Instability thresholds for penetrative porous convection with variable viscosity fluids

A comprehensive investigation discerning different viscosity-temperature variations on the onset of porous penetrative convection modeled through quadratic density-temperature law is undertaken. The fluid viscosity is assumed to vary linearly, quadratically and exponentially with temperature. The variable viscosity results in a coupling between stream function and temperature in the Darcy momentum balance equation. The linear instability of the basic state is analyzed numerically by examining small-amplitude disturbances in velocity and temperature fields. By carrying out numerical computations for a wide range of parametric regimes, it is established that the principle of exchange of stabilities (PES) is valid. The critical values of the Darcy-Rayleigh number and of the wave number are obtained. The analysis reveals that the combined influence of variable viscosity and the density-maximum characteristic is to expedite the onset of porous convection much faster compared to their isolation presence. The size of the convection cell decreases with an increase in the Darcy-modified Rayleigh number linked to the quadratic density variant. The outcomes obtained under the limiting cases are shown to exhibit consistency with previously published findings.