= Sliding-plate flow through a parabolic gap
{title2=$Q=-2Uh_0/3$}
A stationary upper wall $z=h_0+x^2/(2R)$ and a flat lower wall moving at <velocity> $-U$ generate a <Couette-Poiseuille flow> in the <lubrication approximation>. Write $L=\sqrt{Rh_0}$, $X=x/L$, $H=1+X^2/2$ and $p=(\mu UL/h_0^2)\Pi$. Equal endpoint <gauge pressures> give
$$
\Pi(X)=\frac{2X}{H^2},\qquad
Q=-\frac{Uh}{2}-\frac{h^3}{12\mu}\frac{dp}{dx}
=-\frac23Uh_0,
$$
where $Q$ is the signed <volume flux per unit width>. To determine its constant value, the <Reynolds equation> gives $H^3\Pi_X/12+H/2=C$. Integrating $\Pi_X$ over the real line gives $C=I_2/(2I_3)=2/3$, where $I_m=\int_{-\infty}^{\infty}H^{-m}\,dX$ and the substitution $X=\sqrt2\tan\theta$ evaluates the ratio. The <plane Poiseuille flow> and <Couette flow> contributions both point towards negative $x$ exactly for $|x|<\sqrt{2Rh_0/3}$. This is a full-film model with matched equal outer <pressures>.
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