I'm attempting to reimplement the model from this paper
<http://eprints.maths.ox.ac.uk/1519/> in order to build a small
extension. Most of the equations are made up of straightforward
DiffusionTerm or ConvectionTerm or source term elements, some of which
I've been able to solve with FiPy in isolation. However, I've
encountered trouble in encoding the vector-valued momentum balance
equation (eq 19 in the paper) for the low-R flow of mixed viscous
fluids.
Given these definitions (which I may have gotten wrong - please tell
me if I have)
mesh = Gmsh2D( . . . )
thetas = [ CellVariable(mesh = mesh, value = initialT[i]) for i in
species ] # Scalar
pressure = [ CellVariable(mesh = mesh, value = initialP[i]) for i in
species ] # Scalar
fluxes = [ FaceVariable(mesh = mesh, value = [[0.0], [0.0]], rank = 1)
for i in species ] # (x, y) vector
I'm trying to represent the following equation in fipy's terms, and
failing to understand how to express it correctly so that fipy can
solve it:
\theta_i \nabla \dot (c_1 pressures[i] I) + \nabla \dot ( theta_i (
c_2 (\nabla \vec{u}_i + (\nabla \vec{u}_i)^T ) + c_3 ( \nabla \dot
\vec{u}_i) I ) ) = 0
I've written 'I' for the identity matrix, and \vec{u} for the flux.
I've read through the FAQ, and the closest thing I've found in the
examples is the Cahn-Hilliard case, and some additional mailing list
messages [1], but I've not been able to figure out how to put the
pieces together properly. From those, it seems like I may need to
expand some of the derivatives explicitly, and possibly include some
higher-order DiffusionTerms, but I'm unsure how to push that all
through. Can someone offer a pointer on how to accomplish this, or
links to some other resource?
Thanks.
Phil
[1] <http://article.gmane.org/gmane.comp.python.fipy/1317> et seq.
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