Thermomechanical Equations Governing a Material With Prescribed Temperature-Dependent Density With Application to Nonisothermal Plane Poiseuille Flow

Abstract

The standard practice in the literature for modeling materials processing in which changes in temperature induce significant volume changes is based on the a posteriori substitution of a temperature-dependent expression for density into the governing equations for an incompressible material. In this paper we show this ad hoc approach misses important terms in the equations, and by example show the ad hoc equations fail to capture important physical effects. First we derive the three-dimensional equations which govern the deformation and heat transfer of materials with prescribed temperature-dependent density. Specification of density as a function of temperature translates to a thermomechanical constraint, in contrast to the purely mechanical incompressibility constraint, so that the constraint response function (“pressure”) enters into the energy equation as well as the momentum equation. Then we demonstrate the effect of the correct constraint response by comparing solutions of our thermomechanical theory with solutions of the ad hoc theory in plane Poiseuille flow. The differences are significant, both quantitatively and qualitatively. In particular, the observed phenomenon of expansion cooling is captured by the thermomechanically constrained theory, but not by the ad hoc theory.