Long-wave Marangoni instability in a binary-liquid layer with deformable interface in the presence of Soret effect: Linear theory

Abstract

We investigate the long-wave Marangoni instability in a binary-liquid layer in the limit of a small Biot number B. The surface deformation and the Soret effect are both taken into account. It is shown that the problem is characterized by two distinct asymptotic limits for the disturbance wave number k, k∼B1∕4 and k∼B1∕2, which are caused by the action of two instability mechanisms, namely, the thermocapillary and solutocapillary effects. The asymptotic limit of k∼B1∕2 is novel and is not known for pure liquids. A diversity of instability modes is revealed. Specifically, a new long-wave oscillatory mode is found for sufficiently small values of the Galileo number.

Long-Wave Coupled Marangoni - Rayleigh Instability in a Binary Liquid Layer in the Presence of the Soret Effect

“…A comparison of the behavior of the instability thresholds in both long-wave regions k ∼ Bi 1/4 and k ∼ Bi 1/2 with the corresponding results obtained in the limit of pure Marangoni instability [15], [18] shows qualitative differences in the case of negative f , as shown in Fig. 5 in the limit k ∼ Bi 1/4 , and in Fig.…”

“…In the case of positive f = 1 + B, the neutral curves shown in Fig. 4, are similar to those presented in Fig.1 [15] in the limit k ∼ Bi 1/4 and the neutral curves shown in Fig.6 reproduce the results in the limit k ∼ Bi 1/2 presented in Sec.IV A, [18].…”

“…As noted above in contradistinction with the case of a pure fluid, where the consideration of the limit k = O(B 1/4 ) was sufficient for finding the critical Marangoni number at Bi 1, the Marangoni convection in a binary fluid is characterized by two distinct asymptotic limits [18]. Linear stability analysis, carried out in [18] shows that in the region k = O(Bi 1/4 ), the monotonic neutral curve is given by…”

“…This limit was also studied for pure liquids [17], [14], [20], [7]. In addition to the standard long-wave regime k = O(Bi 1/4 ), a novel long-wave asymptotic domain k = O(Bi 1/2 ), special for binary fluids was found and investigated [18]. Therefore, in order to obtain the complete neutral curve, we consider both of these limits.…”

“…Linear stability analysis, carried out in [18] shows that in the region k = O(Bi 1/4 ), the monotonic neutral curve is given by…”