Page 347 - Process Modelling and Simulation With Finite Element Methods

334 Process Modelling and Simulation with Finite Element Methods

Now select the Restart button on the toolbar. The initial condition now has the
pseudosteady velocity and electric fields set up. After the solution, which takes
about six minutes on a PI11 866MHz Intel machine, use Post mode/Plot
parameters to set up the surface color plot of species Y, the contour plot for
potential lines, and the arrow plot for velocity vectors. This should resemble
Figure 9.8. Then animate and watch the streams merge and flow out. We can see
that the final velocity and potential fields are unchanged from those set up in the
initial “fast elliptic solver” step. This would not be true if the conductive
properties of the fluid or viscosity were concentration dependent. The local
concentration profile would then globally affect the potential field, thus
modifying the velocity field, which in turn disperses the fluid differently. Such
highly coupled electrokinetic flows are well suited to numerical analysis by a
modest change to this coding - adding the concentration coupling to the
conductivity according to a mixture rule. Zirnmerman and Homsy treat the
mixture rule for concentration dependency of viscosity, for instance, in the
instability of viscous fingering in porous media [2].

Y-Junction Switcher: An Application of Linkages Through Coupling Variables
As described in Figure 9.6, by including relatively long channels to the
reservoirs of the component species where electric potentials are applied, the
flow and the electric potential can be described by algebraic relations for the
inlet and outlet dependencies. We implement these physical linkages between
nodes a and A, for instance, using the appropriately named coupling variables.
Originally, we set up a pseudo 0-D geometry (geom2 as in Chapters four and
seven), but on advice from COMSOL consultants, realized that the conceptual 0-
D domain was not needed - it can all be done in coupling variables. Using the
model MAT-file of the last section, set up the following twelve coupling
variables:

Select AddEdit Coupling Variables from the Options Menu.

scalar add Qa. Source Geom 1, subdomain 1, boundary 3.
Integrand: zetal *(PHIA-volta)/dsa-Re*(PA-bara)/f/dsa
Integration order: 2
Destination Geom 1 bnd 3 Check “Active in this domain” box.
scalar add Qb. Source Geom 1, subdomain 1, boundary 1.
Integrand: zeta 1 *(PHIB-voltb)/dsb-Re*(PB-barb)/f/dsb
Integration order: 2
Destination Geom 1 bnd 1 Check “Active in this domain” box.
scalar add Qc. Source Geom 1, subdomain 1, boundary 9.
Integrand: zetal “(voltc-PHIC)/dsc-Re*(PC-barc)/f/dsc