Internal constraint theories for the thermal expansion of viscous fluids

Abstract

Viscous fluids in many processes are mechanically incompressible but experience significant thermally induced volume change. We focus here on models for viscous fluids that impose this behavior through a posited internal constraint, following the formalism which introduces constraint responses that produce no entropy. We investigate four different thermal expansion constraints, whereby the independent mechanical variable in the problem formulation (density ρ or pressure p) is assumed to be specified completely by the independent thermal variable (temperature θ or entropy η). The internal constraint approach yields simpler constitutive relations, and therefore easier material characterization; the resulting model equations are alternatives to the compressible Navier–Stokes equations for simulation and analysis of thermal expansion phenomena. However, whereas the internal constraint formalism preserves consistency with the thermomechanical balance laws, second law, and invariance by fiat, there is no a priori guarantee that stability of the rest state and nonnegativity of specific heat and bulk modulus are preserved. Here we first derive the four possible internal constraint models for thermal expansion. Next we show that each of these four models is equivalent to a specific constitutive limit of the compressible theory, namely two of no pressure dependence of density and their duals of no density dependence of pressure. From this connection, we deduce that two constraints, namely ρ= ρ( η) and p= p( θ), offer physically viable candidates for the modeling of thermal expansion, whereas two others (the customary density–temperature constraint ρ= ρ( θ), as well as p= p( η)) are unphysical in the absence of any other conditions on the process, predicting catastrophic instability of the rest state. This analysis reveals that the internal constraint formalism must be further conditioned to preserve fundamental irreversible (second law) and reversible (stability) inequalities.