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

Abstract

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

On the stability of the isoflux Darcy–Bénard problem with a generalised basic state

“…We point out that the solution (8) formally coincides with that discussed by Kimura et al [27] and, more recently, by Gueye et al [31]. Despite the formal congruence of such solutions, their physical meaning is utterly different.…”

“…Despite the formal congruence of such solutions, their physical meaning is utterly different. In fact, both Kimura et al [27] and Gueye et al [31] assume that Eqs. ( 8) are meant to describe the parallel flow in the central core of a single cell formed within a shallow rectangular cavity.…”

“…( 8) are meant to describe the parallel flow in the central core of a single cell formed within a shallow rectangular cavity. The vertical sidewalls of the cavity, placed at suitable x = constant positions, are considered as adiabatic [27,31]. The adiabaticity constraint induces a relationship between the parameters C and Ra, so that these parameters cannot be varied independently.…”