Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 84 / 1 / b / Solution

For a concentrated line release, take $V$ to be actual fluid volume per unit source length and allow symmetric spreading to both sides of $x=0$. PutSubstitution in the porous medium equation gives the exponential drainage transform for porous-medium diffusion:The Barenblatt solution has conserved area ${\int }_{-\infty }^{\infty }Hdx=V/\varphi$. A quadratic cap satisfies this equation:Indeed, inside the current $H_x=-x/(3Ds)$ and $R\propto s^{1/3}$; differentiating verifies $H_s=D(H{H}_x{)}_x$. Its area is $\int Hdx=2R^3/(9Ds)=V/\varphi$, fixing the prefactor. The flux vanishes at the two moving noses because $H=0$ there. Hence the remaining physical volume and occupied width areFor a one-sided current on a reflecting half-line, replace the $2$ in $R^3$ by $1$ when $V$ denotes the total release into that side. Volume alone does not select this exact solution for an arbitrary finite initial patch; it describes an impulsive line release. Its initial central height is singular, so a finite injection region and the constraint $h\ll w$ regularize the near-source early-time behavior.