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 $f^{\prime }(z)\ne 0$, $\mathrm{\forall }z\in \mathrm{\Omega }$ defines a conformal map from $\mathrm{\Omega }$ to $f(\mathrm{\Omega })$. The aim of this homework 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 injective ? Give some examples of such (maximal) set $\mathrm{\Omega }$.

• 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 $R$ (analyze in particular the case $R=1$). What is the image, by $J$, of the outside of the unit circle $|z|>1$.

Hint: you will find useful to write the Cartesian coodinates of $f(z)$ in terms of the polar coordinates of $z$ writing $z=r{e}^{i\theta }$.

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 $z^{\prime }(0)\ne 0$. 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

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

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}, {{1 - R Sin[\[Alpha]] + R Cos[u]}, {R Cos[\[Alpha]] + R Sin[u]}} /. {R -> 1.15, \[Alpha] -> 1.3}}, {u, 0, 2 \[Pi]}]

Harmonic functions and hydrodynamics in the plane

We recall that a function of differentiability class $C^2$, $\phi :\mathrm{\Omega }\to \mathbb{ℝ}$ or $\mathbb{ℂ}$ ($\mathrm{\Omega }$ being an open set of $\mathbb{ℂ}$) is called a "harmonic function" if it satisfies the Laplace equation

$\mathrm{\Delta }\phi =0\text{where}\mathrm{\Delta }\phi \equiv \frac{{\partial }^2\phi }{\partial x^2}+\frac{{\partial }^2\phi }{\partial y^2},$

in all point $z=x+iy\in \mathrm{\Omega }$. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

• Let us consider $g:\mathrm{\Omega }\to \mathbb{ℂ}$ a holomorphic function. Show that $g,\phi =\mathrm{R}\mathrm{e}g$ and $\psi =\mathrm{I}\mathrm{m}g$ are harmonic functions.

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

• Show that, if $\phi :\mathrm{\Omega }\to \mathbb{ℝ}$ is a harmonic function and $f:{\mathrm{\Omega }}^{\prime }\to \mathrm{\Omega }$ a conformal map, then $\mathrm{\Phi }=\phi \circ f$ is also a harmonic function.

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

• Show that $\overrightarrow{v}=(v_x,v_y)$ is the gradient of a scalar potential $\phi (x,y)$ which satisfies $\mathrm{\Delta }\phi =0$ .
• Show that you can construct $\psi (x,y)$ such that $g=\phi +i\psi$ is holomorphic and $v_x+i{v}_y=\overline{g^{\prime }(z)}$. $g(z)$ is the complex potential associated to the 2-dimensional fluid flow.

Back to the Joukovski's transformation

• Consider a constant and uniform flow, parallel to the real axis and with velocity $V_0$. Show that the complex potential writes $g_0(z)=V_{0}z$.
• Consider a fluid in presence of an obstacle. The obstacle is a circle with $R=1$. Far from the circle the velocity is $V_0$. Use the Joukovski's transformation to show that the complex potential writes

$g(z)=V_0(z+\frac{1}{z}).$

• Compute the velocity along the real and the imaginary axis. Draw the streamlines ($\psi (z)=\text{const.}$) of the flow.

ContourPlot[Im[(x + I y) + 1/(x + I y)], {x, -3, 3}, {y, -3, 3}, BoundaryStyle -> Red,  RegionFunction -> Function[{x, y}, x^2 + y^2 > 1], Contours -> 100]

• Explain (without calculation) how you can use the Joukovski's transformation to study the flow if the circle is replaced by the airfoil.