- ia

, outside the physical upper half-plane. A convenient complex potential of a two-dimensional potential flow is

w (z) = \frac{Q}{2 π} [lo g (z - ia) + lo g (z + ia)], w^{'} (z) = \frac{Q z}{π ( z ^{2} + a ^{2} )} .

The two logarithms can be combined locally as

(Q / (2 π)) lo g (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^{'} = u - i v

, the velocity field is

u = \frac{Q}{2 π} (\frac{x}{R _{-}^{2}} + \frac{x}{R _{+}^{2}}), v = \frac{Q}{2 π} (\frac{y - a}{R _{-}^{2}} + \frac{y + a}{R _{+}^{2}}) .

(2)

On the wall

y = 0

, the vertical components cancel:

v (x, 0) = 0, u (x, 0) = \frac{Q x}{π ( x ^{2} + a ^{2} )} .

(3)

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.