The Ellis model describes the apparent viscosity of a shear–thinning fluid with no singularity in the limit of a vanishingly small shear stress. In particular, this model matches the Newtonian behaviour when the shear stresses are very small. The emergence of the Rayleigh–Bénard instability is studied when a horizontal pressure gradient, yielding a basic throughflow, is prescribed in a horizontal porous layer. The threshold conditions for the linear instability of this system are obtained both analytically and numerically. In the case of a negligible flow rate, the onset of the instability occurs for the same parametric conditions reported in the literature for a Newtonian fluid saturating a porous medium. On the other hand, when high flow rates are considered, a negligibly small temperature difference imposed across the horizontal boundaries is sufficient to trigger the convective instability.
The conditions defining a stationary fluid flow may lead to a multiplicity of solutions. This circumstance is widely documented in the literature when mixed convection in a vertical channel or duct is accompanied by an important effect of viscous dissipation. Usually, there are double stationary solutions with a parallel velocity field which satisfy given temperature boundary conditions and with a prescribed mass flow rate. However, in a vertical internal flow, the dual solutions can be determined only numerically as they do not have a closed analytical form. This study shows that, in a horizontal channel, stationary mixed convection with viscous dissipation shows up dual flow branches whose mathematical expressions can be determined analytically. The features of these dual flows are discussed.
The classical Gill's stability problem for stationary and parallel buoyant flow in a vertical porous slab with impermeable and isothermal boundaries kept at different temperatures is reconsidered from a different perspective. A three-layer slab is studied instead of a homogeneous slab as in Gill's problem. The three layers have a symmetric configuration where the two external layers have a high thermal conductivity, while the core layer has a much lower conductivity. A simplified model is set up where the thermal conductivity ratio between the external layers and the internal core is assumed as infinite. It is shown that a flow instability in the sandwiched porous slab may arise with a sufficiently large Rayleigh number. It is also demonstrated that this instability coincides with that predicted in a previous analysis for a homogeneous porous layer with permeable boundaries, by considering the limiting case where the permeability of the external layers is much larger than that of the core layer.
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