On unsteady internal flows of incompressible fluids characterized by implicit constitutive equations in the bulk and on the boundary

TL;DR

This paper develops a mathematical framework for unsteady internal flows of incompressible fluids with implicit constitutive relations, extending analysis to complex boundary slip conditions and providing constructive proofs and uniqueness results.

Contribution

It introduces a robust theory for implicit constitutive models in incompressible fluid flows, including boundary slip conditions, with elementary calculus tools and constructive proofs.

Findings

Established existence of weak solutions for implicit fluid models.

Included nonlinear slip and stick-slip boundary conditions.

Provided a constructive approximation scheme and addressed solution uniqueness.

Abstract

Long-time and large-data existence of weak solutions for initial- and boundary-value problems concerning three-dimensional flows of \emph{incompressible} fluids is nowadays available not only for Navier--Stokes fluids but also for various fluid models where the relation between the Cauchy stress tensor and the symmetric part of the velocity gradient is \emph{nonlinear}. The majority of such studies however concerns models where such a dependence is \emph{explicit} (the stress is a function of the velocity gradient), which makes the class of studied models unduly restrictive. The same concerns boundary conditions, or more precisely the slipping mechanisms on the boundary, where the no-slip is still the most preferred condition considered in the literature. Our main objective is to develop a robust mathematical theory for unsteady internal flows of \emph{implicitly constituted} incompressible fluids with implicit relations between the tangential projections of the velocity and the normal traction on the boundary. The theory covers numerous rheological models used in chemistry, biorheology, polymer and food industry as well as in geomechanics. It also includes, as special cases, nonlinear slip as well as stick-slip boundary conditions. Unlike earlier studies, the conditions characterizing admissible classes of constitutive equations are expressed by means of tools of elementary calculus. In addition, a fully constructive proof (approximation scheme) is incorporated. Finally, we focus on the question of uniqueness of such weak solutions.