Nondimensionalization of a multi-component chemical diffusion equation

Question

Edit I've modified the equations because they were wrong and I added the whole system, as asked by @Wolfgang

I am trying to nondimensionalize a system of partial differential equations similar to 2nd Fick's law, with multicomponents system. The reason is to make it more stable when I will implement it numerically.

$\begin{align} \frac{\partial \textbf{C}}{\partial t} = \frac{\partial (D \frac{\partial \textbf{C}}{\partial x})}{\partial x} \end{align} $

with $\textbf{C}$, the concentrations beeing a vector and $\textbf{D}$, the matrix of diffusion coefficients.

In my case, I have 4 elements, Mg, Fe, Mn and Ca, and Ca is made dependant on the others. So I have 3 equations:

$\begin{align} \frac{\partial C_{Mg}}{\partial t} = \frac{\partial (D_{MgMg} \frac{\partial C_{Mg}}{\partial x})}{\partial x} + \frac{\partial (D_{MgFe} \frac{\partial C_{Fe}}{\partial x})}{\partial x} + \frac{\partial (D_{MgMn} \frac{\partial C_{Mn}}{\partial x})}{\partial x}\\ \frac{\partial C_{Fe}}{\partial t} = \frac{\partial (D_{FeMg} \frac{\partial C_{Mg}}{\partial x})}{\partial x} + \frac{\partial (D_{FeFe} \frac{\partial C_{Fe}}{\partial x})}{\partial x} + \frac{\partial (D_{FeMn} \frac{\partial C_{Mn}}{\partial x})}{\partial x}\\ \frac{\partial C_{Mn}}{\partial t} = \frac{\partial (D_{MnMg} \frac{\partial C_{Mg}}{\partial x})}{\partial x} + \frac{\partial (D_{MnFe} \frac{\partial C_{Fe}}{\partial x})}{\partial x} + \frac{\partial (D_{MnMn} \frac{\partial C_{Mn}}{\partial x})}{\partial x} \end{align} $

with

$\begin{align} \textbf{D} = \begin{bmatrix} D_{MgMg} & D_{MgFe} & D_{MgMn} \\ D_{FeMg} & D_{FeFe} & D_{FeMn} \\ D_{MnMg} & D_{MnFe} & D_{MnMn}\\ \end{bmatrix} \end{align} $

that depends on $x$, $t$ and $C$.

This matrix is computed at each time-step for each point because it depends on the concentration of the points and on experimental data (I ommit the formula here because I don't think it is relevant)

The concentrations are already nondimensionalize as $C_{Mg} + C_{Fe} + C_{Mn} + C_{Ca} = 1$.

I need to nondimensionalize $t$, $x$, and $\textbf{D}$ with these relationships:

$\begin{align} \begin{array}{cc} t^* = \frac{t}{\tau} & x^* = \frac{x}{\chi} & \textbf{D}^* = \textbf{D} \times \frac{\tau}{\chi^2} \\ \end{array} \end{align} $

I think that for the characteristic length, I can take the length of my model.

What is commonly done with 2nd Fick's law is to fix $D^*=1$ so that we can determine the value of $\tau$ from it. The problem is that here D is a matrix. So what should I do? My idea would be to take the maximum value of $\textbf{D}$ at the initial timestep to be equal to 1 and to make $\tau$ depends on that.

Would that work? The problem is that there is in some case quite a big difference between the values in D, so I am not sure it is the way to go. Maybe there is something smarter to do with the eigen values or eigen vectors of $\textbf{D}$? I am also not sure if I can do it like this with $\textbf{D}$ beeing dependant on x.

Do you have other suggestions?

Thanks.

Answer 1

The important part of $D$ are not its entries (because they depend on the choice of basis vectors you choose -- whether you work with concentrations of the three elements individually, or linear combinations thereof) but the eigenvalues (which are independent of the choice of basis vectors). In other words, the nondimensionalization should be done based on information that only comes from the eigenvalues -- say, the middle of the three eigenvalues, or their mean, etc.

Of course, if you can compute eigenvalues, you can also compute eigenvectors, and then diagonalize the whole system. You'd end up with three decoupled diffusion processes in that case -- much simpler to solve than to deal with the three coupled ones. Clearly, this will only work if the eigenvectors do not depend on space and time.