A depth indicator moves vertically downward through
oil-filled gaps at constant speed U. At a distance x
from the origin of the coordinate system, a control surface is shown.
The velocity profile at the control surface is given by,
$$u = \frac{U}{2h^2}(y^2 - y h) \left(\frac{2h}{x} - \frac{x}{2h}\right) - U \frac{x}{2h}$$
a) Determine the velocity at $y = h$.
b) Determine, $\frac{dP}{dx}$ the pressure gradient in terms of $h$,
$U$, $\rho$, $\omega$, $\mu$, $\epsilon$.
c) If $U$ were increased by a factor of two, will the volume
flow rate increase or decrease and by how much will it change.
