Two phase micropolar fluid flow with nonlocal energies: Existence theory, nonpolar limits and nonlocal-to-local convergence

TL;DR

This paper develops a nonlocal phase field model for micropolar fluids, proving existence, convergence to local models, and consistency between solutions, advancing understanding of nonlocal fluid dynamics.

Contribution

It introduces a thermodynamically consistent nonlocal model for micropolar fluids and proves its well-posedness, convergence to local models, and solution consistency.

Findings

Established global weak solutions in 3D

Proved convergence of nonlocal to local models

Provided solution consistency estimates in 2D

Abstract

We study a nonlocal variant of a thermodynamically consistent phase field model for binary mixtures of micropolar fluids, i.e., fluids exhibiting internal rotations. The model is described by a Navier--Stokes--Cahn--Hilliard system that extends the earlier nonlocal variants of the model introduced by Abels, Garcke and Gr\"un for binary Newtonian fluid mixtures with unmatched densities. We establish the global 3D weak existence and global 2D strong well-posedness, followed by the weak convergence of the nonlocal model to its local counterpart as the nonlocal interaction kernel approaches the Dirac delta distribution. In the two dimensional setting we provide consistency estimates between strong solutions of the nonlocal micropolar model and strong solutions of nonlocal variants of the Abels--Garcke--Gr\"un model and Model H.