Skew-Varicose Instability in Two Dimensional Generalized Swift-Hohenberg Equations

TL;DR

This paper investigates the stability of stripe patterns in generalized Swift-Hohenberg equations with mean flow coupling, deriving analytical stability boundaries and analyzing how boundary conditions and parameters affect various instabilities.

Contribution

It introduces a comprehensive analytical and numerical stability analysis of stripe patterns in generalized Swift-Hohenberg models including mean flow effects, extending understanding of instability boundaries.

Findings

Skew-varicose instability boundaries are analytically derived.

Mean flow coupling influences the stability regions and crossover behavior.

Large mean flow coupling eliminates stable stripe regions via cross-roll instability.

Abstract

We apply analytical and numerical methods to study the linear stability of stripe patterns in two generalizations of the two-dimensional Swift-Hohenberg equation that include coupling to a mean flow. A projection operator is included in our models to allow exact stripe solutions. In the generalized models, stripes become unstable to the skew-varicose, oscillatory skew-varicose and cross-roll instabilities, in addition to the usual Eckhaus and zigzag instabilities. We analytically derive stability boundaries for the skew-varicose instability in various cases, including several asymptotic limits. We also use numerical techniques to determine eigenvalues and hence stability boundaries of other instabilities. We extend our analysis to both stress-free and no-slip boundary conditions and we note a cross over from the behaviour characteristic of no-slip to that of stress-free boundaries as the coupling to the mean flow increases or as the Prandtl number decreases. Close to the critical value of the bifurcation parameter, the skew varicose instability has the same curvature as the Eckhaus instability provided the coupling to the mean flow is greater than a critical value. The region of stable stripes is completely eliminated by the cross-roll instability for large coupling to the mean flow.