Kinetic Models of Dilute Polymers: Analysis, Approximation and Computation

We review recent analytical and computational results for macroscopic-microscopic beadspring 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 domain Ω ⊂ Rd, 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 Ω × D × [0, T ], where D ⊂ Rd is the configuration domain and [0, T ] is the temporal domain. The Fokker–Planck equation arises from a system of (Itô) stochastic differential equations, which models the evolution of a 2dcomponent 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 highdimensional 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 ∂D. We present numerical simulations for this coupled high-dimensional micro-macro model and we consider the analysis of the algorithms.