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Two phase micropolar fluid flow with unmatched densities modeled by Navier--Stokes--Cahn--Hilliard systems: Local strong well-posedness and consistency estimates

Kin Shing Chan, Kei Fong Lam

TL;DR

The paper develops a thermodynamically consistent MAGG phase-field model for binary mixtures of micropolar fluids with unmatched densities and proves local-in-time strong well-posedness in 3D under standard BCs. It uses a Faedo–Galerkin approximation coupled with a Schauder fixed-point argument, complemented by uniform a priori and high-order estimates to pass to the limit, yielding a local strong solution with a precise regularity and a strict separation property for the phase field. A central novelty is the nonpolar limit analysis: as the micro-rotation viscosity tends to zero, MAGG consistently reduces to AGG, with quantitative estimates; further, when densities are equal, the model consistently connects to Model H, with explicit bounds in terms of density differences and $oldsymbol{ u}$-viscosity. The results extend the NS–CH framework to micropolar mixtures, quantify the influence of micro-rotations, and link the new model to classical phase-field theories, providing a rigorous foundation for binary micropolar fluids in three dimensions.

Abstract

We study a thermodynamically consistent phase field model for binary mixtures of micropolar fluids, i.e., fluids exhibiting internal rotations. Furnishing with classical no-slip, no-spin and no-flux boundary conditions, in a smooth and bounded three-dimensional domain, we establish the well-posedness of local-in-time strong solutions. Since the model studied is a generalization of the earlier model introduced by Abels, Garcke and Grün for binary Newtonian fluids with unmatched densities, we provide a consistency result between the corresponding strong solutions to both models in terms of a parameter associated to the micro-rotation viscosity.

Two phase micropolar fluid flow with unmatched densities modeled by Navier--Stokes--Cahn--Hilliard systems: Local strong well-posedness and consistency estimates

TL;DR

The paper develops a thermodynamically consistent MAGG phase-field model for binary mixtures of micropolar fluids with unmatched densities and proves local-in-time strong well-posedness in 3D under standard BCs. It uses a Faedo–Galerkin approximation coupled with a Schauder fixed-point argument, complemented by uniform a priori and high-order estimates to pass to the limit, yielding a local strong solution with a precise regularity and a strict separation property for the phase field. A central novelty is the nonpolar limit analysis: as the micro-rotation viscosity tends to zero, MAGG consistently reduces to AGG, with quantitative estimates; further, when densities are equal, the model consistently connects to Model H, with explicit bounds in terms of density differences and -viscosity. The results extend the NS–CH framework to micropolar mixtures, quantify the influence of micro-rotations, and link the new model to classical phase-field theories, providing a rigorous foundation for binary micropolar fluids in three dimensions.

Abstract

We study a thermodynamically consistent phase field model for binary mixtures of micropolar fluids, i.e., fluids exhibiting internal rotations. Furnishing with classical no-slip, no-spin and no-flux boundary conditions, in a smooth and bounded three-dimensional domain, we establish the well-posedness of local-in-time strong solutions. Since the model studied is a generalization of the earlier model introduced by Abels, Garcke and Grün for binary Newtonian fluids with unmatched densities, we provide a consistency result between the corresponding strong solutions to both models in terms of a parameter associated to the micro-rotation viscosity.

Paper Structure

This paper contains 22 sections, 9 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Preliminary results
  5. On the convective Cahn--Hilliard equation with logarithmic potentials
  6. Approximation of the initial data
  7. Global weak solutions to the MAGG model
  8. Existence of local-in-time strong solutions
  9. Approximate problem
  10. Galerkin approximation
  11. Estimates on Galerkin solutions
  12. Fixed point argument
  13. Uniform estimates
  14. Standard estimates
  15. High order estimates
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^3$ be a bounded domain with $C^3$ boundary. Assuming the mass density and the viscosity functions are of the form coeff:defn, and the following technical assumptions on the viscosity coefficients: along with initial data then there exists a time $T_{0} > 0$, depending on the norms of the initial data, and a quintuple of functions $(\bm{u},\bm{\omega}, p, \phi, \mu)

Theorems & Definitions (13)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.3: Two-dimensional setting
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Proposition 3.2
  • ...and 3 more