 Solution

The method of images places an equal line source at $-ia$, outside the physical upper half-plane. A convenient complex potential of a two-dimensional potential flow isThe two logarithms can be combined locally as $(Q/(2\pi ))\log (z^2+a^2)$; their branches of the complex logarithm affect only the potential convention, not the single-valued complex velocity. The point source in this two-dimensional problem has the line source normalization: its outward volume flux is $Q$ per unit span.

Writing $R_{-}^2=x^2+(y-a{)}^2$, $R_{+}^2=x^2+(y+a{)}^2$, and using $w^{\prime }=u-iv$, the velocity field isOn the wall $y=0$, the vertical components cancel:This is the required no-penetration boundary condition. Tangential slip is permitted. At large distance the velocity decays as $1/r$, and near $z=ia$ its singular part is exactly that of the specified source; the reflected source is regular throughout the fluid. Away from the injection point, this is an incompressible flow and an irrotational flow.