Two apparently different interfacial stress formulations constructed from the diffuse interface free energy model

TL;DR

This paper clarifies the distinction between two interfacial stress formulations in diffuse interface models, showing that only one accurately represents capillary stress while the other relates to non-equilibrium processes, with implications for fluid flow simulations.

Contribution

The study theoretically differentiates two common interfacial stress formulations, establishing that only the gradient-based formulation correctly models capillary stress in diffuse interface methods.

Findings

Only the density gradient formulation represents capillary stress.

The density-based formulation corresponds to non-equilibrium irreversible stress.

Both formulations yield identical velocity fields in incompressible one-component flows.

Abstract

To express the capillary stress in the diffuse interface method, there are two different formulations in the literature: one formulation is proportional to the density and the other is to the density gradient. Confusingly, these two apparently different formulations both have been widely used for the same purpose without much attention in the difference. In the present study, we theoretically show that only the latter represents the capillary stress and the former represents the stress due to the non-equilibrium irreversible process in the fluid. The formulations are analyzed not only for the one-component fluid but also for the multi-component fluid. In the prediction of the incompressible flow of one-component fluids, the two formulations turn out to give the identical velocity field, but not the pressure field even in this case.