I'm currently looking at some old questions from my undergraduate studies which I may not have fully understood but would like to understand now.
The initial stage of a dispersal process is very complicated. However, after a time scale of order $h^{2}/k$, the concentration profile in the vertical direction will settle down; for simplicity, assume that for this preliminary model, the concentration is independent of z, so $c(x,y,z,t) = c(x,y,t)$ and that the fall -out is negligible.
Under the prescribed circumstance, c(x,y,z)=c(r,t) where
$r=\sqrt{x^{2}+y^{2}}$
so that it's appropriate to use cylindrical form of the Laplacian and the dispersal is governed by
$\frac{\partial c}{\partial t}=k\left ( \frac{\partial^2 c}{\partial r^2} +\frac{1}{r}\frac{\partial c}{\partial r}\right )$
Question: Use scaling arguments to show that the similarity solution form is given by
$c\left ( r,t \right )=\frac{M}{kht}C\left ( \xi \right )$
I am unfamiliar with scaling arguments. A good explanation to the above question would be very helpful.
Thanks in advance.