Steady and oscillatory side-band instabilities in Marangoni convection with deformable interface

PHYLA ELgEVIER

Physica D 106 (1997) 131-147

Steady and oscillatory side-band instabilities in Marangoni convection with deformable interface A.A. Golovin a,,, A.A. Nepomnyashchy b, L.M. Pismen a, H. Riecke c a Department of Chemical Engineering, Minerva Centrefor Nonlinear Physics of Complex Systems, Technion, Haifa 32000, Israel b Department of Mathematics, Minerva Centrefor Nonlinear Physics of Complex Systems, Technion. Haifa 32000, Israel c Department of Engineering Sciences and Applied Mathematics, Northwestern UniversiO,, Technological Institute 2145 Sheridan Road, Evanston, IL 60208-3125, USA

Received 19 October 1995; revised 6 July 1996; accepted 30 December 1996 Communicated by F.H. Busse

Abstract The stability of Marangoni roll convection in a liquid-gas system with deformable interface is studied in the case when there is a nonlinear interaction between two modes of Marangoni instability: long-scale surface deformations and short-scale convection. Within the framework of a model derived in [ 1], it is shown that the nonlinear interaction between the two modes substantially changes the width of the band of stable wave numbers of the short-scale convection pattern as well as the type of the instability limiting the band. Depending on the parameters of the system, the instability can be either long- or short-wave, either monotonic or oscillatory. The stability boundaries strongly differ from the standard ones and sometimes exclude the band center. The long-wave limit of the side-band instability is studied in detail within the framework of the phase approximation. It is shown that the monotonic instability is always subcritical, while the long-wave oscillatory instability can be supercritical, leading to the formation of either travelling or standing waves modulating the pattern wavelength and the surface deformation. Kevwords: Side-band instability; Modulational waves; Marangoni convection

1. Introduction An important class of instabilities of one-dimensional patterns is the class of side-band instabilities. The most important one among them is the Eckhaus instability [2] which is a monotonic long-wave instability that renders the pattern unstable with respect to infinitesimal compressions and dilatations. Its onset can be characterized by the vanishing phase diffusion coefficient [3]. Near the threshold, where the pattern can be described by a real * Corresponding author. Present address: Department of Engineering Sciences and Applied Mathematics, Northwestern University, Technological Institute 2145 Sheridan Road, Evanston, IL 60208-3125, USA. 0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII SO167-2789(96)00016-X

132

A.A. Golovin et al./Physica D 106 (1997) 131-147

(dissipative) Ginzburg-Landau (GL) equation, the Eckhaus instability arises generically if the deviation q of the wave number of the pattern from that corresponding to the bifurcation point exceeds a certain value, i.e. if

Iql > ~/~¢r~,

(1)

where E2 is the deviation of the control parameter from the threshold. A similar instability of wave patterns (e.g. described by the complex GL equation) is usually referred to as the Benjamin-Felt instability [4-6]. Typically, the Eckhaus instability restricts the range of allowed wave numbers of the pattern, i.e. its nonlinear evolution eventually brings the system back to the stable range of wave numbers. The Eckhaus instability of periodic patterns has been studied extensively, both theoretically and experimentally, in various pattern-forming systems, such as Rayleigh-Brnard convection [7,8], convection in binary mixtures [9], Taylor vortex flow [ 10,11 ], electrohydrodynamic convection in nematics [ 12], and directional solidification [ 13]. Even quite close to the threshold, the boundaries of the Eckhaus instability of stationary periodic patterns can differ from (I) due to various factors such as the system confinement [ 14,15], subcritical character of the instability [16], the shape of the neutral stability curve [17,18], the effect of the nonlinearity [19] and resonances [11], the presence of random fluctuations [20], chiral symmetry breaking (e.g. due to rotation) [21], etc. Although the Eckhaus instability is the most common one-dimensional side-band instability, others are possible as well. In the present paper we show that spatially periodic patterns arising due to Marangoni convection caused by surface tension gradients in a liquid layer with a deformable free surface heated from below 1 can possess additional side-band instabilities: a short-scale steady side-band instability and a long-scale oscillatory side-band instability. We shall perform linear and non linear analysis of these novel types of side-band instabilities.

2. Interaction between two modes of Marangoni convection An important feature of Marangoni convection is the presence of two primary modes of instability (generating the Marangoni convection itself). The one is a l o n g - s c a l e instability (with zero wave number) connected with the deformation of the free surface; it is governed by the coupling between the long-scale surface tension gradients and gravity and capillary forces. The other is a s h o r t - s c a l e convection (with a finite wave number) governed by the short-scale surface tension gradients only. A typical neutral stability curve for this system has two minima, corresponding to two critical Marangoni numbers [22,23]. The one minimum, Maj, is at a zero wave number and corresponds to a long-scale deformational instability, and the other minimum, Mas, is at a non-zero wave number and it corresponds to a short-scale instability (surface deformations are not important in this regime). When the two thresholds are close to each other, the nonlinear interaction between the two modes of Marangoni instability with different scales can lead to various secondary instabilities of the short-scale convection generating various spatially and temporally modulated patterns [ 1,24]. The physical nature of this interaction is as follows: on the one hand, the short-scale convection is more intensive under local elevations of the free surface, due to the local increase of the supercriticality; on the other hand, the long-scale variation of the short-scale convection produces a variation of the mean heat flux from the bottom to the top, and this, in turn, brings about long-scale surface tension gradients which tend to reduce the surface deformation. Coexistence and interaction between the two modes of Marangoni convection has been recently observed in experiment [25]. In the simplest case of 1-D roll Marangoni convection, the effect of the above interaction can be described by a system of two coupled equations in the following scaled form [1]: A t = A -t- A x x - ]AI2 A + A h ,

ht = - r n h x x - w h x x x x + slA]2x •

1This type of convection can be also driven by mass rather than heat transfer [ 1].

(2)

A.A. Golovin et al./Physica D 106 (1997) 131-147

133

Here A is the amplitude of the short-scale convection, h is the deformation of the free surface, x and t are long-scale space coordinate and slow time. The coefficients m, w, s have been determined analytically from the governing Navier-Stokes and convective heat transfer equations [ 1,26]. 2 The parameter m is proportional to the difference between the two thresholds, Mas-Maj. It can be either positive or negative, governing the linear growth (or damping) rate of the long-scale deformational instability. The parameter w > 0 is proportional to the capillary number and characterizes the stabilizing effect of the Laplace pressure, damping the small-scale surface deformations. Note that for the terms in the second equation in (2) to be of the same order, the capillary number (parameter w) has to be large, which is indeed the case under most of experimental conditions. The effect of the long-scale surface deformations on the short-scale convection is described by the additional term Ah in the Ginzburg-Landau equation for A: it is destabilizing when h > 0 and stabilizing when h < 0. The effect of the mean heat flux generated by the short-scale mode is taken into account by the term s lA I~,~ in the equation for the surface deformation h. Since the coefficient s has been computed to be always positive, s > 0 [ 1,26], this term does describe the stabilizing effect of the mean heat flux. This stabilizing influence of the short-scale convection mode allows the difference between the critical Marangoni numbers Mai and Mas to be O(1 ) and still the deformational instability to be restricted to long scales and have small growth rate. However, this stabilization fails if M a s - M a l is too large (see [24]). 3 In [1 ], the effect of the mode interaction on the stability of the short-scale convection was studied only in the vicinity of the band center (A = 1, h = 0). In the following sections we extend the analysis to the full wave number band of the short-scale convection and show that the interaction of the two modes strongly affects the width of the band and can even lead to a change of the type of the instability.

3. Linear stability analysis of short-scale convection rolls We consider the effect of surface deformations on the stability of short-scale convection rolls with wave numbers lying within an O(E) range at small supercriticality (Ma-Mas)/Mas --= e2. This convection pattern is described by the following stationary solution of Eq. (2): At) = ~/1 - q2 eiqX

h = 0,

(3)

where q < 1 is the dimensionless (in units of the characteristic width e of the band of excited wave numbers) deviation from the critical wave number corresponding to the threshold Ma = Mas of the short-scale instability. Consider a perturbed solution in the form A = A0(1 + ae crt+i~x + be~*t-ixx),

h = ce ~rt+iKx + c*e c~*t-i~:x,

(4)

where asterisks denote complex conjugates. Substituting (4) in system (2) yields the following implicit dispersion relation for the side-band instability: 0 .3 + OtO"2 + ~0" + y = O,

(5)

where Ot = --rag 2 + w K 4 + 2 ( 1 -- q 2 + K2), 2 The relevance of this simple model to real systems is discussed in Section 5. 3 It should be noted that the form of the equation for h is due to mass conservation which implies the conservation ofh. In the nonconserved case certain aspects of the interaction between short- and long-scale structures have been discussed in [32].

134

A.A. Golovin et al./Physica D 106 (1997) 131-147

/3 = 2(--rex 2 + wK4)(1 -- q2 + to2) + K4 + 2K2(1 _ 3q2) + 2sx2(1 _ q2), y

=

(_mK2

_[_

wKa)(K4 ÷ 2K2(1 __ 3q2)) + 2SK4(1 _ q2).

We see that the interaction of the two modes leads to a cubic equation for the eigenvalues defining the limits of the side-band instability (similar, say, to the case of the Eckhaus instability of a hexagonal pattern when there is a coupling between three waves [33,34]). In our case, we have the coupling between two side-band modes, which correspond to longitudinal compression-dilatation waves propagating in opposite directions if cr is complex, and the surface deformation. According to the Routh-Hurwitz criterion, it follows from (5) that the stationary solution (3) is stable if ot > 0,

?' > 0,

ot/~ - y > 0.

(6)

The analysis of conditions (6) shows that, depending on the values of the parameters m, w and s, the side-band instability in this system can be either long-wave or short-wave, monotonic or oscillatory, and the range of q where the solution (3) is stable can be either wider or narrower than the standard Eckhaus interval defined by (1). A detailed analysis of various possibilities is very cumbersome since the stability of the system is governed by four parameters, m, w, s, and q. We shall dwell therefore mostly on the case when the solution (3) becomes unstable with respect to perturbations with infinitesimal wave number. The corresponding stability limits can be determined from (6) in the limit x ---> 0. It is, however, more instructive to obtain this result using the phase approximation which is based on the observation that in the long-wave limit the evolution of the amplitude of the convection pattern is slaved to the evolution of its phase [3]. Thus, we set A = (P0 + ~P) exp (iqx + 0), h --- O(~), Ox = O(E), Ot = O(62), and obtain from Eq. (2) for the amplitude correction p P = - q---q~x ÷ P0 ~P0'

P0 -

1 - q2.

(7)

In our case, the amplitude near the instability threshold is slaved to both the phase ¢ and the surface deformation h. The evolution of these two variables is described by the following system of coupled equations: 1 - 3q 2 dPt ~- -1 ~ U

q dpxx ÷ ~ h x '

ht = (s - m ) h x x - 2qSOxxx.

(8)

Thus, solution (3) is stable with respect to perturbations with infinitesimal wave number if 3q 2 - 1 + (m - s)(l - q2) < 0,

(3q 2 - 1)(m - s) + 2qZs > 0.

(9)

It should be noted that this system obtained in the long-wave limit does not give the complete stability information. It can be shown that there always exists an interval of m where the relevant instability is a short-wave instability that can be either monotonic or oscillatory. We shall dwell on the case when the instability is long-wave in the hand center, and no short-wave oscillatory instability arises. This is the case for s < 2w and s < 1 + ~ (see [1]) and if w is larger than a certain value w*(s) (for s = 3, w > w* ~ 8). A typical stability diagram is shown in Fig. 1 for s -- 3 and w = 15. It presents the region in the plane (m, q2) in which the solution (3) is stable. This region is bounded by the lines AB, BC, CD and DE. The lines AB and DE are the loci of q2 =q2

m --s ~ 3m--s'

(10)

and correspond to a monotonic long-wave instability (cf. (9)). The line CD is the locus of q2 = q2 _~ l + s - m

3+s

-m'

(11)

A.A. Golovin et al./Physica D 106 (1997) 131-147

135

q2

0.8 monotonic i ~ __

monotonic short-scale

UNSTABLE

0.6

_ STABLE

0.2

................. -10

-8

-6

-4

-2

~ q

oscillatory " ~ o n g-scale

monotonic \ Iong-caE./Q _ sl e ) D

.

2

6

4

m

Fig. 1. The dependence of the wave number q corresponding to the onset of the various side-band instabilities on the parameter m (w = 15, s = 3). Different sections of the solid line correspond to different types of the instability: AB - long-wave monotonic (Eckhaus); BC - short-wave monotonic; CD - long-wave oscillatory; DE - long-wave monotonic (Eckhaus). The dashed line presents the standard Eckhaus instability threshold q2 = ½.

and it corresponds to an oscillatory long-wave instability (cf. (9)). The line BC corresponds to a monotonic shortwave instability and its equation is q2

2

I

= qms = 3

s m ~wV/ 9w+~w + s(--3m + s +12w).

(12)

1 The horizontal dashed line corresponds to the threshold of the standard Eckhaus instability, q2 = _~"

The coordinates of the points A, B, C, D, and E can be also found analytically: A--+ { - e c , ½}, B = {-~(s Sw). (5s + s w ) / ( 3 ( s + Su,))}, sw = ~/s 2 + 4 8 s w , D = {½s(1 + ~/1 + 4 / s ) , (x/1 + 4 I s - 1)/(3.,/1 + 4 / s + 1)}, E = {s, 0}; the coordinates of the point C can be obtained from the equation q2o(m, s) = qZms(m, s, w) which can be

reduced to the following quartic equation: m 4 --

2(3 + s + 4w)m 3 + m2(9 + 6s + s 2 + 24w + 16sw + 16w 2)

- 8sw(1 + s + 4 w ) m + 1 6 s w ( s w - s

- 3) = 0.

(13)

F o r s = 3, w = 15 in Fig. 1,C = {1.97, 0.5}. Thus, due to the coupling between the phase evolution and the surface deformation, both the interval of wave numbers corresponding to stable stationary convection described by Eq. (3) and the type of its instability depend on the parameter m, i.e. on the difference between the two thresholds of the Marangoni instability, Mas-Mab There are four particular values of m, (ml (s, w), m2(s, w), m3(s), m4(s)), corresponding to the points B, C, E, D in Fig. 1, respectively, which define four regions where qualitatively different instabilities are observed. These regions in the (s, m)-parameter plane are shown in Fig. 2 for w = 15. If m < mj (s, w), solution (3) is stable when q2 < q2 (see Eq. (10)). Since m < 0, the stability interval for q is larger than in the case of the ordinary Eckhaus instability, q2 < ½; part of the energy of phase perturbations is transferred to the damped long-scale deformational mode. This suppresses the Eckhaus instability and makes the stability interval for q larger. Outside this interval of q, solution (3) is unstable and the instability is monotonic and long-wave ( M L ) , like in the standard case with no surface deformation. In the limit m --+ - c ~ , surface deformations are damped very strongly, and the Eckhaus interval tends to its ordinary value. When rnl (s, w) < m < m2(s, w), the stability region for q is determined by q2 < q2 s (see Eqs. (12) and (13)). The threshold qms is still larger than ½. The instability in this case is also monotonic, but short-wave ( M S ) . At the

A.A. Golovin et al./Physica D 106 (1997) 131-147

136 m

4f

~ rn4 m3

nostableregion ~

-2

t

~8

l

~

ml

Fig. 2. Regions in the (m, s)-parameter plane corresponding to different types of the side-band instability (at w --- 15): ML - monotonic long-wave (Eckhaus); MS - monotonic short-wave; OL - oscillatory long-wave. In the region between m 3 and m 4 both OL and ML side-band instabilities can be observed.

point B the modulation wave number ~c goes to zero and the short-wave instability continuously changes over to a long-wave instability. When m2(s, w) < m < ma(s) --- s, the stability interval for q is given by q2 < qo2 (see Eq. (11)). The instability in this case is oscillatory and long-wave (OL) and can lead to slow oscillations of the wave number of the short-scale roll pattern coupled with oscillations of the surface deformation (see Section 3.2 below). l At m = ma(s) = s, the stability region coincides with the ordinary Eckhaus interval, q2 < 3' I f m > m3 (s) = s, the roll pattern corresponding to the band center, q -- 0, becomes unstable. This coincides with the results in [ 1]. Nevertheless, as long as m 3 (S) < m < m4 (S), a stable pattern still exists under these conditions, but with a modified wave number lying in the interval q2 < q2 < q2. The lower boundary of this interval corresponds to the long-wave monotonic instability while the upper one to the long-wave oscillatory instability. Therefore, in this interval of the values of the parameter m, there can be two types of the long-wave side-band instability, both oscillatory and monotonic, depending on the value of the wave number deviation q, see Fig. 2. This splitting of the stable band into two sub-bands is reminiscent of a situation encountered in parametrically excited standing waves: under certain conditions the wave numbers in the center of the band become Eckhaus-unstable while off-center wave numbers remain stable. Such a splitting was found to lead to the formation of stable kinks in the wave number separating domains with different wave numbers [35]. The nonlinear evolution of the present system in this regime will be discussed briefly below. For m > m4(s), solution (3) is unstable at any wave number due to the deformational instability of the surface. Thus, depending on m, i.e. on the difference between the threshold Mas of the short-scale convection and that of the long-scale deformational instability (MAD, the band of stable wave numbers can take on two qualitatively different forms. For m < s the band is contiguous (hatched area in Fig. 3(a)) while for s < m < m4(s) convection is stable within two disconnected wave number bands (hatched area in Fig. 3(b)). It is worth noting that if the parameter w is not large enough, the oscillatory short-wave side-band instability comes into play. In this case, the stability region shown in Fig. 1 slightly changes: there appears a new boundary corresponding to this shOrt-wave oscillatory side-band instability which cuts off the corner at the point C and

A.A. Golovin et al./Physica D 106 (1997) 131-147 m
137 m3
4

4f

i 3.51-

3.5 ...-

..

i

3

''

2.5

....

2.5 t ~1

L.

2 1.5

'.

1.5 l

~

/:

i

1 I

I 0.5 0

0"5 I 0

0.2

i

i

0.4

0.6 k

i

0.8

1

1.2

O' 0

i

0.2

0.4

i

A

i

0.6

0.8

1

1.2

k

Fig. 3. Two qualitatively different types of stable wave number bands for the short-scale Marangoni convection near the instability threshold: (a) m < m3: the stable region (hashed) is larger than in the standard case (dashed line), and its boundary (solid line) can correspond to either a monotonic, long- or short-wave instability, or to an oscillatory long-wave instability; (b) m 3 < m < m4: the band center is unstable, and the stable region (hashed) is split up into two parts; its inner boundary corresponds to a monotonic long-wave instability and the outer one to a long-wave oscillatory instability. The dotted line is the neutral stability curve. connects two boundaries BC and C D corresponding to the short-wave m o n o t o n i c and to the long-scale oscillatory side-band instabilities, respectively.

4. Nonlinear evolution of the long-wave instabilities In this section we shall study the n o n l i n e a r evolution of the long-wave instability near the thresholds given by the curves AB, C D and DE in Fig. 1.

4.1. Eckhaus instability The m o n o t o n i c long-wave instability threshold corresponding to the curves AB and D E in Fig. 1 is given by Eq, (1 1). We use the following scaling: X = Ex, T = E4t, appropriate for the long-wave Eckhaus instability [14], and apply the following ansatz: A = p exp i4~, /9 = Do q- ~:2pl (X, T) + ~73,02(X, T) + • • •,

4 9 = q x +~491(X, T) -+- E2q~2(X, T) -Jr-..., h -----eZhl (X, T ) + e3h2(X , T ) + - . . ,

q = qrn + e2q2 + " " •, where q2 characterizes the deviation of q from the threshold value qm, and it can be chosen to be -t- 1.

(14)

138

A.A. Golovin et al./Physica D 106 (1997) 131-147

Substituting (14) in system (2), one obtains at first order that the change in the local wave number, kj -- Oxdpl, is proportional to the surface deformation h I, s-m

kl - - -

2Sqm

hi

(15)

Proceeding to sixth order, we obtain the following nonlinear evolution equation for the surface deformation h I : (16)

OThl + a l O x x h l + c r 2 O x x x x h l +
where (4m - s)(s - 3m) ~rl = q 2 q m 2 ( _ m 2 + m s + s ) ,

-3m 2 + ms + 4sw cr2= 4 ( - m 2 + m s + s ) '

m ( s - 3m)(5s - 6m)

or3 =

8s(-m 2+ms+s)

The evolution equation for the long-wave modulation of the wave number kl can be obtained from Eq. (16) by a simple renormalization of the coefficient ~r3 --~ ~r32Sqm/(S - m ) (cf. (15)). When m --+ - o o , the surface is undeformable, and the coefficients in the equation for kl tend to those of the nonlinear evolution equation describing an ordinary Eckhaus instability in the real GL equation. It can be shown that the coefficients ~r2 and ~r3 are positive (in the parameter region where the instability is long-wave), and the instability is subcritical. Therefore, all periodic solutions of (16) blow up beyond the stability limit ~rl = 0 indicating a break-down of (16). The same happens in the case of the monotonic short-scale instability, corresponding to the curve BC in Fig. 1; our computations show that it is also subcritical. However, numerical simulations of the full amplitude equations (2) show that different physical processes occur when crossing the lines AC and DE in Fig. 1, respectively. This is related to the fact that along AC the deformational mode is linearly damped (m < s), whereas along DE it is excited (m > s). Figs. 4 and 5 show typical results of numerical simulations of (2). Fig. 4 shows the evolution of the pattern occurring when the line AC is crossed. While the surface deformation (Fig. 4(a)) and the amplitude of the short-scale convection (Fig. 4(b)) remain regular for all times, the wave number goes through a sequence of true singularities, each of which signifies a phase slip. As a result of the phase slips two pairs of convection rolls disappear around t -----50 and another pair around t = 150. Following these phase slips, the pattern relaxes to a pattern with another wave number which lies within the stable band. Only the beginning of this final, slow evolution is shown in Fig. 4. The evolution below the line DE is shown in Fig. 5. Here the deformational mode is excited; it has been shown previously that the instability is subcritical at q = 0 [24]. The simulation of the coupled amplitude equations (2) indicates that the instability does indeed not saturate. This results in a singularity in the surface deformation and in the amplitude of the short-scale convection. Thus, not only the phase equation (8) but also the amplitude equations (2) become invalid. This result suggests a dry-out of the fluid film as has been observed experimentally [25]. Despite the splitting of the wave number band, no stable wave number kinks could be found. This is presumably related to the fact that near E (s = m, q,n ----0) the description using the phase equation becomes insufficient (cf. (15)). This indicates that the higher-order contributions to the phase equation (16), which were not calculated, differ from those arising solely from a nonmonotonicity of the phase diffusion coefficient [35]. It should be noted that at the point D the coefficients al, a2 and a3 tend to infinity, while at the point E the coefficient o"1 vanishes regardless of the value of q2; the system dynamics in the vicinity of these points is more complex; it cannot be described by Eq. (16) and requires separate analysis.

A.A. Golovin et al./Physica D 106 (1997) 131-147

139

200 0.4-

.V: 0 . 2 ._m ®

-1-

O-

-0.2 - -.'ime 90

80

70

60

50

40 ,

30 /

20 ~

7 10 ~ 7

~ 0

0

Position (a)

200

1

~o fl:

-1

(b) Fig. 4. Typical space-time diagrams for the evolution of the long-wave monotonic instability along line AC (see Fig. 1) as obtained by numerical simulation of (2): the surface deformation (a) and the amplitude of the short-scale convection (b) remain regular while the wave number passes through a singularity indicating the phase slip. The parameter values are m = 0, w = 15 and s = 3. The system length is L = 85.5. Note that the spatial resolution of the graph is lower than that used in the numerical simulation.

140

A.A. Golovin et al./Physica D 106 (1997) 131-147

E-1-

.~500

0 (a)

1000

Time

Position

g I1)

rr

~_500

0

1000

Time

Position (b) Fig. 5. Typical space-time diagrams for the evolution of the long-wave monotonic instability along line DE (see Fig. 1) as obtained by numerical simulation of (2). Here the surface deformation (a) and the amplitude of the short-scale convection (b) become singular indicating dry-out of the fluid film. Note the different scales for h and A in Figs. 4 and 5. The parameter values are m = 3.5, w -----15 and s -----3. The system length is L = 150.

A.A. Golovinet al./PhysicaD 106(1997)131-147

141

4.2. Oscillatory long-wave instability Along the line CD in Fig. 1 the oscillatory long-wave instability arises. Its threshold is given by Eq. ( 11 ). In order to study the nonlinear behavior of this long-wave oscillatory instability we set A = p exp i(p, introduce the slow coordinates X = 6x, 7b = E2t, T = ~:4t, and expand P : P0 + ¢Pl (X,

dp=qx

+

To, T) + ~2p2(X , TO, T) + (3p3(X, T0, T) + . . . , ~b0(X, To, T) +¢(ol(X, TO, T) + EZcbz(X, TO, T) + (:3q~3(X,

/]), T) + . . - ,

(17)

h = ~hl(X, To, T) 4- E2hz(X, TO, T) + ,~3h3(X, To, T) + . - . ,

q =qo + ¢2q2 +

"" •

with P0 =

3

-- qo'

Substituting (17) in system (2), we obtain in the first order a linear system for hi and k0 = Ox~0 which, after appropriate diagonalization, can be reduced to a linear free Schrrdinger equation iOTo~ =

--tooOxx~,

(18)

where 'k = ((m - s + io~))hl - k0)/(2ito0),

too = x / - r n 2 +

s(m +

1).

Eq. (I 8) describes waves propagating along the convection pattern and modulating its wave number and the surface deformation. The phase velocity of the waves Up depends on the wave number x of the modulation, as Up = toox. The wave number ko and the surface deformation h 1 are proportional to each other k0 = 3hi,

8 --

s - m + ito0

2sqo

,

(19)

and there is a phase shift between oscillations of the wave number and the surface deformation equal to arg(6). At the onset of the oscillatory long-wave instability, a whole band of wave numbers ~ O (e) is excited. However, a strong (quadratic) dispersion of the waves described by (18) allows only quartic interactions between waves with different wave numbers within the excited interval. The evolution of the wave amplitudes can thus be described in the third order of the perturbation theory by a system of Landau equations. A similar situation occurred in a study of long-scale oscillatory double-diffusive convection [36]. In order to study the wave interaction, we consider two pairs of oppositely propagating waves having wave numbers xl and to2, and present k0 and hi in the following form:

( k° ) = ( ~l) [Al(T)ei°~( + Bl(T)ei°~ + A2(T)ei°+ + B2(T)ei°f ]

(20)

0~,2 = to0K2,zT0 4- tCl,zX, where the amplitudes A 1,2 and Bi,2 undergo slow evolution on the time scale T ~ ~4t. Using (21) and (17), we obtain from (2), as solvability conditions in the fourth and in the fifth orders, the following system of Landau equations for the amplitudes A1,2 and Bl,2:

142

A.A. Golovin et al./Physica D 106 (1997) 131-147

2.25

2.5

2.75

3

~

~

-2

5m

~

-4.

a

-6

stable

\%7,w, \

stable standing wave

\

-8 -1(~

Fig. 6. The real parts of the interaction coefficients ~,~(Xa(m))(curve a) and ,~(~b(m)) (curve b) from Eq. (22) at s = 3, w = 15, as functions of m. The vertical asymptote corresponds to m = m4(s) (point D in Fig. 1). A1 = yIA1 + ~'~l ]AI [2AI + ~-~1[B112A1 + X~21A212A1 + xb21B2[2AI, /~1 = Yl Bl + Z~1 ]B11281 + ~'~l IA1 ]281 q- ~'~2 [821281 + )~2[A212B1,

(21)

,42 = y2Az -t- Z~z[Azl2A2 + ~.2621B212A2 + ~'~11a112A2 + X~I IB112A2, /~2 = )/202 --}-Z;21021202 q- )~b2]a21202 -q- ~ l ]Bl 1282 + )~2blIAII2B2" The linear growth rate coefficients Yn and the Landau coefficients ~a;b (n = 1, 2) have the following form:

Vn =q2ot(m,s) x2 - ~ ( m , s , w)K 4, Xannb = ~a,b(m,s)x 2,

n = 1,2,

(22)

where ,~(~(m, s)) > 0, .~(/~(m, s)) > 0 ifm2 < m < m4. The Landau coefficients Xna~b (n ¢ p) are complicated functions of m, s, xl, x2 satisfying the relationships a,b

a,b

~'12 (KI'K2)~---)~21 (K2, K1), ) ~ b z ( K 1 , K I ) = ) , b I ( K 1 , K I ) = ) ~ b 1,

~-~2(KI,KI) = . ~ I ( K I , K I ) = f ( m , s ) ) ~ 7 1 .

The real parts ~(Xa,6(m, s)) of the functions ~-a,b(m, s) in Eq. (22) are shown in Fig. 6 as functions of m. They become unbounded in the vicinity of the point D (m = ½s(1 + ~ + 4/s)) where system (21) is not valid. System (21) allows to study the competition between travelling and standing waves. Let us consider two waves with a fixed wave number propagating in opposite directions. Setting A2 ~ B2 ~ 0 in (21), we study the stability of the two solutions of system (21) Yl BI = A2 = B2 -----0 (travelling wave); ~'~1'

(1)

a~--

(2)

A2 = B 2 _

~'~1 Y1 -~- ~'bl

A2 = B2 = 0

(standing wave);

(23) (24)

with respect to infinitesimal perturbations of the amplitudes A] and B1. The symbols of the Landau coefficients in (23) and (24) and below denote now only the real parts of the respective coefficients. The stability analysis shows that (a) travelling waves are stable if

x~z<0,

X~l
a X11/Zll < 1;

(25)

(b) standing waves are stable if

~'~1 < 0 ,

a b I > 1; IX11/)~ll

(26)

otherwise, the system undergoes a subcritical instability which may lead to a blow-up and may return the system back to the stable region. We have performed some numerical simulations of (2) which support this expectation.

A.A. Golovin et al./Physica D 106 (1997) 131-147

m=2.35

m=2.8

1

0.8

0.8

0.7 0.6

c~ 0.6

0.5 ~ 0.4

s

0.4

143

m=3.25

! 0.6 I

0,5 0,4

s

~ 0.3

0.3

0.2

0.2

0.2

0.1

0.1 0.2 0.4 0.6 0.8 kl

1

0.2

m=3.4

0.4 kl

0.6

0.8

0.2

m=3.6 0.451 ! 0.4[

0.45 r

0.35

0.35

0.3

0.3f 0.25

0.31 c,O0.25t

0.2

0.2 f 0.15

0.5 0,4

0.1

/

0.2 0.3 kl

0.4

0.2t 0,15~

s

0.1 t

0.05 l

0.05 t 011

0.6

m=3.63 0.41-

0.1 I

0.1 0.2 0.3 0.4 0.5 kl

0.4 kl

012 013 kl

014

0.1

Fig. 7. Regions of stability of standing and travelling waves with a wave n u m b e r x! (lying within the range of linearly excited modes) with respect to perturbations with a wave n u m b e r x 2 (lying within the s a m e region), at different values of the parameter m: s - stable, u - unstable.

We have found (see Fig. 6) that in the case shown in Fig. 1 (w = 15, s = 3), when 2.135 < m < 3.361 standing waves are stable, and when 3.361 < m < 3.745 travelling waves are stable. In the rest intervals of m within the interval m2 < m < m4, close to the points C and D, both types of waves are subcritically unstable. Let us now study the stability of standing and travelling waves with respect to perturbations with different wave numbers, i.e. the stability of solutions (23) and (24) with respect to perturbations of the amplitudes A2 and B2. The stability analysis shows that (a) travelling w a v e s are stable if, besides (25), the following conditions are fulfilled: )~a - }'1 -C~ < O,

~.2bl Y2 - Y177- < O;

(27)

(b) s t a n d i n g w a v e s are stable if, besides (26), the following condition holds: }"2 -- Y1 )~1 a + ~'~1 < O.

(28)

144

A.A. Golovin et aL/Physica D 106 (1997) 131-147

The analysis of conditions (27) and (28) shows that within the intervals defined by (25) and (26) there exist intervals of wave numbers where the waves are also stable to perturbations with an arbitrary wave number. Fig. 7 shows how these intervals change with the variation of the parameter m along curve CD in Fig. 1. It can be seen that standing waves are stable at intermediate wave numbers, while travelling waves can be stable at infinitesimal wave number as well. Thus, according to the weakly nonlinear analysis of the oscillatory long-wave instability along CD the coupling to the deformable interface can generate either standing or travelling compression-dilation waves on the periodic pattern which modulate its wavelength together with the deformation of the free surface. To confirm this result we have also performed direct numerical simulations of (2). The results are presented in Fig. 8(a) and (b). As expected standing compression-dilatation waves are stable for smaller m (m = 3.2 in Fig. 8(a)), whereas the travelling waves are stable for larger m (m : 3.5 in Fig. 8(b)). It is worth noting that the transients are quite long. The steady state shown in Fig. 8(a), for instance, was reached after t = 5 x 105. For smaller m the bifurcation to standing waves becomes subcritical. In simulations in that regime the oscillation amplitude becomes quite large and eventually leads to one or more phase slips. This brings the wave number back into the stable band and the pattern becomes eventually stationary.

5. Discussion and conclusions In the present paper we have studied the side-band instabilities of the short-scale roll Marangoni convection in the case when it interacts with the long-scale deformational Marangoni instability. We have shown that this nonlinear interaction between the long-scale phase perturbations of the roll convection pattern and the additional long-scale deformational mode strongly affects both the interval of the stable wave numbers and the type of the side-band instability of the short-scale roll convection pattern. Due to the coupling between the two modes, the side-band instability of the roll pattern can be either long- or short-wave, either monotonic or oscillatory, the latter generating long standing or travelling compression-dilation waves modulating the roll convection pattern and modulating its wavelength as well as the deformation of the free surface. The presented theory describes the evolution of 1-D roll convection patterns near threshold. However, as far as Marangoni convection is concerned, it should be stressed that near the threshold, Marangoni convection is usually hexagonal, due to specific quadratic resonant interactions typical of convection driven by surface tension gradients. But at some distance from the threshold, the roll pattern rather than the hexagonal one can become preferred [27,28]. The roll Marangoni convection pattern can be selected even near the instability threshold in fluids with low Prandtl numbers for which the quadratic resonant interaction responsible for the appearance of hexagons can vanish [29]. Besides, the roll Marangoni convection pattern can be prescribed by the geometric shape of the container [30,31 ]. Thus, the present 1-D analysis is related to the situation when the preferred planform in Marangoni convection is the 1-D roll pattern, and the system evolution can be described by Eqs. (2). The generalization of system (2) for the case of the interaction between the long-scale deformational instability and the short-scale hexagonal convection pattern is studied in [26].

Fig. 8. Results of numerical simulation of (2) with the parameters in the vicinity of the curve CD in Fig. 1 corresponding to the long-wave oscillatory side-band instability of the periodic short-scale convection. (a) Typical space-time diagram of the standing compression-dilation wave arising from the instability. The parameters are m = 3.2, s = 3 and w = 15. The system length is L = 237.6. (b) Typical space-time diagram of the travelling compression-dilation wave arising from the instability. The parameters are rn = 3.5, s = 3 and w = 15. The system length is L = 285.

~45 A.A. Golovin et al. / Phvsica D 106 (1997) 131-147

5-

-5 250

/

/

/

/

/

200

150

1O0

50

0

2000

Position (a)

cc

Jl 300

~l~f

Time 3

(b)

0

~oo 50

Position

146

A.A. Golovin et al./Physica D 106 (1997) 131-147

Table 1 Various types of the side-band instability of the roll Marangoni convection in the silicon oil-air system [25] Liquid layer depth ( m m )

Temperature difference a (K)

m

w × I 0-3

Type of instability

0.75 0.30 0.25 0.23 0.22

49.4 123.5 148.1 161,0 168.3

-725.6 - 10.9 9.6 15.8 18.6

20.0 8.2 6.8 6.3 6.0

ML MS MS OL,ML No stable state

Gas-to-liquid depths ratio is 1.6;s = 15.46.Instability types: ML- monotonic long-wave, MS - monotonic short-wave, OL- oscillatory long-wave. a The temperature difference, AT, given in the table, is applied across the whole liquid-gas system. The temperature difference across the liquid layer ATI, is 1 + a l l times smaller, where )~ is the gas-to-liquid ratio of heat conductivities and a is the ratio of the phases depths. For the chosen system, A 7~ ~ 0,1.4 T. In real experiments, the Marangoni convection is usually studied in two-layer liquid-gas systems [25]. In this case the governing parameters of Eqs. (2) - m, w and s - can be controlled by varying the depths of the liquid and gas layers, by the intensity of heating as well as by choosing liquids with particular properties [1,26]. As an example, we present the physical parameters of a real liquid-air system where the described different kinds of the side-band instability of roll-cell Marangoni convection could be experimentally observed. These parameters and the corresponding types of the side-band instabilities are given in Table 1 for a silicon oil-air system studied experimentally in [25], where the interaction between the two modes of Marangoni convection was observed. Besides the physical properties of the liquid and gas, we keep the gas-to-liquid depths ratio fixed, varying only the depth of the liquid layer and the overall temperature difference across the system. This allows us to vary the parameters m and w only, keeping the parameter s fixed. The depth of the liquid layer should be very thin in order to induce the deformational instability [25]. The temperature differences and the gas-to-liquid depths ratio are chosen such that the roll pattern would be selected. The details of the calculation of the parameters m, s, w for a two-layer liquid-gas system can be found in [26]. For the chosen value of the gas-to-liquid depths ratio equal to 1.6, we have s = 15.46. One can see that the system is very sensitive to the variation of the liquid layer depths due to a strong dependence of the two thresholds of the primary Marangoni instability, Mas and Mal, corresponding to short- and long-wave convection. The small depths of the liquid layer guarantee also that the Marangoni (surface tension gradients) mechanism of convection is indeed predominant, and the effect of buoyancy is negligible: for the chosen liquid layer depths the ratio of the Rayleigh and the Marangoni numbers is R a / M a = 0.007-0.08. The chosen large temperature differences can be applied for the given system due to a high boiling point and a very low vapor pressure of a silicon oil ( 1 0 - 1 ° p a at room temperature), and they guarantee that the roll convection pattern is selected (see [26] for details). It should be noted, however, that the threshold when the roll convection pattern becomes preferable corresponds to the supercriticality (Ma-Macr)/Macr ~ 2 [26], which is, strictly speaking, not small. The applicability of the developed theory in this case is, therefore, reduced to the applicability of the amplitude equations (2) and should be verified experimentally.

Acknowledgements This research was supported by the Israel Science Foundation. AAG acknowledges the support of the Ministry for Immigrant Absorption. LMP and A A N acknowledge the support by the Fund for Promotion of Research at the Technion. HR gratefully acknowledges a travel grant by the Minerva Center and support by DOE through DE-FG02-92ER 14303.

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147

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