On a Navier-Stokes-Allen-Cahn model with inertial effects

TL;DR

This paper introduces a coupled Navier-Stokes-Allen-Cahn model with inertial effects for two-phase fluid flow, proving existence of solutions and analyzing long-term behavior with energy considerations.

Contribution

It develops a novel inertial-influenced phase transition model coupling Navier-Stokes and Allen-Cahn equations, with mathematical analysis of solutions and energy dissipation.

Findings

Existence of weak solutions in 3D

Existence and uniqueness of strong solutions in 2D

Insights into the energy dissipation over time

Abstract

A mathematical model describing the flow of two-phase fluids in a bounded container $\Omega$ is considered under the assumption that the phase transition process is influenced by inertial effects. The model couples a variant of the Navier-Stokes system for the velocity $u$ with an Allen-Cahn-type equation for the order parameter $\varphi$ relaxed in time in order to introduce inertia. The resulting model is characterized by second-order material derivatives which constitute the main difficulty in the mathematical analysis. Actually, in order to obtain a tractable problem, a viscous relaxation term is included in the phase equation. The mathematical results consist in existence of weak solutions in 3D and, under additional assumptions, existence and uniqueness of strong solutions in 2D. A partial characterization of the long-time behavior of solutions is also given and in particular some issues related to dissipation of energy are discussed.