= Porous-medium transformation of V-channel spreading
{title2=$w=h^2,\quad w_t=\rho g(w^{5/2})_{xx}/(30\mu)$}
The variable $w=h^2$ converts the <shallow V-channel lubrication flux> equation into a <porous medium equation>:
$$
w_t=\frac C5(w^{5/2})_{xx},\qquad C=\frac{\rho g}{6\mu},\qquad\int_{\mathbb R}w\,dx=\alpha V.
$$
Indeed $h^4h_x=w^{3/2}w_x/2=(w^{5/2})_x/5$. Its fixed-mass source-type solution gives $h\propto t^{-1/7}$ and <support> length proportional to $t^{2/7}$. Integrating the similarity equation once gives $Ch^4h_x=-2xh^2/(7t)$, and hence $h=[3(X(t)^2-x^2)/(7Ct)]_+^{1/3}$. The <constant-volume similarity in a V-shaped channel> fixes $X$ from the positive <integral> $J=\int_{-1}^1(1-s^2)^{2/3}ds$: $X=(\alpha V/J)^{3/7}(7Ct/3)^{2/7}$.
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