= Solution
Since the <velocity> tends to zero at infinity, the steady <Bernoulli equation> gives $p+\rho|\mathbf u|^2/2=p_\infty$. On the wall, therefore,
$$
p(x,0)-p_\infty=-\frac{\rho Q^2}{2\pi^2}\frac{x^2}{(x^2+a^2)^2}.
$$
Compute the finite source-induced <force> relative to uniform ambient <pressure>, or equivalently use the <gauge pressure> $p-p_\infty$. The fluid lies above the wall, so its normal <force> on the wall points in $-\mathbf e_y$ for positive <pressure>. The negative excess <pressure> found here consequently produces an upward net force, toward the source.
For the symmetric section $-R\leq x\leq R$, there is no horizontal <force>, and the vertical excess <force> is
$$
\begin{aligned}
F_y(R)&=-\int_{-R}^R(p-p_\infty)\,dx\\
&=\frac{\rho Q^2}{2\pi^2}\left[\frac1a\arctan\frac Ra-\frac R{R^2+a^2}\right].
\end{aligned}
$$
Here an antiderivative of $x^2/(x^2+a^2)^2$ is $(1/(2a))\arctan(x/a)-x/[2(x^2+a^2)]$. Taking $R\to\infty$ gives the <hydrodynamic attraction of a wall to a line source>:
$$
\boxed{F_x=0,\qquad F_y=\frac{\rho Q^2}{4\pi a}.}
$$
Each component is a <force per unit length> of the wall in the out-of-plane direction. The result is an attraction even though the source sends fluid away from itself: faster tangential motion reduces the wall <pressure>.
The excess <torque> about the origin is
$$
M(R)=-\int_{-R}^R x(p-p_\infty)\,dx=0
$$
for every $R$, because the integrand is odd. Thus the symmetric infinite-wall limit is
$$
\boxed{M=0.}
$$
The symmetric prescription matters: the separate one-sided moment integrals grow logarithmically, whereas the <force> integral converges absolutely. This is the appropriate <Cauchy principal value> for the infinite wall. If one instead asks for the absolute fluid force with no balancing ambient <pressure> on the other side, its vertical component on a finite section is $-2Rp_\infty+F_y(R)$ and has no finite infinite-wall limit. The finite answer above is the flow-induced load.
\Image[/past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2015-source-wall-pressure.png]
{title=An image source enforces impermeability while the pressure deficit attracts the wall}
{description=The equal reflected source lies outside the fluid. The right panel shows the dimensionless pressure deficit at the wall, which vanishes directly below the source, peaks at $|x|=a$, and integrates to the finite upward load.}
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