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 $d\geq 3$ 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 $\Omega \in {\mathbb {C} }$, any holomorphic function $f:\Omega \rightarrow {\mathbb {C} }$ such that $\$f'(z)\neq 0\$$, $\forall z\in \Omega$ defines a conformal map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J: \begin{array}[t]{ccc} \mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\ z &\mapsto & z + \displaystyle \frac{1}{z} \end{array} }

• Compute $J'(z)$ and deduce from it the maximal ensemble on which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} is a conformal map. Show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J } is always surjective. Under which condition on the set $\Omega$ the application Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} and making an angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha } with the $x$-axis, (b) the circle centered at the origin of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .

 Hint : it might be useful to use polar coordinates, writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = r e^{i \theta}}