Thermal Expansion Models of Viscous Fluids Based on Limits of Free Energy
S.E. Bechtel, M.G. Forest, F.J. Rooney, Q. Wang
Many viscous fluid flows are mechanically incompressible, yet thermally expand and shrink. Various approximations of the compressible Navier-Stokes equations have been proposed to resolve the diverse phenomena that arise in such fluid systems, with a primary goal to remove rapid timescales associated with sound waves. The Boussinesq model for laboratory-scale, buoyancy-driven thermal convection patterns and the anelastic model for atmospheric-scale, density-stratification phenomena are two important examples. Molten polymer and glass flows exhibit thermal expansion where mechanical compressibility is negligible, yet self-consistent approximate models do not exist. With this motivation, we propose a systematic method to derive thermal expansion models based on mechanical incompressibility conditions applied directly on the free energy formulation of the compressible Navier-Stokes system. The method is distinct from other approaches in that the irreversible physics and second law are preserved, whereas the reversible physical mechanisms governed by the gradient and Hessian of the free energy function take special forms imposed by mechanical incompressibility. The four equivalent free-energy formulations of compressible theory lead to four distinct thermal expansion models. Each model is then evaluated from classical thermodynamical perspectives: convexity inequalities of Gibbs on the free energy function which guarantee local well-posedness near the rest state; non-negativity of sound speed, bulk modulus, and specific heat; and linearized stability of thermo-mechanical equilibrium. Two models preserve these fundamental properties. Two others, however, are ill-posed unless an additional degeneracy condition on free energy is imposed; one model then reduces for isothermal flows to the incompressible Navier-Stokes system with constant density, while the other has limited physical applicability. To elucidate this modeling approach, we parametrize the linearized dispersion relation of compressible theory and follow the wave speeds and eigenmodes in the limits which characterize each thermal expansion model.