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Analysis and approximation of implicitly constituted non-Newtonian fluid flow models

Banff International Research Station for Mathematical Innovation and Discovery

Classical models describing the motion of Newtonian fluids, such as water, rely on the assumption that the Cauchy stress is a linear function of the symmetric part of the velocity gradient of the fluid. This assumption leads to the Navier-Stokes equations. It is known however that the framework of classical continuum mechanics, built upon an explicit constitutive equation for the Cauchy stress, is too narrow to describe inelastic behavior of solid-like materials or viscoelastic properties of materials. Our starting point in this work is therefore a generalization of the classical framework of continuum mechanics, called the implicit constitutive theory, which was proposed recently in a series of papers by K.R. Rajagopal. The underlying principle of implicit constitutive theory in the context of viscous flows is the following: instead of demanding that the Cauchy stress is an explicit (and, in particular, linear) function of the symmetric part of the velocity gradient, one may allow a nonlinear, implicit and not necessarily continuous relationship between these quantities. The resulting general theory therefore admits non-Newtonian fluid flow models with implicit and possibly discontinuous power-law-like rheology. </p>

We develop the analysis of finite element approximations of implicit power-law-like models for viscous in-compressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi-valued, maximal monotone graph. Using a variety of weak compactness techniques, we show that a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the discretisation parameter, measuring the granularity of the finite element triangulation, tends to zero. A key new technical tool in our analysis is a finite element counterpart of the Acerbi-Fusco Lipschitz truncation of Sobolev functions. </p>

The talk is based on a series of recent papers with Lars Diening and Tabea Tscherpel (Bielefeld), Christian Kreuzer (Dortmund), Alexei Gazca Orozco (Erlangen) and Patrick Farrell (Oxford). </p>

Mathematics; Partial differential equations; Fluid mechanics; Fluid dynamics

video/mp4

Author affiliation: University of Oxford

2021-05-24

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/78424

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/