= Pressure-driven flux through a parabolic lubrication gap
{title2=$\Delta p=9\pi\mu q\sqrt{2a}/(2h^{5/2})$}
Two stationary no-slip surfaces with local gap $H(x)=h+x^2/(2a)$ transmit pressure-driven <volume flux> $q=-H^3p_x/(12\mu)$ per unit transverse length. Hence the <pressure> drop is
$$
\Delta p=12\mu q\int_{-\infty}^{\infty}\frac{dx}{H^3}=\frac{9\pi\mu q\sqrt{2a}}{2h^{5/2}}.
$$
The first relation comes from integrating the parabolic <velocity> $u=-p_x y(H-y)/(2\mu)$. Substitution $x=\sqrt{2ah}\tan\theta$ evaluates the resistance <integral> using $\int_{-\pi/2}^{\pi/2}\cos^4\theta\,d\theta=3\pi/8$. The formula gives the leading resistance when $h\ll a$.
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