2 9/8 W

A steady stream of air enter a pipe with a diameter of 1'' at a velocity of 20 m/s at a pressure of 1 bar. The pipe has a gradual contraction in diameter to $\frac12$ ''. What are the velocity and pressure after the contraction? Ignore viscous effects. (This involves the Bernoulli law and mass conservation of undergraduate fluid mechanics.) This is a steady flow, $\partial\vec{v}/\partial{t}=0$ , so explain how it is possible for the fluid to change velocity in a steady flow. In particular, write the material derivative of the velocity in Eulerian coordinates and discuss the various terms. Write in particular also the expression for the acceleration on the axis of the pipe, taking it as your -axis.
You are driving your Miata at 80 mph on a standard day. At a point on the nose just above the boundary layer the flow velocity of the air relative to the car is 120 mph. What is the pressure on the car surface at that point?
Newton's second law per unit volume reads

$\begin{displaymath} \rho \frac{D \vec v}{D t} = \vec f \end{displaymath}$

where is the net force on the fluid per unit volume. Assume inviscid flow, in which case the force per unit volume is minus the pressure gradient plus the force of gravity, so

$\begin{displaymath} \rho \frac{D \vec v}{D t} = -\nabla p + \rho \vec g \end{displaymath}$

Write these equations out fully in Eulerian coordinates for each of the three Cartesian velocity components , , and . Use the Eulerian form of the material derivative as given in class. The equations you should get are known as the Euler equations.
Substitute the Eulerian velocity field of stagnation point flow into the Euler equations that you obtained above. You get three equations for the pressure, one giving its -derivative, one its derivative, and the third its -derivative. More than one equation for a single unknown is usually too much, but show that in this case, there is indeed a solution that satisfies all three equations. Find out what it is. Take gravity to be in the negative -direction, so that and .
The velocity field in Couette flow is given by

$\begin{displaymath} u=cy \qquad v = 0 \qquad w = 0 \end{displaymath}$

Draw the streamlines of this flow and a couple of velocity profiles. Find the velocity derivative tensor , the strain rate tensor , and the matrix .
In Poiseuille flow (laminar flow through a pipe), the velocity field is in cylindrical coordinates given by

$\begin{displaymath} \vec v = {\widehat \imath}_z v_{\rm max} \left(1 -\frac{r^2}{R^2}\right) \end{displaymath}$

where $v_{\rm {max}}$ is the velocity on the centerline of the pipe and the pipe radius. Use Appendices B and C to find the velocity derivative and strain rate tensors of this flow. Do not guess. Evaluate the strain rate tensor at , $\frac12R$ and . What can you say about the straining of small fluid particles on the axis?