Page 352 - Process Modelling and Simulation With Finite Element Methods

Electrokinetic Flow 339

The algebraic relations hold roughly for the electric potential, pressure, current,
and velocity, for instance: u, =L=0.790828. Some are spot on --
Jz
-- '-" 0-7S213 = -1.2535 . In general, the electric potential conditions hold
-
A% 6
exceedingly well, but the velocity field/pressure fields do not hold as well.
Again, this is a strong indicator that the solution is not mesh resolved.
Plot (see Figure 9.9) and Animate the solution. The animation shows
clearly how the front evolves to set up a fully developed upper flow, bypassing
the lower leg. The intermediate voltage in the "b-leg" of the Y-junction tends to
hold back the flow of species Y=l from the merging into the main flow along the
(a)-(c) open switch - only slow diffusion out and in, along with some modest
convection, occur. With concentration dependent conductivity (and viscosity
[2]), it is possible to counteract the diffusion to some extent which sharpens
some fronts.

B. Alternation between voltages

Now for this application to be a Y-junction switch, we need to be able to replace
the constant voltages $a and $b with and $b(t), respectively. The signals
could be arbitrary, however, in practice they are discontinuous level adjustments,
which can be idealized as a square wave. A suitable choice is the alternation
between values do and Qbb0 (PHIA and PHIB), with the signals 90" out of phase.
We coded the square wave in coupling variables in two different ways:

Bl. Logical functions
Under AddEdit Coupling Variables, we made the following changes:
switching
Qa
zetal* (PHIA-PHIA* (sin(Z*pi*t) <O) +PHIB* (sin(Z*pi*t) c0) -volts) /
dsa-Re*(PA-bara)/f/dsa
Qb
zetal*(PHIB-PHIB*(sin(Z*pi*t)<O)+PHIA*(Sin(Z*pi*t)~O) -voltb)/
dsb-Re*(PB-barb)/f/dsb
Ia
sigr* (PHIA-PHIA* (sin(2*pi*t)
<O) tPHIB* (sin(2*pi*t) <O) -volts) /dsa
Ib
sigr* (PHIB-PHIB* (sin(2*pi*t) +PHIA* (sin(Z*pi*t) <O) -voltb) /dsb
<O)
The logical statement (sin(Z*pi*t) <o)takes the value of 1 when true (during
the second half period) and 0 when false (during the first half period), which is
the essence of the square wave that makes a discontinuous switch. It is easy to
code. We substituted this code into the above FEMLAB model without success.
The numerical analysis proceeds smoothly until the end of the first half period, at