 Squeegee lubrication model

A blade moving over a thin gravity-driven viscous film produces a moving-frame flux $\mathbf{q}=-h{\mathbf{e}}_x-h^3\nabla h/3$. Scaling film thickness by $h_0$ and horizontal distance by $\rho g{h}_0^3/(\mu U)$ yields this dimensionless volume-conservation equation. Fluid can escape beneath the blade or around its ends.