Finite Element Methods for Deterministic Simulation of Polymeric Fluids

In this work we consider a nite element approach to solving the coupled Navier Stokes (NS) and Fokker Planck (FP) multiscale model that describes the dynamics of dilute polymeric uids. Deterministic approaches such as ours have not received much attention in the literature because they present a formidable computational challenge due to the fact that the analytical solution of the Fokker Planck equation may be a function of a large number of independent variables. For instance, to simulate a 3-dimensional ow using the dumbbell model for polymers one must solve the coupled NS FP system in which the Fokker Planck equation is posed in a 6-dimensional domain. First we discuss the physical and mathematical foundations of the NS FP model, then we develop our deterministic nite element based approach in detail. Numerical results, obtained using parallel computation, are presented to demonstrate the e cacy of our approach.