ESPCI Wiki - User contributions [en]

T-I-1

2011-10-25T07:48:43Z

Gregory:

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always surjective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is injective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyze in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

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

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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

* Let us consider <math>g: \Omega \to \mathbb{C} </math> a holomorphic function. Show that <math>g, \varphi = \mathrm{Re}\, g </math> and <math> \psi =
\mathrm{Im}\, g </math> are harmonic functions.

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\Delta \varphi=0</math> .
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>V_0</math>. Show that the complex potential writes <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z}) \;.</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T14:26:06Z

Gregory: /* Harmonic functions and hydrodynamics in the plane */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always surjective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is injective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyze in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

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

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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

* Let us consider <math>g: \Omega \to \mathbb{C} </math> a holomorphic function. Show that <math>g, \varphi = \mathrm{Re}\, g </math> and <math> \psi =
\mathrm{Im}\, g </math> are harmonic functions.

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\Delta \varphi=0</math> .
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z}) \;.</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T14:25:33Z

Gregory: /* Harmonic functions and hydrodynamics in the plane */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always surjective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is injective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyze in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

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

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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

* Let us consider <math>g: \Omega \to \mathbb{C} </math> a holomorphic function. Show that <math>g, \varphi = \mathrm{Re}\, g </math> and <math> \psi =
\mathrm{Im}\, g </math> are harmonic functions.

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\Delta \varphi=0</math>.
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z}) \;.</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T14:22:52Z

Gregory: /* Harmonic functions and hydrodynamics in the plane */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always surjective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is injective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyze in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

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

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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

* Let us consider <math>g: \Omega \to \mathbb{C} </math> a holomorphic function. Show that <math>g, \varphi = \mathrm{Re}\, g </math> and <math> \psi =
\mathrm{Im}\, g </math> are harmonic functions.

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\nabla \varphi=0</math>.
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z}) \;.</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T14:21:46Z

Gregory: /* Harmonic functions and hydrodynamics in the plane */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always surjective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is injective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyze in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

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

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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

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

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\nabla \varphi=0</math>.
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z}) \;.</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T14:18:07Z

Gregory: /* Joukovski's transformation */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always surjective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is injective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyze in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

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

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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

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

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\nabla \varphi=0</math>.
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z}) \;.</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T14:17:43Z

Gregory: /* Joukovski's transformation */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always surjective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is injective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyze in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

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

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

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

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\nabla \varphi=0</math>.
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z}) \;.</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T14:15:21Z

Gregory: /* Joukovski's transformation */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always surjective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is injective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyze in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

Hint: it might be useful to use polar coordinates, writing <math>z = r e^{i \theta}</math>. 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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

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

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\nabla \varphi=0</math>.
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z}) \;.</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T14:13:24Z

Gregory: /* Joukovski's transformation */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always surjective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is injective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyse in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

Hint: it might be useful to use polar coordinates, writing <math>z = r e^{i \theta}</math>. 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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

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

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\nabla \varphi=0</math>.
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z}) \;.</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T13:30:57Z

Gregory: /* Harmonic functions and hydrodynamics in the plane */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always injective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is surjective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyse in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

Hint: it might be useful to use polar coordinates, writing <math>z = r e^{i \theta}</math>. 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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

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

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\nabla \varphi=0</math>.
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z}) \;.</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T13:30:26Z

Gregory: /* Back to the Joukovski's transformation */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always injective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is surjective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyse in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

Hint: it might be useful to use polar coordinates, writing <math>z = r e^{i \theta}</math>. 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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

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

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\nabla \varphi=0</math>
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z}) \;.</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T13:27:20Z

Gregory: /* Back to the Joukovski's transformation */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always injective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is surjective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyse in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

Hint: it might be useful to use polar coordinates, writing <math>z = r e^{i \theta}</math>. 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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

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

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\nabla \varphi=0</math>
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z})</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T13:24:17Z

Gregory: /* Joukovski's transformation */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always injective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is surjective ? Give some examples of such (maximal) set <math>\Omega</math>.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyse in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

Hint: it might be useful to use polar coordinates, writing <math>z = r e^{i \theta}</math>. 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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

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

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\nabla \varphi=0</math>
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z})</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-16T13:22:27Z

Gregory: /* Analytical functions: conformal map and applications to hydrodynamics */

__FORCETOC__

=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 <math>d \geq 3</math> 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, <math> d=2 </math>, the space of conformal mappings is
much larger and one can show that, given an open set <math> \Omega \in {\mathbb{C}}</math>, any holomorphic function <math> f : \Omega \rightarrow {\mathbb{C}} </math> such that
<math> f'(z) \neq 0 </math>, <math>\forall z \in \Omega </math> defines a conformal map from <math>\Omega</math> to <math>f(\Omega)</math>. 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

<center><math>
J:
\begin{array}[t]{ccc}
\mathbb{C} \setminus \{ 0 \} &\to &\mathbb{C} \\
z &\mapsto & z + \displaystyle \frac{1}{z}
\end{array}
</math></center>

* Compute <math>J'(z)</math> and deduce from it the maximal ensemble on which <math>J</math> is a conformal map. Show that <math> J </math> is always injective. Under which condition on the set <math>\Omega</math> the application <math>J</math> on <math>\Omega</math> is surjective ? Give some examples of such (maximal) ensembles.

* Give the image by <math>J</math> of the following sub-sets: (a) the half-line passing through the origin <math>O</math> and making an angle <math>\alpha </math> with the <math>x</math>-axis, (b) the circle centered at the origin of radius <math>R</math> (analyse in particular the case <math>R=1</math>). What is the image, by <math>J</math>, of the outside of the unit circle <math>|z| > 1</math>.

Hint: it might be useful to use polar coordinates, writing <math>z = r e^{i \theta}</math>. 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 <math>R</math>:

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

Joukowski showed that the image of a circle passing through <math>z=1</math> and containing the point <math>z=-1</math> 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
<center><math> 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 </math></center>
identifies a circle of radius <math>R</math>, passing through <math>z=1</math>. <math>\alpha</math> being the angle between the real axis and the tangent at <math>z=1</math>. 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 <math> \varphi: \Omega \to \mathbb{R}</math> or <math>\mathbb{C}</math> (<math>\Omega</math> being an open set of <math>\mathbb{C} </math>) is
called a "harmonic function" if it satisfies the Laplace equation
<center>
<math>

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

</math>
</center>
in all point <math> z = x + i y \in \Omega</math>. Similarly to conformal maps, harmonic functions in two dimensions, are closely related to holomorphic functions.

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

* Geometric interpration of <math>\varphi </math> and <math>\psi </math>: show that the streamlines of <math>\nabla \varphi </math> are the level curves of <math> \psi </math>.

* Show that, if <math> \varphi: \Omega \to \mathbb{R} </math> is a harmonic function and <math> f: \Omega' \to \Omega </math> a conformal map, then <math>\Phi = \varphi \circ f </math> 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 <math>\vec{v}=(v_x,v_y) </math> its velocity field.
* Show that <math>\vec{v}=(v_x,v_y) </math> is the gradient of a scalar potential <math>\varphi(x,y)</math> which satisfies <math>\nabla \varphi=0</math>
* Show that you can construct <math>\psi(x,y)</math> such that <math>g=\varphi +i \psi</math> is holomorphic and <math>v_x+i v_y= \overline{g'(z)}</math>. <math>g(z)</math> 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 <math>g_0(z)=V_0 z</math>.
* Consider a fluid in presence of an obstacle. The obstacle is a circle with <math>R=1</math>. Far from the circle the velocity is <math>V_0</math>. Use the Joukovski's transformation to show that the complex potential writes
<center><math>g(z)=V_0(z+\frac{1}{z})</math></center>
* Compute the velocity along the real and the imaginary axis. Draw the streamlines (<math>\psi(z)=\text{const.}</math>) 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.

T-I-1

2011-10-14T15:12:50Z

Gregory: /* Application to hydrodynamics */

T-I-1

2011-10-14T15:12:28Z

Gregory: /* Application to hydrodynamics */

T-I-1

2011-10-14T15:09:57Z

Gregory: /* Harmonic functions */

T-I-1

2011-10-14T15:09:23Z

Gregory: /* Harmonic functions */

T-I-1

2011-10-14T15:08:49Z

Gregory: /* Harmonic functions */

T-I-1

2011-10-14T15:08:20Z

Gregory: /* Harmonic functions */

T-I-1

2011-10-14T15:07:54Z

Gregory: /* Harmonic functions */

T-I-1

2011-10-14T15:07:21Z

Gregory: /* Harmonic functions */

T-I-1

2011-10-14T15:01:49Z

Gregory: /* Harmonic functions */

T-I-1

2011-10-14T14:59:36Z

Gregory: /* Harmonic functions */

T-I-1

2011-10-14T14:56:16Z

Gregory: /* Harmonic functions */

T-I-1

2011-10-14T14:55:37Z

Gregory:

T-I-1

2011-10-14T14:32:08Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:31:36Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:30:39Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:29:18Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:20:38Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:20:17Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:10:05Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:09:44Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:04:57Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:04:33Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:04:06Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:03:15Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T14:02:46Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T13:58:56Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T13:56:41Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T13:56:16Z

Gregory:

T-I-1

2011-10-14T13:55:38Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T13:54:03Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T13:53:43Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T13:53:16Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T13:47:01Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T13:46:34Z

Gregory:

T-I-1

2011-10-14T13:38:56Z

Gregory: /* Joukovski's transformation */

T-I-1

2011-10-14T13:33:42Z

Gregory: