On a regularized family of models for homogeneous incompressible two-phase flows

TL;DR

This paper introduces a unified analysis of a broad family of regularized two-phase flow models, including Navier-Stokes variants, on Riemannian manifolds, establishing existence, stability, attractors, and convergence results.

Contribution

It provides a comprehensive framework for analyzing various regularized two-phase flow models using abstract operator properties, with new results on attractors and convergence.

Findings

Existence, stability, and regularity of solutions established.

Global and exponential attractors proven for the models.

Convergence to equilibrium under specific conditions demonstrated.

Abstract

We consider a general family of regularized models for incompressible two-phase flows based on the Allen-Cahn formulation in n-dimensional compact Riemannian manifolds for n=2,3. The system we consider consists of a regularized family of Navier-Stokes equations (including the Navier-Stokes-{\alpha}-like model, the Leray-{\alpha} model, the Modified Leray-{\alpha} model, the Simplified Bardina model, the Navier-Stokes-Voight model and the Navier-Stokes model) for the fluid velocity suitably coupled with a convective Allen-Cahn equation for the (phase) order parameter. We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parametrizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability and regularity results, and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the {\alpha}->0 limit in {\alpha}-models. Then, we also show the existence of a global attractor and exponential attractor for our general model, and then establish precise conditions under which each trajectory converges to a single equilibrium by means of a LS inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier-Stokes equations and magnetohydrodynamics models which improve and complement the results of Holst et. al. [J. Nonlinear Science 20, 2010, 523-567]. Finally, our analysis is applied to certain regularized Ericksen-Leslie (RSEL) models for the hydrodynamics of liquid crystals in n-dimensional compact Riemannian manifolds.