Analysis and Implementation of Numerical Methods for Simulating Dilute Polymeric Fluids

Abstract

In this thesis we develop, analyse and implement a number of numerical methods for simulating dilute polymeric fluids. We use a well-known model in which the polymeric fluid is represented by a suspension of dumbbells in a Newtonian solvent. This model is governed by a coupled Navier–Stokes–Fokker–Planck system of partial differential equations, in which the Fokker–Planck equation is posed on a high-dimensional domain. We first thoroughly analyse a Galerkin spectral method for the Fokker–Planck equation in \em configuration space, before combining this method with a finite element scheme in \em physical space to obtain an alternating-direction method for the high-dimensional Fokker–Planck equation. Alternating-direction methods have been considered previously in the literature for this problem, but this approach has not been subject to rigorous numerical analysis before. We develop many theoretical results for our numerical algorithms, and we focus particularly on establishing stability and convergence estimates. The numerical methods we develop are fully-practical, and we present a range of numerical results demonstrating their accuracy and efficiency. We also introduce a coupled numerical algorithm for the Navier–Stokes–Fokker–Planck system, which we use to simulate polymeric fluid flow problems of physical interest. The numerical method for the high-dimensional Fokker–Planck equation is the most computationally intensive part of this coupled algorithm, but it is well suited to implementation on a parallel computer, and we exploit this fact to make large-scale computations feasible.