5 HW 5

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. In the upper half plane $y\ge 0$, consider the two-dimensional flow
   
   $$u = c x \qquad v = -c y \qquad p + \rho g y = p_0 -{\textstyle\frac{1}{2}} \rho u^2 - {\textstyle\frac{1}{2}} \rho v^2$$
   
   
   where $c$ and $p_0$ are constants. Find and neatly draw the streamlines, particle paths and streaklines of this flow. Put flow direction arrows on the lines. What is the name for the shape of the streamlines?

2. Find the particle acceleration vector field for the flow of the previous question. Very neatly draw a few acceleration vectors in your graph. Does density times acceleration equal the pressure force per unit volume plus the viscous force per unit volume, plus the downward gravity force per unit volume, assuming a Newtonian fluid?

3. Consider the velocity field
   
   $$\vec v = \frac{t}{r} \hat\imath _r + \frac{1}{r} \hat\imath _\theta$$
   
   
   This represents an ideal circulatory flow around an expanding cylinder of radius t. (So the fluid is restricted to $r>t$). Find and draw the streamlines and particle paths of this flow.

4. For the flow of the previous question, find the expression for the streakline at some time $t$ if the moving smoke generator at some earlier time $\tau$ was at position $r_g=2\tau$, $\theta_g=0$. There is no requirement to draw the curve. But be sure to eliminate $\tau$ to get a relation between $r$ and $\theta$ only.

5. Write the equation that applies for hydrostatics of a Newtonian fluid, in terms of $h$. Now take the curl of the equation (i.e. premultiply by $\nabla\times$). Simplify the expressions using the formulae for $\nabla$ in the vector analysis section of your mathematical handbook. This should show you that $\nabla(\rho g)$ is parallel to $\nabla(h)$ where $h$ is the height above sea-level. Explain why that means that $\rho g$ only varies with $h$, i.e. the density must be everywhere the same in a plane of constant height. In your explanation, use a coordinate system with its $z$-axis upward, so that $z=h$, and write out the gradients.