The influence of thermal relaxation on the oscillatory properties of two-gradient convection in a vertical slot

2013

Proc. R. Soc. A.

There is increasing interest in convection in local thermal non-equilibrium (LTNE) porous media. This is where the solid skeleton and the fluid may have different temperatures. There is also increasing interest in thermal wave motion, especially at the microscale and nanoscale, and particularly in solids. Much of this work has been based on the famous model proposed by Carlo Cattaneo in 1948. In this paper, we develop a model for thermal convection in a fluid-saturated Darcy porous medium allowing the solid and fluid parts to be at different temperatures. However, we base our thermodynamics for the fluid on Fourier's law of heat conduction, whereas we allow the solid skeleton to transfer heat by means of the Cattaneo heat flux theory. This leads to a novel system of partial differential equations involving Darcy's law, a parabolic fluid temperature equation and effectively a hyperbolic solid skeleton temperature equation. This system leads to novel physics, and oscillatory convection is found, whereas for the standard LTNE Darcy model, this does not exist. We are also able to derive a rigorous nonlinear global stability theory, unlike work in thermal convection in other second sound systems in porous media.

“…Papanicolaou et al [28]. Details of these developments may be found in the book by Straughan [4, ch.…”

Section: Introductionmentioning

confidence: 99%

Porous convection with local thermal non-equilibrium temperatures and with Cattaneo effects in the solid

2013

Proc. R. Soc. A.

“…We might point out that nonlinear energy stability methods have been increasingly applied in penetrative convection and in other thermal convection studies, cf. Capone et al (2010Capone et al ( , 2011a, Carr (2004), Falsaperla et al (2012), Falsaperla & Mulone (2010), Hill (2004Hill ( , 2005, Hill & Carr (2010), Hill & Malashetty (2012), Larson (2000Larson ( , 2001, Lombardo et al (2008), Papanicolaou et al (2011), Payne & Straughan (2000), Rionero (2009Rionero ( , 2011Rionero ( , 2012, Shivakumara et al (2010), Straughan (2002aStraughan ( ,b, 2004a and .…”

Section: Global Nonlinear Stabilitymentioning

confidence: 99%

“…In particular, recent studies of linear instability and of nonlinear stability have been developed by Bennacer et al (1993), Capone et al (2010Capone et al ( , 2011a, Carr (2004), Carr & Straughan (2003), Gobin & Bennacer (1996), Gobin & Benoit (2012), Gobin et al (2005), Hill (2004Hill ( , 2005, Krishnamurti (1997), Larson (2000Larson ( , 2001, Normand & Azouni (1992), Papanicolaou et al (2011), Payne & Straughan (2000), Shivakumara et al (2010), Straughan (2002aStraughan ( ,b, 2004a, and Zhang & Schubert (2000, 2002.…”

Section: Introductionmentioning

confidence: 99%

Triply resonant penetrative convection

2012

Proc. R. Soc. A.

A thermal convection model is considered that consists of a layer of viscous incompressible fluid contained between two horizontal planes. Gravity is acting vertically downward, and the fluid has a density maximum in the active temperature range. A heat source/sink that varies with vertical height is imposed. It is shown that in this situation there are three possible (different) sub-layers that may induce convective overturning instability. The possibility of resonance between the motion in these layers is investigated. A region is discovered where a very sharp increase in Rayleigh number is observed. In addition to a linearized instability analysis, two global (unconditional) nonlinear stability thresholds are derived.

“…Here, Da is the Darcy number, Ps is the salt Prandtl number, Ra = R 2 is the salt Rayleigh number, and Sg is a parameter introduced in Papanicolaou et al [25]. For convenience we now drop all asterisks and the non-dimensional perturbation equations become,…”

Section: Overturning Instabilitymentioning

confidence: 99%

Hyperbolic diffusion with Christov–Morro theory

Gentile

Mathematics and Computers in Simulation

2016

We employ recent ideas of C.I. Christov and of A. Morro to develop a theory for diffusion of a solute in a Darcy porous medium taking convection effects into account. The key point is that the solute evolution is not governed by a parabolic system of equations. Indeed, the theory developed is basically hyperbolic. This still leads to a model which allows for convective (gravitational) overturning in a porous layer, but in addition to the classical mode of stationary convection instability there is the possibility of oscillating convection being dominant for a lower salt Rayleigh number, if the relaxation time is sufficiently large.