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 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 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 } is always injective. Under which condition on the 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} the application $J$ on 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} is surjective ? Give some examples of such (maximal) ensembles.

• Give the image 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 J} of the following sub-sets: (a) the half-line passing through the origin $O$ and making an angle 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 } 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 $R$ (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 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} , of the outside of the unit circle $|z|>1$.

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:

- 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 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} :

 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 $z=1$, 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.

Joukowski showed that the image of a circle 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} 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

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 }

identifies a circle 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} , 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 $\varphi :\Omega \to \mathbb {R}$ 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

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

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.

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.

• 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 \vec{v}=(v_x,v_y) } is the gradient of a scalar potential 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(x,y)} which satisfies 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 \nabla \varphi=0}
• Show that you can construct 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(x,y)} 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 g=\varphi +i \psi} is holomorphic 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 v_x+i v_y= \overline{g'(z)}} . 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(z)} is the complex potential associated to the 2-dimensional fluid flow.

Back to the Joukovski's transformation

• Consider a constant a uniform flow, parallel to the real axis and with velocity $V_0$. Show that the complex potential writes 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_0(z)=V_0 z} .
• Consider a fluid in presence of an obstacle. The obstacle is a circle 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 R=1} . Far from the circle the velocity is 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 V_0} . Use the Joukovski's transformation to show that the complex potential writes

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(z)=V_0(z+\frac{1}{z})}

• Compute the velocity along the real and the imaginary axis. Draw the streamlines (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(z)=\text{const.}} ) of the flow.
• Explain (without calculation) how you can use the Joukovski's transformation to study the flow if the circle is replaced by the airfoil.