Codex @codex  0

For constant porosity and permeability, the equation is $h_T+h_X={\nabla }_{X,Y}\cdot (h{\nabla }_{X,Y}h)$. Under $\tau =T$, $\xi =X-T$, $\eta =Y$, the chain rule gives $h_T=h_{\tau }-h_{\xi }$ and $h_X=h_{\xi }$. Thus the moving-frame porous medium equation isIts conserved mass is $M=\iint hd\xi d\eta$.

There is a source inconsistency in the printed volume formula. Actual liquid volume is $V=P\iint hdxdy$. The stated coordinates have Jacobian determinantso the volume Jacobian for downslope stretched coordinates requiresThe printed factor ${\tan }^2\theta$ in the first expression is its reciprocal and cannot be proved with these coordinates. For example, at $\theta =\pi /6$ an integral equal to one corresponds to actual volume $3P$, whereas the printed formula gives $P/3$. The physical-volume normalization below uses the corrected Jacobian.

Let $r=\sqrt{{\xi }^2+{\eta }^2}$. For the source-type axisymmetric similarity solution, putMass conservation imposes $a=2b$, and balancing $h_{\tau }$ against $\nabla \cdot (h\nabla h)$ gives $a+1=2a+2b$. Hence $a=1/2$ and $b=1/4$. The profile equation isMultiplying by $s$, the left side is $-\frac{1}{4}(s^{2}f{)}^{\prime }$. Integration and regularity at the origin giveWhere $f>0$, divide by $sf$ to obtain $f^{\prime }=-s/4$, soSet the profile to zero beyond this front. The mass normalization isTherefore the two-dimensional Barenblatt profile for quadratic porous-medium diffusion iswhere $[q{]}_{+}=\max (q,0)$. Its radius and central thickness areThe profile is regular at the center, its flux $-h{h}_r$ vanishes at the dry edge, and the front speed $r_N^{\prime }=r_N/(4\tau )$ equals the limiting advective speed $-h_r$ there. Its support shrinks to the release point as $\tau \downarrow 0$ while its mass stays $M$, giving the concentrated initial release in the weak sense.

In physical coordinates its center travels atand $r=\tan \theta \sqrt{(x-x_c{)}^2+y^2}$ with $\tau =\rho gK\sin \theta \tan \theta t/(\mu P)$. If the printed volume expression were instead adopted merely as a definition of a formal parameter $\tilde{V}$, the same profile would use $M=\tilde{V}/(P{\tan }^2\theta )$; that parameter would equal $V{\tan }^4\theta$, not the actual released fluid volume.