The laminar flow of a fluid with viscosity µ and density ρ in a circular pipe of radius R has a velocity distribution described by where UCL is the velocity on the centerline where r = 0. (a) Using a control volume analysis, find τw, the shear stress at the wall, in terms of R and the pressure gradient dp/dx. (b) Using the expression for the velocity profile, find τw, the shear stress at the wall, in terms of µ, R, and UCL. (c) Find the average velocity in terms of UCL. (d) Using the results from parts (a)-(c), show that where Re is the Reynolds number based on the diameter D.
The laminar flow of a fluid with viscosity µ and density ρ in a circular pipe of radius R has a velocity distribution described by where UCL is the velocity on the centerline where r = 0. (a) Using a control volume analysis, find τw, the shear stress at the wall, in terms of R and the pressure gradient dp/dx. (b) Using the expression for the velocity profile, find τw, the shear stress at the wall, in terms of µ, R, and UCL. (c) Find the average velocity in terms of UCL. (d) Using the results from parts (a)-(c), show that where Re is the Reynolds number based on the diameter D.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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The laminar flow of a fluid with viscosity µ and density ρ in a circular pipe of radius R has a velocity distribution described by
where UCL is the velocity on the centerline where r = 0.
(a) Using a control volume analysis, find τw, the shear stress at the wall, in terms of R and the pressure gradient dp/dx.
(b) Using the expression for the velocity profile, find τw, the shear stress at the wall, in terms of µ, R, and UCL.
(c) Find the average velocity 
in terms of UCL.
(d) Using the results from parts (a)-(c), show that
where Re is the Reynolds number based on the diameter D.
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