Mathl. Comput. Modelling Vol. 28, No. 10, pp. 45-58, 1998 @ 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain

PII: SO895-7177(98)00154-X 0895-7177/98 $19.00 + 0.00

Pulse Evolution for Marangoni-Bhard Convection

T. R. MARCHANT Department of Mathematics, University of Wollongong

Wollongong, 2522, N.S.W., Australia

N. F. SMYTH Department of Mathematics and Statistics

University of Edinburgh, The King’s Buildings Mayfield Road, Edinburgh, Scotland, U.K., EH9 352

(Received and accepted February 1998)

Abstract-Marangoni-Benard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven, is described by a perturbed Korteweg-de Vries (KdV) equation. For a certain parameter range, this perturbed KdV equation has a solitary wave solution with an unique steady-state amplitude for which the excitation and friction terms in the perturbed KdV equation are in balance. The evolution of an initial sech2 pulse to the steady-state solitary wave governed by the perturbed KdV equation of Marangoni-BBnard convection is examined. Approximate equations, derived from mass conservation, and momentum evolution for the perturbed KdV equation, are used to describe the evolution of the initial pulse into steady-state solitary wave(s) plus dispersive radiation. Initial conditions which result in one or two solitary waves are considered. A phase plane analysis shows that the pulse evolves on two timescales, initially to a solution of the KdV equation, before evolving to the unique steady solitary wave of the perturbed KdV equation. The steady-state solitary wave is shown to be stable. A parameter regime for which the steady-state solitary wave is never reached, with the pulse amplitude increasing without bound, is also examined. The results obtained from the approximate conservation equations are found to be in good agreement with full numerical solutions of the perturbed KdV equation governing Marangoni-B&mrd convection. @ 1998 Elsevier Science Ltd. All rights reserved.

Keywords-Marangoni-Benard convection, Korteweg-de Vries equation, Solitary waves, Conser- vation laws, Approximate solutions.

1. INTRODUCTION

It is well known that if a horizontal liquid with a free surface is heated from below, then past a certain threshold a steady convective pattern of Bdnard cells develops. In contrast, when the heating is from above, Marangoni convection becomes important, with the excitation of solitary waves possible. Marangoni convection is the mechanism by which energy is released due to the change in surface tension with temperature. Typically, the surface tension must decrease with increasing temperature and the heating be from above for the excitation of solitary waves to occur, although for some liquids the surface tension increases with increasing temperature, in which case Marangoni convection can occur when the liquid is heated from below. In the case of Marangoni convection driven by mass-transfer, the liquid’s surface tension varies with the concentration of a surfactant which can either be absorbed or desorbed by the liquid.

Typ-t by 4-G’?S

45

46 T. R. MARCHANT AND N. F. SMYTH

Garazo and Velarde [l] considered a shallow liquid being heated from above or below with both density and surface tension dependent on temperature. The fluid flow was described by the Navier-Stokes equations and the heat-flow by the heat conduction equation. Assuming that the heat transfer at the air-liquid interface is small, they derived

as the evolution equation for the surface displacement of the liquid. The small parameter (Y is related to the Biot number, which is a nondimensional measure of the heat transfer at the liquid’s surface. Equation (1.1) is a perturbed KdV equation where the O(o) terms represent both the excitation, due to the change in surface tension and density, which cause the solitary wave to grow, and the viscosity, which causes the solitary wave to decay. For a certain parameter range, a balance can be achieved between these growth and decay terms, which results in a solitary wave with a fixed amplitude and wavespeed. Therefore, no matter what the initial amplitude, an initial condition will evolve to a solitary wave with an unique steady-state amplitude determined by the parameters cl, ~2, and cz in (1.1). Chu and Velarde [2] derived (1.1) with cz and cs both zero in order to examine the initial excitation of the solitary wave. They found that a certain critical value of the Marangoni number, which is a measure of the rate of change of surface tension with temperature, needs to be exceeded for excitation to occur.

Weidman et al. [3] and Linde et al. [4] presented various experimental results for solitary waves driven by Marangoni-Benard convection. The experiments were of both main types: where the convection was driven by heat transfer and by mass transfer. The experiments examined a range of phenomena, such as head-on interaction of solitary waves and reflection of solitary waves from a wall. It was found that, depending on the angle of incidence, both regular reflection and Mach reflection (with the development of a Mach stem) can occur. Various experimental data were presented, such as the phase shifts the waves undergo after interaction, and the critical angle of incidence at which the Mach stem first appeared.

Kawahara [5] and Kawahara and Toh [S] considered (1.1) with the coefficient cg zero. Kawa- hara [5] derived the steady-state solitary wave amplitude which results in a balance between the excitation and friction terms in (l.l), these terms corresponding to the coefficients cl and cz, respectively. Numerical solutions for various initial conditions showed evolution to solitary waves with amplitude close to the theoretical prediction. Kawahara and Toh (61 assumed a steady trav- eling wave solution and numerically solved the resulting ordinary differential equation to obtain the profiles of solitary waves with steady-state amplitudes. The solitary wave was found to be nonsymmetric with an oscillatory tail. An initial value problem with periodic boundary condi- tions was also examined by solving the original partial differential equation numerically. When dispersion is significant, a periodic lattice of solitary waves was found to evolve, while in the limit of small dispersion, chaotic behaviour was found to occur with solitary wave-like pulses being continuously created and destroyed.

Marchant [7] derived steady-state solitary wave solutions of (1.1) to 0(a2). The evolution of a solitary wave of arbitrary initial amplitude to the steady-state solitary wave was also found, to O(o), by using a perturbation method. While the solitary wave is evolving, it was found that mass leakage results in a tail or shelf forming behind the solitary wave. This in turn causes the asymptotic solution to be nonuniform far behind the solitary wave. To overcome this, the solitary wave solution must be matched to an outer solution in the tail region behind it. In addition, the interaction of evolving solitary waves was examined theoretically, using a perturbation method based on inverse scattering. Good agreement wss obtained between the theoretical results and numerical solutions.

Kath and Smyth [8] considered solution evolution and radiation loss for the KdV equation. The evolution of a initial sech2 pulse into solution(s) plus dispersive radiation was examined by developing approximate evolution equations from the KdV conservation laws. These approximate

Pulse Evolution 47

equations were ordinary differential equations describing the time evolution of the amplitude, width, and velocity of the pulse, and also of the mass of the dispersive radiation. Initial conditions which formed one or two solution(s) were considered and good agreement was found between the solutions of the approximate conservation equations and full numerical solutions. Smyth and Worthy [9] extended the method of [8] to the modified KdV (mKdV) equation. It was found that the evolution behaviour of an initial pulse depended on the power p of the nonlinear term rlp~ in the mKdV equation. For p < 4, the behaviour of the pulse was found to be similar to that for the KdV equation in that the pulse evolved into a number of solitary waves. For p 1 4, the evolving pulse was found to either grow without bound or decay into dispersive radiation alone, depending on its initial amplitude.

The aim of this paper is to apply the method of [8] to the perturbed KdV equation (1.1) in order to describe the evolution of an initial pulse into Marangoni-Benard solitary wave(s) plus associated dispersive radiation. The evolution of an initial KdV solution into a Marangoni-Benard solitary wave can be described using perturbation methods based on inverse scattering. However, when the initial condition is not a KdV solution, or a small perturbation from this, perturbed inverse scattering cannot be used to describe the evolution of the pulse. On the other hand, the method of [8], based on conservation laws, allows such initial conditions to be dealt with. Moreover, the method presented here is simpler than inverse scattering and can be used in cases for which an inverse scattering solution does not exist, such as for the mKdV equation with p > 2

(see 191). In Section 2, the results available for the solitary wave of unique steady-state amplitude are

presented. In Section 3, mass conservation momentum and the moment of momentum equations are used to derive ordinary differential equations describing the approximate time evolution of the amplitude, width, and velocity of an initial sech’ pulse into one or two steady-state Marangoni- Benard solitary waves. The total mass in the dispersive radiation shed by the evolving pulse is also calculated. In Section 4, solutions of the approximate conservation equations are compared with full numerical solutions of (1.1) for initial conditions which evolve into one or two steady- state solitary wave(s). A phase plane analysis is also performed, which determines the stability of the steady-state solitary wave and also shows that the pulse evolves on two time scales. An example for which the initial pulse grows without bound is also examined. Section 5 details the numerical scheme used to solve (1.1) while the Appendix gives details of the stability analysis for the approximate conservation equations.

2. THE STEADY-STATE SOLITARY WAVE The steady-state solitary wave solution of the perturbed KdV equation (1.1) was found by

Marchant [7]. This steady-state wave exists since the terms of O(o) perturbing the KdV equation represent both friction (which causes the solitary wave to decay) and excitation (which causes the solitary wave to grow). Hence, when the friction and the excitation are in balance, a steady solitary wave with an unique amplitude can form. An asymptotic solution of the form

f-f(@) =770(Q) +Qn(e) +&2(~) +... 7

e = 2 - vt, whereV=Ve+02Vs+... , rr<l (2.1)

was assumed (see, for example, [lo]), w h ere 6 is the phase and V is the wavespeed. It was then found that the solution to 0(02) wss given by

q. =ase:h e

0 Ti ’ where a/3’ = 2, Vo = 2a,

71 = :a&-‘S’TL, P-2)

T. FL MARCHANT AND N. F. SMYTH

9 772 =

524 -a s 4

c5c2 - 2I lo2S2c; + 4a2c5c2S2-T + P

+4a2cgs4 2 222

49 + za $3 ;T + 2L2 - 2L - ;a2S2c5c2,

(2.2)(cont.)

v,= l6 -a"Cg(C3 -5&2)-i- 8463~1

-&a3c5c2(c3 - lOc2), 6

where S = sech - 1 0 P'

T = tanh $ , 0

and cg = c3 - 6~. It was further shown that this solution exists for 7Cl

a= lOc2 - 4c3

only. The amplitude of the steady solitary wave to O(cu2) is then

(2.3)

(2.4)

so that a solitary wave solution exists only for the unique amplitude at which the excitation and friction terms in (1.1) are in balance.

Notice that the correction to the solitary wave at O(a) leaves the solitary wave’s amplitude and velocity unchanged. However, the profile is modified, rendering it nonsymmetric, while the correction at 0(cr2) is symmetric. Linde et al. [4] state that the coefficients cl and c2 are both positive for Marangoni-Bensrd convection, while q can be of either sign. Hence, negative c3 stabilises the system by reducing the amplitude of the solitary wave, while positive q < (5/2)c2 causes the amplitude of the solitary wave to increase. For c3 2 (5/2)c2, no solitary wave with a steady-state amplitude exists. In this case, the excitation terms in (1.1) with coefficients cl and g cannot be balanced by the friction term associated with the term in cs for any value of the solitary wave amplitude. This unbounded growth indicates that the approximations made in the derivation of the weakly-nonlinear perturbed KdV equation have broken down and that a higher-order model for Marangoni-BBnard convection than (1.1) is required.

3. APPROXIMATE CONSERVATION EQUATIONS In this section, the approximate conservation equations governing the evolution of an initial

pulse for the perturbed KdV equation (1.1) will be derived using the method developed by Kath and Smyth [B] for the KdV equation. The standard perturbation technique, based on the inverse scattering method, requires the initial condition to be a KdV solution, while the method used here allows more general initial conditions to be used. The present method uses the mass and momentum equations for the perturbed KdV equation and is based on the fact that any dispersive radiation generated by an evolving pulse must lie behind it ss the pulse velocity is positive and linear dispersive radiation has both negative group and phase velocities. The mass conservation, momentum, and moment of momentum equations for the perturbed KdV equation (1.1) are

vt + -4, = f-4

‘I: +2A2z = 2acr$ - 2ac2& + 2cyc317~jz,

(“7p)t + 2A3, = 4q3 - 377; + 2Wiz~; - 2crczz~~z + k327/& where

A1 = 3q2 + t~zz + ~~1712 + [~Czv+zz + (~c3vrlz, 2 (3.1)

A2 = 2q3 +qvz+ - $ +(rc1qqx +acaqqmz +ffc3q2172 -~c2qz%z,

A3 = 2xq3 + xqqzz - x$ mq2

-qqx+ QClw?, 2 - - -ac2q%T

+~c2w%xz -~C22%%, +crcarl: +ac3xq2qz ac3v3

--I 3

Pulse Evolution 49

The second and third of equations (3.1) cannot be put in conservative form; the terms on the right-hand side of these equations describe the growth or decay of the solitary wave to the steady- state.

We wish to consider the evolution of an initial pulse into one or more solitary waves plus dispersive radiation. In the present work, calculations will be done to O(a) only, since including the 0(a2) terms of the solitary wave solution (2.1) leads to a large number of integrals which cannot be evaluated analytically. As the parameter CY is chosen to be small, these extra terms do not have a great effect on the solution and do not add anything to the understanding of the evolution of the initial pulse to the steady-state solitary wave. We then seek a solution of the perturbed KdV equation of the form

np = ase2& f + ~CYQ.CL@-~ Seth f tanh f In P 7 P P

0 = z - E(t), t’(t) = V(i),

where (3.2)

with V(t) being the pulse velocity. The term qp is the time varying pulse which evolves from the initial condition

a(O) = A and p(O) = w. (3.3)

The term Q. is assumed to account for the dispersive radiation shed by the evolving pulse. In general, the dispersive radiation shed by the pulse has small amplitude (see behind the solitary waves in Figure 6), so that ]r,r,.] < ]np]. In (3.2), the amplitude a, width p, position [, and velocity V of the pulse all depend on time t. The assumed form (3.2) allows the evolving pulse to go smoothly from the initial condition (3.3) to the steady-state solitary wave (2.1). Note that it is not required that the initial pulse be a KdV solution (with Aw2 = 2) as is the case when perturbation theory based on the inverse scattering method is used. From the perturbed KdV equation (l.l), it can be seen that to O(1) the phase and group velocities for linear dispersive waves are -k2 and -3k2, respectively. Hence, since the pulse velocity is positive, the shed dispersive radiation is quickly left behind by the evolving pulse. We can therefore assume that any dispersive radiation ahead of and in the vicinity of the pulse will rapidly decay to zero and can be ignored.

The approximate conservation equations are obtained by integrating the conservation equa- tions (3.1); the first is integrated from the pulse position z = E to 2 = co (on noting that there is no radiation ahead of the pulse, as mentioned above), while the others are integrated over the whole domain from z = --co to z = co. These equations are thus

g (aP+ +,a> = 3a2 - Va - 2cbpe2,

The terms on the right-hand side of the second of (3.4) drive the evolution of the pulse. For a certain parameter range, these terms force the pulse to evolve to the unique steady-state solitary wave, independent of its initial amplitude and width. In contrast, for the KdV equation (see [8]) and the mKdV equation (see [9]), the final steady-state amplitude and width depend on the initial amplitude and width of the pulse.

We note that there is no O(a) contribution to the velocity, as was the case for the steady-state solitary wave solution (2.1) of the perturbed KdV equation (1.1). As the velocity is given by an algebraic equation, (3.4) represents two ordinary differential equations for the pulse amplitude a and width p, which are solved numerically using a fourth-order Runge-Kutta scheme.

50 T. R. MARCHANT AND N. F. SMYTH

The mass shed by the evolving pulse also needs to be considered. Integrating the mass con- servation equation (the first of (3.1)) over the whole domain gives the mass of the dispersive radiation as M = 2(Aw - a@), which is just the difference between the initial mass and the current mass of the pulse. When the mass of the initial condition (3.3) is large enough, more than one solitary wave will be produced. The formation of a second solitary wave is signalled by the change in sign of M, the mass in the dispersive radiation. For small enough initial mass, M < 0 and only one solitary wave is produced. However, when M > 0, the mass in the dispersive radiation evolves into one or more additional solitary waves. This is because, since Q is small, the short time evolution of the pulse is nearly the same as for the KdV equation and it is known from inverse scattering theory for the KdV equation that any positive mass will evolve into a solution. When M > 0, the perturbed KdV equation (1.1) generates at least one additional solitary wave. The system (3.4) can be extended to account for these extra solitary waves in the following manner.

A second solitary wave forms out of any positive mass M behind the lead solitary wave. This solitary wave then evolves in a similar manner to the first solitary wave, as was discussed above. The only difference is that this second solitary wave is gaining mass M from the first solitary wave and is itself shedding mass Mz as dispersive radiation. In analogy with the first pulse, let us take this second pulse to be of the form (3.2) with

2 70 = 02 sech f32 -

P2 + 4c~csazP;~ se:h 5 %

7 P2 tanh

P2 In ,

where Qz = x - (z(t), G(t) = b(t).

Hence, the approximate conservation equations for the second pulse are just (3.4) with the vari- ables a, fi, and V replaced by ~22, pz, and V.. This is supplemented by total mass conservation

which describes the transfer of mass from the first solitary wave to the second wave and into dispersive radiation. This mass transfer is described by the extra term M’ on the right-hand side of (3.6). A third solitary wave will form from the radiation behind the second wave if Mz > 0. This will be formed from the second solitary wave in a similar manner to the way in which the second solitary wave was formed from the first wave. This process can then be repeated to form any number of solitary waves.

Initially, the approximate conservation equations do not describe the evolution of the second pulse adequately since this pulse is generated from zero amplitude and the approximate con- servation equations have difficulty coping with a2 NN 0. To overcome this, the second pulse is started from zero amplitude in the same manner as Kath and Smyth [8] used to form a second solution for the KdV equation. Initially it is assumed that the second pulse has the form of a solitary wave (2.1) and that all the mass M goes into forming this solitary wave. Hence, mass conservation gives

&%=$i$$ (3.7)

for the initial evolution of the second pulse. This equation is used to determine the initial evolution of the second pulse, say for a time ti, after which the approximate conservation equations (3.4) are used to evolve the pulse. Like the first pulse, the second pulse will approach the unique steady-state solitary wave.

4. RESULTS AND COMPARISON WITH NUMERICAL SOLUTIONS

In this section, solutions of the approximate conservation equations (3.4) which describe the pulse evolution are considered. A phase plane analysis is performed, which indicates the stability

Pulse Evolution 51

of the steady-state solitary wave, and solutions of the approximate equations are compared with full numerical solutions of the perturbed KdV equation (1.1) which governs Marangoni-Benard convection. The full numerical solutions are obtained using the numerical scheme of [7], which uses centered finite-differences for the spatial coordinate and fourth-order Runge-Kutta for the temporal coordinate. In the KdV limit (cy = D), Kath and Smyth [8] found that the agreement between solutions of the approximate conservation equations and full numerical solutions could be improved by generalising the velocity to

v = (2 + v)a - 2vp-2, (4.1)

where v is a constant. In the solution limit, up2 = 2 and V = 2a, which is the correct KdV solution velocity, regardless of the value of v. Hence, the parameter v is a time constant and only effects the rate of decay to the steady-state without affecting the steady-state itself. For the KdV equation, Kath and Smyth [8] showed that the moment of momentum equation gave (4.1) with v = 1.2, while v = 1.5 gave the best agreement between numerical solutions and solutions of the approximate conservation equations. Hence, all the following examples will use this value of v. The first few examples considered in this section have the parameter values ci = 2.0, cs = 1.8, cs = 1.0, and (Y = 0.1 in the perturbed KdV equation (1.1). For this choice of parameters, the initial pulse evolves to a steady-state solitary wave with first-order amplitude a = 1 and width

1.6

1.6

P

1.4

1.2

1 0.4 0.6 0.8 1 1.2 1.4 1.6 i .a 2

a Figure 1. The pulse amplitude a versus width 0. The parameter values are CI = 2, c2 = 1.8, c3 = 1, and a = 0.1. Sh own are solutions of the approximate conservation equations (-) and the solution line ab2 = 2 (- - -).

Figure 1 shows a phase plane plot of the pulse amplitude a versus pulse width p. Shown are solutions of the approximate conservation equations (- ) and the solution relation ap* = 2 (- - -). The fixed point is the steady-state solution (ad, /3.,) = (1, 4). Each trajectory initially propagates onto a line on which apa w 2, before then evolving to the steady-state. The reason for

52 T. R. MARCHANT AND N. F. SWW

1.8 -

1.6 -

a

P

t Figure 2. The pulse amplitude a and width fl versus time t, up to t = 60. The parameter values are cl = 2, c2 = 1.8, CQ = 1, and a = 0.1. The initial amplitude and width are A = 0.5 and UJ = 2. Shown are the solution of the approximate conservation equations (-) and the theory of Marchant [7] (- - -).

this is that the evolution is occurring on two time scales. Since cy < 1, for small times the pulse is evolving as a KdV solution to the line UP” = 2. However, on a longer time scale of O((Y-I), the O(o) terms in the perturbed KdV equation (1.1) become important and the pulse must evolve to the steady-state solitary wave. The Appendix shows the local analysis of the fixed point. For this example, both the eigenvalues are real and negative; hence, the fixed point represents a stable node. For large time the trajectories approach the fixed point along the straight line with slope g = -0.63. This is close to the slope of the solution line (up2 = 2) for which g = -0.71 at the steady-state. The difference between the slopes of these curves is due to the fact that the steady-state solitary wave varies from a KdV solution at O(o).

Figures 2 and 3 show the pulse amplitude a and width p versus time t up to t = 60. The initial amplitude and width are A = 0.5 and w = 2. Figure 2 shows a and p as given by the approximate conservation equations (-) and the theory of Marchant [7] (- - -). In this case, the pulse amplitude increases in a monotonic fashion ss it evolves to the steady-state solitary wave with amplitude unity. The pulse also narrows with time as the steady-state width fi is approached. In this example, the initial pulse is a KdV solution at first-order (as Aw2 = 2), so the theory of Marchant [7] for an evolving Marangoni-Benard solitary wave is applicable. The pulse amplitudes a, as predicted by Marchant [7] and by the approximate conservation equations, are very similar. The difference between the two solutions is due to the fact that Marchant assumed that the pulse remains a KdV solution at O(1) (with a/3’ = 2) during the evolution, while the approximate conservation equations derived in the present work allow a and p to vary independently.

Figure 3 shows the pulse amplitude a as given by the full numerical solution (-), the ap- proximate conservation equations (- - -), and the 0(cy2) correction (- - - - ). There is a good comparison between the solutions of the approximate conservation equations and the numerical

Pulse Evolution 53

1.1

a 0.8

0.8

I 0.5; 10 20 30 40 50

t Figure 3. The pulse amplitude o verws time t, up to t = 60. The parameter values are cl = 2, cz = 1.8, q = 1, and Q = 0.1. The initial amplitude and width are A = 0.5 and w = 2. Shown ere the numerical solution (-), the solution of the approximate conservation equations (- - -), and the O(cxa) correction (- - - -).

solution. Near the steady-state the numerical solitary wave amplitude is about 6% greater than the first-order pulse amplitude a = 1 as given by the approximate conservation equations. This discrepancy occurs because the amplitude a as given by the approximate conservation equations is merely the first-order approximation, to O(a), to the exact numerical amplitude. The 6% dif- ference is O(cr2). To try to correct for this difference expression (2.4), valid in the limit of a small perturbation from a KdV solution, is used to calculate the second-order steady-state amplitude from the first-order amplitude a. This corrected amplitude, to O(02), is also shown in Figure 3 and it is seen to be very close to the numerical amplitude as the steady-state is approached. Initially however, the approximate conservation equations give a better estimate of the numerical pulse amplitude. This is because the O(02) correction assumes the solution relation ap2 = 2, which is not true in the early stages of the pulse’s evolution. However, as the steady state is approached, ap2 = 2 becomes a good approximation.

Figure 4 shows the pulse amplitude a and width p versus time t up to t = 80 for initial amplitude A = 0.5 and width w = 1, so that the initial condition is not a KdV solution. Shown are the numerical solution ( - ), the solution of the approximate conservation equations ( - - -), and the O(a2) correction (- - - -). The trajectory passing through the point (a, p) = (0.6,l) in Figure 1 qualitatively illustrates the evolution of the pulse for this example. The approximate conservation equations show that initially there is a sharp decrease in pulse amplitude and a sharp increase in pulse width. This is because the pulse evolves to a KdV solution initially. As the forcing terms on the right-hand side of the momentum equation (the second of (3.4)) are O(o), for small times a2P w l/4. Now for a KdV solution ap2 = 2. Hence, if we assume that the pulse is evolving to a KdV solution initially, we have that its amplitude is tending to a = 0.32 and its width is tending to p = 2.5. These values are in accord with the minimum value of a and the maximum value of p in Figure 4. For longer times, the pulse amplitude

54

2.2

2

1.8

1.6

1.4

a 1.2

P 1

0.8

0.6

; :

_: i

_(

I

c

I

T. R. MARCHANT AND N. F. SMYTH

I

1,‘------% I

i

*-.*

%.

%_

--._

%_

--._

%_

--._

---._

---__

----.___

------_____

--------_________ __________

o.20 I I I I I I I 10 20 30 40 50 60 70

t Figure 4. The pulse amplitude a and width 0 versus time t, up to t = 80. The parameter values are cl = 2, cz = 1.8, cs = 1, and LI = 0.1. The initial amplitude and width are A = 0.5 and w = 1. Shown are the numerical solution (-), the solution of the approximate conservation equations (- - -), and the 0(a2) correction (----).

__.

8(

a increases and the width p decreases in a monotonic fashion to the steady-state due to the influence of the O(a) terms on the right-hand side of the momentum equation in (3.4). It can be seen that the numerical solution has the pulse amplitude initially decreasing, as predicted by the approximate conservation equations, before increasing slowly to the steady-state amplitude.

Figures 5a and 5b show the pulse amplitude a and width fi versus time t for an initial amplitude A = 1.5 and width w = 2, so that again the initial condition is not a KdV solution. Shown, up to t = 40, are the numerical solution (- ), the solution of the approximate conservation equations (- - -), and the O(a’) correction (- - - - ). The trajectory starting at (a, @) = (1.5,2) in Figure 1 illustrates the evolution of the pulse. The approximate conservation equations show that the pulse amplitude initially undergoes a sharp increase and the pulse width a sharp decrease. As for case discussed in Figure 4, the pulse first evolves to a KdV solution on a timescale of O(1) before evolving to the steady-state. Hence, from the previous discussion, a + A4/3w2/32-1/3 = 2.16 and p + fi = 0.96 initially, these values being in accord with the maximum value of a and the minimum value of /3 in Figure 5a. At longer times, the width increases and the amplitude decreases to the steady-state values. It can be seen from Figure 5a that the numerical solution gives that the pulse amplitude increases rapidly before decaying to the steady-state solitary wave value. Initially the amplitude, as given by the approximate conservation equations, is close to the numerical amplitude. This is because the pulse has not yet evolved to the solitary wave given by (2.1) and (2.2). Near the steady state, the 0(cr2) correction is a better estimate of the numerical amplitude since this includes the higher-order amplitude terms of the exact steady-state solitary wave. Initially, however, the O(a’) correction significantly over-predicts the pulse amplitude because the pulse is not yet a solitary wave for which the correction terms are valid.

Pulse Evolution 55

For this example, the mass M shed by the solitary wave as it evolves is positive and a second solitary wave will form from this positive mass. Figure. 5b shows the amplitude a of the second pulse up to time t = 70. Shown are the numerical solution ( - ), the solution of the approximate conservation equations ( - - -), and the O(02) correction (- - - - ). The full numerical solution shows that the second pulse can first be identified at time t = 0.7, at which time its amplitude is a = 0.46. Initially, the amplitude of the second solitary wave decreases slightly, before increasing in a monotonic fashion to the steady-state value. As discussed in Section 3, the mass conservation equation (3.7) is used to describe the evolution of the second pulse in the initial stages (up to a = 0.41). After this time, the approximate conservation equations (3.4) are used to describe its evolution. The comparison between the numerical solution and the solution of the approximate conservation equations is again very good.

Figure 6 is a perspective plot showing the evolution of the initial pulse of Figure 5 into two steady-state solitary waves. The figure shows that the pulse initially increases in amplitude and becomes narrower as it evolves into a KdV solution for small times, before decaying to the steady- state solitary wave (with a = 1). A second solitary wave forms from the mass shed behind the first solitary wave and this second pulse also evolves into the steady-state solitary wave. Dispersive radiation of small amplitude can be seen behind the two solitary waves.

A case for which no solitary wave with a steady-state amplitude exists will now be examined. The parameter values chosen are cl = 2, cs = 1.8, cs = 5, and a = 0.1. These values imply that the steady-state amplitude (2.3) is negative. In this case, it is found that the amplitude of the the initial solitary wave grows without bound. Figure 7 shows the pulse amplitude a versus time

2.8)

a

P

(a) The pulse amplitude a and width fl verbs time t, up to t = 40. The parameter values are cl = 2, c2 = 1.8, q = 1, and Q = 0.1. The initial amplitude and width are A = 1.5 and w = 2. Shown is the amplitude of the first (larger) pulse: numerical solution (-), the solution of the approximate conservation equations (- - - ), and the O(a2) correction (- - - -).

Figure 5.

56 T. R. MARCHANT AND N. F. SMYTH

(b) The pulse amplitude a versus time t, up to t = 70. The parameter values are cr = 2, cz = 1.8, cs = 1, and c1 = 0.1. The initial amplitude and width are A = 1.5 and tu = 2. Shown is the second (smaller) pulse: numerical solution (-), the solution of the approximate conservation equations (~- -), and the 0(a2) correction (- - - -).

Figure 5. (cont.)

9

40.

30.

t 20.

10.

L 0.

0. 50. 100. 150. x

Figure 6. Perspective plot of the pulse evolution, up to t = 40. The parameter values are cr = 2, cz = 1.8, c3 = 1, and a = 0.1. The initial amplitude and width areA=1,5andw=2.

Pulse Evolution 57

t up to t = 30. The initial amplitude and width are A = 0.5 and w = 1. Shown are the numerical solution (-), the solution of the approximate conservation equations (- - -), and the O(c?) correction ( - - - - ). It can be seen that the numerical solitary wave amplitude increases without bound. The comparison with the approximate theory is good for small times (up to t M 15), but the numerical amplitude increases much faster than that given by the approximate equations for large times. The approximate theory assumes that the radiation behind the evolving pulse is of small amplitude and that the correction terms in (Y are small. Both of these assumptions break down at large times when the pulse amplitude is increasing without bound. Hence, it is not surprising that there is a variation between the numerical solution and the approximate theory for t 2 15. In this case neglected higher-order terms (of O(ly*)) in the derivation of the perturbed KdV equation (1.1) become important.

3

2.5

2

a 1.5

1

0.5

L OO

I I I I I I 5 10 15 20 25 30

t Figure 7. The pulse amplitude a versus time t, up to t = 30. The parameter values are c1 = 2, c2 = 1.8, c3 = 5, and d = 0.1. The initial amplitude and width are A = 0.5 and w = 1. Shown are the numerical solution (-), the solution of the approximate conservation equations (- - -), and the 0(a2) correction (- - - -).

5. CONCLUSIONS

Pulse evolution for Marangoni-BBnard convection has been described using a particularly sim- ple method involving mass conservation and momentum evolution. The key advantages of this method over the more formal method of inverse scattering are simplicity and the use of a more general initial condition. Inverse scattering assumes the initial profile is a KdV solution, or close to this, while here the evolution of a more general sech* pulse is considered. This can be thought of as the first step in describing the evolution of an arbitrary initial condition into solitary wave(s). A phase plane analysis shows that the pulse evolves on two times scales; first, on a O( 1) timescale to a KdV solution, then on a time scale of O(a-‘) to the unique steady-state solitary wave, which has been shown to be linearly stable.

It has been observed experimentally that Marangoni-Benard solitary waves of an unique am- plitude occur. The results presented here could help in the understanding of the timescale and

58 T. R. MARCHANT AND N. F. SMYTH

nature of the evolution to this steady-state solitary wave from an initial deformation of the sur- face. In particular, it would be of interest to observe experimentally the evolution of pulses to confirm the theoretical prediction that the pulse amplitude can initially rapidly increase or decrease away from the steady-state value, before a slower evolution to the steady-state at long time. In addition, it would be of interest to observe the nature of the waves in the parameter range where blow-up is predicted.

APPENDIX STABILITY OF THE STEADY-STATE SOLITARY WAVE

The linear stability of the steady-state solitary wave (the fixed point of the approximate equa- tions) can be examined by perturbing (3.4) (with the velocity given by (4.1)) about this steady solution,

a = a, + h, P = Ps + 8, where a, = 7Cl 1ocz - 4c3 ’

a& = 2, (A.11

and ]a] CC a, and $1 CC &. Substituting (A.l) into (3.4) and (4.1) gives the linear system

( > T

Av’ = Bv, wherev= &,,8 ,

;cla,/3;1 96 32 24 16 4 cfj = + 35”3 - -72 > a%l, CT = -C2 - 7 35c3 > a: - pat.

Substituting the form v = veeXt into (A.2) gives JAX - BI = 0 as an equation for the eigenvalues. When QI = 0, this eigenvalue equation reduces to

x = 3(1- I$, s

so that X is negative and hence stability results if v > 1. This corresponds to the result found by Kath and Smyth [8] for the KdV equation. For nonzero (y: numerical solutions of the eigenvalue equation were found. For the case with parameters cl = 2, cs = 1.8, and ca = 1, the eigenvalues are negative, and hence, the steady-state solitary wave is stable, for CY ranging from zero up to CYX 1.

REFERENCES 1. A.N. Garazo and M.G. Velarde, Dissipative Korteweg-de Vries description of Marangoni-Benard convection,

P&s. Fluids A 3, 2295-2300, (1988). 2. X-.L. Chu and M.G. Velarde, Korteweg-de Vriea excitation in Benard-Marangoni convection, Amer. Phys.

Sot. 43, 1094-1096, (1991). 3. P.D. Weidman, H. Linde and M.G. Velarde, Evidence for solitary wave behaviour in Marangoni-Bknard

convection, Phya. Fluids A 4, 921-926, (1992). 4. H. Linde, X.-L. Chu, M.G. Velarde and W.W. Waldhelm, Wall reflections of solitary waves in Merangoni-

BBnard convection, Phya. Fluids. A 5, 31623166, (1993). 5. T. Kawahara, Formation of saturated eolltons in 8 nonlinear dispersive system with instability and dispersion,

Phys. Rev. Lett. 61, 381983, (1983). 6. T. Kawahara and S. Toh, Pulse interactions in an unstable dissipative-dispersive nonlinear system, Phya.

Fluids 31, 2103-2111, (1988). 7. T.R. Marchant, The evolution and interaction of Marangoni-Benard solitary waves, Wave Motion 23, 307-

320, (1996). 8. W.L. Kath and N.F. Smyth, Soliton evolution and radiation loes for the Korteweg-de Vrlea equation, Phys.

Rev. E 51 (l), 661-670, (1995). 9. N.F. Smyth and A.L. Worthy, Solitary wave evolution for mKdV equations, Wove Motion 21, 263-275,

(1995). 10. R. Grimahaw, Slowly varying waves. I. Korteweg-de Vriea equation, Pmt. Roy. Sot. Land. A 386, 359-375,

(1979).