= Self-similar spreading of a viscous oil slick
For external pressure $\Delta\rho\,gh$ and edge condition $h\sigma_{rr}=-\Delta\rho\,gh^2/2$, the fixed-volume <similarity solution> is a uniformly thick disc:
$$
h=\frac{6\mu}{\Delta\rho\,gt},\qquad u=\frac r{2t},\qquad R^2=\frac{\Delta\rho\,gVt}{6\pi\mu}.
$$
<Conservation of mass> fixes the similarity velocity. The radial equation then forces a differentiable thickness profile to be constant, and the edge <stress> fixes that constant. The singular point-release solution describes the spreading scaling; a finite initial uniform disc corresponds to shifting time.
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