The Langevin and Fokker–Planck Equations in Polymer Rheology

Abstract

Publisher Summary This chapter discusses the applications of Langevin and Fokker–Planck equations in polymer rheology. It presents the stochastic simulation techniques for solving the Langevin equation. It introduces the stochastic differential equations for dilute polymer solutions modeled by dumbbells. Micro-macro techniques for simulating flows of polymeric fluids are discussed in the chapter. These methods are based on coupling macroscopic techniques for solving the conservation equations with microscopic methods for determining the polymeric stress in the fluid. Some of the early attempts to reduce the statistical error in the stochastic simulations without increasing the number of realizations are described in the chapter. Some of the major advances in the development and implementation of micro-macro techniques presented, such as the method of Brownian configuration fields of Hulsen, van Heel, and van den Brule. The chapter also describes efficient implicit schemes for micro-macro simulations developed by Laso, Ramirez, and Picasso. These schemes give rise to a large nonlinear system of algebraic equations for both the macroscopic and microscopic degrees of freedom at each time step with efficiency being achieved using size reduction techniques. A brief account of the solution of stochastic differential equations for linear polymer melts based on the Doi–Edwards model is discussed in the chapter. The deterministic numerical methods based on the Fokker–Planck equation for several kinetic theory models of polymer fluids are discussed in the chapter.