Page 243 - Process Modelling and Simulation With Finite Element Methods

230 Process Modelling and Simulation with Finite Element Methods

It is rather difficult to analyze a two front problem, especially with one an
effective point source that is moving. It makes sense to transform our coordinate
system to remove one, if not both, of the moving fronts. We experimented with
nonlinear coordinate transforms in time to remove both fronts to a fixed domain.
Surprisingly, this was possible, adding some greater complexity to the PDE
(6.9), but it is not physical, as the variation is not monotonic for the transform
coordinate. Better to stick to one front (the internal compaction front) and
transform away the top front to a fixed domain.

<=1
TI
c=+

j=O <=O

Figure 6.10 Coordinate transformation: one front.

The tranformation that achieves this is simple:

(6.1 1)

and results in a new specification for the compaction front:
I-aT
h -- (6.12)
<- 1-7

Differentials are expressed in the new coordinates according to the chain rule:
a 1 a a
-
a - la. - +- (6.13)
ay 1-tag’ at (1-t)ag aT
which results in a transformed PDE for surfactant transport:

(6.14)
The terms are now representative of, on the LHS: accumulation, pseudo-
advection, and quasi-diffusion. On the RHS, the point source remanifests itself.
1
.
The solution is “sensible” for 0 5 < - The boundary and initial conditions
a