Page 240 - Process Modelling and Simulation With Finite Element Methods

Geometric Continuation 227

Exercise 6.3: Platelet geometric continuation
(a) Change the top boundary condition of the orifice plate to be a symmetry
boundary condition. This models a two-dimensional platelet with viscous flow
past it. Try geometric continuation.
(b) Alter your m-file to use the solution to the last geometric configuration as the
initial condition for the next. Does your m-file finish executing faster?
(c) This example isn’t really multiphysics. Try adding the streamfunction-
vorticity equation as in the buoyant convection example so as to compute
streamlines.
The platelet problem was studied by Kim [I31 with an analytically determined
long perturbation series that was summed to yield the singular behavior of the
drag force as the gap width becomes small.

6.3 Transient Geometric Continuation: Film Drying

In the previous section, geometric continuation did not require using the
previous solution with a different geometry, varied slightly, as an initial
condition for the new solution. Geometric continuation was carried out for the
obvious reason of exploring the model for a range of geometric parameters that
alter the domain. In this section, the solution of a transient problem is posed in
the case that domain is changing over time, so the solution at the previous time is
essential for the prediction of the solution at the current time. The application is
to film drying. The model here is an idealization of experiments on film drying
reported by Mallegol et al. [14].
Figure 6.8 gives the definition sketch of the film drying process. A thin film
of liquid containing particles at an initial volume fraction of is subject to
evaporation from the top surface at a constant rate. If diffusion of particles
throughout the film is small an accumulation at the top surface is observed, with
particles packing at a volume fraction &. Over time the thickness of the packed
layer above the still fluid layer increases. The overall film thickness decreases
linearly with time, and scaling time with the evaporation rate and initial film
thickness allows the film surface to be described by = 1 - 7. A simple
top
mass balance gives that the compaction front moves at a velocity a, given by
a= @m
@m -@o
It follows that no further compaction can take place after time 7 = 1 /a , in
which case a steady film thickness is reached.