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

Joukowski showed that the image of a circle passing through $z=1$ and containing the point $z=-1$ 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

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

identifies a circle of radius $R$, passing through $z=1$. $\alpha$ being the angle between the real axis and the tangent at $z=1$. You can now visualize the Joukowski airfoil using the following code:

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 $\varphi :\Omega \to \mathbb {R}$ or $\mathbb {C}$ ($\Omega$ being an open set of $\mathbb {C}$) 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 $z=x+iy\in \Omega$. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

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

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

• Show that, if $\varphi :\Omega \to \mathbb {R}$ is a harmonic function and $f:\Omega '\to \Omega$ a conformal map, then $\Phi =\varphi \circ f$ 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 ${\vec {v}}=(v_{x},v_{y})$ its velocity field.