= Draining viscous film on a finite horizontal plate
With $K=\rho g/(3\mu)$ and formal edge conditions $h(\pm L,t)=0$, the <lubrication theory> equation is $h_t=K(h^3h_x)_x$. Its positive separated long-time profile has
$$
h(x,t)=\left(\frac{L^2}{Kt}\right)^{1/3}H(x/L),\qquad
(H^3H')'=-\frac H3.
$$
Writing $C=B(4/5,1/2)$ in terms of the <beta function>, the centre height is $h_0^3=10\mu L^2/(\rho gC^2t)$. The <edge region of a draining viscous film> determines how the singular outer profile matches the flow over the physical edge.
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