Flux average for a translating lubrication gap ($2Q=\frac{\int h^{-2}dx}{\int h^{-3}dx}$)

In dimensionless Couette-Poiseuille flow in a thin gap with lower-wall velocity one and stationary upper wall, the conserved volume flux satisfies $Q=h/2-h^3{p}_x/12$. Equal pressures at the two ends imply $\int p_{x}dx=0$ and give the displayed expression. It is a weighted average of the positive gap height with weight $h^{-3}$, so $\min h\le 2Q\le \max h$. The local pressure gradient has the sign of $h-2Q$, making the flux constraint useful for locating pressure extrema without solving for the pressure explicitly.

For a symmetric gap $h(x)=\sqrt{1+x^2}$ on $[-1,1]$, unit lower-wall speed and equal zero gauge pressures at the ends, the flux average for a translating lubrication gap gives $Q=\pi /(4\sqrt{2})$. Integrating $p_x=6(h-2Q)/h^3$ gives the displayed odd pressure. It has a positive maximum at $x=-\sqrt{{\pi }^2/8-1}$ and a negative minimum at the opposite point. The converging and diverging portions of a symmetric gap therefore have opposite pressure lobes even though the geometry itself is even: the directed wall motion breaks reflection symmetry of the flow.