Well-posedness for a class of compressible non-Newtonian fluids equations

TL;DR

This paper establishes local-in-time existence and uniqueness of strong solutions for certain non-Newtonian fluid equations, specifically the Power Law and Bingham models, highlighting differences in dimensional applicability due to constitutive law properties.

Contribution

It proves well-posedness for the Power Law model in three dimensions and for the Bingham model in one dimension, addressing challenges posed by discontinuous constitutive laws.

Findings

Existence and uniqueness of solutions for Power Law model in 3D.

Existence and uniqueness of solutions for Bingham model in 1D.

Discontinuity in Bingham law limits higher-dimensional analysis.

Abstract

The purpose of this paper is to deal with the issue of well-posedness for a class of non-Newtonian fluid dynamics equations. These sets of equations are commonly used to describe various complex models that appear in nature, industry, and biology. The equations describing the motion of such fluids are characterized by a non-linear constitutive law relating the state of stress to the rate of deformation. We show the local-in-time existence and uniqueness of strong solutions to two important models: the Power Law model and the Bingham model. While our result for the first model holds over a periodic domain $\Omega=\mathbb{R}^3,$ the result obtained on the second model is limited to the one-dimensional case. This is because Bingham's constitutive law is discontinuous due to phase transition that may appear during the time when flows change nature, particularly from liquid motion to rigid motion and vice-versa. This property reduces the probability of showing smooth solutions to such a system in higher dimension space.