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{\bf EML 5709 \hfill Homework 1 \hfill Spring 1996 }
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\item
At sea level, the number of air molecules per cubic meter
$n=2.5\ 10^{25}$ and the free path length is $\lambda=6.6\ 10^{-8}$
meter.
For what sort of bodies can we describe the motion of the air as
a continuum instead of as the motion of individual molecules?
How about at 150 kilometers, where
$n=4.1\ 10^{16}$ and $\lambda=41$ meter?
How about the starship Enterprise?

\item
The velocity of a fluid such as air or water is given by
$$V= 10 x^2 \hat i + xz \hat j - 20 xz \hat k.$$
Find and draw the surface forces acting on a small
$\epsilon\times\epsilon\times\epsilon$ cube of air
located at $\vec r =(1,3,3)^T$.  Does it depend on Stokes' hypothesis?

\item
Assume that the temperature in a fluid such as air or water
varies with position as
$$T = 10 e^y \sin(x+z).$$
Find the heat fluxes through the sides of a small
$\epsilon\times\epsilon\times\epsilon$ cube of air
located at $\vec r =(1,3,3)^T$.
Are the heat fluxes through, for example, the left and right hand
surfaces of the cube exactly the same?

\item
In questions 2 and 3, are the velocity and temperature fields steady or
unsteady?  Does the temperature of the fluid at $\vec r =(1,3,3)^T$
vary with time?  What are the Eulerian and Lagrangian time derivatives
of the temperature at $\vec r =(1,3,3)^T$?  What is the acceleration
of the fluid at $\vec r =(1,3,3)^T$ in the $x$-direction?
Does the fluid at this position
experience a net force per unit volume?

\item
Make question 1.6 in the book.

\item
A jet-engine can be roughly modelled as a piece of circular pipe of radius
$R$ with its axis along the z-direction.
At one end, air enters the engine with density $\rho$ and uniform velocity
$w=V_\infty$,
at the other end the air exits with velocity $w=3V_\infty (1 - (r/R)^2)$.
The velocity field inside the pipe is steady.
Draw the air which was at time $t$ within the engine at time $t+Dt$.
The  mass of this air did not change during time interval $Dt$,
neglecting the added fuel.  Explain from the figure
why this means that the density
at the entrance of the engine must be different from that at the exit.
Assume the density is constant at both the entrance and at the exit.
Show that the momentum of the air has changed during the time interval
$Dt$, so that the engine produces a thrust force.
Compare the results you get using this Lagrangian approach with the
control volume expressions derived in class.

\item
Repeat question 6 for the case that the engine itself moves toward the
left with a velocity $V_2$.  Show that the force exerted on the
air in the jet engine, (which is the negative of the thrust force),
is now given by
$$F = \int\int_{\hbox{exit}} \rho (u + V_2) u dA -
\int\int_{\hbox{entrance}} \rho (u + V_2) u dA.$$
In other words, the relative velocity is used in $\vec V \cdot \vec n$,
but the absolute velocity in the momentum per unit volume $\rho \vec V$.

\item
From the energy equation derived in class, show that the sum
of kinetic energy and enthalpy per unit mass of
air that passes through a constriction in a pipe is unchanged.
To do so, ignore gravity,
assume that the viscous stresses on the air entering and leaving the
pipe can be ignored (although they are important inside the pipe),
that there is no heat conduction through the surface of the pipe,
and that the flow inside the pipe is steady.
Take the flow velocity and thermodynamic properties to be constant at
both the entrance and the exit. Discuss each term in the energy equation.

\item
Write the five conservation equations in differential form for the case
that all flow quantities depend on only one spatial dimension.
(In other words, everything depends on $y$ and $t$ only.)
Write out completely in terms of $u$, $v$, and $w$ and thermodynamic
quantities.
Assume an isotropic Newtonian viscous fluid satisfying Fourier's law,
where $\mu>0$ and $k>0$ depend on temperature. Ignore gravity.

\item
Assume in addition to question 8 that the pressure $p$,
density $\rho$, viscosity $\mu$, and heat conduction coefficient $k$
are constant, and that $v$ and $w$ are zero.
Write the five resulting equations for $u$ and $T$.
Based on the results, explain why you expect this class will be a
fun continuation of Analysis in Mechanical Engineering.

\item
A velocity field is locally approximated by
$u = x$, $v=y$, $w=z$.
Consider the fluid that is initially at $x=y=z=1$.
How long does it take for the density of this fluid to decrease by a factor 8?
Where is the fluid when that happens?

\item
Consider a little cube of fluid ${\rm d} x {\rm d} y {\rm d} z$.
Show that the force due to shear stresses on the outside faces of the
cube is $F_i = {\rm d} x{\rm d} y{\rm d} z\partial\sigma_{ij}/\partial x_j$.
So $\partial\sigma_{ij}/\partial x_j$ is the force per unit volume.
What is the force per unit volume for inviscid fluids?
Express using nabla.
Write Newton's second law per unit volume for an inviscid fluid.
Give the name of this equation.

\item
The air inside a balloon is expanding uniformly in all directions.
Give the rate of strain tensor and the viscous stress tensor.

\item
Show that the dissipation for a Newtonian fluid satisfying Stokes'
hypothesis with $\mu > 0$ is always positive.

\item
Consider an incompressible fluid extending over entire space.
Ignore gravity.
Show that if the flow is inviscid, the total kinetic energy of the fluid
is preserved.

\item
Show that in the absence of heat conduction, the entropy of
a Newtonian fluid increases with time.

\item
A metal sphere has been dropped in the sea and is sinking at a
speed $V$. Give the boundary conditions for the flow of the sea water
around this sinking sphere in a spherical coordinate system
with origin at the center of the sphere.
What would the inviscid boundary conditions be?

\end{enumerate}

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