Consider steady laminar fully developed flow of an incompres

Consider steady, laminar, fully developed flow of an incompressible, Newtonian fluid in a straight pipe of constant radius, R. As shown in the figure, we will use cylindrical coordinates where the z axis is aligned with the centerline of the pipe. The fluid is driven by a pressure drop across the pipe where P_1 is the pressure at the entrance of the pipe and P_2 is the pressure at the exit. The length of the pipe is L. The effect of gravity is negligible. Assume the flow is axisymmetric and has no swirl initially. (a) From the continuity equation, show that V_r = 0 everywhere. (b) Show that the Navier-Stokes equations can be simplified for this flow as follows. mu1/r/r(r V_z/r) - P/z = 0. (Clearly justify all simplifications) (c) Show that the pressure P is function of only and dP/dz = const. (d) Show that the velocity profile can be obtained from the simplified continuity and Navier-Stokes equations and boundary conditions as follows. V_z(r) = -R^2/4mu (dP/dz) [1 - (r/R)^2] (e) Show that the relationship between the pressure drop, Delat P = P_1 - P_2 and volume flow rate, Q can be described as Q = piR^4Delta P/8muL. (f) Using the above result, show that friction factor f = 64/Re_D where Re_D = rho_UD/mu, U = Q/A, A is the cross-sectional area of the pipe and D is the diameter of the pipe.

Solution

A 30-mm bore series 02 angular-contact ball bearing carries a 700-lb radial load and a 1200-lb thrust load. The inner ring rotates at 1500 rpm. Determine the rating life in hours.(Note 1 lbf=4.45N) (answer: 250 hrs) I am using the Shigley S Mechanical Engineering Design 9th edition textbook and need a detailed explanation.