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Unformatted text preview: (R2 r2) (6.40) The laminar-flow profile is thus a paraboloid falling to zero at the wall and reaching a maximum at the axis umax R2 4 d (p dx gz) (6.41) It resembles the sketch of u(r) given in Fig. 6.10. The laminar distribution (6.40) is called Hagen-Poiseuille flow to commemorate the experimental work of G. Hagen in 1839 and J. L. Poiseuille in 1940, both of whom established the pressure-drop law, Eq. (6.1). The first theoretical derivation of Eq. (6.40) was given independently by E. Hagenbach and by F. Neumann around 1859. Other pipe-flow results follow immediately from Eq. (6.40). The volume flow is R Q R u dA 0 1 umax R2 2 0 umax 1 R4 8 r2 2 r dr R2 d (p dx gz) | v v Thus the average velocity in laminar flow is one-half the maximum velocity | e-Text Main Menu | Textbook Table of Contents | Study Guide (6.42) Chapter 6 Viscous Flow in Ducts Q A V For a horizontal tube ( z iment, Eq. (6.1): Q R2 1 umax 2 (6.43) 0), Eq. (6.42) is of the form predicted by Hagen’s exper8 LQ R4 p (6.44) The wall shear is computed from the wall velocity gradient du dr w rR 2 umax R 1d R (p 2 dx gz) (6.45) This gives an exact theory for laminar Darcy friction factor f 8w V2 or 8(8 V/d) V2 flam 64 Vd 64 Red (6.46) This is plotted late...
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This note was uploaded on 10/27/2009 for the course MAE 101a taught by Professor Sakar during the Spring '08 term at UCSD.

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