Sharp global nonlinear stability for a fluid overlying a highly porous material

Abstract

The stability of convection in a two-layer system in which a layer of fluid with a temperature-dependent viscosity overlies and saturates a highly porous material is studied. Owing to the difficulties associated with incorporating the nonlinear advection term in the Navier-Stokes equations into a stability analysis, previous literature on fluid/porous thermal convection has modelled the fluid using the linear Stokes equations. This paper derives global stability for the full nonlinear system, by utilizing a model proposed by Ladyzhenskaya. The nonlinear stability boundaries are shown to be sharp when compared with the linear instability thresholds.

Convection in a Coupled Free Flow-Porous Media System

“…The thermal conductivity κ m and specific heat capacity (ρ 0 c p ) m of the porous medium are defined as averages of the fluid and solid components. Many references simply use an arithmetic average [28,29,30] φ m = χφ f + (1 − χ) φ s , where φ represents either thermal conductivity or heat capacity. However, we point out that homogenization theory gives the harmonic…”

“…Given the urgent need to understand water resources more fully, investigating the interaction between surface-and groundwater is particularly timely [51]. To gain useful insight into the nature of these coupled fluid-porous systems, both linear and nonlinear stability arguments have been conducted and analyzed [28,29,30]. However, the presence of nonlinear advection (u · ∇) u can hinder nonlinear stability analysis since, when coupled to non-trivial interface conditions, it produces a sign-indefinite term in the energy bound.…”

A thermo-mechanical model for flows in superposed porous and fluid layers with interphasial heat and mass exchange

“…Therefore, a single equation, describing the energy balance of the mixture, is needed to determine the temperature inside the porous region (one-equation model). Within this context, various authors have studied the stability of thermal convection in superposed fluid and porous layers; see, for example, [22][23][24][25][26][27][28][29][30][31][32][33] and references therein. More recently, the authors of [34] presented Reynolds-averaged simulations for statistically stationary turbulent heat transfer in such domains.…”

Nonlinear stability of the one-domain approach to modelling convection in superposed fluid and porous layers

“…Nield & Bejan 2006) in the porous region, coupled with the Navier-Stokes *Author for correspondence (antony.hill@nottingham.ac.uk). equations in the fluid region. Recent contributions include Vafai (2005), Chang (2006), Mu & Xu (2007), Straughan (2008), Hill & Straughan (2009) and Hill & Carr (2010).…”

“…In both these papers, owing to the difficulties associated with incorporating the nonlinear v · Vv advection term in the Navier-Stokes equations into a stability analysis, the fluid was modelled using Stokes equations. Hill & Carr (2010) used a viscoelastic model proposed by Ladyzhenskaya (Ladyzhenskaya 1969;Straughan 2008) as an alternative to Navier-Stokes, and constructed nonlinear stability thresholds.…”