Fokker-Planck-based methods for flows of dilute polymeric solutions described by dumbbell models

Some viscoelastic constitutive models, such as the Oldroyd B model, may be described in two ways: a closed-form differential constitutive equation and a kinetic theory description. There are, however, many other constitutive models of polymeric liquids that allow only the latter form. These (mesoscopic) models are generally regarded as being potentially more realistic than their closed-form counterparts but their numerical simulation may require much more work. Mathematically, these models can be written in two equivalent forms: either as stochastic differential equations or as deterministic Fokker-Planck (FP) equations. The first option gives rise to stochastic numerical methods, which have become very popular during the last 10 years (CONNFFESSIT method, Brownian configuration fields etc.) The second option is relatively unexploited. In this seminar we will consider the application of FP-based methods to the solution of flows of dilute polymeric solutions where the polymers are represented as FENE dumbbells. The seminar is divided into two parts: (i) We begin by describing some FP-based methods with the usual homogeneous flow assumption (over an ensemble of dumbbells) that enables the velocity of a fluid at any point in the flow domain to be written as the linear part of a Taylor series about a reference point (ie the velocity gradient is a constant). We note the considerable saving in CPU time over conventional stochastic techniques that is realisable for low-order configurational space. (ii) We then consider the consequences of a departure from the usual homogeneous flow assumption. This leads to an FP equation for the configurational distribution function (cdf) with diffusion terms in both real space and configurational space. Thus, unlike the case of homogeneous flows, boundary conditions on the cdf must be found. The modified Fokker-Planck equation for the cdf is solved in both physical and configurational space with appropriate boundary conditions and proper account is taken of the fact that configurational space will change as a function of physical position. On this latter point, it has usually been assumed that for dumbbells in two-dimensional flow the configurational space is a disc with radius the maximum extensibility of the dumbbell. However it is clear that within a molecule distance of a physical solid boundary the dumbbell (or, more realistically, the chain) is restricted in the configurations that it may assume. The seminar will conclude with a brief overview of extensions of the above methods to high-order configurational space and to the simulation of flows of melts and concentrated polymeric solutions.