Pressure Reconstruction for Weak Solutions of the Two-Phase Incompressible Navier--Stokes Equations with Surface Tension

TL;DR

This paper develops a weak formulation for the two-phase incompressible Navier--Stokes equations with surface tension, enabling pressure reconstruction under certain regularity assumptions.

Contribution

It introduces a divergence-free test function-based variational formulation and proves a consistency result for pressure reconstruction.

Findings

Validated the weak formulation for two-phase flow with surface tension.

Established conditions for pressure reconstruction from weak solutions.

Provided theoretical justification for the proposed formulation.

Abstract

For the two-phase incompressible Navier--Stokes equations with surface tension, we derive an appropriate weak formulation incorporating a variational formulation using divergence-free test functions. We prove a consistency result to justify our definition and, under reasonable regularity assumptions, we reconstruct the pressure function from the weak formulation.