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 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 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} (analyse in particular the case 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=1} ). What is the image, by $J$, of the outside of the unit circle 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} .

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}} .

• Get a better idea of this Joukowski's transformation using the following code in Mathematica:

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

• 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 $z=1$, 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 $\gamma$ 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 $z'(0)\neq 0$. 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 $J$ 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.

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 $\mathbb {C}$ (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

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

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 $g,\varphi =\mathrm {Re} \,g,\psi =\mathrm {Im} \,g$ 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.