Phys. Rev. E 84 (2011) 036201.
Skew-varicose instability in two dimensional generalized Swift-Hohenberg
Department of Applied Mathematics,
University of Leeds, Leeds LS2 9JT, UK
We apply analytical and numerical methods to study the linear stability of
stripe patterns in two generalizations of the two-dimensional SwiftHohenberg
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.
preprint version of this paper (1MB)