Non‐linear stability for convection with quadratic temperature dependent viscosity

Abstract

SUMMARYIn this paper, we study the non-linear stability of convection for a Newtonian uid heated from below, where the viscosity of the uid depends upon temperature. We are able to show that for Rayleigh numbers below a certain critical value, Rac, the rest state of the uid and the steady temperature distribution remains non-linearly stable, using the calculations of Diaz and Straughan (Continuum Mech. Thermodyn. 2004; 16:347-352). The central contribution of this paper lies in a simpler proof of non-linear stability, than the ones in the current literature, by use of a suitable maximum principle argument.

Spectral numerical schemes for time-dependent convection with viscosity dependent on temperature

“…More recently [10,11] have considered the hyperbolic tangent or the arctangent as viscosity laws for they model a viscosity transition in a narrow temperature gap. Other studies have treated other weaker dependencies such as linear [12,13] or quadratic ones [14,15].…”

Bifurcations and dynamics in convection with temperature-dependent viscosity in the presence of the O(2) symmetry

“…Convection in fluids with temperature-dependent viscosity is of interest because of its importance in engineering and geophysics. Linear and quadratic dependencies of the viscosity on temperature have been discussed [47,52,22,65], but in order to address the Earth's upper mantle convection, in which viscosity contrasts are of several orders of magnitude, a stronger dependence with temperature must be considered. This problem has been approached both in experiments [53,13,6,67] and in theory [45,7,50,44,60,61].…”

A note on convection with nonlinear heat flux

“…Nonlinear stability and linear instability studies of flow with variable thermal expansion coefficient and kinematic viscosity, in particular, have occupied much attention, see, e.g. Ames and Payne [1,2], Budu [7], Capone [8], Capone and Gentile [9,10], Capone and Rionero [11], Diaz and Straughan [12], Flavin and Rionero [14,15], Payne and Song [22], Qin et al [23], Straughan [32][33][34], Vaidya and Wulandana [37], Wall and Wilson [38], Webber [39], and the references therein. Furthermore, analyses of structural stability, i.e.…”