Investigation of simplified thermal expansion models for compressible Newtonian fluids applied to nonisothermal plane Couette and Poiseuille flows

In this paper six different theories of a Newtonian viscous fluid are investigated and compared, namely, the theory of a compressible Newtonian fluid, and five constitutive limits of this theory: the incompressible theory, the limit where density changes only due to changes in temperature, the limit where density changes only with changes in entropy, the limit where pressure is a function only of temperature, and the limit of pressure a function only of entropy. The six theories are compared through their ability to model two test problems: (i) steady flow between moving parallel isothermal planes separated by a fixed distance with no pressure gradient in the flow direction, and (ii) steady flow between stationary isothermal parallel planes with a pressure gradient. The incompressible theory admits solutions to these problems of the plane Couette/Poiseuille flow form: a single nonzero velocity component in a direction parallel to the bounding planes, and velocity and temperature varying only in the direction perpendicular to the planes. The compressible theory admits a solution of this special form to problem (i) but not problem (ii). We find that the other four constitutive limits have Couette-form solutions to (i), but only the limits of density a function of temperature and pressure a function of entropy join the incompressible fluid in admitting the Poiseuille-form solutions to (ii); the limits of density a function only of entropy and pressure a function of temperature, as with the compressible theory, do not have solutions of that form. Based on the predictions of the fully compressible theory and its limits, we assess the usefulness of the limits as simplified models of thermal expansion.