Answer the following:

$($a$)$ The flow inside a pipe, spinning along its axis, takes place entirely in cylindrical shells so that

there is no radial velocity component. Write down an explicit expression $($in terms of

cylindrical coordinates $r,,z$ for a streamfunction that describes one set of streamsurfaces of

this flow.

$($b$)$ For the flow in part $($a$),$ derive an expression for the non$-$zero cylindrical velocity components

in terms of the derivative of the second streamfunction. You may assume that the flow is

compressible and steady.

$($c$)$ Consider now water being stored in a $2$ m $-$diameter spherical tank, described using spherical

coordinates $(r,,)$ with an origin at the center of the tank, and the latitude $$ measured

from the vertical axis rising out of the top of the tank. $($See the coordinate system defined in

our Vector Algebra class slides$).$ Gravity exerts a downward force per unit mass of $9\mathrm{.}8\frac{m}{s^2}$ on

the water. Write down the $($spherical$)$ component representation of the gravity vector at $($i$)$

$(0,0,0),($ii$)(1,0,0),($iii$)(1,\frac{}{4},).$

$($d$)$ The tank in part $($c$)$ contains a stirrer that rotates about the vertical axis twice every second.

Estimate maximum vertical vorticity of the water as it is drained through a $1$ cm diameter hole

at the bottom of the tank.