Weliwita J A, Rucklidge A M, Tobias S M
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Sep;84(3 Pt 2):036201. doi: 10.1103/PhysRevE.84.036201. Epub 2011 Sep 6.
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 crossover from the behavior 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.
我们应用解析和数值方法来研究二维Swift-Hohenberg方程的两种推广形式中条纹图案的线性稳定性，这两种推广形式包括与平均流的耦合。我们的模型中包含一个投影算子，以允许存在精确的条纹解。在广义模型中，除了通常的埃克豪斯（Eckhaus）和锯齿形不稳定性外，条纹还会对斜曲张、振荡斜曲张和交叉滚动不稳定性变得不稳定。我们在各种情况下，包括几个渐近极限，解析推导了斜曲张不稳定性的稳定性边界。我们还使用数值技术来确定特征值，从而确定其他不稳定性的稳定性边界。我们将分析扩展到无应力和无滑移边界条件，并注意到随着与平均流的耦合增加或普朗特数减小，从无滑移边界的行为特征到无应力边界的行为特征会出现转变。接近分岔参数的临界值时，若与平均流的耦合大于临界值，则斜曲张不稳定性与埃克豪斯不稳定性具有相同的曲率。对于与平均流的大耦合，交叉滚动不稳定性会完全消除稳定条纹区域。
