Long-wave instabilities of a power-law fluid flowing over a heated, uneven and porous incline: A two-sided model

The stability conditions of a two-dimensional gravity -driven flow of a thin layer of a power -law fluid flowing over a heated, uneven, inclined porous surface are investigated. A two-sided model is employed to account for the bottom filtration in the porous layer. The governing equations are reduced under the long -wave approximation and the cross -stream dependence is eliminated by means of the Integral Boundary Layer technique. Floquet-Bloch theory is used to investigate at linear level how the porous bottom waviness influences the thermocapillarity stability of the flow in a shear -thinning fluid. Differently from the even case, the linear stability analysis suggests that for flow over sufficiently wavy undulations the thermocapillarity may stabilize the equilibrium flow, depending on the values of dimensionless governing numbers and parameters. This stabilizing phenomenon is enhanced by the shear -thinning rheology of the fluid while it is reduced by the permeability of the layer. Numerical simulations, performed solving the reduced nonlinear model through a second order Finite Volume scheme, confirm the results of the linear stability analysis.