Unconditional non‐linear stability for a fluid of third grade

Abstract

SUMMARYThe thermal convection in a layer of a third grade uid is investigated, with viscosity being a general function of temperature. We develop a non-linear stability analysis and prove that unconditional nonlinear stability criterion is achieved using a natural energy approach. This shows that, in some sense, the equations for a uid of third grade are preferable to those for a uid of second grade or a dipolar uid.

Nonlinear stability analysis for double-diffusive convection when the viscosity depends on temperature

“…A nonlinear energy stability analysis is really beneficial when it can be used to provide stability thresholds close to those of linear theory and valid for all initial conditions,as emphasized by Straughan (p. 157 of [19]). Many recent studies have focussed upon this problem (see Payne and Straughan [17], Budu [20], Lombardo et al [21], Straughan [22]). Flavin and Rionero [12][13][14] utilized the 'natural' transformation to deal with when the thermal conductivity is a nonlinear function of temperature, at least with regard to a particular class of nonlinearity.…”