On three dimensional flows of viscoelastic fluids of Giesekus type

TL;DR

This paper proves the existence of weak solutions for three-dimensional Giesekus viscoelastic fluid flows over long times and large data, extending results to higher dimensions and more complex models.

Contribution

It provides the first complete proof of existence for weak solutions of 3D Giesekus fluid flows, including generalizations to higher dimensions and complex viscoelastic models.

Findings

Existence of weak solutions for 3D Giesekus fluids established.

Extended existence results to higher dimensions and models with multiple relaxation mechanisms.

Introduced new auxiliary tools for analyzing biting limits in parabolic settings.

Abstract

Viscoelastic rate-type fluids are popular models of choice in many applications involving flows of fluid-like materials with complex micro-structure. A well-developed mathematical theory for the most of these classical fluid models is however missing. The main purpose of this study is to provide a complete proof of long-time and large-data existence of weak solutions to unsteady internal three-dimensional flows of Giesekus fluids subject to a no-slip boundary condition. As a new auxiliary tool, we provide the identification of certain biting limits in the parabolic setting, presented here within the framework of evolutionary Stokes problems. We also generalize the long-time and large-data existence result to higher dimensions, to viscoelastic models with multiple relaxation mechanisms and to viscoelastic models with different type of dissipation.