Weak Solutions for a Non-Newtonian Diffuse Interface Model with Different Densities

TL;DR

This paper proves the existence of weak solutions for a complex coupled model describing two non-Newtonian fluids with different densities, using advanced mathematical techniques in a bounded domain.

Contribution

It establishes the existence of weak solutions for a non-Newtonian diffuse interface model with different densities, extending previous results to more general power-law fluids.

Findings

Existence of weak solutions for p > (2d+2)/(d+2)

Application of the $L^inity$-truncation method

Model includes coupled Navier-Stokes and Cahn-Hilliard systems

Abstract

We consider weak solutions for a diffuse interface model of two non-Newtonian viscous, incompressible fluids of power-law type in the case of different densities in a bounded, sufficiently smooth domain. This leads to a coupled system of a nonhomogenouos generalized Navier-Stokes system and a Cahn-Hilliard equation. For the Cahn-Hilliard part a smooth free energy density and a constant, positive mobility is assumed. Using the $L^\infty$-truncation method we prove existence of weak solutions for a power-law exponent $p>\frac{2d+2}{d+2}$, $d=2,3$.