Long-wave Marangoni convection in a thin film heated from below

Abstract

We consider long-wave Marangoni convection in a liquid layer atop a substrate of low thermal conductivity, heated from below. We demonstrate that the critical perturbations are materialized at the wave number K ∼ √ Bi, where Bi is the Biot number which characterizes the weak heat flux from the free surface. In addition to the conventional monotonic mode, a novel oscillatory mode is found. Applying the K ∼ √ Bi scaling, we derive a new set of amplitude equations. Pattern selection on square and hexagonal lattices shows that supercritical branching is possible. A large variety of stable patterns is found for both modes of instability. Finite-amplitude one-dimensional solutions of the set, corresponding to either steady or traveling rolls, are studied numerically; a complicated sequence of bifurcations is found in the former case. The emergence of an oscillatory mode in the case of heating from below and stable patterns with finite-amplitude surface deformation are shown in this system for the first time.

Oscillatory Longwave Marangoni Convection in a Binary Liquid. Part 2: Square Patterns

“…In contrast to [12] and several other papers [9,10,11,15,19,20] where longwave oscillatory convection is described by a pair of partial differential amplitude equations, the nonlinear dynamics for the thermocapillary and solutocapillary convection in a layer of binary mixture with a nondeformable free surface is governed by the solvability conditions for a certain set of linear nonhomogeneous equations [22]. In other words, the nonlinear dynamics is described by a set of nonlocal amplitude equations.…”

Oscillatory thermocapillary instability of a film heated by a thick substrate

“…Multi-layer film configurations do not, however, guarantee oscillatory modes: for example, such instabilities were not obtained by Pototsky et al (2005) who investigated the dewetting dynamics of isothermal, ultrathin bilayers. Of particular interest to the present work are oscillatory instabilities reported by Shklyaev et al (2012) in a model of thin-film thermocapillary destabilization from below. While there are similarities between that work and the present, we point out one important difference: in Shklyaev et al (2012), the instability is driven by imposing a heat flux at the film-substrate interface; instead, in the present work we consider the full time-dependent heat-transfer in the substrate.…”

Controlling of longwave oscillatory Marangoni patterns on a rhombic lattice

“…We have recently considered the influence of the feedback control on the oscillatory Marangoni instability in a thin film heated from below. We have shown that a linear control gain can delay the onset of instability [3] and a quadratic control gain can eliminate the subcritical excitation of instability [4]. The analysis of pattern formation was done for an infinite region, nonlinear interaction of the traveling waves was considered.…”

“…2. There we present a set of coupled amplitude equations which governs the evolution of the layer thickness and the temperature deviation under nonlinear feedback control [4]. In Sec.…”