Weakly nonlocal thermodynamics of binary mixtures of Korteweg fluids with two velocities and two temperatures

TL;DR

This paper develops a thermodynamic framework for binary Korteweg fluids with two velocities and temperatures, incorporating nonlocal effects and gradient dependencies, and provides a complete solution in one dimension.

Contribution

It introduces a comprehensive thermodynamic model for complex binary mixtures with nonlocal effects and solves the restrictions explicitly in one dimension.

Findings

Derived constitutive relations for partial heat fluxes and stress tensors.

Determined the entropy flux including classical and nonlocal contributions.

Ensured compatibility with the second law of thermodynamics.

Abstract

We provide a thermodynamic framework for binary mixtures of Korteweg fluids with two velocities and two temperatures. The constitutive functions are allowed to depend on the diffusion velocity and the specific internal energy of both constituents, together with their first gradients, as well as on the mass density of the mixture and the concentration of one of the constituents, the latters together with their first and second gradients. Compatibility with second law of thermodynamics is investigated by applying a generalized Liu procedure. In the one-dimensional case, a complete solution of the set of thermodynamic restrictions is obtained by postulating a possible form of the constitutive equations for the partial heat fluxes and stress tensors. Taking a first order expansion in the gradients of the specific entropy, the expression of the entropy flux is determined. This contains the classical terms (namely, the sum of the ratios between the heat fluxes and the temperatures of the constituents) and some additional contributions accounting for nonlocal effects.