Unsteady flows of heat-conducting non-Newtonian fluids in~Musielak-Orlicz spaces

TL;DR

This paper proves the existence of weak solutions for unsteady heat-conducting non-Newtonian fluids with complex rheology in Musielak-Orlicz spaces, accommodating general growth conditions without small initial data assumptions.

Contribution

It introduces a novel framework using Musielak-Orlicz spaces to handle general stress tensor growth in non-Newtonian fluid flow, extending previous models.

Findings

Existence of weak solutions established for complex non-Newtonian flows.

No restrictions on initial data size for long-time existence.

Application of advanced mathematical techniques like Young measures.

Abstract

Our purpose is to show the existence of weak solutions for unsteady flow of non-Newtonian incompressible nonhomogeneous, heat-conducting fluids with generalised form of the stress tensor without any restriction on its upper growth. Motivated by fluids of nonstandard rheology we focus on the general form of growth conditions for the stress tensor which makes anisotropic Musielak-Orlicz spaces a suitable function space for the considered problem. We do not assume any smallness condition on initial data in order to obtain long-time existence. Within the proof we use monotonicity methods, integration by parts adapted to nonreflexive spaces and Young measure techniques