= Gravity-driven spreading of a planar viscous drop
{title2=$\ell(t)=[5gA^3t/(36\nu I^3)]^{1/5}$}
A wide, shallow drop with cross-sectional area $A$ has the gravity-driven <volume flux> $q=-gh^3h_x/(3\nu)$ when substrate no-slip and zero surface shear are imposed. <Mass conservation> yields a <porous medium equation>, $h_t=[g/(3\nu)](h^3h_x)_x$. Its symmetric source-type <similarity solution> has half-width $\ell$ in the display and height $h=[9\nu(\ell^2-x^2)/(10gt)]_+^{1/3}$, where $I=\int_0^{\pi/2}\cos^{5/3}\theta\,d\theta$. Integrating the profile recovers $A$. The contact-edge slope diverges, so the bulk <lubrication approximation> requires an inner edge regularization if contact-line microphysics or <surface tension> is retained.
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