First-Order System Least Squares and the Energetic Variational Approach for Two-Phase Flow

TL;DR

This paper introduces a first-order system least-squares formulation for two-phase flow equations, demonstrating its effectiveness with numerical techniques and its ability to preserve energy laws based on an energetic variational approach.

Contribution

It develops a novel FOSLS framework for two-phase flow that maintains energy laws and integrates advanced numerical methods for complex fluid simulations.

Findings

FOSLS formulation effectively solves two-phase flow equations.

Numerical techniques like multigrid and adaptive refinement improve efficiency.

Energy law preservation is achieved within the FOSLS framework.

Abstract

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law. We discuss the energy law and inherent structure for two-phase flow using the Allen-Cahn interface model and indicate how it is related to other complex fluid models, such as magnetohydrodynamics. Finally, we show that, using the FOSLS framework, one can still satisfy the appropriate energy law globally while using well-known numerical techniques.