Page 340 - Process Modelling and Simulation With Finite Element Methods

Electrokinetic Flow 327

they contribute to the stiffness matrix and residual instead of the constraint
matrices [l]. So one expects that the outlet species concentration, though
nonlinear, may be treated satisfactorily by the standard handling of constraints.
Very simply, as a Neumann condition, it does not count as a weak boundary
constraint - it is naturally in FEM (see chapter 2) so it automatically is treated
correctly.
(2) Implementing constraints using derivatives
In FEMLAB 2.2 and later, the derivatives of the dependent variables are
available also on the boundary. Constraints on only the tangential component of
the derivative work when using standard constraints, whereas here it is necessary
to use a weak constraint to be able to handle non-tangential constraints (the
velocity BCs on the walls).

Condition (3) is clearly satisfied, yet condition (2) is not violated with the caveat
that Neumann constraints do not count. We should not need to use a weak
boundary constraint on the outlet boundary (bnd 4) for the species transport
equation, we did not, and it works. When we tried a weak boundary constraint,
it failed.
From (9.6), we defined for our species general mode

as the straightforward way of dealing with the electrophoretic term.
Consequently, our boundary condition on the outlet takes the form

(9.9)
Equation (9.9) is a non-zero Neumann condition with regard to the flux I-. But
since Neumann conditions do not count as constraints, the standard BC works
fine.
Figure 9.5 shows convincingly that the expected value with uniform Y and phix
on the outlet boundary is achieved by the model at all times.

9.3.4 Links to physical boundaries

Current microchannel devices may consist of many distinct channel segments
joined at several junctions. Future ones may well comprise hundreds of
segments joined at a similar number of junctions. Detailed computation of the
flow in such a system is unlikely to be feasible for some time to come and,
indeed, is probably not desirable. Rather, an approach in which a particular
junction of interest or perhaps an evolving mixing zone such as that considered
in MacInnes et al. (2003) is probably appropriate. The approach emerges from