= Hydrodynamic attraction of a wall to a line source
{title2=$F_y=\rho Q^2/(4\pi a)$}
In the upper <half-plane>, a <line source> of outward <volume flux> $Q$ at height $a$ above an impermeable wall has an equal image source below it. On the wall $u=Qx/[\pi(x^2+a^2)]$ and the normal <velocity> vanishes. The steady <Bernoulli equation> gives $p-p_\infty=-\rho Q^2x^2/[2\pi^2(x^2+a^2)^2]$. Since
$$
\int_{-\infty}^{\infty}\frac{x^2}{(x^2+a^2)^2}\,dx=\frac\pi{2a},
$$
the ambient-relative <force per unit length> pulls the wall toward the source with the displayed magnitude. The <torque> about the source's projection onto the wall is zero in the symmetric <Cauchy principal value> sense. Individual one-sided moments diverge logarithmically. This planar result differs from the three-dimensional <hydrodynamic attraction of a plane wall to a point source>.
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