Cao et al. previously derived a thermomechanically constrained theory for materials with temperature-dependent density and applied it to the illustrative problem of plane Poiseuille flow between isothermal walls. Here the theory is applied to geometries and thermal boundary conditions of practical importance in polymer processing: flows through planar, circular and annular dies with heat loss through the die walls. How geometry and thermal boundary conditions combine with material properties, especially thermal expansivity, to effect velocity and temperature profiles, and mass and volume flow rates is explored. A comparison is made with predictions that follow if temperature-dependence of density is either ignored or handled (as is the standard practice) by a posteriori insertion of a temperature-dependent expression for density into equations derived for constant density.