Finite Element Methods for Deterministic Simulation of Polymeric Fluids

In this work we consider a finite element approach to solving the coupled Navier–Stokes (NS) and Fokker–Planck (FP) multiscale model that describes the dynamics of dilute polymeric fluids. 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 non-homogeneous flow one must solve the coupled NS–FP system in which (for a 3-dimensional flow, using the dumbbell model for polymers) the Fokker–Planck equation is posed in a 6-dimensional domain. In this work we seek to demonstrate the feasibility of our deterministic approach. We begin by dis-cussing the physical and mathematical foundations of the NS–FP model. We then present a literature review of relevant developments in computational rheology and develop our deterministic finite element based approach in detail. Numerical results demonstrating the efficacy of our approach are then given, including some novel results for the simulation of a fully 3-dimensional flow. We utilise parallel computation in performing the larger-scale computations. Key words and phrases: Polymer fluid dynamics, FENE dumbbell, finite element methods, multiscale modelling, parallel computin