Show all assumptions derivations integrals and calculations

Show all assumptions, derivations, integrals and calculations We have steady, laminar, incompressible and fully-developed flow of a fluid through the pipe shown in the figure. Using Navier-Stokes equations in cylindrical coordinates (given in appendix E in the text) show that the velocity profile v_z can be written as V_z/V_zmax = l-(r/R)^2. Where V_zmax is the centerline velocity and given as V_zmax = [-R^2(4 mu)][d/dz(p+ rho gz)]. Show that the average velocity V in the pipe V= V_zmax/2. Using conservation of momentum and energy between points 1 and 2 show that the losses hf =[2 tau_w/(rho g)]L/R. In a fully-developed laminar pipe flow wall shear stress tau_w is a function of density rho, kinematic viscosity v, average velocity V and pipe diameter D. Using Buckingham phi theorem show that tau_w/(rho V^2) = f(Re) where Re=(VD)/v. Define function f(friction factor) as f = 8 tau_w/(rho V^2) and using results obtained in parts a, b and c show that in a fully developed pipe flow h_f = f(L/D)v^2/(2g) and for a fully-developed laminar pipe flow f = 64/Re

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