Nonisothermal Cahn-Hilliard Navier-Stokes system

TL;DR

This paper develops a structure-preserving approximation method for the non-isothermal Cahn-Hilliard-Navier-Stokes system, incorporating stabilization techniques to ensure energy conservation and entropy production.

Contribution

It extends existing finite element methods by integrating stabilization techniques into the non-isothermal model, ensuring physical consistency.

Findings

The method preserves total energy conservation.

It maintains entropy production in the numerical scheme.

The approach effectively handles coupling in the energy equation.

Abstract

In this research, we introduce and investigate an approximation method that preserves the structural integrity of the non-isothermal Cahn-Hilliard-Navier-Stokes system. Our approach extends a previously proposed technique [1], which utilizes conforming (inf-sup stable) finite elements in space, coupled with implicit time discretization employing convex-concave splitting. Expanding upon this method, we incorporate the unstable P1|P1 pair for the Navier-Stokes contributions, integrating Brezzi-Pitk\"aranta stabilization. Additionally, we improve the enforcement of incompressibility conditions through grad div stabilization. While these techniques are well-established for Navier-Stokes equations, it becomes apparent that for non-isothermal models, they introduce additional coupling terms to the equation governing internal energy. To ensure the conservation of total energy and maintain entropy production, these stabilization terms are appropriately integrated into the internal energy equation.