\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 2 \hfill Spring 1996 }
\end{center}

\begin{enumerate}

\setlength{\leftmargin}{0pt}

\item
Find the streamline that passes through the origin
for the flow with velocity field
$$u = 4 (1+t) y,\qquad v = - 2 (1+t) (x-1), \qquad w = t (z-1),$$
at time $t=0$.
Eliminate any unnecessary parameters from the solution.
Sketch the solution in a Cartesian coordinate system.

\item
For the same flow of the previous question,
find the pathline of the particle that is at the origin at time $t=0$.
Eliminate any unnecessary parameters from the solution:
express the curve explicitly in terms of $z$.
Describe and sketch this pathline.

\item
For the same flow of the previous question,
find the streakline that a smoke generator at the origin
generates at time $t=0$.
Assume that the generator has been working for a long time.
Eliminate any unnecessary parameters from the solution.
Describe and sketch the streakline features.

\item
Find the streamlines, pathlines, and streaklines for
the flow with velocity field
$$u_r = r, \qquad u_\theta = r.$$
Draw two typical streamlines, two pathlines, and two streaklines.

\item
A fluid is rotating as a solid body:
$$\vec u = \vec \gamma \times \vec r.$$
Compute the vorticity field of the flow.
Describe the vortex tubes of this flow and verify that the circulation
along each tube is indeed constant.
Verify Stokes theorem for this flow.

\item
Make question 2.5 in the book.
Compare your results to Stokes' theorem and comment on that.

\item
Discuss where you would find vorticity in the flow about a car.
Must the vortex lines always meet the surface of the car tangentially?
If not, give cases in which they do not.
What numerical advantage could there be in describing the flow field
about the car by means of its vorticity, instead of its velocity?

\end{enumerate}

\end{document}
