= Hydrodynamic attraction of a plane wall to a point source
{title2=$F=\rho Q^2/(16\pi a^2)$}
An incompressible <point source> of volume <flux> $Q$ at height $a$ above an impermeable plane has <velocity potential> $\phi=-Q(1/r_++1/r_-)/(4\pi)$, with an equal image source below the plane. On the wall, $u_s=Qs/[2\pi(s^2+a^2)^{3/2}]$. The <Bernoulli equation> gives $p-p_\infty=-\rho Q^2s^2/[8\pi^2(s^2+a^2)^3]$. Integrating the <pressure> deficit over the plane gives the displayed attraction force relative to ambient <pressure>. If source strength instead denotes the coefficient $Q/(4\pi)$ of radial inverse-square velocity, the force is $\pi\rho m^2/a^2$.
Back to article page
