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/m^{3} 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)(V^{2}/2g)