Good afternoon may you please provide me with an in detailed workout of this question with necessary explanations but please don't let it over power the workout. The use of partial derivatives in this question is giving me problems so I would appreciate it if you show that workout also.

A water pump is used to transport water with a density of 1000 kg/m and dynamic viscosity of 0.001 Pas. The pump has a diameter of 0.1 m and rotates at a speed of 600 rpm, producing a pressure of 2 bar at the outlet. The inlet diameter of the pump is 0.05 m and the inlet velocity of the water is negligible. Assuming steady-state, incompressible flow and neglecting losses, what is the velocity of the water at the outlet of the pump?

Parameters:

: density of the water = 1000 kg/m

: dynamic viscosity of the water = 0.001 Pas

D: diameter of the pump = 0.1 m

N: rotational speed of the pump = 600 rpm

P_out: pressure at the outlet of the pump = 2 bar

D_in: diameter of the inlet of the pump = 0.05 m

u_out: velocity of the water at the outlet of the pump

P/x: pressure gradient along the length of the pump

u/r: radial velocity gradient inside the pump