On the Effect of Sidewalls in Cellular Convection

Abstract

Much progress in understanding the stability of continuous systems has come from intensive investigation of the prototype situations of an infinite layer of fluid heated from below (Bénard problem) and of fluid contained between concentric infinite rotating cylinders (Taylor problem) (see the survey by Segel [1]). Implicit in these investigations of unbounded regions of fluid is the assumption that the results are meaningful for sufficiently broad convecting layers or sufficiently long rotating cylinder apparatuses. The present paper reports on work which begins the tasks of verifying this assumption and estimating the effects of bounding walls. The approach is to introduce slow spatial modulation of the solutions derived from the nonlinear stability theory of the unbounded region. Although illustrated on the Bénard problem, it should be clear that the analysis is of general applicability. The reader should consult the paper by Newell in this volume for a report on closely related work and should refer to a paper of the author [2] for a number of details of the basic analysis which are omitted here.