Equations for Polymeric Materials

Abstract

Systems coupling fluids and polymers are of great interest in many branches of sciences and engineering (applied physics, chemistry, biology,. . .). These systems attempt to describe the behavior of complex mixtures of particles and fluids, and as such, they present numerous challenges, simultaneously at three levels: at the level of their derivation, the level of their numerical simulation, and that of their mathematical treatment. This chapter is devoted to the mathematical treatment after a brief discussion of the derivation of such models. Recent results about existence and uniqueness of strong solutions as well as global existence of weak solutions will be discussed. At the mathematical level, one of the main difficulties comes from the coupling of the Navier-Stokes system with a transport equation for the density of polymers.

Finite Amplitude Stability of Internal Steady Flows of the Giesekus Viscoelastic Rate-Type Fluid

“…In particular, we do not consider the perturbations that solve the governing equation only in the weak sense, although it is an important issue worth of further investigation. The reader interested in the discussion of the state-of-the-art rigorous mathematical theory of equations governing the motion of viscoelastic fluids is kindly referred to the discussion in Masmoudi (2018) or Barrett and Süli (2018).…”

On the stabilizing effect of swimming in an active suspension

“…Complex fluids. Although there is a vast PDE literature on related models without swimming (see the survey [37]), to the authors' knowledge, there is a single PDE work on the model (1.1)- (1.3). In [15], Chen and Liu establish the existence of global weak entropy solutions to (1.1) coupled with the Navier-Stokes equations or the Stokes equations (1.2)-(1.3) for the velocity.…”

“…The first treats the polymers as elastic dumbbells (two beads attached by a spring). We refer to [33,37] for an overview of well-posedness results here. In the second class of models (sometimes called Doi-type), to which the system (1.1)-(1.3) belongs, the immersed polymers are treated as rigid rods.…”