= Advection limit of an inclined porous current
{title2=$h=h_0\exp[-(t-\phi x/u)/\tau]$}
With exponentially decreasing injection, the advective approximation $\phi h_t+uh_x=0$ and inlet condition $uh(0,t)=Qe^{-t/\tau}$ give $h_0=Q/u$ and the displayed solution along <characteristic curves>. It applies behind the advective nose $X=ut/\phi$. The relative size of the discarded nonlinear spreading term for this profile is $2\phi h\cot\theta/(u\tau)$. Smallness of that quantity, and matching through a narrow leading layer, are necessary qualifications: elapsed time alone does not make the approximation uniform at the nose. In a translating frame the full equation remains a <porous medium equation> with coefficient $u\cot\theta/\phi$.
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