Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 38B (Fluid Dynamics II)

Let $x$ point down the plane and $z$ normal to it, with velocity $(u,w)$. The two-dimensional incompressible Navier-Stokes equation isThe lubrication theory limit has film thickness much smaller than its streamwise length. Transverse viscous derivatives dominate, inertia is negligible at late times, and the leading equations areAt the wall, $u=w=0$ by the no-slip boundary condition. At the free surface $z=h(x,t)$, take $p=p_{\mathrm{atm}}$ and $u_z=0$, neglecting surface tension. The normal balance gives the hydrostatic pressureIf the current has thickness scale $H$ and length scale $L$, thenThus the streamwise pressure gradient is asymptotically smaller than the downslope body force for a fixed nonzero inclination. The streamwise equation becomesUsing $u(0)=0$ and $u_z(h)=0$ givesThe volume flux isIntegrating incompressibility across the film and using the kinematic boundary condition gives $h_t+q_x=0$, hence

WriteConservation of the two-dimensional volume $V=\int hdx$ suggestsSubstitution into the scalar conservation law $h_t+(C{h}^3{)}_x=0$ givesThe integration constant is zero at the dry rear, soThusThe jump from $h_N$ to the dry region is a shock. Its Rankine-Hugoniot condition gives ${\dot{x}}_N=C{h}_N^2$, which is also satisfied by the similarity profile.

Volume conservation determines the front:Therefore the finite-volume gravity-driven thin-film current hasand its head height is

Let the supplied volume per unit width be $V(t)$. The governing balance gives the scalesEliminating $x$ yieldsFor the head height to approach a nonzero constant $h_{\infty }$, the total supplied volume must therefore grow linearly:This is the constant-head supply for a gravity-driven thin film.