Longwave Marangoni convection in a binary liquid layer heated from above: weakly nonlinear analysis

Abstract

Nonlinear regime of longwave surface-tension-driven convection in a layer of binary mixture heated from above is considered. Under the assumption of the small Biot number, which corresponds to the almost heat insulated free surface, we derive the nonlocal amplitude equation. Analysis of pattern selection demonstrates that hexagons emerge subcritically and up-hexagons are stable within the entire domain of their existence. Squares become stable if the absolute value of the Marangoni number exceeds a certain value.

Longwave convection in a layer of binary mixture with modulated heat flux: weakly nonlinear analysis

“…For small b the effect of the heat modulation is nontrivial as well: at the zeroth order with respect to b the coefficients of (32) are reduced to (43)-( 45) at b = 0. Thus, Landau equations (32) coincide with the equations derived by Shklyaev and Nepomnyashchy (2014) and differ from those found by Oron and Nepomnyashchy (2004) in the absence of modulation; although α, γ, and Q H are the same in both papers, the coefficients of nonlinear interaction K 0 and K kq differ. This difference stems from the different mathematics: at b = 0 at the stability boundary ∂ = τ 0 and the operators  1 and  2 contain 2 only, therefore, * m 0 does not depend on K at all and at the stability boundary  1 and  2 coincide up to a constant factor. Hence, the solvability condition for (30) and ( 31) at b = 0 provides the local partial differential equation:…”

“…To summarize, in both limits, ≪ K 1 and ≪ b 1, amplitude equations (32) with the coefficients given by ( 43)-( 45) govern the nonlinear dynamics; the only difference is that the parameter b has to be taken equal to zero in the corresponding expressions in the latter case. As we have already mentioned, the pattern selection within this equation has been studied by Shklyaev and Nepomnyashchy (2014). In fact, most of the those results are the consequence of (33) only and they remain valid even for finite K and b unless the self-interaction coefficient K 0 becomes negative.…”

“…forms the hexagonal lattice. Below we analyze the superlattice which comprises six pairs of the base vectors tilted by the angle π 6 one to another, see figure 3(a) in Shklyaev and Nepomnyashchy (2014); this superlattice combines two hexagonal lattices with the base vectors tilted by π 2 to each other. Recall that dealing with (32), we mainly keep in mind the confined layer with the sidewalls adiabatic and impermeable for the solute.…”

“…Since we are not aware of the binary mixtures where the condition * χ χ < is met, stabilization is the effect to be expected. At b = 0, Landau equation ( 32) with the coefficients of nonlinear interaction given by ( 43)-( 45) and ( 33) was derived by Shklyaev and Nepomnyashchy (2014) in the limiting case…”

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

“…Here we use the term in its original meaning, "equation(s) lacking a simple relation between a function and its time and/or spatial derivatives of a low order." A particular class of problems, where the solvability conditions cannot be reduced to ordinary or partial differential equations, is wide; see [7,13,14,21,22].…”