Abstract

We review recent analytical and computational results for macroscopic-microscopic bead-spring models that arise from the kinetic theory of dilute solutions of incompressible polymeric fluids with noninteracting polymer chains, involving the coupling of the unsteady Navier–Stokes system in a bounded $d$-dimensional domain $\Omega$, $d=2$ or 3, with an elastic extra-stress tensor as right-hand side in the momentum equation, and a (possibly degenerate) Fokker—Planck equation over the $(2d+1)$-dimensional region $\Omega \times D \times [0,T]$, where $D \subset \mathbb{R}^d$ is the configuration domain and $[0,T]$ is the temporal domain. The Fokker—Planck equation arises from a system of (It$\hat{\rm o}$) stochastic differential equations, which models the evolution of a $2d$-component vectorial stochastic process comprised by the $d$-component centre-of-mass vector and the $d$-component orientation (or configuration) vector of the polymer chain. We show the existence of global-in-time weak solutions to the coupled Navier—Stokes—Fokker—Planck system for a general class of spring potentials including, in particular, the widely used finitely extensible nonlinear elastic (FENE) potential. The numerical approximation of this high-dimensional coupled system is a formidable computational challenge, complicated by the fact that for practically relevant spring potentials, such as the FENE potential, the drift term in the Fokker—Planck equation is unbounded on $\partial D$. The talk is based on joint work with John W. Barrett (Imperial College London) and David J. Knezevic (Massachusetts Institute of Technology).