Repulsion of Eigenvalues in the Rayleigh–Bénard Problem

Abstract

The mathematical origin of the repulsion of eigenvalues in the Rayleigh–Benard problem in a rectangular cavity is explored by considering a partially nonslip boundary condition at the side walls. It is known that the eigenvalues, i.e., the neutral Rayleigh numbers, intersect each other at critical aspect ratios under the stress-free boundary condition at the side walls when the eigenvalues are plotted against the aspect ratio of the cavity, but repulse each other under the nonslip boundary condition. The linear eigenvalue problem for the occurrence condition of convection is solved by double expansions in the deviation of the aspect ratio from one of the critical values and the nonslip parameter. It is found that the repulsion of eigenvalues arises from a structural instability of the transform of matrices into a Jordan canonical form, and the magnitude of the gap between two eigenvalues is evaluated.