Marangoni convection in a binary liquid layer with

PHYSICS OF FLUIDS 21, 054101 共2009兲

Marangoni convection in a binary liquid layer with Soret effect at small Lewis number: Linear stability analysis S. Shklyaev,1 A. A. Nepomnyashchy,2,3 and A. Oron4 1

Department of Theoretical Physics, Perm State University, Perm 614990, Russia Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel 3 Minerva Center for Nonlinear Physics of Complex Systems, Technion, Israel Institute of Technology, Haifa 32000, Israel 4 Department of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel 2

共Received 28 July 2008; accepted 7 April 2009; published online 6 May 2009兲 We consider surface-tension-driven convection in a layer of a binary mixture. A linear stability problem is studied in the presence of both thermocapillary and solutocapillary effects. Assuming the Lewis and Biot numbers to be small, we develop the long wave theory and find both monotonic and oscillatory modes. Three various modes of oscillatory convection exist depending on the ratio between the small parameters. In the case of finite but sufficiently small values of the Biot and Lewis numbers, linear stability thresholds are determined numerically. The numerical results agree well with those found analytically. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3127802兴 I. INTRODUCTION

The problem of surface-tension-driven convection is valuable in the context of microtechnologies. Indeed, at small scales surface effects become more important than the volumetric ones. Therefore, investigation of flows in thin, nonuniformly heated films shows that one can safely neglect the buoyancy convection in favor of the Marangoni effects. A discussion of methods, main ideas and results in this field is given, for instance, in Refs. 1 and 2. In particular, dynamics of binary mixtures is important for many applications: often there is a need to collect solute in a certain part of fluid or, oppositely, to homogenize the fluid. Most of liquid binary mixtures are characterized by small values of the Lewis numbers 共L兲, i.e., the typical characteristic time of the diffusion is larger than that of the heat transfer processes. It is well known that for a buoyancy convection in binary mixtures a large disparity in the characteristic times results in an unusual behavior of the flow field, such as emergence of blinking waves, dispersive chaos, localized waves, or coexistence of traveling and standing waves under certain conditions. These were found both experimentally and numerically, see Refs. 3–7, and the references therein. Moreover, even weakly nonlinear analysis for short-wave perturbations is different: a conventional complex Ginzburg– Landau equation8 must be supplemented by an additional equation for the mean field of concentration.9 A regular procedure accounting for the smallness of L was used by Riecke.10 A similar analysis for the Marangoni convection, especially for the longwave mode has not been carried out yet. Note that a small value of the mass diffusivity can result in the emergence of a boundary layer near the free surface. This boundary layer plays an important role for the surface-tension-driven convection, thus the analysis of Marangoni convection may differ from that for the buoyancy convection. An analysis of the particular case of a small Lewis 1070-6631/2009/21共5兲/054101/18/$25.00

number is also important in the context of particle-laden fluids. As known from Ref. 11, a binary mixture is the simplest model of a nanofluid, with particles considered as a solute. Indeed, the mass diffusivity is remarkably small for nanoparticles. Recent papers12,13 demonstrate that surface tension depends on the concentration of particles, therefore an analog of a solutocapillary effect takes place. Linear stability problem for Marangoni convection in a thin layer of a binary liquid is a subject of many papers.14–19 Both the linear and nonlinear problems in a confined cavity are considered in Ref. 20. Linear stability of a layer of a binary liquid flowing down an inclined plate is carried out in Ref. 21. However, all the papers referred to above are concerned with the situation, when the constant temperature is prescribed at the bottom of the layer. For this type of boundary condition the longwave oscillatory instability emerges only in presence of the surface deformation. In the presence of a thermally insulated substrate another type of the long wave instability sets in. It is known since Pearson,22 that this instability emerges in a pure liquid even with the nondeformable free surface. As it was shown by Oron and Nepomnyashchy,23 the longwave thermo- and solutocapillary convection demonstrates a more complicated behavior under conditions of a fixed heat flux at the substrate. Both monotonic and oscillatory modes are possible in a binary liquid layer. It turns out that the former is similar to the Pearson’s mode. Moreover, a weakly nonlinear study of the monotonic mode23 results in the same set of evolution equations as derived by Shtilman and Sivashinsky24 for a pure liquid, and only the coefficients of these equations change. Two-dimensional convective patterns are studied in Ref. 23 in the case of the oscillatory mode. A detailed bifurcation analysis for the three-dimensional patterns is carried out in Ref. 25. It is shown that alternating rolls are selected on both square and rhombic lattices. The effect of deformability of

21, 054101-1

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054101-2

Phys. Fluids 21, 054101 共2009兲

Shklyaev, Nepomnyashchy, and Oron

the free surface on the Marangoni instability of binary liquid layers is investigated by Podolny et al.26–28 in the context of linear and weakly nonlinear analyses. In particular, a new long wave mode with k = O共B1/2兲 different from the conventional scaling k = O共B1/4兲, exists even for a nondeformable interface. Here k is the wavenumber of critical perturbations and B is the Biot number. The longwave convection at finite values of the Lewis number is considered in Refs. 23, 25, and 26. In fact, this is equivalent to an assumption of B Ⰶ L. One can of course set L = 0 in those theories, but this would correspond to the case B Ⰶ L Ⰶ 1 only, whereas the important cases L = O共B兲 and L Ⰶ B would be left out by this. The purpose of this paper is a detailed study of these cases in the framework of linear stability theory only. Furthermore, we perform a numerical and analytical study of short-wave mode 共with finite values of k兲, as such an analysis has not been done yet. This paper is outlined as follows. Starting with the formulation of the problem in Sec. II, we proceed to analysis of the monotonic mode in Sec. III. Then, in Secs. IV and V the oscillatory mode of the longwave convection is analyzed. Section VI is devoted to the numerical study of a short-wave oscillatory convection in the case of finite but sufficiently small values of both the Lewis and the Biot numbers. These results provide the typical values of L and B for which the predictions of the longwave theory work well. The summary of results is presented in Sec. VII. II. PROBLEM FORMULATION

We consider a planar layer of an incompressible binary liquid lying on a solid heated substrate. The layer is of infinite extent in both the directions along the substrate. Assuming the heat conductivity of the substrate to be much smaller than that of the liquid, a uniform temperature gradient −a is specified at the solid surface z = 0. It is well known that according to Fourier’s law the gradient of the temperature ␽ produces the heat flux Jh = −kth ⵜ ␽, while Fick’s law couples the gradient of the concentration c with the mass flux Jm = −␳D ⵜ c. Here ␳, kth, and D are the density, the thermal conductivity, and the mass diffusivity, respectively. However, often cross effects are also of importance: the gradient of the temperature leads to the emergence of the mass flux 共the Soret effect兲, whereas the concentration gradient generates the heat flux 共the Dufour effect兲.29 The latter is negligible for liquids,30 therefore, we employ the expressions Jh = − kth ⵜ ␽,

Jm = − ␳D共ⵜc + ␣ ⵜ ␽兲,

共1兲

where ␣ is the Soret coefficient of the binary mixture. If the solute does not penetrate both the gas phase and the solid substrate, the temperature gradient at the boundaries generates the gradient of solute concentration. We assume that the ambient gas has such viscous and thermal properties that allow to decouple the dynamics of the gas and liquid phases and to neglect the gas dynamics in the further analysis 共see the corresponding conditions in Ref. 31兲. The surface tension ␴ depends on the temperature and the solute concentration, i.e., both the thermocapillary and

solutocapillary effects may be important. At sufficiently small variations of the temperature and the concentration from their equilibrium values ␽0 and c0, respectively, ␴共␽ , c兲 can be presented as a linear function of its arguments

␴ = ␴共␽0,c0兲 − ␴t共␽ − ␽0兲 + ␴c共c − c0兲.

共2兲

Both the surface tension and the gravity are assumed to be large, i.e., the free surface remains nondeformable. On the other hand, the layer thickness d is assumed to be small, which allows to neglect the buoyancy convection in comparison with the surface-tension-driven one. We choose d2 / ␬, d, ad, ad␴t / ␴c, ␬ / d, and ␳␯␬ / d2 as the scales for time, length, temperature, the solute concentration, the velocity, and the pressure fields, respectively. Here ␬ is the thermal diffusivity and ␯ is the kinematic viscosity of the mixture. Thus, the dimensionless set of governing equations and boundary conditions takes the following form:23 ⵜ · v = 0,

共3a兲

P−1关vt + 共v · ⵜ兲v兴 = − ⵜp + ⵜ2v,

共3b兲

Tt + v · ⵜT = ⵜ2T,

共3c兲

Ct + v · ⵜC = L共ⵜ2C + ␹ⵜ2T兲,

共3d兲

v = 0,

Tz = − 1, Cz = ␹

w = 0,

Tz + BT = 0,

uz = − Mⵜ2共T − C兲,

at

共4a兲

z = 0,

Cz − ␹BT = 0, 共4b兲 at

z = 1,

where w is a z-component of the velocity, u is a twodimensional 共2D兲 projection of the velocity vector onto the x − y plane, ⵜ2 ⬅ 共⳵x , ⳵y兲 and subscripts denote partial derivatives with respect to the corresponding variables. Note that Eqs. 共3兲 and 共4兲 differ from the corresponding Eq. 共7兲 in Ref. 23 in choice of the time scale. The boundary value problem 共3兲 and 共4兲 is characterized by the following five independent dimensionless parameters: M=

␴tad2 , ␳␯␬

B=

qd , kth

␹=

␣␴c , ␴t

␯ P= , ␬

L=

D , 共5兲 ␬

that are the Marangoni, Biot, Soret, Prandtl, and Lewis numbers, respectively. Here, q is the heat transfer rate according to Newton’s law of cooling. In this paper, we study only the case of heating from below, i.e., it is assumed that M ⬎ 0 for the standard case of ␴t ⬎ 0. It should be emphasized that values of the Lewis number are small for most of mixtures. Moreover, the assumption of a small B is quite reasonable for many practical cases. The boundary value problem 共3兲 and 共4兲 possesses a base state corresponding to the conductive state

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Phys. Fluids 21, 054101 共2009兲

Marangoni convection in a binary liquid layer

v0 = 0,

B+1 − z, T0 = B

C0 = ␹z + const,

p0 = const. 共6兲

The following boundary value problem was derived in Ref. 23 to study linear stability of conductive state 共6兲 共note a change in the time scale兲 −

␭ ˜ 共␺⬙ − k2˜␺兲 = ˜␺IV − 2k2˜␺⬙ + k4˜␺ , P

共7a兲

˜ + ik˜␺ = ˜T⬙ − k2˜T , − ␭T

共7b兲

␹ ␭˜ ˜ ⬙ − k 2C ˜ + ␹共T ˜ ⬙ − k2˜T兲, − C − ik ˜␺ = C L L

共7c兲

˜␺ = 0,

˜␺⬘ = 0,

˜T⬘ = 0,

˜␺ = 0,

˜T⬘ + BT ˜ = 0,

˜⬘ = 0 C

at

z = 0,

共8a兲

˜ ⬘ − ␹BT ˜ = 0, C 共8b兲

˜␺⬙ = − ikM共T ˜ −C ˜ 兲,

at

z = 1.

˜ , and ˜␺ are z-dependent amplitudes of perturbaHere, ˜T, C tions of the temperature, the solute concentration, and the stream function, respectively. Obvious symmetry properties of the system allow us to consider a 2D linear stability problem. All perturbed fields are presented in the form f共x , z , t兲 =˜f 共z兲exp共−␭t + ikx兲, where ␭ is the complex decay rate and k is the disturbance wavenumber. The derivatives with respect to z are denoted by primes, and the superscript “IV” denotes the fourth derivative. This problem was considered in the longwave approximation by Oron and Nepomnyashchy23 for finite values of the Lewis number and small Biot numbers; both monotonic and oscillatory modes were revealed. However, the analysis of short-wave instability was not carried out for this type of boundary conditions. Most of earlier works address the stability problem for a substrate of perfect thermal conductivity,

¯ = M ⴱ

i.e., a uniform substrate temperature was prescribed.14–19 We present here the analytical and numerical 共for the oscillatory mode兲 study of the short-wave instability in order to extend the linear analysis of Ref. 23. We also investigate the long wave mode for the case of a small Lewis number. In fact, one can readily consider small values of the Lewis number within the framework of the model developed in Ref. 23. Based on the analysis of the general case, one may conclude that such theory is valid only in the case B Ⰶ L Ⰶ 1. However, the particular case L = O共B兲 is also of importance. This investigation for the monotonic mode, as well as the analysis of the short-wave instability are presented in Sec. III. Furthermore, it is shown that two other important cases emerge for the oscillatory mode: L = O共B兲 Ⰶ 1 共case I兲 and L = O共B3兲 Ⰶ 1 共case II兲. These two “rays” divide the B − L plane into three regions: 共i兲 L Ⰷ B; 共ii兲 B Ⰷ L Ⰷ B3; 共iii兲 L Ⰶ B3. We find that an analysis of cases I and II is sufficient for the description of oscillatory neutral curves in the entire B − L plane. Cases I and II, and their subcases are examined in Sec. IV and V, respectively. Section VI presents the numerical study of short-wave oscillatory convection at finite values of L and B and contains a comparison between the analytical and numerical results. III. MONOTONIC INSTABILITY A. General case

To carry out the analysis of the monotonic mode for small L, it is useful to introduce the “solutal” Marangoni ¯ = ML−1. Indeed, in this limiting case the solutonumber M capillary convection dominates over the thermocapillary one and the stability threshold is determined by the mass diffusivity D rather than the thermal diffusivity ␬. The temperature gradient itself generates a gradient of the solute concentration via the Soret effect, but it does not influence directly the emergence of convection, at least at leading order. Straightforward but tedious derivations not shown here result in the following expression for the neutral stability curve:

4k2 sinh k共sinh 2k − 2k兲共B cosh k + k sinh k兲

␹B cosh2 kq2共k兲 + 21 k sinh 2k关␹Lq2共k兲 − ␹Bq1共k兲兴 − ␹Lk2 sinh2 kq1共k兲

where ␹L = ␹ + L + ␹L, q1共k兲 = 2 + k2, q2共k兲 = k2 + sinh2 k. It is easily seen that for L Ⰶ 兩␹兩, the Marangoni and the Soret numbers are of the same sign. Below we are concerned with the case of heating from below only, i.e., both the Marangoni number and Soret number are assumed to be positive in Secs. III A and III B. The limiting case of 兩␹兩 = O共L兲 is analyzed in Sec. III C; both positive and negative values of the Soret number are considered there. Equation 共9兲 can be easily recast in terms of the generalized Marangoni and Soret num-

,

共9兲

bers, M g = M共1 + ␹兲 and ␹g = ␹L−1 / 共1 + ␹兲, respectively,17 ¯ ␹ and ␹ / ␹ = 1 + ␹−1. based on the relationships M g␹g = M L g ¯ 共k兲 is given in An example of numerical evaluation of M ⴱ Fig. 1. For small values of L there exists a longwave instability, whereas with increase of L, the critical wavenumber kc ¯ decreases. grows and M c ¯ and The critical values of the Marangoni number M c wavenumber kc are shown for the short-wave mode in Fig. 2. Note, that in accordance with the results of Ref. 23, the

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054101-4

Phys. Fluids 21, 054101 共2009兲

Shklyaev, Nepomnyashchy, and Oron 2

500

480

1

1

_ M* 460

_

mβ

2

0 440

3 420 0

0.2

0.4

0.6

-1

1

0.8

k

0

¯ 共k兲 for short-wave monotonic perturbaFIG. 1. Marginal stability curves M ⴱ tions. ␹ = 0.1, B = 0.01. Dashed, solid, and dashed-dotted curves correspond to L = 0.001, 0.05, and 0.01, respectively.

冋 冉

2 ¯ = 48 1 + k 1 − Bⴱ M ⴱ 15 ␹ B + k2

B. Long wave instability, positive ␹

共10兲

or Mg ⬇

48关B + 共1 +

␹ gB 共 1 +

4 2 5k

+

兲

共

兲 兴

13 2B 8 2 4 15 B k + 7 + 15 k 17 4 7 2 2 75 k + 共␹g + 1兲k 1 + 15 k

兲

共

兲

For L = O共B兲 = O共k2兲 and ␹ = O共1兲 we obtain 480

2 460 _ Mc

¯␤= m

3

0.2

(a)

0.4

0.6 B

0.8

1

0

0.2

(b)

0.4

共14兲

5␹ 共M ⴱ − M 0兲 16BL

A2 ¯2 + 1 K ␤

¯ = 冑A − 1, K ␤c

1 0 0.2

,

0.6

冊

,

A2 ⬅

Bⴱ . B

共15兲

Typical neutral stability curves m␤共K␤兲 are shown in Fig. 3. These curves are qualitatively similar to those shown in Fig. 1. It is clear that depending on the parameter A, the minimal ¯ = 0 for A ⱕ 1, or at some finite ¯ ␤c is attained at K value m ␤c value

2

0

冊册

which corrects the critical value M 0 = 48L␹−1 corresponding to the well-known result22 for the Marangoni number defined via the mass diffusivity D and the concentration gradient ␣a. Substituting both the correction to the Marangoni number

冉

kc 0.4

420

冋 冉

L k2 Bⴱ 1+ 1− 15 ␹ B + k2

¯2 1 − ¯␤=K m ␤

0.6

stable

共13兲

¯ = k / 冑B, we arrive at and the rescaled wavenumber K ␤

3

440

共12兲

,

15共␹ + 1兲L , ␹

M ⴱ = 48

0.8

1

冊册

¯ L, or, turning back to M = M

共11兲

.

2

¯ 兲 for the monotonic mode, as obtained ¯ ␤共K FIG. 3. Neutral stability curves m ␤ from Eq. 共15兲. Curves 1, 2, and 3 correspond to A = 0, 1, and 2, respectively.

Bⴱ =

13 2B 8 2 4 ¯ ⬇ 48关B + 共1 + 15 B兲k + 共 7 + 15 兲k 兴 M ⴱ 17 4 7 2 ␹B共1 + 54 k2 + 75 k 兲 + ␹Lk2共1 + 15 k兲

1.5

Kβ

wavenumber of critical perturbations is small, but nonzero, for small B. The longwave mode with k = 0 becomes critical for B exceeding some critical value Bⴱ共L兲.27 The expression for Bⴱ is determined in Sec. III B.

In Secs. III B and III C, we discuss various cases of long ¯ 共k兲 for small wave instability based on Eq. 共9兲. Expanding M 4 k and keeping terms up to k , we obtain

1 _

0.5

0.8

1

B

¯ and 共b兲 critical FIG. 2. Variation of 共a兲 the critical Marangoni number M c wavenumbers kc with the Biot number B in the case of the monotonic instability for ␹ = 0.1. Curves 1, 2, and 3 correspond to L = 0.001, 0.005, and 0.01, respectively. The solid curves represent the results of numerical computations, whereas the dashed ones are obtained from the long wave approximation, Eq. 共17兲.

¯ ␤c = − 共A − 1兲2 m

共16兲

for A ⬎ 1. Therefore, the critical Marangoni number M c and wavenumber kc are L M c = 48 , ␹

kc = 0

for

B ⬎ Bⴱ

or

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共17a兲

054101-5

M c = 48

Phys. Fluids 21, 054101 共2009兲

Marangoni convection in a binary liquid layer

冋

册

共冑Bⴱ − 冑B兲2 L 1− , 15 ␹

k2c = 冑BBⴱ − B

for

For the critical value of the Marangoni number we obtain 共17b兲

B ⬍ Bⴱ .

A comparison of Eq. 共17兲 with the numerical results is presented in Fig. 2. It is clearly seen that the long wave asymptotic expressions shown by dashed lines qualitatively agree with the numerical data even for L = 0.01. For smaller values of the Lewis number the difference between the analytical results and those of numerical computations becomes very small. Equation 共14兲 matches both asymptotics of small B, Ref. 23, and the expression for finite B, Ref. 27. Indeed, assuming k2 Ⰷ B, similarly to Ref. 23, where the case B = O共k4兲 was considered, we obtain M ⴱ = 48

冋

册

共␹ + 1兲L k2 共␹ + 1兲L B L + . 1− + 15 k2 ␹ ␹ ␹

共18兲

The first two terms represent the expansion of the critical Marangoni number M 0 = 48关1 + ␹共1 + L−1兲兴−1 at small L, while the others provide the corrections to the Marangoni number at small B, cf. Eqs. 共19兲 and 共24兲 in Ref. 23, respectively. The factor P in k4 / 15 appears as a misprint in Eq. 共24兲 there. On the other hand, in the case B Ⰷ k2, one can omit k2 in comparison with B, which leads to

冋 冉 冊册 2

M ⴱ = 48

L k Bⴱ 1+ 1− 15 ␹ B

共19兲

,

obtained by Podolny et al.27 for finite B 关Eqs. 共59兲, 共61兲, and 共62兲 at L Ⰶ 1, G → ⬁ to suppress the surface deflection, and b = 0 for a thermally insulated solid substrate兴. Thus, the intermediate asymptotics is obtained. As it matches the two opposite known limits, there is no need in a different analysis to study the stability of the system. As noted above, the only case which should be considered separately, is the case of small 兩␹兩. C. Longwave instability, small 円␹円

Equation 共14兲 becomes invalid as the absolute value of the Soret number 兩␹兩 decreases. A closer look at Eq. 共10兲 shows that at 兩␹兩 = O共L兲, additional terms appear in the ex¯ preventing the divergence of the Marangoni pression for M number at ␹ = 0. However, in the case of L = O共兩␹兩兲 Ⰶ 1, the analysis can be carried out based on Eq. 共11兲. Indeed, in this case ␹g ⬇ ␹ / L = O共1兲 and, consequently, B is the only small parameter in Eq. 共11兲. In other words, two limits, B Ⰶ 1 and L = O共兩␹兩兲 Ⰶ 1, are interchangeable and one can consider small L and ␹ in the framework of the conventional case B Ⰶ 1.23 This analysis yields the critical value

冋

2 LB ¯ = 48 1 + k + M ⴱ 15 共␹ + L兲k2 L+␹

册

which for ␹ Ⰷ L matches Eq. 共18兲 for ␹ Ⰶ 1.

共20兲

Mc =

冋 冑

48L 1+2 L+␹

册

LB , 15共␹ + L兲

for

k4c =

15LB , ␹+L 共21兲

i.e., the critical value of the wavenumber is proportional to B1/4, as for the conventional instability mode.23

IV. OSCILLATORY MODE FOR SMALL LEWIS NUMBERS. CASE I: L = O„B… A. Long wave expansion

We now proceed to the investigation of the oscillatory mode. In order to consider large-scale oscillatory convection we introduce the scaling ␭ = ⑀2⌳,

k = ⑀K,

L = ⑀2l,

B = ⑀ 2␤ ,

共22兲

where ⑀ Ⰶ 1, and expand the perturbation fields in powers of ⑀2 ˜ = C + ⑀ 2C + ¯ , C 0 1

˜T = T + ⑀2T + ¯ , 0 1

共23a兲

˜␺ = ⑀共⌿ + ⑀2⌿ + ¯兲. 0 1

共23b兲

There is no need to expand the Marangoni number with respect to ⑀, as the sought result is obtained at the lowest order. The notation m ⬅ M / 48 is used and for the sake of brevity in this section we omit the word “rescaled,” referring to m as the “Marangoni number.” Substituting the expansions given by Eqs. 共22兲 and 共23兲 into the boundary value problem 共7兲 and 共8兲, we obtain at zero order T0⬙ = 0,

共24a兲

⌳C0 = − l共C0⬙ + ␹T0⬙兲 − iK␹⌿0 ,

共24b兲

⌿IV 0 = 0,

⌿0 = 0,

⌿0⬘ = 0,

T0⬘ = 0,

⌿0 = 0,

T0⬘ = 0,

C0⬘ = 0,

⌿0⬙ = − 48iKm共T0 − C0兲,

C0⬘ = 0,

at

z = 0, 共24c兲

共24d兲 at

z = 1.

The solution of Eqs. 共24a兲, 共24c兲, and 共24d兲 is T0 = F,

⌿0 = 12miKz2共1 − z兲h.

共25兲

Here h = F − C0共z = 1兲,

共26兲

and therefore, −h is the dimensionless perturbation of the surface tension. At first order we need only the energy balance equation and the corresponding boundary conditions T1⬙ = − ⌳T0 + iK⌿0 + K2T0 ,

共27a兲

T1⬘ = 0,

共27b兲

at

z = 0,

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054101-6

Phys. Fluids 21, 054101 共2009兲

Shklyaev, Nepomnyashchy, and Oron

0.16

0.04

S=

4 3 0.02 Si

2

Sr 0.08

1

4 3

0

S = − z2共z − 1兲.

2

-0.02 0

0.5 z

(a)

1

0 (b)

0.5 z

1

FIG. 4. A transverse variation of 共a兲 the real and 共b兲 imaginary parts of S proportional to the concentration perturbations, see Eq. 共30兲. ⌳ = i⍀, l = 1. Curves 1, 2, 3, and 4 correspond to ⍀ = 0.1, 10, 100, and 1000, respectively. An approximation in the limit of large ⍀, Eq. 共33兲, is shown by the dashed curves for ⍀ = 1000.

T1⬘ = − ␤T0,

at

z = 1.

共27c兲

The solvability condition of this boundary value problem yields an equation for F. Integrating Eq. 共27a兲 across the layer and accounting for Eqs. 共25兲, 共27b兲, and 共27c兲, we obtain − ⌳F + ␤F − mK2h + K2F = 0.

共28兲

This equation should be supplemented with the boundary value problem for the perturbations of concentration, Eqs. 共24b兲–共24d兲. The latter can be rewritten using Eq. 共25兲 lC0⬙ + ⌳C0 = 12␹mK2z2共1 − z兲h,

共29a兲

C0⬘ = 0,

at

z = 0,

共29b兲

C0⬘ = 0,

at

z = 1.

共29c兲

Note that not as in other work on this subject, the considered case has an unusual property: the equation for the concentration 共29兲 is not an algebraic relation, but an ordinary differential equation. It is also important to note that h is a constant now, which allows us to integrate Eq. 共29a兲. Accounting for the boundary conditions we obtain the solution C0 =

bh S, q2

S = q−3

冋

共30兲

共q + 6兲cosh qz − 6 cosh q共1 − z兲 − q3z2共z − 1兲 sinh q

册

− 2q共3z − 1兲 , where q2 = −⌳l−1 and b = −12m␹K2l−1. Both real and imaginary parts of S are depicted in Fig. 4, we set q = 共1 − i兲冑⍀ / 2 presenting this figure. This ansatz corresponds to a pure imaginary decay rate ⌳ = i⍀ and l = 1, see remark in the beginning of Sec. IV B concerning the last expression. In the limit of small ⍀, Eq. 共30兲 yields

共32兲

Equation 共32兲 represents a first term of expansion in ⍀−1, hence it works well only at extremely large ⍀. However, exponentially small terms can be safely neglected even at finite ⍀. Such approximation 共“physical limit”兲, based upon disregarding the exponentially small terms, when all the powers of q−1 are kept, results in the expression S = − 关z2共z − 1兲 + 2q−2共3z − 1兲兴.

共33兲

Both real and imaginary parts of S given by Eq. 共33兲 are shown by dashed lines in Fig. 4. It is clearly seen, that the exact solutions coincide with the approximate ones even at ⍀ = 1000 共curves 4兲 in large part of the layer. The difference between the two expressions is substantial only in a thin boundary layer near the free surface, where the impermeability condition has to be met. In this boundary layer, the first term in Eq. 共30兲 becomes important. This leads to S=

q2 + 6 q共z−1兲 2 2 − z 共z − 1兲 − 2 共3z − 1兲 3 e q q + e.s.t.,

for

z ⬇ 1,

共34兲

where e.s.t. denotes “exponentially small terms.” Evaluating C0 at z = 1 and using the relationship C0共z = 1兲 = F − h, we derive the second algebraic equation determining, along with Eq. 共28兲, the decay rate ⌳ F−h=−

bl f共q兲h, ⌳

共35兲

where f ⬅ S共z = 1兲, i.e., f = q−3关共q2 + 6兲coth q − 6 sinh−1 q − 4q兴.

共36兲

Variation of both real 共f r兲 and imaginary 共f i兲 parts of f with 兩q兩 is shown in Fig. 5 where the relationship q = 共1 − i兲兩q兩 / 冑2 with 兩q兩 = 冑⍀ is again employed. Function f共q兲 has the asymptotic properties f共q兲 ⬇

2

共31兲

which agrees well with the results shown by curves 1 in Fig. 4. It is noteworthy that there is a good quantitative agreement even for ⍀ = 1. For large ⍀, the concentration profile reads

1

0

1 q2 + 共18z5 − 30z4 + 15z2 − 2兲, 12 360

1 12

+ O共q2兲,

f共q兲 ⬇ q−1 + O共q−2兲,

q → 0,

at at

q → ⬁.

共37兲 共38兲

However, in what follows it will be necessary to consider also sufficiently large, but finite values of 兩q兩. In this case the physical limit with the correction near the free surface, Eq. 共34兲, is good, and we obtain f共q兲 = q−3共q2 − 4q + 6兲.

共39兲

This approximation will be also used at 兩q兩 ⱖ 16 to ensure the relative error of calculations to be less than 10−6. Note that the physical limit corresponds to splitting the layer into three regions with a qualitatively different behavior of the concentration perturbations: a core region, where the diffusive mass flux serves as a small correction to the

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054101-7

Phys. Fluids 21, 054101 共2009兲

Marangoni convection in a binary liquid layer

0.12

mⴱ = 0.08

fr,i

1 0.04

0

2 -0.04 0

10

20

30

40

|q| FIG. 5. Variation of the real and imaginary parts of f共q兲 shown by curves 1 and 2, respectively, with the modulus of q, q = 共1 − i兲兩q兩 / 冑2. Solid, dotted, and dashed curves show the results obtained from Eqs. 共36兲, 共38兲, and 共39兲, respectively.

1 , 1+␹

⍀=

冑

−

␹ K2 . 1+␹

共41兲

These expressions determine the longwave minimum of the marginal stability curve mⴱ共K兲, which takes place only at ␤ = 0. Moreover, it will be shown below, that the longwave minimum is always local, whereas the global minimum is attained at larger values of K. Equations 共41兲 coincide with Eqs. 共25兲 and 共26兲 in Ref. 23 at L = 0, i.e., this longwave mode is reminiscent of the conventional one. Case 共ii兲 共large ␤兲 is more interesting. Note that it follows from Eq. 共40兲 that both K and ⍀ are large. Indeed, as the complex-valued expression in the square brackets of Eq. 共40兲 does not vanish, other values, such as ⍀, K, or/and m have to be large in order to compensate for a large ␤. The limiting case of large ⍀ is analyzed in the Appendix, it is shown there that Eq. 共A1兲 is appropriate at ⍀ Ⰷ 1 instead of the exact relation, Eq. 共40兲. Introducing the stretched wavenumber and frequency of perturbations K = ␤5/8K␤,

⍀ = ␤⍀␤ ,

共42兲

expanding mⴱ into power series of ␤−1/4 as advection equation 关see Eq. 共33兲兴 and the two boundary layers near both the free 关Eq. 共34兲兴 and solid surface accommodating the boundary conditions. Disregarding the exponentially small terms, we neglect the influence of a boundary layer near the substrate on the concentration at the free surface and, hence, on the velocity of a solutocapillary flow. The set of two linear algebraic equations, Eqs. 共28兲 and 共35兲, is solvable if the following relation is satisfied:

冋

mK2 = 共␤ + K2 − ⌳兲 1 +

册

12m␹K2 f共q兲 , ⌳

mⴱ = m␤共0兲 + ␤−1/4m␤共1兲 ,

and substituting Eqs. 共42兲 and 共43兲 into Eq. 共A1兲, we obtain

冋

m␤共0兲 + ␤−1/4m␤共1兲 = 1 +

册

12␹K␤2 共− i⍀␤兲3/2

册

⫻共m␤共0兲 + ␤−1/4m␤共1兲兲 .

共44兲

Collecting terms with equal powers of ␤ results in m␤共0兲 = 1,

m␤共1兲 = K␤−2 −

冉 冊 72␹2

1/5

K␤2

共45兲

,

⍀␤ = 共72␹2K␤8 兲1/5 .

共46兲

The neutral stability curve, given by Eq. 共45兲, is shown in Fig. 6 by the dashed-dotted line which attains the minimal value m共1兲 c =−

B. Limiting cases

We now analyze the oscillatory mode setting ⌳ = i⍀ for the neutral stability taking place at m = mⴱ. In this section we set l = 1, assuming thereby ⑀ ⬅ 冑L. This assumption does not lead to loss of generality, only the wavenumber K, the frequency ⍀, and ␤ are rescaled, whereas mⴱ remains unchanged, see also Sec. IV D for details. We begin with two limiting cases of either small or large values of ␤ defined at the end of Sec. II, namely cases 共i兲 and 共ii兲, respectively. For ␤ = 0, there exists a longwave asymptotics with K = 0:

冋

␤−1/4 共1 − i⍀␤兲 K␤2

⫻ 1 − ␤−1/4

共40兲

The obtained transcendental equation determines the complex decay rate as a function of the remaining parameters of the problem. First, we show that it does not describe a new mode of monotonic instability. Indeed, at the marginal stability curve, ⌳ = 0, i.e., q = 0. However this is not the case, as the denominator tends to zero in the second term in the square brackets, while other terms remain finite. The only possibility to avoid this divergence is to assume m = 0 along with ⌳ = 0. Therefore, for the monotonic mode we return to the scaling similar to one considered in Sec. III: m = O共⑀2兲, ␭ = o共⑀2兲.

共43兲

冉 冊

4 72␹2 5 5

1/4

,

at

K ␤c =

冉 冊 55 72␹2

1/8

.

共47兲

Returning to the initial variables using Eqs. 共42兲 and 共43兲, we obtain mⴱ = 1 +

冉 冊

72␹2 ␤ 2 − K K2

1/5

,

⍀ = 共72␹2K8兲1/5 .

共48a兲 共48b兲

Equation 共48兲 is valid for ␹ ⬍ 0. For this mode, the limitation ␹ ⬎ −1, see Ref. 23, does not play any role. The critical values of the Marangoni number and the frequency are given by

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054101-8

Phys. Fluids 21, 054101 共2009兲

Shklyaev, Nepomnyashchy, and Oron 0

-0.32

60

-0.36

50

-0.4

40

-0.4

1 m(1) β

m1c

-0.8

K0c

-0.44

30

-0.48

20

2 3

-1.2

-0.52

10 0

40

1

1.5

2

2.5

3

Kβ FIG. 6. Neutral stability curves m共1兲 ␤ 共K␤兲 for ␹ = −0.8 in the limit of large ␤. The dashed-dotted curve represents the analytical solution given by Eq. 共47兲, whereas the solid curves 1, 2, and 3 show the numerical results for ␤ = 103, 105, and 108, respectively. The dashed curves are obtained from Eq. 共A2兲 for the respective values of ␤.

mc = 1 −

⍀c = 5␤,

冉 冊

4 72␹2 5 5␤ for

1/4

,

冋 册

共5␤兲5 Kc = 72␹2

1/8

共49兲

冋 冉 冊册 冋 册

M c = 48 1 −

4 72␹ L 5 5B

␻c = 5B,

at

1/4

共5B兲5 kc = 72␹2L

160

200

FIG. 7. Variation of the first-order correction m1c to the Marangoni number 共solid curve兲 and the critical wavenumber K0c 共dashed curve兲 with ␤ in the case of small ␹, see Eqs. 共51兲 and 共52兲.

Similar results are valid in the limit of small 兩␹兩 as well. Indeed, for 兩␹兩 Ⰶ 1, the second term in the square bracket in Eq. 共40兲 represents a small correction to unity which can be taken into account using the perturbation technique. Substituting the scaling for the Marangoni number and the wavenumber mⴱ = 1 + 冑兩␹兩m1 + ¯ ,

K = 兩␹兩−1/4K0

共51兲

into Eq. 共40兲 and collecting the terms of order 冑兩␹兩, we obtain

or, returning to the original Marangoni number M = 48m and the wavenumber k = 冑LK 2

120

β

-1.6 0.5

80

,

共50兲

1/8

,

where ␭ = i␻ at the stability threshold. This expression seems to be rather intriguing, as the critical Marangoni number for the oscillatory convection in binary mixture given by Eq. 共50兲 is close to the threshold for the monotonic mode in a pure liquid. The detailed analysis of this mode including the explanation of this coincidence, is carried out in Sec. V. In some sense, a similar phenomenon was found in Ref. 32, where the influence of the surfactant on Marangoni convection in a two-layer system was studied. The main effect of surfactant is that in the case of heating from above, the monotonic mode is replaced by the oscillatory one, while the critical Marangoni number slightly changes. Here we also note that the marginal stability curve and the frequency of neutral perturbations, Eq. 共48兲, as well as the critical parameters given by Eq. 共49兲, represent the first terms of the expansion in ␤−1/4, see Eq. 共44兲. Therefore, this asymptotics is expected to be valid only for extremely large ␤.

m1K20 = ␤ − i⍀ −

12K40 f共q兲. i⍀

共52兲

Variation of the critical value of m1 and the corresponding wavenumber K0 with the rescaled Biot number ␤ is presented in Fig. 7. It is evident that m1 ⬍ 0 for any ␤, i.e., mⴱ ⬍ 1. Other limiting cases are considered in the Appendix. Here we just note in passing that Eq. 共48兲 holds for large K even for finite ␤, in this case the second term should be omitted for m0共K兲 关cf. Eq. 共A4兲兴. It means that mⴱ ⬍ 1 at K Ⰷ 1 and, consequently, mc ⬍ 1 for any ␹ and ␤. C. Computation of neutral curves

For arbitrary values of ␤ and ␹, Eq. 共40兲 is solved numerically. In fact, separating the real and imaginary parts of this equation, one can obtain the parametric representation of mⴱ共K兲 with ⍀ serving as a parameter, see Eq. 共A2兲 as example for specific particular case. However, for arbitrary values of q 共i.e., ⍀兲, it leads to cumbersome expressions, which are not presented here. Several typical neutral stability curves mⴱ共K兲 are depicted in Fig. 8. For ␹ ⬎ −1, they consist of two branches, which can either form a single curve, for example, for ␹ = −0.6, as shown in Fig. 8共a兲, or be represented by separate curves with a common asymptote, as shown in Fig. 8共b兲 for ␹ = −0.8. The behaviors of these two branches are quite dif-

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054101-9

Phys. Fluids 21, 054101 共2009兲

Marangoni convection in a binary liquid layer

10

40

10

m* 5

m* 20

m* 5

1

6

4

1

mc

2

0.9

3

3.2

5

lgKc

lgΩc

3 2

2.4

3

4

2

1

0

0 0

2 (a)

4

6

K

0.8

0 0

2 (b)

4 K

6

4

6 (c)

8

ferent, namely, the left one, corresponding to smaller values of K, demonstrates a very strong dependence on both ␤ and ␹. There exists a well-pronounced stabilization of this branch with increase of ␤ and for ␹ → −1. The branch corresponding to larger values of K slightly varies with the Soret number and the rescaled Biot number in a wide interval of these parameters. Another interesting feature of this curve is that it attains a minimum at sufficiently large 共but finite!兲 values of K: Kc ⬎ 100, see Fig. 9 for details. With either a further decrease in ␹ or increase in ␤, the left branch raises and disappears at a certain value ␹c共␤兲 ⬎ −1, see, for instance, the dashed line in Fig. 8共a兲. For ␹ ⬍ −1, only the “short-wave” branch survives, as shown in Fig. 8共c兲. Therefore, the left branch is associated with the oscillatory mode studied before,23,26 which exists only within the interval −1 ⬍ ␹ ⬍ 0. Note that the global minimum of mⴱ共K兲 always belongs to the right branch, and the critical values of Marangoni number are smaller than one. We now discuss the results of minimization of mⴱ with respect to the wavenumber K. Variations of the critical

1

2

3

4

5

0

lgβ

(a)

K

FIG. 8. Neutral stability curves mⴱ共K兲 for 共a兲 ␹ = −0.6, 共b兲 ␹ = −0.8, and 共c兲 ␹ = −1.2. Lines for ␤ = 0, 1, and 10, are shown by the solid, dashed-dotted, and dashed curves, respectively. The domains of instability are located above the respective curves. The dashed-dotted curves are not shown in 共b兲 for the right branch and in 共c兲. They are located between the dashed and the solid curves.

3

1.6 0

10

1 1

2

(b)

3

4

0

5

lgβ

1

2

3

4

5

lgβ

(c)

FIG. 10. Variation of the critical values of 共a兲 the Marangoni number, 共b兲 the wavenumber, and 共c兲 the frequency with lg ␤. Curves 1, 2, and 3 correspond to ␹ = −0.2, ⫺0.8, and ⫺2, respectively. The dashed curves present the results of the analysis in the limit of large ␤, Eq. 共49兲: 关共a兲 and 共b兲兴 ␹ = −0.2 and ␹ = −2, 共c兲 for any ␹.

Marangoni number mc, wavenumber Kc, and frequency ⍀c with ␤ for different values of the Soret number are depicted in Fig. 10. One can see that the value of mc slightly grows with ␤, tending to unity at large values of ␤. The critical wavenumber grows with the increase in ␤. A comparison with the approximations given by Eq. 共49兲 is also presented in this figure. It is clear that these asymptotic expressions provide a good approximation at extremely large, and in fact, unachievable values of ␤, thus, they are of a limited value only. Note that ⍀c ⬎ 300 for all values of ␹ and ␤, i.e., the inequality 兩q兩 ⬎ 16 is satisfied for critical perturbations, and one can use approximation 共39兲 when calculating f共q兲 in Eq. 共40兲. Finally, we show in Fig. 11 the variation of the critical Marangoni number with the Soret number for different ␤. The results slightly depend on ␤ in a wide interval of this parameter. The critical value of the Marangoni number slowly decreases with increase of 兩␹兩, from unity at ␹ = 0 to the limiting value of mc ⬇ 0.8284, prescribed by Eq. 共A7兲. The critical wavenumber has minimum at some ␹ and linearly increases with a further increase of 兩␹兩. Due to the fact that ⍀c is large, approximation 共39兲 is valid.

0.96

D. Intermediate asymptotics for B ™ L ™ 1, case „i…

Keeping in mind that in the case of small B and finite values of L, the critical wavenumber k1 = O共B1/4兲, whereas for L = O共B兲 Ⰶ 1, the critical wavenumber is k2 = O共L1/2兲, we infer that three different limits should be now investigated: B Ⰶ L2, B = O共L2兲, and L2 Ⰶ B Ⰶ L.

0.92

m* 0.88

1

2

mc

10000

K

0.8

-2

(a)

FIG. 9. Neutral stability curves mⴱ共K兲 in the limit of large K for ␹ = −0.8. The curves 1 and 2 共both solid and dashed兲 correspond to ␤ = 0 and ␤ = 103, respectively. The solid and dashed curves are obtained from the numerical computations and from Eq. 共A2兲, respectively. The dashed-dotted curve represents the asymptotics in the limit of large K, Eqs. 共A4b兲. The domains of instability are located above the respective curves.

0

0

-3

χ

-1

0

3 1,2

2

2 1

1,2

1000

4

10-4Ωc

200

3

1 100

3

Kc

0.9

0.84

6

400

-3

-2

(b)

χ

-1

0

-3

-2

(c)

χ

-1

0

FIG. 11. Variation of the critical value of 共a兲 the Marangoni number, 共b兲 wavenumber, and 共c兲 frequency with ␹. Curves 1, 2, and 3 correspond to ␤ = 0, 10, and 103, respectively. The dashed curves present the analytical results for large ␹, Eq. 共A7兲. Curve 2 visually coincides with curve 1 on the scales of 共a兲 and 共c兲.

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054101-10

Phys. Fluids 21, 054101 共2009兲

Shklyaev, Nepomnyashchy, and Oron

We begin with the first one, setting k1 Ⰶ k2. At first sight, this limiting case has to be considered based on Ref. 23, just setting L Ⰶ 1 in Eqs. 共25兲, 共26兲, 共27a兲–共27c兲, and 共28兲 there. The critical Marangoni number for this mode is 48/ 共1 + ␹兲 + O共冑B兲 ⬎ 48. However, as shown above, in the case of small L, there exists “novel” mode of long wave instability with a typical wavenumber k2. For this mode, one can repeat the derivation of Sec. IV A in order to obtain the set of Eqs. 共28兲 and 共35兲, but with ␤ = 0, as ␤ = lBL−1 Ⰶ 1 共l = 1, as discussed above兲. It is important that the critical Marangoni number for this mode is lower than 48 and does not depend on the Lewis number. Consequently, this mode causes instability. The linear stability theory based on the conventional asymptotics with k = k1 only describes the behavior of the marginal stability curve near k2 = 0, i.e., near the local minimum. The situation is even simpler in the second case, B = O共L2兲, when k1 is comparable with k2. The scaling typical for the conventional mode becomes insufficient: in Ref. 23, the zero-order approximation for the critical Marangoni number is independent of the wavenumber; k1 is determined from the minimization of the first 关O共冑B兲兴 correction to M 0 = 48/ 共1 + ␹兲. However for B = O共L2兲, the critical Marangoni number depends on the wavenumber even at zero order, as shown by solid curves in Fig. 8, and the 冑B-order corrections are unimportant to determine the minimum of the neutral stability curve. Of course, the same is valid for the third case, L2 Ⰶ B Ⰶ L, where the stability is described by Eq. 共40兲 in the limiting case ␤ = 0. Therefore, even at B Ⰶ L, the main conclusion of Secs. IV B and IV C remains valid: the oscillatory instability sets in at M ⬍ 48, see curve 1 in Fig. 11. An increase in L leads primarily to the increase in the critical wavenumber according to the law kc ⬃ Kc冑L, where Kc ⬇ 100, see Fig. 11共b兲. When L becomes finite, the critical wavenumber leaves the domain of validity for the longwave approximation. This will be discussed in Sec. VI below. Apart of this, an interesting competition of the two modes considered here, with k = k1 and k = k2, is expected for ␹ → 0−, when the two minima become close to each other. This analysis is outside the scope of this paper. We only refer to Sec. VI, where the results of numerical solution of stability problem 共7兲 and 共8兲 are presented. V. OSCILLATORY MODE FOR SMALL LEWIS NUMBERS. CASE II: L = O„B3…

˜T = ⑀2Tˆ,

˜ = Cˆ, C

共53b兲 ˆ. ˜␺ = ⑀3⌿

共53c兲

Substituting these expressions into boundary value problem 共7兲 and 共8兲 we arrive at ˆ 兲 − ⑀ 4⌳ ˆ Tˆ , ˆ 2Tˆ + iKˆ⌿ Tˆ⬙ = ⑀2共K

共54a兲

ˆ + ˆl⑀8Cˆ⬙ + ˆl⑀10共␹Tˆ⬙ − Kˆ2Cˆ兲 − ˆl␹K ˆ 2⑀12Tˆ , ˆ Cˆ = iKˆ␹⌿ −⌳ 共54b兲 ˆ ⬙ − ⑀4Kˆ4⌿ ˆ, ˆ IV = 2⑀2Kˆ2⌿ ⌿ ˆ = 0, ⌿

ˆ ⬘ = 0, ⌿

Tˆ⬘ = 0,

ˆ = 0, ⌿

Tˆ⬘ = − ␤ˆ ⑀4Tˆ,

共54c兲 ˆ ⬘ = 0, C

at

z = 0,

共54d兲

ˆ ⬘ = ␹␤ˆ ⑀6Tˆ , C 共54e兲

ˆ ⬙ = − 48iKˆm共Tˆ − ⑀−2Cˆ兲, ⌿

at

z = 1.

ˆ enters the tangential stress condition, Despite the fact that C Eq. 共54e兲, with a negative power of ⑀, the set of Eqs. 共54兲 is self-consistent. It will be shown below that at leading order Cˆ vanishes at the free surface. We expand the Marangoni number and the perturbation fields in powers of ⑀2 as M = 48共m0 + ⑀2m1 + ¯兲,

共55a兲

Tˆ = T0 + ⑀2T1 + ⑀4T2 + ¯ ,

共55b兲

ˆ = ⌿ + ⑀ 2⌿ + ¯ . ⌿ 0 1

共55c兲

The expansion of Cˆ with decoration omitted in the rest of Sec. V A, is not necessary now, and this issue will be discussed below. At zero order we obtain T0⬙ = 0,

⌿IV 0 = 0,

共56a兲

ˆ C = − iK ˆ ␹⌿ , ⌳ 0

共56b兲

⌿0 = 0,

⌿0⬘ = 0,

T0⬘ = 0,

⌿0 = 0,

T0⬘ = 0,

C⬘ = 0,

C⬘ = 0

at

z = 0,

共56c兲

共56d兲

We now consider the limiting case of L = O共B3兲 Ⰶ 1, which should be closely related, at least for LB−3 Ⰷ 1, to case 共ii兲, i.e., the limit ␤ Ⰷ 1 of case I, see Eqs. 共48a兲, 共48b兲, 共49兲, and 共50兲. It is also clear from the previous analysis that the solute concentration drastically changes near the free surface, as follows from Eq. 共30兲 taking into consideration that 兩q兩 ⬃ 冑⍀ Ⰷ 1 at ␤ Ⰷ 1. We begin with the following scalings: B = ⑀4␤ˆ ,

ˆ, ␭ = ⑀ 4⌳

⌿0⬙ = − 48m0iKˆT0,

A. Derivation of the decay rate

L = ⑀12ˆl,

k = ⑀Kˆ,

共53a兲

at

z = 1.

Note that we have neglected the solutocapillary force in Eq. 共56d兲, assuming C共z = 1兲 = O共⑀4兲. The validity of this step will be proved immediately. In this case, the temperature and stream function disturbances are both independent of the concentration field. The solution for the former is given by Eqs. 共25兲 where h is replaced by F: T0 = F,

⌿0 = 12iKˆm0z2共1 − z兲F.

共57兲

The concentration is governed by the algebraic Eq. 共56b兲 and its solution is

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054101-11

C=

Phys. Fluids 21, 054101 共2009兲

Marangoni convection in a binary liquid layer

ˆ 2␹ m 12K 0 2 z 共1 − z兲F. ˆ ⌳

共58兲

ˆ ⫽ 0, as the considered mode can be only oscilNote that ⌳ latory, at least near the marginal stability border. It is obvious that the solution given by Eq. 共58兲 satisfies the boundary condition at the bottom, Eq. 共56c兲, but it is inconsistent with the boundary condition at the free surface, Eq. 共56d兲. Indeed, neglecting the diffusive terms we lower the order of the equation and some boundary conditions become “excessive.” In fact, one has to introduce a thin boundary layer near the free surface in order to satisfy the no-flux condition. This layer is of crucial importance, since even small changes in the concentration near the free surface trigger the solutocapillary effect. Based on this we introduce a “fast” coordinate ␰ near the free surface, assuming

␰ = ⑀−4共1 − z兲,

共59兲

and expand the concentration field in the boundary layer as C = Co共z兲 + ⑀4Ci共␰兲 + ¯ .

共60兲

Here, subscripts o and i denote “outer” and “inner” solutions, respectively. The outer solution is given by Eq. 共58兲, while the inner solution describes fast change in the concentration inside the boundary layer. It is taken into account that in order to satisfy the Neumann boundary condition, one needs to add only a small correction depending on ␰. Indeed, because of the relationship C⬘共z = 1兲 = Co⬘共z = 1兲 −

dCi 共␰ = 0兲 + ¯ , d␰

共61兲

even a small correction to the concentration field changes its gradient by a finite value and allows to satisfy the boundary conditions of zero mass flux. The concentration inside the boundary layer is determined from the following problem: 2

ˆ C = − ˆl d Ci , ⌳ i d␰2

共62a兲

ˆ 2␹ m F 12K dCi 0 , = Co⬘共z = 1兲 = − ˆ d␰ ⌳ dCi → 0, d␰

at

at

␰ = 0,

␰ → ⬁.

共62b兲

共62c兲

The boundary value problem 共62兲 has a solution Ci =

ˆ 2␹ m F 12K 0 e−qˆ␰ , ˆ ˆq⌳

共63兲

ˆ ˆl−1 关cf. Eqs. 共30兲 and 共34兲兴. Thus, one finds the where qˆ2 = −⌳ value of the concentration at the free surface C共z = 1兲 = Co共z = 1兲 + ⑀4Ci共␰ = 0兲 = ⑀4G, where

共64兲

G⬅−

冑

12Kˆ2␹m0 ˆl F. ˆ 兲3/2 共− ⌳

共65兲

Therefore, the disturbances of the concentration are of order ⑀4 at the free surface, which confirms neglecting C in Eq. 共56d兲, see the remarks following Eqs. 共54兲 and 共56兲. It is worth noting that the concentration Co in the core 共outer兲 region satisfies the equations similar to Eq. 共56b兲 ˆ C共n−2兲 + ¯ , ˆ C共n兲 = iKˆ␹⌿ + ⌳ −⌳ 0 o n 2 o

共66兲

in each order in ⑀2 until n = 4, where the diffusive terms start influencing the outer solution. Here, the expansions 2 共1兲 2ˆ ˆ ˆ Co = C共0兲 o + ⑀ Co + ¯, ⌳ = ⌳0 + ⑀ ⌳1 + ¯ are used. Due to nondeformability of the free surface ⌿n共z = 1兲 = 0 for any n, i.e., C共n兲 o 共z = 1兲 = 0 for n = 0, 1, 2, 3 and Co共z = 1兲 = O共⑀8兲. The same expansion in ⑀2 should be written for Ci, but it only corrects Eq. 共64兲, which serves as a zeroorder approximation. We do not use such a formal expansion of Co,i here because there is no need in higher-order terms. Finalizing the discussion of the concentration field we note that the solution C = Co + ⑀4Ci is a particular case of Eq. 共30兲. Indeed, based on Eqs. 共32兲 and 共34兲 we can rewrite Eq. 共30兲 for large 兩q兩 as C0 ⬇

12m␹Kˆ2 2 hz 共1 − z兲 + O共q−2兲 + e.s.t., ⌳

共67兲

in the outer region and C0 ⬇

12m␹Kˆ2 2 h关z 共1 − z兲 + q−1e−q共1−z兲兴 + O共q−2兲 + e.s.t., ⌳ 共68兲

near the free surface. It is obvious that Eq. 共67兲 coincides with Eq. 共58兲 共with h instead of F兲, while Eq. 共68兲 is equivalent to Co + ⑀4Ci with Co and Ci defined by Eqs. 共58兲 and 共63兲, respectively. Indeed, ˆ 共lˆ⑀10兲−1 = −⑀−8qˆ2, i.e., qˆ = ⑀4q, that, along q2 = −⌳l−1 = −⑀2⌳ with the relation h = F, makes these solutions almost identical. Thus, the field of concentration in case II corresponds to the limiting case of “high-frequency” 共兩⌳兩 Ⰷ l兲 in case I. We now proceed to the next order of Eq. 共54兲 with the expansions given by Eq. 共55兲. The problem at first order is T1⬙ = Kˆ2T0 + iKˆ⌿0 ,

共69a兲

ˆ2 ⌿IV 1 = 2K ⌿0⬙ ,

共69b兲

⌿1 = 0,

⌿1⬘ = 0,

T1⬘ = 0,

at

z = 0,

共69c兲

⌿1⬙ = − 48iKˆ关m0共T1 − G兲 + m1T0兴, ⌿1 = 0,

T1⬘ = 0,

共69d兲 at

z = 1.

The solvability condition for Eq. 共69a兲 and the corresponding boundary conditions yields

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054101-12

Phys. Fluids 21, 054101 共2009兲

Shklyaev, Nepomnyashchy, and Oron 4

共70兲

m0 = 1,

i.e., the critical value for the instability in a pure liquid, as it was stated above. This result now does not seem to be unexpected, as Eq. 共57兲 suggests that the temperature disturbances F trigger the fluid flow and, hence, may cause instability of the binary mixture layer. The concentration has only a weak impact on it. It is interesting that the perturbations of concentration are large in comparison with the perturbations of the temperature due to Eq. 共53c兲. However, the opposite is valid at the free surface, where C = O共⑀4兲 in accordance with Eq. 共64兲. The solution of Eq. 共69兲 is T1 = Q − 共−

3 5 5z

4

+z −

1 2 2z

+

1 15

兲K F, ˆ2

2

1

^ m β 0

2 -2

3

共71a兲 -4

⌿1 = − z2共 56 z3 − 2z2 +

4 5

兲iKˆ3F + 12iKˆ

0.5

ˆT , T2⬙ = Kˆ2T1 + iKˆ⌿1 − ⌳ 0

T2⬘ = − ␤ˆ T0,

共72b兲

at

共72c兲

z = 1.

Integrating Eq. 共72a兲 in z over 0 ⱕ z ⱕ 1, and accounting for the boundary conditions, we obtain an additional to Eq. 共65兲 relation coupling F and G ˆ4 ˆ 2G. ˆ F = − ␤ˆ F + m K ˆ 2F − K F − K −⌳ 1 15

␻ = 共72␹2k8L兲1/5 .

共73兲

冑

ˆ4 ˆ ˆ4 ˆ 2 + K − 12K ␹ l . ˆ = ␤ˆ − m K ⌳ 1 15 共− ⌳ ˆ 兲3/2

共74兲

ˆ␤= m

ˆ 兲 for the To obtain the marginal stability curve m1 = m1ⴱ共K ˆ oscillatory mode, we set ⌳ = i⍀. Separation of the real and imaginary parts of Eq. 共74兲 results in 1/5

,

ˆ 8 2ˆ

⍀ = 72K ␹ l . 5

共75兲

Note that this mode exists only for negative values of ␹. Turning back to the unscaled Biot, Marangoni, Lewis numbers, and the wavenumber, we obtain

冋

M ⴱ = 48 1 +

冉 冊册

72␹2L B k2 − + k2 15 k2

1/5

,

共76a兲

冉 冊

ˆ2 1 K + ␤ − ¯AKˆ␤−2/5, Kˆ2 15

2 ¯A = 72␹ L 3 B

␤

1/5

.

共77兲

ˆ 兲 for various values of ¯A ˆ ␤共K Marginal stability curves m ␤ are presented in Fig. 12. It is seen that only one minimum exists for any value of ¯A. This minimum cannot be found analytically except for the limiting cases of small and large ¯A. In case 共iii兲 共i.e., L Ⰶ B3 or ¯A = 0兲, we obtain ˆ ␤c = m or

2

冑15 ,

at

ˆ = 151/4 K ␤c

冉 冑冊

M c = 48 1 + 2

B. Linear stability threshold

共76b兲

ˆ ␤ = m1ⴱ␤ˆ −1/2 Substituting the rescaled Marangoni number m −1/4 ˆ ˆ ˆ and wavenumber K␤ = K␤ into Eq. 共75兲 results in

Substituting the definition of G, Eq. 共65兲, yields the expression for the complex decay rate

冉 冊

2.5

ˆ ␤共Kˆ␤兲 in case II, given by Eq. 共77兲, for FIG. 12. Marginal stability curves m ¯A = 0, 2, and 4 shown by curves 1, 2, and 3, respectively.

共72a兲

z = 0,

72␹2ˆl ␤ˆ Kˆ2 − + m1ⴱ = ˆ 2 15 ˆ2 K K

2

Kβ

共71b兲

which can be obtained from Eq. 共A8兲 in Ref. 23. The last step in derivation of the neutral stability curve is the analysis of the boundary value problem at second order for the temperature field

at

1.5

^

⫻共m1F + Q − G兲z2共1 − z兲,

T2⬘ = 0,

1

B 15

at

kc = 共15B兲1/4 .

共78a兲

共78b兲

This reproduces a known result for the monotonic Marangoni instability in a pure liquid.33,34 The only impact of solute at extremely small Lewis numbers is the emergence of oscillations with the frequency

␻c = 共16200␹2B2L兲1/5

共79兲

关cf. Eq. 共76b兲 with k = kc兴. Note that ␻c → 0 for L → 0, i.e., the oscillatory mode transforms into the monotonic one if there is no diffusion and the boundary layer does not develop near the free surface. In case 共ii兲 共L Ⰷ B3 or ¯A Ⰷ 1兲, the critical wavenumber Kˆ␤c tends to zero, and the second term in Eq. 共77兲 becomes small. Thus, the neutral stability curve becomes the same as that given by Eq. 共48a兲. Using Eq. 共49兲 one obtains

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054101-13

Phys. Fluids 21, 054101 共2009兲

Marangoni convection in a binary liquid layer

4

2

0

1.6

^

^ m βc

Kβc 1

-4

1.2

2

-8

0.8

-12

0.4 0

2

4

6

_

8

10

A ˆ ␤c and waveFIG. 13. Variation of the rescaled critical Marangoni number m ˆ with ¯A = 共72␹2LB−3兲1/5 in case II, shown by curves 1 and 2, number K ␤c respectively. The solid curves represent the numerical results, whereas the dashed curves display the results obtained from the asymptotic limit of large A, Eq. 共80兲.

importance for a further derivation of the amplitude equations, governing the weakly nonlinear dynamics of the perturbation fields. First, we examine case 共ii兲, B = O共L␥兲, where 1 / 3 ⬍ ␥ ⬍ 1, i.e., the case intermediate with respect to cases I and II. As noted above, the critical Marangoni number in case I for ¯ Ⰷ 1兲 given ␤ Ⰷ l given by Eq. 共50兲 and in case II for ˆl Ⰷ ␤ˆ 3共A by Eq. 共81兲 are the same. It thus follows that the two asymptotics match. To reiterate, M = 48+ O关共L / B兲1/4兴, while the critical wavenumber is O关共B5 / L兲1/8兴. Furthermore, note that when considering case I for ␤ Ⰷ l, we set 兩q兩 to be large, which is equivalent to introduction of a boundary layer in accordance with Eq. 共30兲. Case II entirely deals with the diffusive boundary layer near the free surface. Therefore, it is reasonable to expect the emergence of a boundary layer in the intermediate case. Taking into account the ratio between the Biot and Lewis numbers, as well as the critical values of the Marangoni number, the wavenumber and the frequency, see Eq. 共50兲, we scale the variables as L = ⑀8˜l,

冉冊

¯A ˆ ␤c = − 4 m 5

5/4

,

at

冉冊

ˆ = 5 K ␤c ¯A

5/8

.

共80兲

冑

ˆ ␤c兲, kc = ⑀␤ˆ 1/4K␤c, and ␻c = ⑀2⍀c, the As M c = 48共1 + ⑀2 ␤ˆ m critical Marangoni number and the frequency in the limit of ˆl Ⰷ ␤ˆ 3 共or equivalently, L Ⰷ B3兲 are

冋 冉 冊册 冋 册

4 72␹ L M c = 48 1 − 5 5B 2

␻c = 5B

at

kc =

B = ⑀8␥˜␤ ,

˜, k = ⑀5␥−1K

˜, ␻ = ⑀ 8␥⍀

M = M 0 + ⑀2−2␥M 1 + ¯ .

共82a兲 共82b兲 共82c兲

We find that the scaling for the perturbation fields ˜ = C 共z兲 + ⑀4共1−␥兲C 共␰兲, C o i

共83a兲

˜T = ⑀2−2␥共T + ⑀10␥−2T + ⑀8␥T + ¯兲, 0 1 2

共83b兲

˜␺ = ⑀3␥+1共⌿ + ⑀2−2␥⌿ + ¯兲 0 1

共83c兲

1/4

共81a兲

,

共5B兲5 72␹2L

1/8

.

共81b兲

This asymptotics coincides with that given by Eq. 共50兲, i.e., the result for L Ⰶ B. Therefore, Eq. 共81兲 关or Eq. 共50兲兴 provides the intermediate asymptotics, valid for case 共ii兲, L Ⰶ B Ⰶ L1/3, see Sec. V C for details. ˆ 兲 is found numeriˆ ␤共K For finite ¯A, the minimum of m ␤ cally and the variations of both the critical wavenumber Kˆ␤c ˆ with ¯A are depicted in Fig. and the Marangoni number m ␤c

13. It is clearly seen from this figure, that the asymptotics in the limit of large ¯A provides a fairly good approximation even for ¯A = 2. C. Intermediate asymptotics for B š L š B3, case „ii… and L ™ B3, case „iii…

Consider now two intermediate cases, which are the opposite limits of case II. This will complete the analysis, filling all the gaps on the plane L − B at small values of both Lewis and Biot numbers. In addition to the stability boundaries, i.e., solutions of the corresponding eigenvalue problems, we are also interested in the ratio between the amplitudes, i.e., the form of eigenfunctions. The latter is of crucial

is necessary in order to obtain the closed set of equations. Again, ␰ is the fast coordinate inside the boundary layer near the free surface defined as

␰ = ⑀4共␥−1兲共1 − z兲.

共84兲

Note that the scalings, given by Eqs. 共82a兲–共82c兲, 共83a兲– 共83c兲, and 共84兲, coincide with Eqs. 共22兲 and 共23兲 for ␥ = 1 and with Eqs. 共53兲 and 共55兲 for ␥ = 1 / 3, i.e., the intermediate asymptotics matches cases I and II. Substituting these expansions into the boundary value problem 共7兲 and 共8兲 and collecting terms of lower orders, we arrive at the boundary value problems very similar to Eqs. 共56兲, 共62兲, 共69兲, and 共72兲. The only difference is the absence of the right-hand side in Eq. 共69b兲, the term −48m0K2T1 in the corresponding boundary condition, Eq. 共69d兲, and the term K2T1 in the right-hand side of Eq. 共72a兲. The solutions of the corresponding boundary value problem are obtained from Eqs. 共57兲, 共58兲, 共63兲, and 共71兲, respectively. One needs only to neglect the K3-term and Q in Eq. 共71b兲. The solvability condition for Eq. 共72兲 without K2T1 in the right-hand side leads to

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054101-14

Phys. Fluids 21, 054101 共2009兲

Shklyaev, Nepomnyashchy, and Oron 72

80 64

70

80

2

75 56 0

50

0.1

M* 70

0.2

3

0

1

2

3

2 -2

1 (b)

2

3

k

FIG. 14. Marginal stability curves M ⴱ共k兲 of the oscillatory mode for P = 2, ␹ = −0.2, 共a兲 L = 0.001 and 共b兲 L = 0.03. Curves 1, 2, and 3 correspond to B = 0, 10−4, and 0.01, respectively. 共a兲 Curve 4 corresponds to B = 0.1; the dashed curves are obtained from the longwave asymptotics: case I for B = 0 in the figure, and the asymptotics in the limit of small B, finite L, as given by Eqs. 共25兲 and 共27兲 in Ref. 23, for B = 10−4 in the inset.

ˆ F = − ␤ˆ F + m K F − K G, −⌳ 1 2

2

共85兲

with the above-mentioned Eq. 共65兲 coupling F with G. This, in turn, leads to Eq. 共75兲 without the second term in the expression for m1. The critical parameters are given by Eq. 共81兲 after accounting for the scaling 共82兲. We now proceed to case 共iii兲, B = O共L␥兲, with ␥ ⬍ 1 / 3, starting from the following scalings: L = ⑀20¯l, 5␥ ¯

k = ⑀ K,

B = ⑀20␥¯␤ ,

␻=⑀

共86a兲

4+8␥ ¯

⍀,

共86b兲

M = M 0 + ⑀10␥M 1 + ¯ ,

共86c兲

and expanding the perturbation fields as ¯ = C 共z兲 + ⑀4共2−␥兲C 共␰兲, C o i

␰ = ⑀−4共2−␥兲共1 − z兲,

共87a兲

¯T = ⑀4−2␥共T + ⑀10␥T + ⑀20␥T + ⑀4+8␥T + ¯兲, 0 1 2 3

共87b兲

¯␺ = ⑀4+3␥共⌿ + ⑀10␥⌿ + ⑀4−2␥⌿ + ¯兲. 0 1 2

共87c兲

For ␥ = 1 / 3, Eqs. 共86兲 and 共87兲 reduce to Eqs. 共53兲 and 共55兲. Substituting these expansions into Eqs. 共7兲 and 共8兲 and collecting the terms with equal powers in ⑀, we repeat the analysis of Sec. V A for the concentration field and the first three orders of expansion for the temperature and the stream function. However, the concentration G should be omitted now in the boundary condition at the free surface, Eq. 共69d兲. Consequently, one has to set G = 0 in the solution of the first-order problem, Eq. 共71兲, and in the solvability condition of the second-order problem, Eq. 共72兲. This condition for the case L Ⰶ B3 reduces to m1ⴱ =

¯␤ K ¯2 , + ¯ 2 15 K

1

40

1 0

k

(a)

1

2

1

65 60

3 2

kc

50

40

2

3

60 Mc

4

1 3

4

2

1

70 M* 60

4 4

thus, again the standard result for the thermocapillary con2 vection in a pure liquid33,34 is recovered. For ␥ ⬎ 11 , the next correction to the critical Marangoni number is caused by the

-1 χ

-0.5

0

-2

-1.5 (b)

-1 χ

-0.5

0

FIG. 15. Variation of the critical 共a兲 Marangoni number M c and 共b兲 wavenumber kc for the oscillatory mode with ␹ for B = 0, P = 2. Curves 1, 2, 3, and 4 correspond to L = 10−5, 0.001, 0.01, and 0.03, respectively. The solid curves represent the results of numerical computations, whereas the dashed curves are obtained from the conventional long wave approximation, Eq. 共41兲. The dashed-dotted curve corresponds to case I of the long wave analysis.

¯ −2兲1/5. perturbations of the concentration, which is −共72␹2¯lK 2 In the opposite case, ␥ ⬍ 11 , this correction originating from the presence of solute is smaller than the term proportional to ¯ 4, originating from the thermocapillary convection. K Therefore, one can conclude that additional terms with even powers of k should be added in the right-hand side of Eq. 共76兲 for L Ⰶ B3. These terms are the expansion of the Marangoni number in k2 for thermocapillary convection in pure liquid. VI. RESULTS OF COMPUTATIONS FOR THE OSCILLATORY MODE

In the case of the oscillatory mode, there is no explicit expression for the marginal stability curve. However, the problem of finding the critical Marangoni number and the frequency ␻ can be reduced to the solution of a single algebraic equation for ␻. Based on this solution M ⴱ is expressed via ␻. Some of the results of the numerical solution are presented in Figs. 14–16. For L = 0.001, there are two minima of M ⴱ共k兲 at small values of the Biot number, curves 1 and 2 in Fig. 14共a兲. The conventional longwave theory with k = O共B1/4兲 works well for these two values of B, see inset in 80

2

60 55

70 Mc 60

4

4 4

4

3

3

50 -0.08

-0.04

2

kc 1

3 0

1 2

50 1

4

40

0 -2

共88兲

0

-1.5 (a)

-1.5 (a)

-1 χ

-0.5

0

-2

-1.5 (b)

-1 χ

-0.5

0

FIG. 16. Variation of the critical 共a兲 Marangoni number M c and 共b兲 wavenumber kc for the oscillatory mode with ␹. B = 0.01, P = 2. Curves 1, 2, 3, and 4 correspond to L = 10−5, 0.001, 0.01, and 0.03, respectively. The solid curves represent the short-wave minimum, whereas the dashed curves show the minimum at smaller k.

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054101-15

Phys. Fluids 21, 054101 共2009兲

Marangoni convection in a binary liquid layer

Fig. 14共a兲. However, these long wave perturbations do not materialize the global minimum and the critical perturbations belong to another, short-wave minimum. With increase in B, the minimum corresponding to the long wave perturbations disappears. The transformation of the marginal stability curves presented in Fig. 14共a兲 with an increase in the Biot number is qualitatively similar to that of the long wave instability in case I 共cf. Fig. 8兲. However, as clearly seen in Fig. 14共a兲, there is no quantitative agreement: the longwave theory predicts larger values for the critical wavenumber and smaller values for M c. It is not surprising because according to case I, Kc冑L ⬇ 2.09 for L = 0.001, B = 0, and ␹ = −0.2, i.e., for this parameter set the minimum is far away from the range of applicability of the longwave approximation. For larger values of the Lewis number, the qualitative behavior of the neutral stability curves is similar, as shown in Fig. 14共b兲. However, in this case the mode with smaller values of k can provide the global minimum of the marginal stability curve, see curves 1 and 2. The results of minimization of the Marangoni number with respect to k are presented in Fig. 15 for a thermally insulated free surface, B = 0. The competition between two types of perturbations takes place at sufficiently small values of 兩␹兩, the long wave mode is critical only for small values of 兩␹兩. It is seen that with a decrease in the Lewis number the curves M c共␹兲 tend to the asymptotic curve typical for case I of the long wave instability: L = 0.001 is not small enough for a good agreement of the results, whereas for L = 10−5 the numerical results are close to the asymptotical law. Note that for L = 10−5, the physical limit is used in order to compute M c共␹兲 similar to that described in Sec. IV A. We neglect the exponentially small terms for the concentration perturbations and introduce the diffusive boundary layer near the free surface. For B = 0.01 the variations of M c共␹兲 and kc共␹兲 shown in Fig. 16 are quite similar to the case B = 0. The only distinction is that for small L, the marginal stability curve becomes unimodal and no competition between two modes takes place. However, this does not suggest the disappearance of the long wave instability, in fact, there is a continuous growth of the critical wavenumber with a transition from the longwave mode to the short-wave one. For example, at L = 0.01 and small values of 兩␹兩, the variation of M c共␹兲 is close to the longwave mode, as seen in the inset of Fig. 16. It should be also noted that within a very thin interval of ␹, 兩␹兩 = O共L2兲, the monotonic long wave mode becomes critical. The corresponding stability boundary is defined by Eq. 共21兲. On the scale of Figs. 15 and 16, these curves appear as vertical lines M c ⱖ 48 at ␹ = 0. Figure 17 presents the curves separating between the long wave and short-wave instabilities in the ␹ − L plane. Above these lines the global minimum of the neutral stability curve belongs to the minimum with small values of k, whereas below them the perturbations with larger k are critical. It should be noted, that this classification is quite relative. In fact, the former case corresponds to the conventional mode with k = O共B1/4兲, while the latter, at least for L ⱕ 10−5, corresponds to the novel mode with k = O共冑L兲, case 共i兲.

0

-0.1

Long Waves -0.2

χ -0.3

Short Waves -0.4

-0.5 0

0.02

0.04

0.06

0.08

L FIG. 17. The boundary between the short-wave and longwave instabilities for P = 2. The solid and dashed curves correspond to B = 0 and B = 0.01, respectively. The conventional mode with k = O共B1/4兲 is critical above the curves.

Moreover, these curves have end points, where the confluence of the two minima with a disappearance of one of them takes place: for smaller values of 兩␹兩, the neutral stability curves are unimodal. As the values of the Lewis number are sufficiently small for the majority of binary mixtures, L ⱕ 0.01, the conventional long wave theory with k = O共B1/4兲, as in Refs. 23 and 25, can be applied only for small absolute values of the Soret number. VII. SUMMARY AND CONCLUSIONS

A linear theory of thermocapillary and solutocapillary convection in a layer of binary mixture is developed here. The paper focuses on the case of small values of L typical for the majority of binary liquids. Long wave instability is studied in detail in the limit of asymptotically small Lewis and Biot numbers. For finite but sufficiently small values of these parameters, a numerical investigation is carried out. A long wave monotonic mode sets in for positive values of the Soret number ␹, it is characterized by small critical Marangoni numbers, given by Eq. 共17a兲, i.e., M c ⬇ 48␹L−1. This value corresponds to the standard stability threshold22 evaluated for pure solutocapillary convection under the concentration gradient ␹. Thermocapillary effect provides only a small correction to the critical Marangoni number which, in turn, determines the critical wavenumber. According to Eq. 共17兲 the latter depends on the ratio Bⴱ / B, or L / B, and can be either zero for B ⬎ Bⴱ or proportional to 共BL兲1/4 for B Ⰶ 1. This equation also matches the known asymptotics considered in Ref. 23, i.e., B Ⰶ 1, L = O共1兲, and in Ref. 27, i.e., L = O共1兲, B = O共1兲. For small 兩␹兩, the critical Marangoni number and wavenumber for the monotonic mode are expressed by Eq. 共21兲 which is valid for ␹ ⬎ −L, but the corresponding value of M c is larger than 48. The results of numerical evaluation show similar effects; namely, for small and finite values of the Biot number,

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054101-16

Phys. Fluids 21, 054101 共2009兲

Shklyaev, Nepomnyashchy, and Oron

TABLE I. Summary of the results for oscillatory instability in various subdomains of small values of the Lewis and Biot numbers. Case 共i兲, L Ⰷ B mⴱ共k兲 ␻共k兲

Eq. 共40兲 Eq. 共40兲

mc

Fig. 11共a兲

kc ␻c

Fig. 11共b兲 Fig. 11共c兲

Case 共ii兲, B Ⰷ L Ⰷ B3 1 + k2 − 共 k2 兲 共72␹2k8L兲1/5

72␹2L 1/5

B

1− 5共

4 72␹2L

兲

1/4 5B 共5B兲5 1/8

关 72␹ L 兴 2

5B

ln B

ln B

ln B

Case 共iii兲, B3 Ⰷ L B

k2

1 + k2 + 15 共72␹2k8L兲1/5

ln L

1 + 2冑 15 B

共15B兲1/4 共16 200␹2B2L兲1/5

(a)

B ⬎ Bⴱ, the longwave perturbations are critical, whereas for B ⬍ Bⴱ, the instability is caused by short-wave disturbances. Moreover, the agreement between the numerical and analytical results is excellent, namely, even for L = 0.01, the discrepancy in the critical values of the Marangoni number is less than 1%, as shown in Fig. 2. The results for the longwave oscillatory mode, which takes place at ␹ ⬍ 0, are summarized in Table I and in Fig. 18. There exist three different regions in the plane L − B, namely: case 共i兲: 1 Ⰷ L Ⰷ B; case 共ii兲: B Ⰷ L Ⰷ B3; case 共iii兲: L Ⰶ B3, and two “transition” zones located near the lines L = B 共case I兲 and L = B3 共case II兲. Case 共i兲 corresponds to the limiting case ␤ = 0 of case I, see Secs. IV B and IV C. The novel long wave oscillatory mode with the critical wavenumber proportional to 冑L is revealed. The critical value of the Marangoni number is below 48 and independent of both B and L; the frequency of the critical perturbations is proportional to L. The variations of mc = M c / 48, Kc = kc / 冑L, and ⍀c = ␻c / L with the Soret number ␹ are presented by the solid curves 共␤ = 0兲 in Figs. 11共a兲–11共c兲, respectively. It is important that the conventional longwave mode with k = O共B1/4兲, studied in Refs. 23 and 25, is not critical for asymptotically small L. Furthermore, a novel long wave instability sets in for any negative value of ␹, cf. the condition −1 ⬍ ␹ ⬍ 0 for the conventional mode.23 Case 共ii兲 corresponds to the formation of the diffusive boundary layer near the free surface. The critical parameters for this mode are given by Eq. 共50兲: note that M c is slightly lower than 48, which represents the value of the critical Marangoni number for the monotonic mode in a pure liquid. The mechanism of the oscillatory instability emerging in this case can be explained as follows. Because the Marangoni number is less than 48, an initial disturbance triggers a slowly decaying thermocapillary flow. The solute concentration at this stage behaves as a passive scalar advected by the flow. This advection leads to accumulation of the solute near the stagnation point at the free surface and to the slow diffusion of the solute directly to the surface. This, in turn, gives rise to an intensive solutocapillary flow sweeping the spot of a high solute concentration. Recall that the concentration gradient is potentially stable for negative ␹. At the last stage, the pure thermocapillary decay of the flow occurs and the cycle is repeated.

(b)

(c)

FIG. 18. Variation of 共a兲 ln共1 − mc兲, 共b兲 ln kc, and 共c兲 ln ␻c with ln B and ln L in the limit of small L and B. The three intermediate regions are separated by the straight lines, corresponding to case I 共the upper lines兲 and case II 共the lower lines兲. The white and black shades correspond, respectively, to the maximal and minimal values of the corresponding function.

Case 共iii兲 has even more similarities with the thermocapillary convection in a pure liquid: M c and kc are given by Eq. 共78b兲 found in Refs. 33 and 34. However, for the binary mixture this mode is oscillatory with the frequency given by Eq. 共79兲. A very narrow domain near L = B 共case I兲 matches the cases 共i兲 and 共ii兲. This limiting case is studied in detail in Sec. IV. The stability boundaries are determined by the transcendental complex-valued Eq. 共40兲 and the results of the numerical and analytical solutions of this equation are presented in Figs. 8–11. The area in the vicinity of the line L = B3 共case II兲 matches cases 共ii兲 and 共iii兲 and is analyzed in Sec. V. The marginal stability boundary for this case is given by Eq. 共76兲. Typical neutral curves are presented in Fig. 12 and the results of numerical minimization of M ⴱ共k兲 are shown in Fig. 13. It is worth noting that for L Ⰶ 1, the critical value of the Marangoni number is below 48, i.e., for negative values of the Soret number, oscillatory convection is dominant. The results of numerical computations for the shortwave oscillatory instability are qualitatively similar to case 共i兲 of the long wave analysis. However, a quantitative agreement takes place only at L ⱕ 10−5. Finally, a numerical study of the short-wave convection provides the domain of parameters, where the conventional mode with k = O共B1/4兲 becomes critical. For typical values of the Lewis number, L ⱕ 0.01, this mode causes the instability only for small values of 兩␹兩, as shown in Fig. 17. Note that for the conventional mode M c ⬎ 48, and the competition between the monotonic and oscillatory modes takes place for negative ␹ near zero, 兩␹兩 = O共L2兲. ACKNOWLEDGMENTS

This work was partially supported by joint grants of the Israel Ministry of Sciences 共Grant No. 3-5799兲 and the Russian Foundation for Basic Research 共Grant No. 09-01-92472兲, and by the European Union via the FP7 Marie Curie scheme 关Grant No. PITN-GA-2008-214919 共MULTIFLOW兲兴. S.S. is partially supported by the Foundation “Perm Hydrodynamics.” A.O. acknowledges the partial support of the Fund for Promotion of Research at the

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054101-17

Phys. Fluids 21, 054101 共2009兲

Marangoni convection in a binary liquid layer

Technion and of the Technion President and Technion Research Vice-President Funds. A.A.N. is supported by the Israel Ministry of Science through the Grant No. 3-3570. APPENDIX: CASE I, LARGE Ω

First, we consider Eq. 共40兲 in the limiting case of large ⍀. In this case one can use the asymptotics of large q, Eq. 共38兲, which leads to

冋

mⴱK2 = 共K2 + ␤ − i⍀兲 1 +

册

6冑2␹mⴱK2 共1 − i兲 . ⍀3/2

共␤ + K 兲 =

m ⴱK 2 =

⍀5/2

6冑2兩␹兩

−⍀ ,

共A2a兲

2

⍀5/2

6冑2兩␹兩共␤ + K2 + ⍀兲

共A2b兲

.

These equations, following an elimination of K from Eq. 共A2b兲, serve as a parametric representation for the neutral stability curve mⴱ共K兲. A further simplification of Eq. 共A2兲 is necessary. Indeed, while using Eq. 共38兲 we have neglected the term q−1 ⬀ ⍀−1/2 in comparison with unity. Thus, the same accuracy should be retained in Eq. 共A2兲. We now refine Eq. 共A2兲 in several particular cases, starting from the asymptotics of large ␤ + K2. Neglecting the second term in the right-hand side of Eq. 共A2a兲 and substituting ⍀ in Eq. 共A2b兲, we obtain ⍀ = 关72␹2共␤ + K2兲4兴1/5 , mⴱ =

1 + ␤K−2 1+

共 ␤72+K␹ 兲1/5 2

2

冋 冉 冊册

72␹2 ⬇ 共1 + ␤K 兲 1 − ␤ + K2 −2

再

mⴱK2 = 共␤ + K2 − i⍀兲 1 −

共A1兲

Here we used the relation i−1共−i兲−1/2 = 共1 − i兲 / 冑2. Separating between the real and the imaginary parts we obtain after some algebraic simplifications 2 2

discrepancy is obvious: the neglected terms O共⍀−1/2兲 remain sufficiently large even at K = 104. It has been mentioned above that Eq. 共A2兲 is not rigorous, we have neglected some terms of order of ⍀−1/2, but retained the same in the righthand parts. Nevertheless, the “bad approximation” provides a better agreement than the “correct” asymptotics, Eq. 共A4兲, does. Using the physical limit for estimation of f共q兲, with Eq. 共39兲 instead of Eq. 共38兲, we obtain

共A3a兲

冉 冊

共A5兲

instead of Eq. 共A1兲. It leads to more cumbersome expressions than Eq. 共A2兲 but provides the results with an exponentially small error; the corresponding curves in Fig. 9 coincide with the results obtained numerically. A similar situation is found at large ␤. A comparison of the approximations given by Eqs. 共45兲 with the numerical results is shown in Fig. 6. Again, there is no good agreement between the results, as the expansion in power series in ␤−1/4 works well only at extremely large values of ␤. Moreover, a poor approximation Eq. 共A2兲 gives a better agreement than the valid asymptotics. For example, the dashed and solid curves 3 in Fig. 6 are indistinguishable. The physical limit agrees very well with the results of numerical computations. Finally, we consider the limiting case of large 兩␹兩. It seems to be out of interest from physical point of view, but can be easily obtained from Eq. 共A2兲. Assuming that 冑⍀ ⬀ 兩␹兩 Ⰷ 1, we introduce the auxiliary variable ␮ = 冑⍀ / 共6冑2兩␹兩兲. Neglecting ␤ assumed to be finite in addition with an assumption of large K, ⍀, and 兩␹兩, we obtain K = 6冑2兩␹兩␮共␮ − 1兲1/4 ,

1/5

共A6a兲

.

The second term in the square brackets presents a small correction to unity, but this term has to be retained. Neglecting it results in the neutral stability curve with the minimum at K → ⬁. 共Note that this term is proportional to ⍀−1/4, i.e., it can be retained while ⍀−1/2 is disregarded.兲 First, Eq. 共A3兲 provides the asymptotics of large values of K, K2 Ⰷ ␤:

72␹2 mⴱ = 1 − K2

冎

⫻关− i⍀ − 2共1 − i兲冑2⍀ + 6兴 ,

mⴱ =

共A3b兲

⍀ = 共72␹2K8兲1/5 ,

12mⴱ␹K2 共− i⍀兲5/2

共A4a兲

␮

␮ − 1 + 冑␮ − 1

共A6b兲

.

Minimization of the expression for m0 yields mc =

4 + 2冑2

3 + 2 冑2 + 冑3 + 2 冑2

⬇ 0.8284,

Kc ⬇ 90.03兩␹兩. 共A7兲

These results are shown by the dashed curves in Fig. 11. 1

1/5

.

共A4b兲

Therefore, at any set of parameters, the neutral stability curve tends to the limiting value mⴱ = 1 from below, i.e., the critical perturbations have a finite value of K. The approximate law given by Eq. 共A4兲 is compared with the numerical results in Fig. 9. One can readily see that the numerical results and analytical expressions display a qualitative agreement only. The explanation of such a large

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Phys. Fluids 21, 054101 共2009兲

Shklyaev, Nepomnyashchy, and Oron

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