d, which yields
yd(3iE+E2 -2)=-C2 k 21 d1 2.
(122)
Comparing imaginary parts, we find that
+ (E 2 -2)Im(yd) =0,
3E Red)
(123)
(117)
2
8(IM(yd))I
2 Jimmy)
(124)
The requirement that the left-hand side in ( 122) must have
negative real part forces us to choose the plus sign in front
of -the square root in (124). Having found E, we can now
determine k by taking real parts in the first equation of
(121). This yields
2
)r,+2Eyi]±Ur.+
6,(
Cg
2 A -8eAB,
+6AO(e,,IA 12, [A' I2,AA') =0.
(119)
where C and E are real. Note that the expression given by
(120) decays to zero at infinity. Inserting this into (119)
yields the equations
k 2 [(le
_
-io)A+dIAI 2A=0,
where d=P-b6/(cg-co).We now try to solve (119) with
the ansatz2 2
(116)
A' =ED,
2
5K\
(+ cKc
-3 Re(yd)+ 19(Re (d))
where K is an arbitrary constant of integration, which we
henceforth regard as fixed and given.
In (115), we may introduce D=A' as a new variable.
If, in addition, we rescale the independent variable with a
factor e, then the first equation of (115) and (116) lead to
the new system
yD'=-ceD- (a-ico)eA-13elAI
(118)
which is solved by
2) =0.
2
2
And
For IA
We can integrate the second equation to obtain
2
K-8IA12
Eq.(117)g-cO ru
t2,iAt)
= 0.
C0
(125)
If k2 as determined by (125) is positive, then Eq. (120)
yields a solution, since we can now easily determine o0 by
taking the imaginary part in the first equation of (121) and
we can determine C from (122).
Another ansatz (see Refs. 20 and 22) for a solution is
A = C tanh(k¢)exp(iE ln cosh(kg) ).
(126)
This solution behaves like C exp(iEt) as g_ + oo and like
-Cexp(-iEg) as - - o . When inserted into (119),
this leads to the equations
-k 2 y(E 2 +3iE-2) +C2 d=O,
(127)
+K
:k2y(-2+3iE) +f+_
L
-io=O.
~~~~~~~~cg,-co
The first equation of (127) differs from the second of
(121) only in a minus sign. We therefore obtain the same
expression (124) for E, but we must choose the minus sign
in front of the square root. From the second equation of
(127), we obtain, by comparing real parts,
M. Renardy and Y. Renardy
2757
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k2
-2,yy)
o,+
Cg
'K=0.
CO
(128)
The ansatz (126) yields a solution if k2 as determined by
(128) is positive.
We can think of (117) as a dynamical system. Since
the equation is of second order and A is complex, it is a
dynamical system in four dimensions. The solution given
by (120) is a homoclinic orbit connecting the fixed point
A = 0 to itself. The solution given by (126) is a heteroclinic
orbit which approaches two different periodic orbits for
An--o and c- oo. We shall now consider the question
whether solutions with the same qualitative behavior exist
for small nonzero e.
In general the stationary point A =0 of (117) has a
two-dimensional stable and a two-dimensional unstable
manifold. A homoclinic orbit requires the two manifolds to
intersect, and, due to the symmetry under the transformation A -A exp (io), the two manifolds actually coincide in
this case. The coincidence of these two two-dimensional
manifolds in four-dimensional space will not persist under
general perturbations of the equations; it is a codimension
two phenomenon. However, we have the two parameters co
and c at our disposal. If certain generic transversality conditions hold [analogous to condition (viii) of Theorem 3.4
in Ref. 22], there will therefore be a homoclinic orbit for
specific values co(e), c(e).
It can also be shown (see Ref. 22, Proposition 3.7) that
the limiting periodic orbit belonging to (126) for
-.
o0
has a two-dimensional stable and a three-dimensional unstable manifold in four-dimensional space. For the limiting
periodic orbit as
o-o, the numbers are reversed. Hence
*c offhand:
Plot
tile.
CM-kCookd.?
C..'&.
A...r
ARidtT=-R.B
dBU.dT=--flU..
.013
FILE:
o.t oF
R -I
bo... A.t
t
= -1.756860
FILE2Z.OE
d/dT=-Rtt
dB/dT=-g - BBt
X -1
5
FIG. 3. Phase plane portrait for the system
R'=-RB,
B' - B+ B2 +R -1. The R axis is horizontal and the B axis is vertical.
The range of R is from -0.5 to 1.5, and the range of B is from -2 to 2.
a heteroclinic orbit again requires the coincidence of two
two-dimensional manifolds, which, under generic conditions, occurs for specific values of co(e) and c(e).
We note that in Ref. 22 the symmetry of the equations
under reversal in g is exploited. This allows reducing the
number or adjustable parameters from two to one; the parameter c does not occur in Ref. 22. In the present situation, however, the symmetry under reversal in ¢ is destroyed for e#k 0.
The homoclinic and heterocinic solutions considered
here are only the simplest such solutions for (119). There
are many others, at least for coefficients in a certain range,
and there are also spatially chaotic solutions.2 1 Again this
behavior can be expected to persist for nonzero e. Thus the
sideband stability of periodic waves, investigated in the
sections above, is only one aspect of predicting the possible
behavior of experiments. Some of the solutions with more
complicated spatial behavior might also be observed. In
general, little is known about the stability of such solutions.
For the solutions considered in this section, there are some
limited results of a negative nature, i.e. showing
instability. 2 0-2 2
dN/AT=
P1t
can1 .
..
t
o
X
-X
bound at
t
=
t.45I0
FILE: FILE1.OE
d/dT1-RtI1
dB/dT=B'B * R
-I
2
FIG. 2. Phase plane portrait for the system R'= - RB, B'= B +R - 1.
The R axis is horizontal and the B axis is vertical. The range of R is from
-0.5 to 1.5, and the range of B is from -2 to 2.
2758
Phys. Fluids A, Vol. 5, No. 11, November 1993
The homoclinic and heteroclinic solutions discussed
above arise essentially from the Ginzburg-Landau part of
Eq. (2). In the following we discuss other homoclinic solutions which involve long-wave modes. We make the ansatz A =A (+ (cg-cO-ec)r), B= B(+ (Cg-CO-E)T).
Note that A is the amplitude factor for a traveling wave,
while B represents a mean interface shift. The solutions
considered below are of heteroclinic type; they either approach different constant interface heights at ±i 0o, or they
M. Renardy and Y. Renardy
2758
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NC o..ntd
Plot fti.
a number of other combinations lead to similar qualitative
behavior. We first consider the case &Y=0. In that case,
(133) has the three equilibrium points B=0, R=K/g and
R = 0, B= zE SK//. In the (B,R) -phase plane, the first of
these points is a center; the other two are saddle points.
The two saddle points are connected by two heteroclinic
orbits: one is part of the line R =0, the other is an arc on a
parabola (cf. Ref. 22, Proposition 1.1). The phase plane
portrait is displayed in Fig. 2.
For c==0 the quantity
I
jC..rA.&.7
IA...
Lii
H(RB)=R1+a~flB2+&jaR-K]
(134)
is constant along orbits of (132). Here we have set
dN/dT--RU
3S
U
J3.J.TS-U
.El]
FILE:
-. t
.
R -I
OF )....A.
.t
rfl(CgCO)
t = -1.U500f00
i75r
FILE3.SE
d*IdT=-R3d
dB/dT4 * DB3
R
f -1
FIG. 4. Phase plane portrait for the system R'=-RB,
B'=B+B2 +R-1. The R axis is horizontal and the B axis is vertical.
The range of R is from -0.5 to 1.5, and the range of B is from -2 to 2.
approach a constant interface height at one end and a traveling wave at the other.
The ansatz above leads to the system (again we suppress the hats):
eyA"+ (co-cg+e)A'+±EA+E8IA
fB" +B'+
I2A +8AB0,
(B 2 ) +6(IA 2 ) =0.
(129)
If co:Cg, then, for small e, there is an invariant slow manifold of the form A' =0 (eAB); for e=O we evidently have
q(eA,B) =5AB!(cg-co). Moreover, we can integrate the
second equation of (129). We thus obtain the new system
A'=0(eA,B),
jB'+,FB+,B 2 +81A 2 =K.
(130)
q(eA exp(ith),B)
the
invariance
We
have
expire4') 0(eA,B). This allows us to split the first equation in (130) into separate equations for the modulus and
phase of A. For the following, we are interested only in the
equation for the modulus. With R denoting IA 12, we obtain the new system
R'=0(e,R,B):=2rR Re 0(eV,B),
j7B'+cB+t3B2 +
(131)
R =K.
25, RB, 7B'±+FB+fB
The two orbits connecting the saddle points lie on the level
set H= 0. For F> 0, H increases along orbits, while for
c<0 it decreases. As a result, the phase plane pictures in
these cases look like Figs. 3 and 4. There is a heteroclinic
orbit connecting one of the saddle points with the critical
point at B= 0 R = K/. These heteroclinic orbits are structurally stable and hence persist for e*&
0. The orbit connecting the two saddle points occurs for a set of codimension one, i.e. for a specific choice of cCe).
Even the classical Ginzburg-Landau equation allows a
rich variety of solutions which is only partially understood,
and the more complicated system (1) leads to even more
possibilities. The analysis of sideband instability and the
homoclinic and heteroclinic solutions studied in this section are only a beginning; we are far from an exhaustive
description of physically realizable patterns. Clearly, there
is much potential for further research on the existence and
stability of solutions to our amplitude equations.
ACKNOWLEDGMENT
We thank H. C. Chang for a colloquium lecture and
discussions, which motivated us to undertake this project,
and we thank Dan Farkas for telling us about Sturm's
theorem. We thank Gregg Lee for helping us with the plots
for the figures in Sec. VI.
This research was supported by the National Science
Foundation under Grant No. DMS-9008497 and by the
Office of Naval Research under Grant No. N00014-92-J1664.
APPENDIX: EXPLICIT FORM OF THE OPERATORS IN
SEC. III
Let u= (u,v,wp,h). Then
For e=0, Eq. (131) reads
Rt-
(135)
2
+±
4 R=K.
(132)
M(R)u= (U( 1 )
,V(1 ) ,W(l) ,U( 2 ) ,V( 2 ) ,W( 2 ) ,0,
0,0,0,0,0,0,h).
The qualitative behavior of solutions to (132) depends on
the signs of the coefficients. For the following, we consider
the case where
f>0, 13<0, 5<0, K<0, 6/(Cg-co) <0;
2759
Phys. Fluids A, Vol. 5, No. 11, November 1993
(133)
Here the subscripts refer to fluids I and 2. Since M does
not depend on any of the parameters, we have of course
M'(R) =0. This would change if the equations were written slightly differently, e.g. if the Navier-Stokes equations
M. Renardy and Y. Renardy
2759
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were multiplied by the Reynolds number. Our development in Sec. III allows for this possibility. Next, we write
L(R,Dx,Dy)u in components which we label f, through
f 14 We have
f 8 =f 9 =f1 0 =O,
du
1
ap
fl =-Uy
d-wU +RIAU dx
f12 =0,
I AU dy
f2=-UdxRv
dw
1
2T
f13=- mR, ikh,
dp
aXW +RIAiw azf
f3=-U
f4=- U
dv
1
5=Ux
R2
dw
1
We have
dp
L 3 (R,ik,0)
U-ayX
-
arj-
rp,0,VA0,0,0
I
du dv dw
f7=YX-+5y7+
u= (0, -POO
3
+aw Iw
f 6 =-U
f 1 4 =-U(11 )h- fou dz.
du
1
dp
dx- xwU'+R-Au-r
R2AU
ax'
L 22 (Rjik,O) u=z L 3 3 (Rik,0) u
Z-[W],
(2
fs=h[U'I±Iu],
=
2
2
Uji- UI-
f94[v,
W,
mR
2 R2
R2
flo= [WI
L 23 (Rjik,O)u=O.
[:; dz ax )
f12=
f14=-
The quadratic term N 2 (R,D,,Dy;u,u), written in components fi through f 14, is as follows:
Lt(8u+ dw)3
R3z-t
f13=
f=-u
-mR Ah-h'
YXU(1 1)h+
dz)
Ju
dz]< j
0
-v
-
av
8v
8v
8w
8w
8w
au
8u
au
av
8v
v dz.
Next, we write L 2 (R,ik,O)u, again in the form of components f 1 through f 14 . We have
2
f 1 =-Uu-p+R-iku,
8v
2
A
Uv+i-ikv,
f=-
2
f 3=-Uw+RI
ikw,
U~X_-
8w
V
~--W-,~
8w
f 6=--U _j- -VTy-
8w
-W-_Z
2
Uu-rp+- iku,
f4=-
2
f
=-
Uv+- ikv,
2
f 6= -Uw+K-
2760
ikw,
Phys. Fluids A, Vol. 5, No. 11, November 1993
f8=hW+-[U"],
f9=
vz~l
M. Renardy and Y. Renardy
2760
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h2 r 21L u 33w
Dw~
flo=h
f13= _i_
Fa I
2
Lt(a
U+ D2W
f 1l=h~
D~
fAk
pi
~
-
dydh2tI 2
--l ta4
- u
S
p
Dh
Dv'~
a(u
dhDd 2
DW\ 1
+dx
_-
ap
D 2
2mR
D
t(D
,//'h\2
X
~t(Dv Dw\
DyXR1
1
~
2u
atD
a 2w
+± D~
dz
D
2 z- --
1P
RT AI dyJI
2h(V)2a
I-ah
(Dh \2
-dx
(ahD20
_~~_ dy
_
dx
2
2
Dxtty)
d (h2 du(l)
Dh 2
(
dh
DA
1
y
P
g du Dlv
Dy+xI
R;DL
T a2h
2Dx)Ji'
D x I jY
A Dau
Dhk~ (DV
Dy -pl\y-
1+{h
± Il
I dx
[lR
+x
2
2W
4t(D2 V+ D
f 12 =h
~2+
-a z~ 1
2r
)
(Dh\ 2 TLt (D2 v d2w
Dkl (au 8aw\)1
2
x[I kx~D
x z)
~
2
~
R,
)
T d2h
2mR 1 Dy2
2T d2h dh dh
+ m;R 1dxdy dx
__x~~5
dy'
h38U"
6 I)
a h a((hl)
dy' 2 dzY(
Having defined N2 and N 3 for equal arguments, we can
extend them uniquely as symmetric functions, e.g.
DyJ
(aDz
1
N 2 (RDxDy;uv) =-(N2 (RDxDy;u+vu+v)
The cubic term N 3 (R,Dx,Dy;u,u,u) is:
f1 =12=f3=f4=f5=f6
87(h2 au~ h3 1
1 Dx 2 [ d + 1
1
_N2(RD~xDy;u-v,u-v) 3.
Finally, we list the bilinear operators M2 and Q2 appearing in (51). The components of M 2 (D ;u,ug) are
=0,
8( 2 Dvh
+ D 2 Dz )
\
du
A =-u YE '
f
Dv
=-uTI
1w
f3 = - u 4,g
h2ia2VI
A=-
2
21-az-11 I
14= -U
h1a2WI
h2
a(3
3
Ak
uaD
w
u ) -2-hi--IhL
D2 wM
wII
at u
2
1
&
(au
8w)~
ADAh.Lifv
-,
Dv
f5
16=
(a
D
au
-u Ui
--
-
~
8w)~
f 7 =~jIuJ+h -[uJ+h
f
12=
JlI-2 -h I -I
-I -l - '!
-h aL&(au
AX
Dh2
2761
~7,U'J,
_-
2
Dw\
(-au
D
Dxh-(
u
w
Phys. Fluids A, Vol. 5, No. 11, November 1993
M. Renardy and Y. Renardy
2761
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A
12=
ht av
Ah~j av
ko'~ k0agL~'T'~
--
2 Ah~~(au+ aw)
f
ahgu1) h aU(
ahL~taw1
2
hah
1)
The components of Q2 ( D~;u^u7) read
-vT
A=
-
au
T?,
aw
f6
=
-
V j_
a~i
2
ah
hal2
2762
,7[vl~
davit
T
Phys. Fluids A, Vol. 5, No. 11, November 1993
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