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.
