= Solution
For an infinite blade, the steady moving-frame flux is constant and its upstream value is $-1$. The <infinite-blade gravity pile> therefore satisfies $-h-h^3h_x/3=-1$, or $h_x=-3(h-1)/h^3$. Integrating gives
$$
\boxed{X(h)=\frac{h^3}{9}+\frac{h^2}{6}+\frac h3+\frac13\log(h-1)=x_0-x,\qquad h>1.}
$$
An additive constant in $X$ simply shifts $x_0$. The head at the blade, and hence $x_0$, must supply the pressure difference that makes the flux beneath the blade equal to the incoming flux. For $x_0\gg1$ the deep part has $h\simeq[9(x_0-x)]^{1/3}$, followed by a smooth transition to $h=1$ with $h-1$ exponentially small, proportional to $e^{-3x}$. Thus $x_0$ is an approximate nose coordinate, not a sharp front of the exact solution.
Near the rounded edge the <parabolic lubrication gap> is $d(x)=\epsilon+\delta-\sqrt{\delta^2-x^2}\simeq\epsilon+x^2/(2\delta)$. In the blade frame the floor moves at $-1$ and the blade is stationary. The <Couette-Poiseuille flow in a thin gap> has flux $q_x=-d/2-d^3p_x/12$. Let $J=-q_x$ be positive leakage towards the rear, and let $\Delta p=p_{\rm ahead}-p_{\rm rear}>0$. The narrow-gap integrals are
$$
I_2=\int_{-\infty}^{\infty}\frac{dx}{d^2}=\frac{\pi\sqrt{2\delta}}{2\epsilon^{3/2}},\qquad I_3=\int_{-\infty}^{\infty}\frac{dx}{d^3}=\frac{3\pi\sqrt{2\delta}}{8\epsilon^{5/2}}.
$$
Integrating the pressure gradient gives $\Delta p=12JI_3-6I_2$, so the <squeegee gap leakage flux> is
$$
\boxed{J\simeq\frac{2\epsilon}{3}+\frac{2\epsilon^{5/2}}{9\pi\sqrt{2\delta}}\Delta p.}
$$
The first term is the floor-driven contribution. For the steady infinite blade, $J=1$, so $\Delta p\simeq9\pi\sqrt{2\delta}/(2\epsilon^{5/2})$. The <hydrostatic pressure> difference is the height difference at the two sides. A fully developed rear layer of thickness $h_-$ has moving-frame flux $-h_-$; hence \b[$h_-=1$], and
$$
\boxed{h(0^+)\simeq\frac{9\pi\sqrt{2\delta}}{2\epsilon^{5/2}}.}
$$
The large head eventually forces all the arriving film beneath an infinite blade. Taking a perfectly closed gap instead would eliminate this steady state.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-329-squeegee-flow.png]
{title=Infinite-blade height profile and schematic finite-blade pile with sideways drainage}
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