John W. Barrett

Existence and Approximation of  Global Weak Solutions to some Regularized Dumbbell Models for Dilute Polymers

We consider the existence of global-in-time weak solutions to a coupled macroscopic-
microscopic bead-spring model with microscopic cut-off, which arises from the kinetic the-
ory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model
consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ Rd,
d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as
the right-hand side in the momentum equation. The extra-stress tensor stems from the ran-
dom movement of the polymer chains and is defined through the associated probability density
function ψ that satisfies a Fokker–Planck-type parabolic equation, crucial features of which are
the presence of a center-of-mass diffusion term and a cut-off function βL(∙) := min(∙, L) in the
drag term, where L ≫ 1. We establish the existence of global-in-time weak solutions to this
model for a general class of spring-force potentials including, in particular, the widely used
finitely extensible nonlinear elastic (FENE) potential.

We construct a fully discrete Galerkin finite element method for the numerical approxi-
mation of this model, which mimics the energy law in the continuous case. We show that
a (sub)sequence of numerical solutions converges to a weak solution of this coupled Navier–
Stokes–Fokker–Planck system as the spatial and temporal discretization parameters tend to zero.

We prove similar existence and approximation results for a corresponding regularized Oldroyd-B model.

The work on the regularized macroscopic/microscopic model is joint with Endre Süli, OUL, University of Oxford.

The work on the regularized Oldroyd-B model is joint with Sébastien Boyaval, CERMICS, Ecole Nationale des Ponts et Chaussées.