Influence of lateral boundaries on the Eckhaus instability in anisotropic pattern-forming systems
type: Item
instance of: scholarly article
title: Influence of lateral boundaries on the Eckhaus instability in anisotropic pattern-forming systems
zbMATH Open document ID: 0646.76057
DOI: 10.1016/0167-2789(87)90016-9
author: Eberhard Bodenschatz
author: Lorenz Kramer
published in: Physica D
publication date: 1987-01-01T00:00:00Z
review text: This paper examines the influence of lateral boundaries upon the existence and stability of roll type instabilities in two-dimensional systems having axial symmetry, the lateral boundaries being perpendicular to the axis of the rolls. A weakly nonlinear analysis, incorporating slow space and time variables, is used to investigate stationary solutions of the amplitude equation which arises from a solvability condition. Solutions are shown to exist only when \(\epsilon \equiv (R-R_ c)/R_ c\) exceeds a certain critical value \(\epsilon_ c\), where \(R_ c\) is the linear threshold value. Both analytical and numerical methods are employed in performing a linear stability analysis of the nonlinear solution. It is found that deviations from the universal Eckhaus ratio of 1/\(\sqrt{3}\), due to the finite width of the system, are small. However the presence of lateral boundaries gives rise to an increase in the modulation wave number of the fastest growing linear mode and this effect can be considerable as the ratio \(\epsilon_ c/\epsilon\) becomes large. In concluding, a quantitative comparison is made with results concerning electrohydrodynamic instabilities in nematic liquid crystals. This paper should be of interest to researchers concerned with nonlinear convective instabilities in anisotropic systems.
Mathematics Subject Classification ID: 76E15
Mathematics Subject Classification ID: 76E25
Mathematics Subject Classification ID: 76E30
zbMATH DE Number: 4055241
zbMATH Keywords: lateral boundaries
zbMATH Keywords: existence
zbMATH Keywords: stability of roll type instabilities
zbMATH Keywords: two- dimensional systems
zbMATH Keywords: weakly nonlinear analysis
zbMATH Keywords: stationary solutions
zbMATH Keywords: solvability condition
reviewed by: P. J. Barratt
MaRDI profile type: Publication
full work available at URL: 0167-2789(87)90016-9
OpenAlex ID: W2056672037
cites work: Q5330852
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cites work: On Thermal Convection in a Large Box
cites work: Q5558293
description: scientific article; zbMATH DE number 4055241