On models for viscoelastic materials that are mechanically incompressible and thermally compressible or expansible and their oberbeck-boussinesq type approximations

摘要

Viscoelastic fluid like materials that are mechanically incompressible but are compressible or expansible with respect to thermal stimuli are of interest in various applications ranging from geophysics and polymer processing to glass manufacturing. Here we develop a thermodynamical framework for the modeling of such materials. First we illustrate the basic ideas in the simpler case of a viscous fluid, and after that we use the notion of natural configuration and the concept of the maximization of the entropy production, and we develop a model for a Maxwell type viscoelastic fluid that is mechanically incompressible and thermally expansible or compressible. An important approximation in fluid mechanics that is frequently used in modeling buoyancy driven flows is the Oberbeck-Boussinesq approximation. Originally, the approximation was used for studying the flows of viscous fluids in thin layers subject to a small temperature gradient. However, the approximation has been used almost without any justification even for flows of non-Newtonian fluids induced by strong temperature gradients in thick layers. Having a full system of the governing equations for a Maxwell type viscoelastic mechanically incompressible and thermally expansible or compressible fluid, we investigate the validity of the Oberbeck-Boussinesq type approximation for flows of this type of fluids. It turns out that the Oberbeck-Boussinesq type approximation is in general not a good approximation, in particular if one considers "high Rayleigh number" flows. This indicates that the Oberbeck-Boussinesq type approximation should not be used routinely for all buoyancy driven flows, and its validity should be thoroughly examined before it is used as a mathematical model.