A variational formulation for dissipative fluids with interfaces in an inhomogeneous temperature field

TL;DR

This paper develops a variational framework for modeling dissipative fluids with interfaces in inhomogeneous temperature fields, deriving equations of motion and entropy relations consistent with thermodynamics.

Contribution

It introduces a novel variational formulation that incorporates entropy constraints, generalizing Noether's theorem for dissipative fluid systems with interfaces.

Findings

Derived equations of motion for fluids with interfaces in inhomogeneous temperature fields.

Clarified cross effects between entropy flux and phenomena like vaporization, diffusion, and liquid crystal rotation.

Validated the approach through examples illustrating the interplay of entropy and fluid dynamics.

Abstract

We propose a formalization for dissipative fluids with interfaces in an inhomogeneous temperature field from the viewpoint of a variational principle. Generally, the Lagrangian of a fluid is given by the kinetic energy density minus the internal energy density. The necessary condition for minimizing an action with subject to the constraint of entropy yields the equation of motion. However, it is sometimes to know the proper equation of entropy. Our main purpose is to obtain it by using the three requirements, which are a generalization of Noether's Theorem, the second law of thermodynamics, and well-posedness. To illustrate this approach, we investigate several phenomena in an inhomogeneous temperature field. In the case of vaporization, diffusion and the rotation of a chiral liquid crystals, we clarify the cross effects between the entropy flux and these phenomena via the internal energy.