 Pressure-driven flux through a parabolic lubrication gap

Two stationary no-slip surfaces with local gap $H(x)=h+x^2/(2a)$ transmit pressure-driven volume flux $q=-H^3{p}_x/(12\mu )$ per unit transverse length. Hence the pressure drop isThe 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 $h\ll a$.