= Uniform gravity-driven Newtonian film
{title2=$v(x)=\rho g\sin\theta\,(dx-x^2/2)/\eta,\quad q=\rho g d^3\sin\theta/(3\eta)$}
A uniform steady film of thickness $d$ on a plane inclined by $\theta$ above horizontal has downslope coordinate $s$ and outward normal coordinate $x$. Neglecting air shear, the pressure is $p_{\rm atm}+\rho g\cos\theta(d-x)$ and its downslope gradient vanishes. The <Newtonian fluid> momentum equation is $\eta v''+\rho g\sin\theta=0$. The <no-slip boundary condition> $v(0)=0$ and shear-free <free surface> $v'(d)=0$ give the displayed parabolic profile. Integrating over thickness gives volume flux per unit width $q$. The wall shear $\rho gd\sin\theta$ balances the film's downslope weight, and the vertical-wall case is $\theta=\pi/2$.
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