10.4. Hydrodynamics of an ideal fluid in two dimensions Consider the stationary motion of an incompressible and irrotational fluid in two dimensions. Denoting by v(x, y) = (v1, v2) the vector field of its velocity at the point

(x, y) of the plane, it satisfies

∇

. v = 0, ∇ ∧ v = 0

a. Show that these conditions implies the existence of a potential  that satisfies the Laplace equation. Moreover, show that, by introducing the conjugate function 

and defining f (z) = +i (the so-called complex potential), we have df dz

= ∂

∂x

+i

∂

∂x

= ∂

∂x

−i

∂

∂y

= v1 −iv2 = ¯v.

The complex vector field of the velocity is then given by v = df dz

.

b. Study the flux lines of the velocity associated to the analytic function f (z) = iγ

2π

lnz and show that the vector field of the velocity corresponds to a vortex, localized at the origin. Give an interpretation of the parameter γ .

c. Study the flux lines of the velocity relative to the potential f (z) = v0z+ a2 z

+ iγ

2π

lnz.

Determine the points where the velocity vanish and study their location by varying the parameter γ .