Nonlinear convection in a porous layer with finite conducting boundaries

Abstract

The problem of finite-amplitude thermal convection in a porous layer with finite conducting boundaries is investigated. The nonlinear problem of three-dimensional convection is solved by expanding the dependent variables in terms of powers of the amplitude of convection. The preferred mode of convection is determined by a stability analysis in which arbitrary infinitesimal disturbances are superimposed on the steady solutions. Square-flow-pattern convection is found to be preferred in a bounded region [Gcy ] in the (γ b , γ t )-space, where γ b and γ t are the ratios of the thermal conductivities of the lower and upper boundaries to that of the fluid. Two-dimensional rolls are found to be the preferred pattern outside [Gcy ]. The qualitative features of the convection problem appear to be essentially symmetric with respect to γ b and γ t . The dependence of the heat transported by convection on γ b and γ t is computed for the various solutions analysed in the paper.