Natural convection in a fluid saturating an anisotropic porous medium in LTNE: effect of depth-dependent viscosity

Abstract

Thermal convection in a fluid saturating an anisotropic porous medium in local thermal nonequilibrium (LTNE) is investigated, with specific attention to the effect of variable viscosity on the onset of convection. Many fluids show a remarkable dependence of viscosity on temperature that cannot be neglected. For this reason, we take into account a fluid whose viscosity decreases exponentially with depth, according to Straughan (Acta Mech. 61:59–72, 1986), Torrance and Turcotte (J. Fluid Mech. 47(1):113–125, 1971). The novelty of this paper is to highlight how variable viscosity coupled with the LTNE assumption affects the onset of convection. A numerical procedure shows the destabilising effect of depth-dependent viscosity. Moreover, it comes out that the LTNE hypothesis makes the influence of viscosity more intense. Linear instability analysis of the conduction solution is carried out by means of the Chebyshev-tau method coupled to the QZ algorithm, which provides the critical Rayleigh number for the onset of convection in a straightforward way. The energy method is employed in order to study the nonlinear stability. The optimal result of coincidence between the linear instability threshold and the global nonlinear stability threshold is obtained. The influence of anisotropic permeability and conductivity, weighted conductivity ratio, and interaction coefficient on the onset of convection is highlighted.

Stability of penetrative convective currents in local thermal non-equilibrium

“…In this section, we provide some details useful to the implementation of the method. For a deeper discussion, we refer to [39][40][41].…”

Penetrative convection in a bi‐disperse porous medium

“…Introducing a perturbation {u 𝑓 , u p , 𝜋 𝑓 , 𝜋 p , 𝜃} to the conduction solution, with u s = (u s , v s , w s ) for s = {𝑓 , p}, the perturbation equations arising from (5) are…”

“…More precisely, the aim of the linear instability analysis is to provide criteria by which one can predict whether a given basic flow is unstable: developing a normal‐mode analysis, one determines a critical threshold, $$$ {\mathcal{R}}_L $$$, such that the convection occur if $$$ \mathcal{R}>{\mathcal{R}}_L $$$. Concerning the nonlinear stability of the model, the energy method is employed [1, 4, 5]. More specifically, the condition for which a suitable weighted energy functional is decreasing along the solutions of the system gives rise to a nonlinear threshold $$$ {\mathcal{R}}_N\left(\le {\mathcal{R}}_L\right) $$$ such that if $$$ \mathcal{R}<{\mathcal{R}}_N $$$, the basic steady motion is asymptotically stable; that is, convection cannot occur.…”

Effects of LTNE on Two-Component Convective Instability in a Composite System with Thermal Gradient and Heat Source

“…Meften and Ali [15] studied two-component convection with variable viscosity. Capone et al [16,17] investigated the convection in a rotating, viscous, anisotropic porous medium in the LTNE regime. Using the Brinkmann-Forchheimer model, Meften et al [18] examined the LTNE effects for dual-component convection with rotation.…”