Page 338 - Process Modelling and Simulation With Finite Element Methods

Electrokinetic Flow 325

Ths now defines the mode wcu and wcv and dependent variables Imu and lmv.
We could do this slightly tidier with one weak boundary constraint with two
variables. Back in the FEMLAB GUI main window, select Boundary Mode
and Boundary Settings for mode wcu and then mode wcv.

Boundary Mode and Boundary Settings
Mode wcu. Select domain 2 and 3, check
active in this subdomain, type ‘u’ into the
constraint variable entry box, and Apply
Mode wcv. Select domain 2 and 3, check
active in this subdomain, type ‘v’ into the
constraint variable entry box, and Apply
OK
NOW it is safe to click on the Solve (=) toolbar button. It still takes some
substantial time to make progress in this model - the coupling does not help the
sparseness of the matrix assembled - but timestepping does proceed to solution
in 7 minutes. A Pentium I11 866Hz produced the first output time step in 4
minutes.
Figure 9.3 shows all the information rolled up into one plot for the final
time t=l. By this time, all streamlines are parallel and velocity vectors
uniform - flat profile. The spreading of concentration and speeding up of
the flow are all driven by the electric field, which is now apparently uniform
in magnitude. A few cross plots (see Figure 9.4) show that the steady state
electric field relaxes its transients within the first output time and remains
constant thereafter (phi is linear for all times after t=0.1). As expected,
electrokmetic flow is dragged along by its boundary layer coupling to the
electric field.
But why did this recipe work? Of course we tried everything we could think
of. For instance, we tried adding an additional time dependence in the
electrostatic potential equation, da=0.001, as an attempt to overcome the
stiffness of instantaneous relaxation to electrostatic equilibrium. But the final
result uses weak boundary conditions for the side wall Navier-Stokes velocities
which are linearly coupled to the electric field, but not for the outlet species
condition which is nonlinearly coupled to concentration and electric field. We
tried some variations on the species mode:
Trial 1 : No weak boundary constraint (general form) - apparently fine
Trial 2: Weak boundary constraint (general form) - does not work
Trial 3: Weak boundary constraint (weak form) - does not work