A simple thermodynamic framework for heat-conducting flows of mixtures of two interacting fluids

TL;DR

This paper develops a thermodynamic model for heat-conducting mixtures of two interacting fluids, capturing their behavior through energy storage, entropy production, and minimal material parameters, applicable to both compressible and incompressible cases.

Contribution

It introduces a unified thermodynamic framework for modeling two-fluid mixtures with heat conduction, using minimal parameters and addressing boundary condition identification.

Findings

Model derived for both compressible and incompressible mixtures.

Different variants based on definitions of mixture velocity analyzed.

Guidelines provided for boundary condition formulation for constituents.

Abstract

Within the theory of interacting continua, we develop a model for a heat conducting mixture of two interacting fluids described in terms of the densities and the velocities for each fluid and the temperature field for the mixture as a whole. We use a general thermodynamic framework that determines the response of the material from the knowledge of two pieces of information, namely how the material stores the energy and how the entropy is produced. This information is expressed in the form of the constitutive equations for two scalars: the Helmholtz free energy and the entropy production. Additionally, we follow the goal to determine the response of a mixture from a small (minimal) set of material parameters, including shear viscosity, bulk viscosity and heat conductivity associated with the mixture as a whole and the drag coefficient connected with the interaction force between the constituents. The same thermodynamic approach is used to obtain the model when the mixture as a whole responses as an incompressible material. For both the compressible and incompressible mixtures, we investigate three variants stemming from different definitions of the (averaged) velocity associated with the mixture as a whole. We also address the issue of identification of boundary conditions for the individual constituents from the standard boundary conditions formulated in terms of the quantities associated with the mixture as a whole.