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Analytical and Numerical study of the primary and secondary instabilities of viscoelastic fluids saturating an horizontal porous layer heated by a constant flux is proposed in the present work. The mathematical formulation of the problem is based on the phenomenological law of Darcy generalized to a viscoelastic fluid verifying the Boussinesq approximation. This formulation introduces two additional parameters related to viscoelasticity, namely the relaxation time λ1 and the ratio Rv between the viscosity of the solvent and the total viscosity of the polymer solution.
When the horizontal walls are considered perfectly conductive of heat, the linear and weakly nonlinear stability study has been carried out by Hirata et al. [1]. In the case when the horizontal boundaries are maintained at a constant flux, the linear stability analysis conducted in this work shows that the state of conduction loses its stability in favor of convective structures with different spatio-temporal properties depending on the viscoelastic parameters λ1 and Rv. In the (Rv, λ1) plane, we identified two distinct regions. In a weakly viscoelastic regime, these convective structures have a stationary character where the most amplified mode is characterized by a long wavelength and the viscoelastic fluid behaves like a Newtonian fluid. However, it is shown that for strongly viscoelastic fluids, a Hopf bifurcation appears leading to oscillatory convective flows with a finite wave number and finite frequency.
For weakly viscoelastic fluids, the parallel flow approximation is first used to determine the analytical solution of the unicellular flow convection in the nonlinear regime. These analytical solutions are then corroborated by direct numerical simulations in a second part of this study. In a third part, we perform a stability analysis of the unicellular flow convection for both Newtonien fluids and viscoelastic fluids against perturbations in the form of moving transverse rolls (TR), whose axis is perpendicular to the direction of the main flow. For Newtonian fluids, it is found that the critical Rayleigh number and the critical frequency corresponding to the appearance of multicellular flow in the form of TR are in excellent agreement with the values found in [2]. In the case of viscoelastic fluids, the threshold of the appearance of the secondary instabilities is determined and demonstrates the destabilizing effect of the viscoelastic character of the fluid. The effect of the secondary bifurcation on heat transfer is also discussed by evaluating numerically the Nusselt number for a large set of the viscoelastic parameters λ1 and Rv.
[1] S. Hirata, G. Ella et M. N. Ouarzazi, Nonlinear pattern selection and heat transfer in thermal convection of a viscoelastic fluid saturating a porous medium, Int. J. Thermal Sciences, 95 (2015) 136-146.
[2] S. Kimura, M. Vynnycky and F. Alavyoon. Unicellular natural circulation in a shallow horizontal porous layer heated from below by a constant flux. J. Fluid Mech, 294 (1995), p. 231-257.
