Sliding-plate flow through a parabolic gap ($Q=-2U{h}_0/3$)

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 $L=\sqrt{R{h}_0}$, $X=x/L$, $H=1+X^2/2$ and $p=(\mu UL/h_0^2)\Pi$. Equal endpoint gauge pressures givewhere $Q$ 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 $X=\sqrt{2}\tan \theta$ evaluates the ratio. The plane Poiseuille flow and Couette flow contributions both point towards negative $x$ exactly for $|x|<\sqrt{2R{h}_0/3}$. This is a full-film model with matched equal outer pressures.