Onset of Darcy--Bénard convection under throughflow of a shear-thinning fluid

We present an investigation on the onset of Darcy--B'enard instability in a two--dimensional porous medium saturated with a non--Newtonian fluid and heated from below in presence of a uniform horizontal pressure gradient. The fluid is taken to be of power--law nature with constant rheological index $n$ and temperature dependent consistency index $mu^*$. A two--dimensional linear stability analysis in the vertical plane yields the critical wave number and the generalised critical Rayleigh number as functions of dimensionless problem parameters, with a non monotonic dependence from $n$ and with maxima/minima at given values of $gamma$, a parameter representing the effects of consistency index variations due to temperature. A series of experiments are conducted in a Hele-Shaw cell of aspect ratio $H/b=13.3-20$ to provide a verification of the theory. Xanthan Gum mixtures {(nominal concentration from 0.10% to 0.20%)} are employed as working fluids with a parameter range $n=0.55-0.72$ and $mu^*_0=0.02-0.10,mathrm{Pa},mathrm{s}^n$. The experimental critical wave number corresponding to incipient instability of the convective cells is derived via image analysis for different values of the imposed horizontal velocity. Theoretical results for critical wave number favourably compare with experiments, systematically underestimating their experimental counterparts by $10%$ at most. The discrepancy between experiments and theory is more relevant for the critical Rayleigh number, with theory overestimating the experiments by a maximum factor less than two. Discrepancies are {attributable} to a {combination of factors: nonlinear phenomena, possible subcritical bifurcations, and unaccounted-for disturbing effects such as approximations in the rheological model, wall slip, ageing and degradation of the fluid properties.}