Kevin -
- Should I model the last term in the temperature (T) equations as a convection
term or explicit source term?
The best coupling would be obtained by treating this term as a convection term
on n with a velocity proportional to \partial T/\partial x. Unfortunately,
(D/n) sits outside the derivatives and so FiPy has no implicit way to represent
it. I think you're stuck with the source.
- What about the "S_Z" term in the Z equation?
Again, as written, these are just sources.
- Is it required that I declare diffusivity (D) as a cell variable, as it is
now?
In order to write an equation to solve for D, it must be a CellVariable,
however, there's no advantage in adding this as a solution equation. I would
just define D as a function of Z. As such, it's better to define it as a
FaceVariable for use in the DiffusionTerms.
Note:
The expressions Z*(Z.grad) and (Z.grad)**2 amount to multiplying a scalar by a
vector and a vector by a vector, but D needs to be scalar. I recommend working
out what your equations would be using either divergence and gradient operators
or Einstein notation, as premature introduction of d/dx in 1D equations leads
to equations that are difficult to debug and render properly.
You may find that when you go back to the source equations in general vectorial
form that the sources in the T and Z equations are amenable to being written as
convection terms or implicit sources, which would be good.
- Jon
> On Dec 11, 2017, at 2:42 PM, Kevin Blondino <kablond...@gmail.com> wrote:
>
> Hello,
>
> I have a somewhat complicated system of highly nonlinear, 1D PDE's which I am
> having trouble modeling. The issue is that I am not sure how to represent
> certain terms.
>
> Here is a the following LaTeX code for the system, as to make sure there is
> no ambiguity:
> \begin{align}
> \frac{\partial n}{\partial t} \,&=\, \frac{\partial}{\partial x}\left(D
> \frac{\partial n}{\partial x}\right) \\
> \frac{\partial T}{\partial t} \,&=\, \frac{\partial}{\partial x}\left(D
> \frac{\partial T}{\partial x}\right) + \frac{D}{n}\,\frac{\partial
> n}{\partial x}\,\frac{\partial T}{\partial x} \\
> \frac{\partial Z}{\partial t} \,&=\, \frac{\partial}{\partial x}\left(D
> \frac{\partial Z}{\partial x}\right) + \frac{T}{n^2}\frac{\partial
> n}{\partial x} + \frac{1}{n}\frac{\partial T}{\partial x} + G(Z) \\
> G(Z) \,&=\, a + b(Z - Z_S) + c(Z - Z_S)^3 \\
> D \,&=\, 1 + \frac{1}{1 + Z^2 + Z\cdot\frac{\partial Z}{\partial x} +
> \left(\frac{\partial Z}{\partial x}\right)^2}
> \end{align}
>
> I have purposefully squelched many coefficients as possible to make simple.
>
> The main code I have been using to model them is as follows:
>
> # ----------------- Variable Declarations -----------------
> density = CellVariable(name=r"$n$", mesh=mesh, hasOld=True)
>
> temperature = CellVariable(name=r"$T$", mesh=mesh, hasOld=True)
>
> Z = CellVariable(name=r"$Z$", mesh=mesh, hasOld=True)
>
> ## Diffusivity
> D = CellVariable(name=r"$D$", mesh=mesh, hasOld=True)
>
> ... (initial and boundary conditions are here) ...
>
> # ----------------- PDE Declarations ----------------------
> # Diffusivity Equation (D)
> diffusivity_equation = ImplicitSourceTerm(coeff=1.0, var=D) == 1.0 +
> 1.0 / (1 + Z**2 + Z*(Z.grad) + (Z.grad)**2)
>
>
> # Density Equation (n)
> density_equation = TransientTerm(var=density) == DiffusionTerm(coeff=D,
> var=density)
>
>
> # Temperature Equation (T)
> S_T = (D/density) * numerix.dot(density.grad,temperature.grad) # ???
> S_T_conv = ConvectionTerm(coeff=(D/density)*density.grad,
> var=temperature) # ??
>
> temp_equation = TransientTerm(var=temperature) ==
> DiffusionTerm(coeff=D, var=temperature) + S_T
>
>
> # Z Equation
> G = a + b*(Z - Z_S) + c*(Z - Z_S)**3
> S_Z = G + (1 / density)*temperature.grad.mag + (temperature /
> density**2)*density.grad.mag # ???
> Z_equation = TransientTerm(coeff=1.0, var=Z) == DiffusionTerm(coeff=D,
> var=Z) + S_Z
>
> # Fully-Coupled Equation
> full_equation = density_equation & temp_equation & Z_equation &
> diffusivity_equation
>
> Should I model the last term in the temperature (T) equations as a convection
> term or explicit source term?
> What about the "S_Z" term in the Z equation?
> Is it required that I declare diffusivity (D) as a cell variable, as it is
> now?
>
> I appreciate any help that can be given.
>
> Thank you,
> Kevin
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