Thermal expansion models of viscous fluids based on limits of free energy

Abstract

Many viscous flows are mechanically incompressible, yet thermally expand and shrink. Approximations of the compressible Navier–Stokes equations are routinely utilized to model diverse phenomena that share these properties, with a primary goal to remove rapid timescales associated with sound waves. Most models are derived from thermodynamic assumptions coupled with application-specific, scale separation assumptions in time and space. The Boussinesq model for laboratory-scale, buoyancy-driven thermal convection patterns [Spiegel and Veronis, Astrophys. J. 131, 442 (1960)] and the anelastic model for atmospheric-scale, density-stratification phenomena [Emanuel, Atmospheric Convection (Oxford University Press, New York, 1994)] are two important examples. Some engineering models of thermal expansion are fluid specific, e.g., for molten glasses and polymers, and rest upon thermodynamic assumptions alone. These models postpone specification of flow conditions, since applications range from slow confined flows for mold filling, to film flows, to high-speed, free surface fiber flows. The aims of this paper are to systematize the implementation of thermodynamic assumptions to derive thermal expansion models; to benchmark this class of models with respect to near-equilibrium physical behavior; and to pose candidate models for thermal expansion which may be used for a diverse set of flow conditions. By casting thermodynamic assumptions as limits of free energy, we reduce the full Navier–Stokes system in a straightforward way, and parametrize the limits of linearized compressible modes and their propagation properties. The thermodynamic assumptions are uncoupled from flow-dependent scale separation conditions, which can be subsequently augmented for specific applications. Indeed, one conclusion of this analysis is that the traditional thermal expansion assumption, where density is slaved to the temperature field, must be coupled with additional flow or thermal assumptions or else the model equations possess unphysical properties (e.g., instability of thermodynamic equilibrium or negative specific heat). Two new models for thermal expansion are deduced solely from a limit condition on free energy, which pass the minimal benchmark of non-negativity of bulk modulus, squared sound speed, and specific heat. We close with an indication of viscous flow applications of one of the new thermal expansion models.