\documentstyle{article}
\setlength{\textwidth}{6.6truein}\setlength{\oddsidemargin}{-.2truein}
\setlength{\evensidemargin}{-.2truein}\setlength{\textheight}{9truein}
\setlength{\topmargin}{-.4truein}\setlength{\headsep}{.2truein}
\setlength{\footskip}{.3truein}\pagestyle{empty}
\begin{document}

\begin{center}
{\bf EML 5709 \hfill Homework 3 \hfill Spring 1996 }
\end{center}

\begin{enumerate}

\setlength{\leftmargin}{0pt}

\item
As a model for the formation of a tornado, assume that in
cylindrical coordinates $(r,\theta,z)$, the radial and axial
velocity components near the axis are are
$$u_r = - r \qquad u_z =  2 z.$$
Find the radial position of the fluid particles as a function of time.
If the {\it initial} azimuthal velocity is
$$u_{\theta0} = f(r) \hbox{ at $t=0$, }$$
find the azimuthal velocity at later times $t$ using only
Kelvin's theorem for rings of fluid around the axis.
(Answer: $u_\theta= e^t f(e^tr)$.)
Draw the pathlines in the plane $z=0$.
Why does the tornado become stronger?

\item
Show that the flow of the previous question satisfies the
continuity equation for incompressible flow.

\item
Show that the flow of the previous questions satisfies the three momentum
equations of incompressible inviscid flow.
Note: the momentum equations for incompressible flow may be derived
using Appendix A; or you may want to study the book somewhat more closely.

\item
Which of the two forms of the Bernoulli equation, if any, applies to
the flow of the previous questions?

\item
Water is flowing through an open channel.  Explain why the water surface
goes down when the fluid speeds up using the Bernoulli law.
Also explain it directly from energy considerations.

\item
A circular cylinder of unit radius at $x_0(t)$,
$$(x-x_0)^2 + y^2 =1,$$
moves towards the right with speed $\dot x_0 \hat i$.
The fluid flow velocity potential around this cylinder is given by
$$\phi = - \dot x_0 {x-x_0\over (x-x_0)^2+y^2}.$$
Give the fluid flow velocity components on the cylinder.
Show that the inviscid flow boundary condition is satisfied.

\item
For the flow of the previous question, compare the pressure on the
cylinder for the case that the cylinder is moving with unit velocity
$\dot x_0=1$ to the case that the cylinder is accelerating with speed
$\dot x_0=t$.
Use the Bernoulli law, ignoring gravity, and assuming the pressure
vanishes at large distances.

\item
Find the drag forces on the cylinder for the flows of the previous question.

\end{enumerate}

\end{document}
