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{\bf EML 5709 \hfill Homework 5 \hfill Spring 1996 }
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\item
Solve impulsively started Poiseuille flow in a plane duct.
In particular, assume that the gap between two flat plates
has height $\pi$,
that the flow is unsteady, that the pressure gradient is nonzero and constant,
and that the initial velocity $u(y,0)$ is zero.

\item
For the previous question,
by using one of the Fourier series
in the Mathematical Handbook, show that for long time, the velocity profile
in the gap is parabolic.

\item
Make question 7.3 in the book.

\item
Solve the axisymmetric Stokes problem: initially the fluid inside a circular
pipe of unit radius is at rest,
but at time $t=0$ the pipe is given a rotational velocity $\Omega$
around its axis.
You can find the Navier-Stokes equations in cylindrical coordinates
in the book; assume that $u_\theta$ depends only on radial position
$R$ and time $t$,
and that $u_R = u_z = 0$.
$u_\theta(R,0)=0$, $u_\theta(1,t)=\Omega$.
Solve using separation of variables.
To make the boundary conditions homogeneous,
substract the steady solution given in the book.

\item
For the flow of the previous question,
find the rate of decay of the slowest decaying mode.
Also draw the infinite time velocity distribution in the pipe.
Further, sketch the Stokes layer that exists at small times.
Roughly how long does it take until this Stokes layer fills the entire cross
section of the pipe?

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