Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 329 / 2 / a

Put $b=\Delta \rho g>0$, so $p_{\mathrm{ext}}=bh$. Balancing the viscous and pressure terms in the axisymmetric viscous-sheet stretching equations, together with fixed volume, suggests a similarity solutionThe conservation of mass equation becomesThere is no ongoing source at the origin, so the integration constant vanishes and $U=\eta /2$ wherever $H>0$. Thus $u=r/(2t)$. Substitution into radial force balance givesA differentiable $H$ cannot have nonzero derivative on an interval while being fixed there at $3\mu /b$. Hence $H^{\prime }=0$: the spreading sheet has uniform thickness.

At the material edge, $R^{\prime }=u(R,t)=R/(2t)$. Since ${\sigma }_{rr}=-bh+3\mu /t$, the edge condition $h{\sigma }_{rr}=-b{h}^2/2$ fixes $h=6\mu /(bt)$. Fixed volume $V=\pi R^{2}h$ then fixes the radius. The self-similar spreading of a viscous oil slick isThe point-release idealization is singular at $t=0$; a finite initial uniform slick gives the same solution with a time shift.