On May 26, 2006, at 5:30 AM, Damm, Edward F. (E. Buddy) wrote:
If it is, I'd like to understand this better. I can't seem to find
anything in the manual or reference guide that explains that FiPy
includes the concentration in the below equation, and it needs to
therefore be factored out in some cases. If this is something that
is known to one skilled in the art, please say so (or point me to a
reference that might explain it). If it is a unique feature of how
FiPy solves things, could you perhaps elaborate further in future
manual and reference materials. Is this true for all ways of
solving convection (i.e. PowerLaw, Hybrid, etc.). Does the
diffCoeff also need to have concentration factored out?
\begin{equation}\nabla\cdot \left[ConvCoeff\nabla \phi \right] \end
{equation}
There's a \phi in $\partial \phi / \partial t$. Do you put \phi in
the coefficient of a TransientTerm? There's a \phi in $D \nabla^2 \phi
$. Do you put a \phi in the diffusion coefficient? I don't understand
why a ConvectionTerm is such a hangup.
The general equation that FiPy can solve is:
\[
\frac{\partial \phi}{\partial t} = \nabla\cdot(\vec{u} \phi) +
[\nabla\cdot(\Gamma_i)]^n \phi + S_1 phi + S_0
\]
or
\[
\frac{\partial \phi}{\partial t} - \nabla\cdot(\vec{u} \phi) -
[\nabla\cdot(\Gamma_i)]^n \phi - S_1 phi = S_0
\]
or, factoring out everything that *operates* on phi:
\[
\left[\frac{\partial}{\partial t} - \nabla\cdot(\vec{u}) -
[\nabla\cdot(\Gamma_i)]^n - S_1 \right] phi = S_0
\]
*This* is now in the form of a FiPy equation:
>>> TransientTerm() - ConvectionTerm(coeff=u) - DiffusiontTerm
(coeff=(Gamma1, ...)) - ImplicitSourceTerm(coeff=S1) == S0
\phi doesn't appear explicitly in *any* of these expressions; it's
built into the Terms.
