Sharp global nonlinear stability for temperature-dependent viscosity convection

Abstract

With regard to the dependence of viscosity on temperature, an unconditional nonlinear energy-stability analysis for thermal convection according to Navier-Stokes theory has not yet been developed. We here analyse three models of fluid behaviour proposed by O. A. Ladyzhenskaya. We show that by using these theories we can develop an unconditional analysis directly. Two of the models lead directly to an unconditional development employing L 2 theory, while the third necessitates the introduction of a generalized energy also involving an L 3 term. The nonlinear stability boundaries in all three cases are sharp when compared with the instability thresholds of linear theory.

Triply resonant penetrative convection

“…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.…”

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

“…Although we only consider liquids that have a viscosity which decreases with increasing temperature, the analysis can be easily generalized to a more general viscosity-temperature relationship. The governing model for the fluid layer we select is (Antontsev et al 2001;Straughan 2002Straughan , 2004 where v i , p, t and ρ 0 are velocity, pressure, time and reference density and κ f , g, c p and α are the thermal conductivity, acceleration due to gravity, specific heat at a constant pressure and coefficient of thermal expansion. A variation of this model was suggested by Ladyzhensakaya (1967Ladyzhensakaya ( , 1968Ladyzhensakaya ( , 1969 as an alternative to the NavierStokes equations and is a generalization of a well-known model in viscoelasticity (Antontsev et al 2001).…”

“…A detailed review is given by Nield & Bejan (2006), with current highly relevant literature including Chen & Chen (1988), Ewing & Weekes (1998), Blest et al (1999), Straughan (2002Straughan ( , 2008, Carr (2004), Chang (2004Chang ( , 2005Chang ( , 2006, Hirata et al (2007), Hoppe et al (2007), Mu & Xu (2007) and Hill & Straughan (2009).…”

“…This paper utilizes a model proposed by Ladyzhenskaya (1967Ladyzhenskaya ( , 1968Ladyzhenskaya ( , 1969Straughan 2002Straughan , 2004Straughan , 2008, which is used as an alternative to Navier-Stokes. This allows for the development of an unconditional nonlinear energy stability analysis for thermal convection with temperature-dependent viscosity in a fluid/porous system, without the need to remove the nonlinear advection term v · ∇v.…”

On the Modulation of Oscillation in Thermohaline Convection Problems Using Temperature Dependent Viscosity

“…Straughan [8]). In the majority of the cases, viscosity becomes the only property which may have considerable effect on the heat transfer and the temperature variation and the dependence of other thermo-physical properties to temperature is often negligible.…”