May I have a proper workout of this question showing all the necessary calculations I would greatly appreciate it.

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