Mass transfer, marangoni effect, and instability of interfacial longitudinal waves

Mass Transfer, Marangoni Effect, and Instability of Interfacial Longitudinal Waves I. Diffusional Exchanges M. HENNENBERG,* P. M. BISCH, M. VIGNES-ADLER, AND A. SANFELD Laboratoire d'Adrothermique du Centre National de la Recherche Scientifique, 4 ter, route des Gardes 92190-Meudon, France; and Service de Chimie Physique H, Facult~ des Sciences, C. P. 231, Universitd Libre de Bruxelles, 1050-Bruxelles, Belgique Received June 23, 1978; accepted October 4, 1978 A general formalism is developed to study interracial instability o f two immiscible incompressible fluids. M a s s diffusion fluxes across the interface are the determining step. The surface m a s s balance equation d e p e n d s u p o n the surface diffusion and convection and on the net flux. Discussion is restricted to longitudinal perturbations. Using the concept o f surface elasticity, n e c e s s a r y and sufficient instability conditions for oscillating and n o n oscillating regimes are given for long wavelengths, T h e obtained criteria are e x t e n s i o n s o f the Sternling and Scriven ones.

sorption-absorption and chemical reactions occur at the interface (3). It is based on the generalization of the surface momentum balance equations, as given by Aris (4), and on the surface mass balance equations derived from flux-force relations at the interface as discussed by many authors (5-7). However this general formulation is still difficult to deal with. Only exchange of stability conditions are easy to find. Another approach was developed by Levich (8), Lucassen (9-11), and van den Tempel (12) to study wave propagation at the interface as small perturbations to the equilibrium reference state. They distinguished between capillary waves induced by normal deformations and longitudinal waves depending upon the tangential gradient of surface tension. They have introduced the concept of surface elasticity in connection with exchanges of mass and momentum between the bulks and the interface. With this complex coefficient they described the response to oscillatory disturbances. The purpose of the present paper is to give a linear stability analysis, when mass transfer oc-

INTRODUCTION

In recent years much attention has been devoted to the spontaneous convective motion at the interface during mass transfer between two immiscible fluid phases. A theoretical approach has been developed by Sternling and Scriven (1) to study the hydrodynamic stability when diffusion-determined fluxes cross the interface. They ascribed this phenomenon to gradients of surface tension (Marangoni effect) due to variations in surface excess concentration of the diffusing solute. They considered the interface as a two-dimensional Newtonian fluid, leading to the appropriate boundary conditions for bulk equations. Recently a more general approach has been developed to study the interfacial instability when diffusion fluxes cross the interface (2) or when nonequilibrium ad* A u t h o r to w h o m c o r r e s p o n d e n c e should be sent. P e r m a n e n t address: Service de Chimie P h y s i q u e II, Facultd des Sciences, C.P.231, Universitd Libre de Bruxelles, C a m p u s Plaine, Boulevard du T r i o m p h e , 1050-Bruxelles, Belgique. 128 0021-9797/79/040128-10502.00/0 Copyright © 1979by AcademicPress, Inc. All rights of reproduction in any form reserved.

Journal of Colloid and lnterjace Science, Vol. 69, No. 1, March 15, 1979

129

DIFFUSIONAL CHANGES IN INTERFACIAL WAVES

curs between two immiscible fluids, starting from the general formulation developed previously. However, we will uncouple longitudinal waves from transversal ones, and focus our attention on the Marangoni problem. We will compare our formalism with that developed by Lucassen-Reynders (9) Lucassen and van den Tempel (10, 11) and use the concept of surface elasticity to study the stability of oscillatory regimes of the longitudinal waves. In Section 1, we consider only diffusional exchange. Later on, we will show the relative influence of adsorption-desorption processes and diffusion fluxes. Here we will assume an instantaneous equilibrium between the variation of surface adsorbed matter and of the sublayer concentration. The reference state is supposed to be at rest and the interface is flat. We neglect all temperature fluctuations. First, we give a general derivation of the compatibility equation, showing how our generalization of the Sternling and Scriven problem is related to the Lucassen and van den Tempel approach. Later on we develop the study of oscillatory and nonoscillatory regimes under a long wavelength approximation, giving the necessary conditions for instability. These results are compared to the Sternling and Scriven criteria. 1. LINEARIZED HYDRODYNAMICAL EQUATIONS AT THE INTERFACE

Let us call 1 the lower fluid and 2 the upper one. The reference steady state is supposed to be at rest, and the two phases are separated by a flat interface. In the perturbed state, the incompressibility condition is div v = 0,

[1]

where _v is the perturbed velocity. The linearized Navier-Stokes equations are given by Ov p -- Ot

grad (6p) + /xA_v,

[2]

where p is the total bulk density, 6p is the perturbed pressure, and the body forces contribution is neglected. The complete solution of Eqs. [1] and [2] in each bulk requires the appropriate boundary conditions at the common limiting surface (z = zs). No-slip condition at the interface and incompressibility of bulk media give _vllzs = _

= _Vs.

[31

Momentum balance equations at the surface must also be introduced. We consider the surface a two-dimensional Newtonian fluid (7, 13). 2. SURFACE MOMENTUM BALANCE EQUATIONS

(a) Momentum Balance Normal to the Interface For a small deviation from a plane surface the generalization of the Laplace equation is given by: 0 FT~VZ s

[

°

= 2~ - 6 p + 2/x~z

vz+ og6z

1

+ o-°Vs26Z, [4] where 6z is the normal deformation of the surface, pg is the gravitational force, o-° is the unperturbed surface tension, and FT is the total unperturbed adsorption. The operator A applied on a function gives the jump from bulk 2 to bulk 1 (Af = f e - f l ) the last term of the r.h.s, of [4] is the variation of curvature. The normal deformation 6z is related to the velocity vz by 0 Ot

--

~z

=

v~ ~.

[5]

(b) Tangential Momentum Balance The tangential part of the momentum balance depends upon the two-dimensional gradient of surface tension along the interJournal of Colloid and lnterjace Science, Vol. 69, No. 1, March 15, 1979

130

HENNENBERG

ET AL.

face (Marangoni effect) and reads

°

FT-'~

V~s = A tz

0

V¢ + - - ~

Vz

)]

0

0

+--8o"

-- ' r l a - - - -

oj

Oj Oz

where ~a and ~s are the intrinsic surface shear and dilatational viscosities, respectively. Assuming that the fluctuation of the surface tension is related only to the variation of the adsorption, we may write 80. = -oeSF,

[7]

where F is the adsorption of the surfactant. ot is taken as a positive quantity.

Ot

[8]

where Ja is the bulk diffusion flux, j_s is the surface diffusion flux, n is the outward directed unit normal. The perturbed fluxes 8 J d are determined by the mass balance equations in each bulk, [9]

where D is the bulk diffusion coefficient of the solute. To determine the constants involved in the solution of Eq. [9] a new boundary condition is required. Here we assume instantaneous equilibrium between the perturbed adsorption and concentration at the sublayer. 8F = K S c z=zs= K[Sc z=0 + dc°--dTz8z] . [10] In the unperturbed reference state we assume a linear concentration profile solution of the one-dimensional Fickian law

c o = c ° ° - flz

[6]

fllnl

-~ f12D2 .

[12]

The coefficients fl are positive quantities if transfer is from 1 to 2 and negative if transfer is the other way round. Now in the perturbed state the diffusion fluxes are [13]

where c o is given by Eq. [11] and 8c by the solution of Eqs. [10] and [11]. 4. S E C U L A R E Q U A T I O N

0 - - 8F + F ° divs _vs + divs (S J_s)

Ot

(j = x, y),

0 8c], 8J_a'n_ = - D [-~z

For a diffusion controlled flux of tensioactive solute through the interface, the surface mass balance equation reads

0 - - 8c = - v , grad, c o + DV28c,

vfl + "o~V~2vfl

interface, which means

3. S U R F A C E M A S S B A L A N C E E Q U A T I O N

= A[84q-n_,

0

[11]

and no accumulation of matter along the Journal of Colloidand Interface Science, Vol. 69, No. 1, March 15, 1979

We investigate the linear stability of the system. Each perturbed quantity is developed into normal modes, using the classical Fourier integral. Two different approaches are used: (a)

k = kR = 2~-/X,

[15]

09 = OgR + io9I,

where h is the wavelength and to the frequency. The real part ofog, OgRis the damping (<0) or amplification (>0) coefficient. The marginal state corresponds to oga = 0. In this case we study the instability at a given position with increasing time (13) (b)

k =kR-iki

= Ik] e x p - i a ,

[16]

o9 = io9I,

where 8 is the phase angle of k, and ki is the damping (>0) or amplification coefficient (<0). The marginal oscillating state corresponds to ki = 0. This expansion enables us to study the instability at a given time with increasing distance from the origin (x = y = 0). It is a particularly useful development when one analyzes the response of the s y s t e m to an oscillatory

DIFFUSIONAL

CHANGES IN INTERFACIAL

external disturbance (9). The Fourier components of bulk Eqs. [1], [2], and [9], are vanishing at _+~. The remaining constants

131

WAVES

are determined by the boundary conditions at the surface (Eqs. [3], [4], [6], [8], and [10]). The compatibility condition tends to the following dispersion relation:

[17]

A ' B + C = 0,

[18] 1[ k2Ds ~ D3"rrl B = T o9 + + K~ J

X

[FTo9 + (~s + "oa)k 2 + ~3" tx3"(q3" + k)]

+ o~k{F° - - 121 o9

o9

Lq+rJj

1l,

[191 [20]

3" qz, + z

where qr = (k z + w/v7) l/z,

Re(qy) >~ 0,

r3" = (k 2 + o9/D3") "2,

Re(ry)/> 0,

The left-hand side of Eq. [17] is the product of two terms describing different effects together with a coupling term. Indeed, term A describes the well-known problem of the stability of an adverse gradient of density in a gravitational field. The quantity (k2cr - gAp)/o9 is related to the Laplace condition and thus to the normal deformation of the surface. It is the R a y l e i g h - T a y l o r problem (13) and we will not discuss it further. The term B is related to the Marangoni effect, due to the variation of the surface tension along the interface. Both problems are coupled through the term C. 5. L O N G I T U D I N A L W A V E S A N D SURFACE ELASTICITY

To focus our attention on the Marangoni effect we will uncouple the two problems. This corresponds to taking C = 0, or /.zl(ql -- k)

=

/z2(q2 - - k ) .

[22]

When this condition is satisfied, we are able to treat the two problems separately. For the longitudinal modes, we have

k2c~

[21] y = 1, 2.

I F + (~lDJo9)A[(q + k)/(q + r)]] k2DS y

+ Fvo9 + (~s + ~d)k 2 + ~2 /z3"(q3"+k) = 0.

[23]

3'

Since, we consider that A ¢ 0, from assumption [22] the normal momentum balance equations reduces to A.vz = 0

[24]

so no normal deformation at the interface should be taken into account. Then Eq. [23] is a generalization of the Marangoni problem as studied by Sternling and Scriven (7). Indeed, here we include the variations of adsorbed matter along the surface through the r.h.s, of Eq. [8] which were neglected in their treatment. L e t us introduce now the concept of surface elasticity, as given by Lucassen and van den Tempel [10], go- = e~(ln A), Journal of Colloid and lnterjace Science,

[25]

Vol. 69, No. 1, March 15, 1979

132

HENNENBERG ET AL.

where 1

0

~(ln A ) = - - - ~ ( - - ~ z V z S ) .

[26]

Then the tangential m o m e n t u m balance [6] becomes e

0

0 = -r~o~ o---Zvzs - E

~(q3" + k)

Y

0 × -Vzs Oz

A [ / x ( q - k ) ] k v z ~.

[27]

To study longitudinal waves we take Vz~ = O, and from Eq. [27] we have the following compatibility condition:

+ ~ /x3"(q3"+ k ) + FrO = 0.

[28]

3"

In this equation e is a phenomenological coefficient and must be related to the process leading to the variations of surface tension. In the present approach these processes are related to the mass exchanges between bulks and surface. In the previous paragraph we found the general dispersion relation, for a diffusion controlled flux crossing the interface. The comparison between Eq. [28] and Eq. [23] gives the explicit form of the elasticity for this situation 1 + (fllD1/ogF°) × A[(ql + k ) / ( q l + ra)] eo

[31]

We have thus a viscoelastic behavior of the interface (9, 17). As in the three-dimensional theory of viscoelasticity, en is a storage modulus and e~ a loss modulus. There are thus two kinds of dissipative processes at the interface. The first one is linked to the intrinsic viscosity and does not depend on the exchange of matter. The other is solely due to these exchanges and tends to an important change in the rheological properties of the interface. We will now turn to the stability analysis. For oscillatory regimes (overstable and unstable oscillatory states) we use Eq. [16] and for nonoscillatory regimes (exchange of stability) Eq. [15]. 6. NONOSCILLATORYREGIME

3'

where dcr° = ozF° d In F r=vo

e = ~rt + i~i = le[ expi0.

[29]

o~ + k2Ds + ~ D r r J K r

e0 =

are looking at the response of the system to an oscillatory disturbance. The elasticity coefficient is a function neither of the intrinsic surface viscosities, nor the surface acceleration. The last quantity gives only a small contribution to the surface stability. The values of "0s + ~d are expected to be small for soluble surfactants, as compared to the dissipative contribution of e. It should be noted that the explicit derivation of Eq. [29] d o e s not depend upon the s t r e s s - s t r a i n model (14). They are related only to the exchange of matter between bulks and surface, and to the choice of the surface state equation. Then e is a complex quantity, and may be written as

[30]

is the restricted definition of the Gibbs elasticity. As k or w are complex quantities (see [15] or [16]) the phase angle of e is generally not a multiple of 2~-, especially if we Journal o f Colloid and Interface Science, Vol. 69, No. I, March 15, 1979

Combining Eq. [15] with Eq. [17] up to a first-order development in oJ, we obtain an analytical solution in the region surrounding the point of marginal stability without oscillations (1, 3) OJn ~

0,

¢OI ~

0.

[32]

At this limit, the term C vanishes. The Marangoni effect may be studied independently of the Rayleigh-Taylor problem. The

133

DIFFUSIONAL CHANGES IN INTERFACIAL WAVES

neutral stability is given by ill(1 - D1/D2) 4ker2F °

=1+

[2(tLa + /x2) + ker(~s + "0d)][kerOs + D 1 / K 1 + D2/K2] c~F°

This last e q u a t i o n a d m i t s only one real solution if fil - /32 = ill(1 - D1/D2) > 0.

[34]

On the o t h e r hand, since for k ~ 0, we have shown

o J ~ 2kZ(1/Da + I/D2) ~o ~ 0+

[35]

which m e a n s that there is always a region of instability for v e r y large wavelengths. In the limit [32] there are no other positive roots b e t w e e n k = 0 and k = ker. We have thus instability for

diffusion coefficient. F u r t h e r m o r e t h e r e exists a m o d e of m a x i m u m instability (2). L e t us now suppose the existence of unstable nonoscillating states for the longitudinal modes. In the limit of small wavenumbers Ir~[ 2 ~> ]q~l 2 -> k 2, [35] we have q~

--

[( D 1 ] 1/2

filD2 - -

TABLE I

O a /2b

< l(a = 1,2;b

Nonoscillatory instability Marginal stability:exchange of stability

v_..~a > --Da > 1 Vb Ob

Nonoscillatory instability and oscillatory instability Marginal stability: overstability

Da > Va > 1 Db ub

Oscillatory instability Marginal stability:overstability

v__%> 1 > Da vb Ob

= 2, l)

[38]

which is true if one of the three following inequalities is satisfied. /2a

--> Vb

D a

1 >--,

/2a

D a

/2b

Db

1 >-->--,

Db /2a

Types of instability

De < D~

vb

[37]

Pb/2a

[39]

Possibility of instability with respect to oscillation has not been excluded in a definitive way

D a

-- > /2b

- -va

<0, ]

and

Condition on

diffusionand

1 >

--

\ /22 /

so we have nonoscillatory unstable states in the small w a v e n u m b e r s limit, for a transfer f r o m a to b if

Instability Conditions after Sternling and Scriven (1) for a Transfer by Diffusion a --~ b

Do

[36]

(D2/1/21

-

L\ Pl /

A n e c e s s a r y and sufficient condition of interracial instability is that m a s s transfer b y diffusion occurs f r o m the liquid with the smallest value of the diffusion coefficient towards the fluid with the largest value of

--Da >

r~-

With these assumptions Eq. [23] has a solution only if

and stability for

kinematic viscosity

and

\/2~/

0 < k < ker

ker < k.

[33]

> 1

[40]

Ob

T h e s e r e s u l t s a r e in a c c o r d a n c e w i t h Sternling and Scriven analysis (1). The first two c o r r e s p o n d to the exchange of stability condition Db > D e , and the last one Eq. [40] to the simultaneous a p p e a r a n c e of nonoscillatory and oscillatory unstable regimes (see Table I). 7, LONG WAVELENGTH APPROXIMATION AND LONGITUDINAL OSCILLATORY PERTURBATION

L e t us now look to oscillatory disturbances and use Eq. [16]. When the w a v e Journal of Colloid and lnterjace Science, Vol. 69, No. 1, March 15, 1979

134

HENNENBERGETAL.

length induced by an oscillatory disturbance is much larger than the penetration depth of the perturbation as in some real situations (12), we then have

I rl 2, Iqrl >> Ikl 2

The marginal overstable case corresponds to =0,

rr=(l

03

,

tan 0 =

~1/2

+ l)t"~--~--~v) .

b e y o n d which the exchange of m o m e n t u m can be neglected. Within this layer o f width dv ~ the oscillatory m o v e m e n t of the interface is taken into account. B e y o n d this layer the bulk viscosity force restores the steady state. A mass penetration depth may be defined in the same way dr ° =

[43]

b e y o n d which diffusion has no practical importance. Since D r < vr, the mass penetration depth is smaller than the m o m e n t u m penetration depth. Neglecting the contribution o f surface acceleration and o f surface intrinsic viscosities, and taking into account the approximations [41] and the definitions [16] and [31], Eq. [27] reduces to

and

r 8 = ~-/8 + 0/2.

[44]

The space damping or amplification o f the disturbance is related to 8, and from Eq. [44] we have: 0 < 8 < ~r/2 -~-/2 < 8 < 0 Journal of Colloid and Interlace Science,

or or

~+M~

1 (D111,2 K, t2o i)

,

[461

a

+

1 K2 t2 i)

'

and

M-

r°

ii

[_..~1 ( E l tl/2 +

LK, t 2 :

I ( 0 2 )I/212

K t--T) I '

[471

where we have neglected the term k2Ds/o~i < 1, since Ds is of the same order of magnitude as the bulk diffusion. Each ~r is specific to mass exchange between bulk and the interface. It is a ratio o f two lengths. The first one is the penetration depth determined by dr D, the other is related to the instantaneous equilibrium between layer and sublayer, and it can be assimilated to the sublayer depth. Furthermore, ~2 can be seen as the ratio o f two times, aJ~-~ which depends on the external c o n d i t i o n and T , = 1 / ( ~ r (Dr/2Kr)l/2) 2 which is characteristic o f the media and yields a time scale. The quantity M is the ratio of T . and a time related to the nonequilibrium fluxes crossing the interface flrDr, n o r m a l i z e d to the r e f e r e n c e adsorption F °. T h e s e fluxes, as s h o w n by Eq. [9], are carried out by the convective

-~'/4 < 0 < 37#4, -57#4 < 0 < -7#4,

Vol. 69, No. 1, March 15, 1979

Mg2( 1 + 0

[45]

where ~ and M are dimensionless quantities defined by

--

[42]

g1 +

[41]

The problem becomes formally similar to the flow a b o u t a plate w h i c h oscillates longitudinally (15). L e t us define a momentum penetration depth dr ~ =

tan0 = -1.

(1 + M2~g) 1/2 I el = eo (1 + 2~ + 2~2)v2 ,

£0 ~ 1/2

• [

-~r/4,

The explicit form of the elasticity in the approximation [41] is derived from Eq. [29]

or

qr ---- (I + i ) t ~ - ~ )

0=

stable solutions, unstable solutions•

135

D I F F U S I O N A L C H A N G E S IN I N T E R F A C I A L W A V E S

~M =-10" 4

,,,..;--- ...~ \

Ir/4 /

0

\\M=-IO "= \\

M = 16-4

'

/ o2

~

j'/

M- 10 / /

/

/

'

f ,"

/

/

/

1/

~ ~ -~-~- ~ . ~._jJ"

....

/

. . . . . -4

-3

-2

-1

~

/M=tO

/

t---Zf~

f

/

M:I /

//

- 1T//4

/

stable region marginal stability unstable region ;

0

1

,

,

3

4

-2JOg

'~

Fro. l. Elasticity phase angle 0 versus logarithm of dimensionless frequency 1/~2.

motion of adjacent fluids. So the difference between the reverse of the square root of the Schmidt number (D/u) 1/2 gives the efficiency and direction of these c o n v e c t i v e trans-

log 16/

\ \

IMI = 10

\

ported fluxes. Figs. 1 and 2 show the role of nonequilibrium processes through the quantity M. In Fig. 1, we plot the elasticity phase angle 0 versus the logarithm of the dimensionless frequency 1/~2. For M = 0, corresponding to equilibrium steady state (zero fluxes) or to equal Schmidt numbers, the values of 0 are always in the stable region, and we obtain the same result as Lucassen and van den Tempel (10). For M > 0, we find unstable solutions for > ~cr,

\

1

\\IMI = 1 ,,

~,\IMI = 10-t

D

x

\

1 + (1 + M) 1/2 ~er =

x ~\

"''~-~

x

\ IMI=10.2 \

''.

....

" ....

1

2

~ /"

-1" x IMI=tO.:~ j z

Ml:tog( -2~4

~3

-2

-1

0

- 2xiOg ~

F~G, 2. Amplitude of coefficient of elasticity log l e/Cot versus logarithm of dimensionless frequency 1/~.

M

with

~cr > 0.

[48]

H o w e v e r for M < 0 we have always stable solutions. In Fig. 1, we give the results corresponding to some negative and positive values of M. The amplitude of the coefficient of elasticity is plotted in Fig. 2. It depends only on the absolute value of M, so that we cannot distinguish stable and unstable solutions in Fig. 2. For M = 0, we find exactly the results of Lucassen and van den Tempel (10). When [M[ > O, however the result is quite different. In the region 1/~2 --~ 0 (small frequency), we find the solution corresponding to the Sternling and Scriven problem (1), in the limit of long Journal of Colloid and lnterjace Science, V o l . 69, N o . 1, M a r c h 15, 1979

136

HENNENBERG ET AL. TABLE H

wavelength, 21/2

[49]

and when 1/~~ --> ~ (high frequency) the result tends to the equilibrium wave. The effect of the imposed diffusion profiles is to increase the amplitude of the elasticity coefficient. To summarize this analysis, in the case of the long waves approximation: (a) M ~< 0 is a sufficient condition for stability, (b) M > 0 is a necessary condition for instability, (c) M > 0 and ~ > ~cr > 0 is a necessary and sufficient Condition for instability. Indeed, instable states are obtained for all ~ > ~cr provided these values of ~ belong to the domain of validity of the long waves approximation. All the physical coefficients of the two phases can be conveniently combined into two parameters. The discussion is solely based on these two parameters and gives a complete description about the instability of the system. Finally, let us make explicit this condition M>0 or //D1 \1/2 1/2)

If transfer is from a to b the last inequality is satisfied if one of the three following conditions is satisfied: Da

Pa

>-->1,

hb

Pb

Da > 1 > v._.~(a = 1, 2 and b = 2, 1), Db 1 >

[51]

Vb Oa

Va

>--. Db Vb

The first corresponds to the SternlingScriven analysis and to Linde and Kffnkel experiments with transfer from the gas Journal of Colloid and Interface Science, Vol. 69, No. 1, March 15, 1979

Necessary and Sufficient Instability Conditions for Transfer a --->b at the Long Wavelength Approximation Condition on diffusion and kinematic viscosities

Type of instability

Da < Db

Nonoscillating instability Marginal stability: exchanges of stabilities

Dav-------Lb< 1

Nonoscillating instability

PbVa

Special cases D__L>I > v_2_b ] Oa

Va

D---L> v-2-~> 1 Da

Va

Marginal stability: exchange of stabilities

1 < Da < Va Db Vb Dbv____~_a< 1

Da vb Special cases

Oscillating instability for ~ > ~¢r > 0 Marginal stability: overstability at ~ = ~c~

D__5~>f_~_" > 1 Do vb --Da > 1 > --Va Db Vb

v--Lb> D--~ > 1 va

Da

phase to the liquid phase (1, 16). The second one is not excluded from their analysis but they overlooked it. The third possibility was not considered in any previous analysis (see Tables I and II). If we compare the above results with the previous development due to Gouda and Joos (17) we notice the following. They also make a stability analysis in the long wavelength approximation using the method developed by Lucassen and van den Tempel (11), as we do here. However they do not decouple explicitly the longitudinal and capillary waves. They neglected the potential part of the velocity

DIFFUSIONAL CHANGES IN INTERFACIAL WAVES field as c o m p a r e d to the rotational part. T h e y had thus to ignore all the b o u n d a r y conditions linked to normal deformations. So they were not able to make an explicit comparison with the results of Sternling and Scriven. This contrasts with our present approach. Indeed, from the assumption of u n c o u p l e d m o d e s , or v a n i s h i n g n o r m a l deformation, we can easily show that Eq. [23] is a generalization of the Sternling and S c r i v e n f o r m u l a t i o n (1), w h e n the variations of adsorbed matter along the interface are taken into account. In fact, we arrive at the same formal definition of e (Eqs. [45] and [46]) as G o u d a and Joos however, the explicit form of M is quite different and leads to completely different physical results. 8. CONCLUSIONS Starting from a general formulation, we have u n c o u p l e d longitudinal w a v e s from capillary ripples. The study o f longitudinal waves, related to the Marangoni effect, shows that the imposed diffusion profiles modify the elasticity coefficient and may be responsible for the interfacial instability. The present development leads b o t h to the L u c a s s e n - v a n den T e m p e l results for equilibrium situations and to the S t e r n l i n g - S c r i v e n stability criteria. These latter results are summarized in Table I, and can be compared to those of our approach given by Table II. ACKNOWLEDGMENTS We thank Prof. J. J. Bernard, director of the Laboratoire d'A6rothermique C.N.R.S., Bellevue

137

(France) for many stimulating discussions. Part of this work was performed during the sejour of Dr. M. Hennenberg at the Laboratoire d'A6rothermique under the sponsorship of the CIES (French Foreign Ministry). Dr. P. M. Bisch gratefully acknowledges support from the CNPq .(Brazil). This work was partially supported by Solvay Society. REFERENCES 1. Sternling, C. V., and Scriven, L. E., A.I.Ch.E.J. 5, 514 (1959). 2. Scrensen, T. S., Hennenberg, M., and Sanfeld, A., J. Colloid Interface Sci. 61, 62 (1977). 3. Hennenberg, M., S~rensen, T. S., Steinchen, A., and Sanfeld, A., J. Chim. Phys. 72, 1202 (1975). 4. Aris, R., "Vectors, Tensors, and the Basic Equations of Fluid Mechanics," Prentice-Hall, Englewood Cliffs, N. J., 1962. 5. Dudeck, M., and Prud'homme, R., J. Appl. Phys. 48, 1 (1977). 6. Hennenberg, M., Sorensen, T. S., and Sanfeld, A., J. Chem. Soc., Faraday Trans. 73, 48 (1977). 7. Scriven, L. E., Chem. Eng. Sci. 12, 98 (1960). 8. Levich, V. G., and Krylov, V. S., Ann. Rev. Fluid Mech. 1, 293 (1962). 9. Lucassen-Reynders, E. H., and Lucassen, J., Advan. Colloidal Interface Sci. 2, 347 (1969). 10. Lucassen, J., and van den Tempel, M., Chem. Eng, Sci. 27, 1283 (1972). 11. Lucassen, J., and van den Tempel, M., J. Colloid Interface Sci. 41, 491 (1972). 12. van den Tempel, M,, J. Non-Newtonian Fluid Mech. 2, 205 (1977). 13. Chandrasekhar, S., "Hydrodynamic and Hydromagnetic Stability," Oxford Univ. Press (Clarendon), 1961. 14. Wasan, D. J., Gupta, L., and Vora, M. K., A.I.Ch.E.J. 17, 287 (1971). 15. Schlichting, H., "Boundary Layer Theory," McGraw-Hill, New York, 1968. 16. Linde, H., and Kiinkel, M., Wi~rme und Stoffi~bertragung 2, 60 (1969). 17. Gouda, J. H., and Joos, P., Chem. Eng: Sci. 30, 521 (1975).

Journal of Colloid and lnterjaee Science, Vo|. 69, No. 1, March 15, 1979