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{\bf EML 5709 \hfill Homework 4 \hfill Spring 1996 }
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\begin{enumerate}

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\item
The radius $R_0$ of a circular cylinder is expanding in time.
The cylinder is surrounded by an incompressible fluid extending to infinity
in all directions.
Write the partial differential equation and boundary
conditions for the velocity potential
in the incompressible fluid in terms of $R_0$ and its derivatives.

\item
Solve the partial differential equation of the previous question
using separation of variables.
Compare with the basic solutions.

\item
Use the result of the previous question to find the pressure and
velocity components in the incompressible fluid.  What can you say about
the pressure at large distances?

\item
Is the solution derived in the previous questions an irrotational
flow?  Is it a solution of the Euler equations satisfying the right
boundary conditions?  Is it a solution of the Navier-Stokes equations
satisfying the right boundary conditions?

\item
Using real variables only,
describe the velocity field corresponding to a vortex of strength $\Gamma$
located at $x=0,\  y=h$, and one of opposite strength at $x=0,\ y=-h$.
In particular, find $\phi$ and $\psi$ and sketch the streamlines.
In what direction is the velocity on the x-axis (y=0)?
What can you say about the flow corresponding to a single vortex
located a distance $h$ above a solid wall?
Mirroring an image of the features of a flow
into a solid region is called
the mirror method.
Give the pressure on the wall.
Explain why the effects of wingtip vortices are much less pronounced
when the plane is flying close to the ground.

\item
Repeat question 5, but now use complex variables everywhere:
Describe the velocity field corresponding to a vortex of strength $\Gamma$
located at $x=0,\  y=h$, and one of opposite strength at $x=0,\ y=-h$.
In particular, find $F$ and from it $\psi$ and sketch the streamlines.
Show that $F$ is real on the $x$-axis.
Find the complex conjugate velocity and show that it is horizontal on the
$x$-axis.
Find the pressure on the wall from $W$.

\item
Using complex variables,
write down the combined potential of
a uniform flow in the $x$-direction of unit velocity,
a strong sink of strength $-m=2\pi/\epsilon$ at the origin,
and a strong source of strength
$m=2\pi/\epsilon$ at $(x,y)=(-\epsilon,0)$.
Show that for small $\epsilon$, we get potential flow past a circular
cylinder.
Draw the streamlines.
Where does the fluid coming out of the source at
the $(-epsilon,0)$ end up?

\item
Using complex variables,
write down the combined potential of
a uniform flow in the $x$-direction of unit velocity
and a source of strength $m=2\pi$ at the origin.
Draw the streamlines.
What is the velocity at large distances?
Where does the fluid coming out of the source at
the origin end up?

\item
Make question 4.3 in the book.

\item
Make question 4.7 in the book.

\item
Repeat question 4.7 in the book, but
now use the Blasius theorem to find the force.  Compare the results.

\item
A complex velocity potential in a $\zeta$-plane is given by
$$F = U\left(\zeta e^{-i \alpha} + {e^{i\alpha}\over\zeta}\right)
+ {i \Gamma \over 2 \pi} \ln \zeta$$
where $\Gamma = 4 \pi U \sin\alpha$.
Show that $F$ is real on the unit circle, so that it is a streamline.
Find $F' = dF/d\zeta$ on the unit circle $\zeta=e^{i\theta}$
in terms of the angular coordinate $\theta$.
Show that stagnation points occur at $\zeta=1$ and $e^{i(\pi+2\alpha)}$.
Identify $F$ at large distances.
Now draw the streamlines and describe the flow.

\item
For the $\zeta$-plane of the previous question, assume that the corresponding
physical $(x,y)$-plane is found from
$$z = \zeta+\epsilon + {(1-\epsilon)^2\over \zeta - \epsilon}$$
Identify the critical points where $z' = dz/d\zeta = 0$.
Locate these also in the $z$-plane.
Now sketch and describe the curve in the $z$-plane corresponding to
the unit circle $\zeta=e^{i\theta}$ in the $\zeta$-plane.
Show that for small $\epsilon$, this curve is approximately equal to
$(x,y)\approx(2\cos\theta,0)$.  Describe this approximate curve.
Draw the streamlines in the $z$-plane.

\item
Using the results of the previous questions, show that the lift coefficient
of a thick airfoil can be more than $2 \pi \sin\alpha$.

\item
Using the results of the previous questions, find
the complex conjugate velocity $W= dF/dz$
on the unit circle $\zeta=e^{i\theta}$.
Approximate this for small values of $\epsilon$.
Rewrite this in terms of $x=2\cos\theta$ to get
$$W \approx U \left( \cos\alpha \pm \sin\alpha \sqrt{2-x\over 2+x}\right)$$

\item
Using the result of the previous question to draw the pressure distribution
on the top and bottom of the plate $-2\le x\le2$.  Comment on the result.

\end{enumerate}

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