Oscillatory Longwave Marangoni Convection in a Binary Liquid: Rhombic Patterns

Abstract

Longwave oscillatory Marangoni convection in a planar binary liquid layer is studied. Finite-amplitude regimes with perturbations of temperature and solute concentration of order unity are considered. Both thermocapillary and solutocapillary effects are taken into account. One-and two-dimensional patterns on a rhombic lattice are studied and their stability to the perturbations which do not necessarily belong to the lattice is analyzed. In the framework of the two-dimensional problem, traveling rolls are stable, whereas for a full three-dimensional problem alternating rolls are selected.

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

“…Dynamical system. We refer to [22,24] where the mathematical problem is formulated and a set of nonlocal amplitude equations is derived. These nonlocal equations were reduced to a dynamical system for the Fourier amplitudes without any prescribed symmetry of solution [24].…”

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php amplitudes evolve on a "slow" time scale τ = √ BT . (The Biot number B defined in [24], see (2.8) there, is assumed to be small.) The leading, uniform across the layer, components of the temperature F and solute concentration G = F − h can be easily expressed via A nj ; see (2.18) in [24].…”

“…(The Biot number B defined in [24], see (2.8) there, is assumed to be small.) The leading, uniform across the layer, components of the temperature F and solute concentration G = F − h can be easily expressed via A nj ; see (2.18) in [24]. All fields, h, F , and G, are assumed to be 2π/k-periodic in both directions X and Y in the layer plane.…”

“…The analysis of finite-amplitude solutions started in [24], where the nonlocal set of amplitude equations is rewritten as a dynamical system for the Fourier amplitudes. Two simplest cases of finite-amplitude perturbations, namely, one-dimensional regimes (two-dimensional convective flows) and patterns emerging on a rhombic (in the Fourier space) lattice, were studied there.…”

“…However, the analysis of periodic patterns for an infinite layer is incomplete and external perturbations breaking the lattice symmetry have to be investigated. This yields the so-called Busse balloons [1,2], although the mathematics used there is different from the method developed in [24]. The purpose of the present paper is to develop a natural extension for the analysis presented in [24] and to consider patterns emerging on a square lattice and their stability; for the square lattice the situation is more complicated because of multiple resonant interactions involving an infinite sequence of the Fourier harmonics.…”