Existence theorem for homogeneous incompressible Navier-Stokes equation with variable rheology

TL;DR

This paper proves the long-time existence of weak solutions for a variable rheology Navier-Stokes system modeling pyroclastic flows, combining analytical methods with numerical simulations to explore flow behavior.

Contribution

It introduces a novel existence proof for a complex Bingham-type fluid with variable rheology, relevant to volcanology and gas-particle flows.

Findings

Existence of weak solutions over long times

Numerical simulations illustrating fluidization effects

Insights into stress tensor behavior in variable rheology fluids

Abstract

We look at a homogeneous incompressible fluid with a time and space variable rheology of Bingham type, which is governed by a coupling equation. Such a system is a simplified model for a gas-particle mixture that flows under the influence of gravity. The main application of this kind of model is pyroclastic flows in the context of volcanology. In order to prove long time existence of weak solutions, classical Galerkin approximation method coupled with a priori estimates allows us to partially solve the problem. A difficulty remains with the stress tensor, which must satisfy an implicit constitutive relation. Some numerical simulations of a flow of this type are given in the last section. These numerical experiments highlight the influence of the fluidization phenomenon in the flow.