Page 254 - Process Modelling and Simulation With Finite Element Methods

Geometric Continuation 24 1

Exercise 6.4: Asymptotic sugactant concentration

An interesting feature of the film drying application that is industrially relevant is
the concentration of surfactant on the bottom surface when the front arrives, as a
function of Pe. This has been commented on by many authors and they try to
control it by varying the substrate chemistry, but our model suggests it is actually
the particle/surfactant adsorption isotherm that controls this. This has many
important ramifications for how to formulate industrial coatings. Write a
MATLAB m-file script by altering fi1mdry.m to store only the final (7 = 1 /a )
surfactant concentration value at k=O with parametric variation from Pe in the
range [1:5:100]. Try the isotherm parameters m=O.7, y0=2 and m=0.7, yo=l.
Plot your data of u at c=O versus Pe for both cases. Is the bottom surfactant
concentration sensitive to the particlehurfactant adsorption isotherm?

6.4 Summary
In this chapter, we explored how FEMLAB can be used to set up simulations
where the geometry model changes smoothly over either a parametric range or
smoothly due to transient evolution of a front. The groundwork for these two
situations was laid with previous discussion of parametric continuation. In
particular, this chapter introduces an operator splitting technique to deal with
transient geometric continuation, with geometry modification occurring during
the first part of the time step, and a PDE being solved during the second part.
The technique was shown to be self-consistent with asymptotic convergence
tested for some parametric values and the simulation parameter - the increment
over which the geometry is modified.
Although not novel, the transient model required re-starting the solution at
one time step with the old solution at the last time increment. Yet, in order to do
this, the old solution must be interpolated onto the new mesh, with potentially
different numbers (and relevance) of the degrees of freedom. asseminit 0 was
found to have sufficient power to do this, with the code supplied by the
FEMLAB GUI programming interpreter, through a model m-file translation. By
now, this should be a common technique for programming MATLAB routines
calling FEMLAB functions - let the FEMLAB GUI provide the right commands.
The transient models for film drying were examples of two types of
geometric continuation. The noncumulative model merely moved the initial
position of the point source compaction front. Each front position was solved
independently for surfactant concentration. The cumulative model read in the
previous solvent concentration profile as its initial condition - the essence of
parametric continuation and of transient integration. The operator splitting
scheme developed here uses the best features of both types of continuation.