The effect of temperature-dependent viscosity and thermal conductivity on the onset of compressible convection

The linear equations of thermal convection in a compressible fluid with non-constant transport coefficients are derived. The criterion for the onset of convection is established, based on linear stability analysis, for a range of different temperature-dependent profiles of thermal conductivity and viscosity. Temperature-dependent transport coefficients are shown to lead to a more complex behaviour than their constant counterparts, and modifies the stability condition of the fluid. When the Rayleigh number is defined in terms of the mid-layer physical properties and the temperature gradient at the top is held constant, increasing the temperature-dependence of thermal conductivity is found to raise the critical Rayleigh number dramatically, as the convective disturbance is then concentrated mainly at the top of the layer. In contrast, for viscosity a more subtle effect on stability is identified.