= Flux average for a translating lubrication gap
{title2=$2Q=\dfrac{\int h^{-2}\,dx}{\int h^{-3}\,dx}$}
In dimensionless <Couette-Poiseuille flow in a thin gap> with lower-wall <velocity> one and stationary upper wall, the conserved <volume flux> satisfies $Q=h/2-h^3p_x/12$. Equal pressures at the two ends imply $\int p_xdx=0$ and give the displayed expression. It is a weighted average of the positive gap height with weight $h^{-3}$, so $\min h\leq2Q\leq\max h$. 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.
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