THERMOMECHANICAL EQUATIONS GOVERNING A MATERIAL WITH PRESCRIBED TEMPERATURE-DEPENDENT DENSITY WITH APPLICATION TO NONISOTHERMAL PLANE POISEUILLE FLOW

The standard practice in the literature far modeling materials process ing in which changes in temperature induce significant volume changes is based on the aposteriori substitution of a temperature-dependent ex pression for density into the governing equations for an incompressibl e material. In this paper we show this ad hoc approach misses importan t terms in the equations, and by example show the ad hoc equations fai l to capture important physical effects, First we derive the three-dim ensional equations which govern the deformation and heat transfer of m aterials with prescribed temperature-dependent density. Specification of density as a function of temperature translates to a thermomechanic al constraint, in contrast to the purely mechanical incompressibility constraint, so that the constraint response function (''pressure'') en ters into the energy equation as well as the momentum equation. Then w e demonstrate the effect of the correct constraint response by compari ng, 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 phe nomenon of expansion cooling is captured by the thermomechanically con strained theory but not by the ad hoc theory.