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 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 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, 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 d=2 } , the space of conformal mappings is much larger and one can show that, given an open set 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 \in {\mathbb{C}}} , any holomorphic 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 f : \Omega \rightarrow {\mathbb{C}} } such 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 f'(z) \neq 0 } , 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 \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 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

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 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'(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 $\Omega$ the application $J$ on $\Omega$ is injective ? Give some examples of such (maximal) set $\Omega$.

• 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 $z=re^{i\theta }$.

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 $z=1$. 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} being the angle between the real axis and the tangent at 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} . 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 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 C^2 } , $\varphi :\Omega \to \mathbb {R}$ 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 $g,\varphi =\mathrm {Re} \,g$ and $\psi =\mathrm {Im} \,g$ 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.

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.

• Show that ${\vec {v}}=(v_{x},v_{y})$ is the gradient of a scalar potential $\varphi (x,y)$ which satisfies $\Delta \varphi =0$ .
• Show that you can construct $\psi (x,y)$ such that $g=\varphi +i\psi$ is holomorphic and $v_{x}+iv_{y}={\overline {g'(z)}}$. $g(z)$ 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.