Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 38C (Fluid Dynamics II)

Balancing the downslope gravitational force density $\rho g\sin \alpha$ against the viscous term $\mu u_{yy}$ gives the velocity scaleIf the streamwise variation scale is $L\sim h/|h_x|$, the lubrication theory requirement is firstThe ratio of streamwise inertia $\rho U^2/L$ to transverse viscous stress $\mu U/h^2$ isThus self-consistency also requires

The leading lubrication momentum equations and incompressibility condition areAt the solid plane $y=0$, no slip and no penetration give $u=v=0$. At the free surface $y=h(x,t)$, neglecting surface tension and ambient viscous stress givesto leading order. The kinematic boundary condition for a free-surface graph is

Integrating the normal momentum equation from the free surface gives the hydrostatic pressureThus $p_x=\rho g\cos \alpha h_x$. Twice integrating the downslope equation and applying $u(0)=0$ and $u_y(h)=0$ gives

The volume flux per unit transverse width isIntegrating incompressibility across the layer and using the wall and free-surface kinematic conditions gives local mass conservation $h_t+q_x=0$. Hence the gravity-driven thin film on an incline obeys

Put $z=x-ct$ and $h(x,t)=f(z)$. The conservation law becomesOne integration gives the required first-order autonomous equationwhere $C$ is a constant fixed by the far-field film heights or other boundary data.