Squeegee lubrication model ($h_t-h_x=\nabla \cdot (h^3\nabla h)/3$)

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.

A long blade with negligible under-blade leakage diverts incoming fluid laterally. A deep-pile profile $h^3=9[x_N(y)-x]$ and integrated flux conservation give $d(\int h^{4}dx)/dy=-12y$, hence $x_N=[126({\alpha }^2-y^2){]}^{3/7}/9$. This leading approximation excludes the nose and blade-tip transition regions.

For a rounded blade with local parabolic lubrication gap $d=ϵ+x^2/(2\delta )$, $ϵ\ll \delta$, the Couette-Poiseuille flow in a thin gap yields a floor-driven flux $2ϵ/3$ and pressure-driven leakage proportional to ${ϵ}^{5/2}\Delta p/\sqrt{\delta }$. Positive $J$ is from the upstream pile towards the rear.

The steady upstream film of an infinite blade has $h_x=-3(h-1)/h^3$. Thus $X(h)=h^3/9+h^2/6+h/3+\log (h-1)/3$. The head is fixed by under-blade leakage balancing the incoming film. The deep pile has a cubic profile, while the actual upstream transition has an exponential tail towards $h=1$.