T-I-1

Analytical functions: conformal map and applications to hydrodynamics

This homework deals with the application of conformal maps to the study of two-dimensional hydrodynamics. A conformal map is a geometrical transformation which preserves all (oriented) crossing angles between lines. In dimension $d\ge 3$ a conformal map is necessarily composed from the following limited number of transformations: translations, rotations, homothetic transformation and special conformal transformation (which is the composition of a reflection and an inversion in a sphere). However in two dimensions, $d=2$, the space of conformal mappings is much larger and one can show that, given an open set $\mathrm{\Omega }\in \mathbb{ℂ}$, any holomorphic function $f:\mathrm{\Omega }\to \mathbb{ℂ}$ such that $\mathrm{\$}{f}^{\prime }(z)\ne 0\mathrm{\$}$, $\mathrm{\forall }z\in \mathrm{\Omega }$ defines a conformal map from $\mathrm{\Omega }$ to $f(\mathrm{\Omega })$. The aim of this HW is to exploit this property to study some hydrodynamic flows in two spatial dimensions.

Joukovski's transformation

The Joukovski's transformation is defined by the following application

$J:\begin{array}{ccc}\mathbb{ℂ}\setminus \{0\}& \to & \mathbb{ℂ}\\ z& \mapsto & z+\frac{1}{z}\end{array}$

• Compute $J^{\prime }(z)$ and deduce from it the maximal ensemble on which $J$ is a conformal map. Show that $J$ is always surjective. Under which condition on the set $\mathrm{\Omega }$ the application $J$ on $\mathrm{\Omega }$ is surjective ? Give some examples of such (maximal) ensembles.

• Give the image by $J$ of the following sub-sets:

(a) the half-line passing through the origin $O$ and making an angle $\alpha$ with the $x$-axis. 
(b) the circle centered at the origin of radius $R$ 

 Hint : it might be useful to use polar coordinates, writing $z=r{e}^{i\theta }$