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The effects of sidewall heat loss on convection in a saturated porous vertical slab

The onset of natural convection is considered in a vertically oriented, thin, finite slab of saturated porous media when sidewall heat transfer exists. First, a linear stability analysis is carried out for a system with impermeable boundaries. The sidewall temperature increases linearly with depth while the smaller-area endwalls are insulated. Convection occurs when the Rayleigh number R is asymptotically large relative to the inverse square of the horizontal aspect ratio, H2 [Lt ] 1. The convection pattern is composed of an integer number of vertically oriented three-dimensional, finger-like cells. The wavelength of each cell, relative to the larger horizontal dimension of the slab, is proportional to H2½. This somewhat surprising type of modal configuration is also found when there is a specified vertical mass flux through the slab. In this second example one considers the characteristics of the 3-dimensional fully developed solution for the thin vertical-slab problem which is compatible with a linear temperature increase on the vertical walls. When R is like that found in the first problem, closely spaced finger-like cells are found superimposed on the generally upward fluid flow. It is concluded that sidewall heat loss has a very strong stabilizing effect on the initiation of buoyancy-induced convection relative to the more traditional situation where side- and endwalls are insulated. Furthermore the appearance of slender, finger-like convection cells is characteristic of motion in a narrow vertical-slab configuration. Finally it is noted that the precise modal configuration selected by a system is extremely sensitive to the value of the Rayleigh number.