Page 230 - Process Modelling and Simulation With Finite Element Methods

Geometric Continuation 217
method of implementing boundary stress conditions with grid adaptation.
Goodwin and Schowalter [2] have successfully implemented their simultaneous
solution for the position of the mesh with the solution of the flow equations and
boundary conditions using Newton iteration in the treatment of a capillary-
viscous jet using finite element methods. In principle, FEMLAB could also do
the latter, but in practice, the equations for the FEMLAB application modes
would need to be augmented with the residuals for the movement of the grid
positions. Standardizing the methodology for including these extra terms in all
application modes whenever the grid is “active,” i.e. there is a free boundary,
would be a substantial re-write of FEMLAB. Given that the number of models
that require free boundary computations, even in surface tension dominated
flows, is rather few, such a general alteration to the package would not seem
warranted. FIDAP, which does treat free boundary flows, uses the iterative flow
solutiodelliptic mesh regeneration methodology, rather than the simultaneous
Newton iteration. In our transient model in a shrinking domain 46.3, we adopt
the iterative approach to the variation of the geometric domain over time.

6.2 Stationary Geometric Continuation: Pressure Drop in a Channel
with an Orifice Plate

In this section, we consider two related models that require geometric
continuation. They are the orifice plate and the platelet in a duct filled with
viscous fluid. They are related, as in fact there is only a slight change in the
model from one case to the other.

Figure 6.1 Mesh generated for the orifice plate in a duct filled with viscous fluid. The parameter
representing the percentage of blockage is ~40% (aspect ratio).
Although it is possible to consider the calculation of the flow around the orifice
plate at arbitrary Reynolds number, the major effects in laminar flow are similar
to those with artificially vanishing Reynolds number - the Stokes equations. The
fundamental reason for this is that most of the dissipation occurs in the opening
of the orifice plate, where flow is accelerated yet the small gap leads to strong
viscous friction dominating the flow. So to a first approximation, we will model
the momentum transport by the Stokes equations: