Consider a fully developed laminar flow of a fluid through a 200 m long and 4 cm diameter horizontal and circular pipe. The density and the viscosity of the fluid are 1400 kg/m³ and 0.3073 kg/(m.s). The velocity profile at a cross section is given by u(r) = 8 [1 - (r/R)²] in m/s, where r is the axial distance from the center and R is the radius of the pipe. Determine the following:

(i)   the maximum velocity at a cross section of the pipe, u_max

(ii)  the average velocity at a cross section of the pipe, V_avg

(iii) the volume flow rate, ý

(iv) Reynolds number, Re  of the flow

(v)  Friction factor, f

(vi) Head loss, h_L

(vii)Pressure loss, DP

(viii) Pumping power required, ω

(ix)  For the same pumping power calculated above, the percent decrease of the flow rate if the pipe is inclined 15^o upward (assume the head loss, h_L calculated in part (vi) does not change)

Useful formulae:   f = 64/Re and  h_L = f (L/D)(V²/2g)