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 $\Omega$ to $f(\Omega )$. 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

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

• Compute $J'(z)$ and deduce from it the maximal ensemble on which $J$ is a conformal map. Show that $J$ is always injective. Under which condition on the set $\Omega$ the application $J$ on $\Omega$ 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 $R$ (analyse in particular the case $R=1$). What is the image, by $J$, of the outside of the unit circle $|z|>1$.

Hint: it might be useful to use polar coordinates, writing $z=re^{i\theta }$. Get a better idea of this Joukowski's transformation using the following code in Mathematica:

- for the half-line passing through the origin:

 Jouk[z_] := z + 1/z
 Jouk[R Cos[u] + I  R Sin[u]];
 ParametricPlot[{{Re[%], Im[%]} /. {u -> 0.5}, {R Cos[u], R Sin[u]} /. {u -> 0.5}}, {R, .01, 10}]

- for the circle centered at the origin of radius $R$:

 Jouk[R Cos[u] + I  R Sin[u]];
 ParametricPlot[{{Re[%], Im[%]} /. {R -> 0.79}, {R Cos[u], R Sin[u]} /. {R -> 0.79}}, {u, 0, 2 \[Pi]}, 
 PlotRange -> {{-3, 3}, {-1.5, 1.5}}]

• Study the conformal map $J$ in the vicinity of 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 = 1} : we consider a "smooth" curve 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 \gamma} passing through 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=1} , with a well defined tangent. Show that the image of 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 \gamma} exhibits a cusp in 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(1)} . In this purpose, we parametrize this curve 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 \gamma} by 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(t) } with 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(0)=1} and 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'(0) \neq 0} . Write then the Taylor expansion of 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} in 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 t=0} up to first order and the expansion of 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} close 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 1} up to second order.

Joukowski  showed that the image of a circle passing through  and containing the point  is mapped onto a curve shaped like the cross section of an airplane wing.  We call this curve the Joukowski airfoil. Convince yourself that the parametric curve

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 1 - R \left( \cos(u) + \sin(\alpha) \right) + i R \left( \cos(\alpha) + \sin(u) \right) \quad \quad \text{with} \quad 0<u<2 \pi }

Jouk[z_] := z + 1/z
Jouk[1 - R Sin[\[Alpha]] + R Cos[u] +   I (R Cos[\[Alpha]] + R Sin[u])];
ParametricPlot[{Re[%], Im[%]} /. {R -> 1.15, \[Alpha] -> 1.3}, {u, 0, 2 \[Pi]}, PlotRange -> {{-3, 3}, {-1, 1}}, AspectRatio -> 1/3]

Harmonic functions

We recall that a function 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 \varphi: \Omega \to \mathbb{R}} or 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 \mathbb{C}} (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} being an open set of 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 \mathbb{C} } ) is called a "harmonic function" if it satisfies the Laplace equation

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 \Delta \varphi = 0 \qquad \text{where} \qquad \Delta \varphi \equiv \frac{\partial^2 \varphi}{\partial x^2}+\frac{\partial^2 \varphi}{\partial y^2} }

in all point 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 = x + i y \in \Omega} . Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

• Let us consider 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 g: \Omega \to \mathbb{C} } a holomorphic function. 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 g, \varphi = \mathrm{Re}\, g, \psi = \mathrm{Im}\, g } are harmonic functions.

• Geometric interpration of 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 \varphi } and 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 \psi } : show that the streamlines of $\nabla \varphi$ are the level curves of 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 \psi } .

• Show that, if 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 \varphi: \Omega \to \mathbb{R} } is a harmonic function and 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' \to \Omega } a conformal map, then 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 \Phi = \varphi \circ f } is also a harmonic function.

Application to hydrodynamics in the plane

We now consider the irrotational flow of a non-viscous and incompressible fluid in some region of the plane. We denote by 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 \vec{v}=(v_x,v_y) } its velocity field.