Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux

We analyze the thermal convection thresholds and linear characteristics of the primary 1 and secondary instabilities for viscoelastic fluids saturating a porous horizontal layer heated from 2 below by a constant flux. Galerkin method is used to solve the eigenvalue problem by taking 3 into account the elasticity of the fluid, the ratio between the viscosity of the solvent and the total 4 viscosity of the fluid and the lateral confinement of the medium. For the primary instability, we 5 found out that depending on the rheological parameters, two types of convective structures may 6 appear when the basic conductive solution loses its stability: stationary long wavelength instability 7 as for Newtonian fluids and oscillatory convection. The effect of the lateral confinement of the 8 porous medium by adiabatic walls is to stabilize the oblique and longitudinal rolls and therefore 9 selects transverse rolls at the onset of convection. In the range of the rheological parameters where 10 stationary long wave instability develops first, we use a parallel flow approximation to determine 11 analytically the velocity and temperature fields associated to the monocellular convective flow. The 12 linear stability analysis of the monocellular flow is performed, and the critical conditions above 13 which the flow becomes unstable are determined. The combined influence of the viscoelastic 14 parameters and the lateral confinement on the characteristics of the secondary instability is 15 quantified. The major new findings concerning the secondary instabilities may be summarized 16 as follows: (i) For concentrated viscoelastic fluids, computations showed that the most amplified 17 mode of convection corresponds to oscillatory transverse rolls which appears via a Hopf bifurcation. 18 This pattern selection is independent of both the fluid elasticity and the lateral confinement of the 19 porous medium; (ii) For diluted viscoelastic fluids, the preferred mode of convection is found to be 20 oscillatory transverse rolls for a very laterally confined medium. Otherwise stationary or oscillatory 21 longitudinal rolls may develop depending on the fluid elasticity. 22 Results also showed the destabilizing effect of the relaxation fluid elasticity and the stabilizing effect 23 of the viscosity ratio for the onset of both primary and secondary instabilities. 24