The effect of rigid sidewalls on nonlinear two-dimensional Bénard convection

Abstract

Consider a rectangular box filled with a fluid and heated from below. A two‐dimensional model is used to describe the effects of rigid sidewalls. For some aspect ratios the linear theory cannot predict which mode is preferred; thus a study of the bifurcating solutions of the nonlinear problem around these special ratios is made. The stability of the steady solutions is discussed when the competing modes are either the one and two roll modes or the three and four roll modes. In the first case as the Rayleigh number is increased, the flow prefers either the one roll mode for almost all Prandtl numbers or a mixed mode for a small range of low Prandtl numbers. In the second the flow always prefers the four roll mode.