Corotational Hookean Models of Dilute Polymeric Fluids: Existence of Global Weak Solutions, Weak-Strong Uniqueness, Equilibration, and Macroscopic Closure

Abstract

We prove the existence of global weak solutions to the corotational Hookean dumbbell model, a system of PDEs arising in the kinetic theory of dilute polymers, involving the unsteady incompressible Navier–Stokes equations in a bounded domain coupled to a Fokker–Planck type parabolic equation including a center-of-mass diffusion term, satisfied by the probability density function, modeling the evolution of the configuration of noninteracting polymer molecules in a viscous incompressible solvent. The micro-macro interaction is manifested by the presence of a corotational drag term in the Fokker–Planck equation and the divergence of a polymeric extra-stress tensor on the right-hand side of the Navier–Stokes momentum equation. We also analyze certain properties of weak solutions to this system of PDEs: we use the relative energy method to deduce a weak-strong uniqueness type result, and derive the macroscopic closure of the kinetic model; a corotational Oldroyd-B model with stress-diffusion. Finally, we discuss the existence and uniqueness of global weak solutions to this class of corotational Oldroyd-B models with stress-diffusion.