Effects of variable fluid properties and boundary conditions on thermal convection

The onset of convection in a fluid layer heated from below is considered. The fluid properties are allowed to vary with temperature according to a linear relation. The fluid layer is bounded by two horizontal rigid plates, and the lower boundary is considered a perfect thermal conductor whereas the upper one is taken with a finite thermal conductivity. With these asymmetric boundary conditions linear variations in density, thermal conductivity and specific heat yield first-order corrections to the critical value of the Rayleigh number whereas linear variations in viscosity and thermal expansion coefficient give rise to second-order corrections to the critical Rayleigh number. The greatest effect is due to thermal conductivity variation. When the boundary conditions are symmetric the corrections are second-order.