Consider the one-dimensional non-linear diffusion equation: $$\frac{∂n}{∂t}=D_0\frac{∂}{∂x}\Bigg\{n^4\frac{∂n}{∂x}\Bigg\}$$
where $t$ is time, $x$ is the spatial coordinate, $D_0$ is the coefficient of diffusion and $n$ is the concentration with $[n] = ML^{−3}.$
- Determine the dimension of $D_0$.
- Bring the equation into non-dimensional form, if it is known that there is a characteristic value of the concentration $n_0$, and a characteristic length scale $L.$
I've been working on these 2 questions for the last couple of hours with no luck. So any help for both answers would be highly appreciated.
For the first part:
$T^{-1}ML^{-3}=[D_0]L^{-1}M^4L^{-12}ML^{-3}L^{-1}$
$T^{-1}ML^{-3}LM^{-4}L^{12}M^{-1}L^3L=[D_0]$
$T^{-1}M^{-4}L^{14}=[D_0]$
For the second:
$\bar D_0=D_0\dfrac{n_0^4}{L^2}$ with $\bar n=n/n_0$ and $\bar x=x/L$
But not completely non-dimensional:
$[\bar D_0]=T^{-1}$