Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter D or radius R = D/2 inclined at angle a. There is no applied pressure gradient (@P/x = 0). Instead, the fluid flows down the pipe due to gravity alone. We adopt the coordinate system shown, with x down the axis of the pipe. Derive an expression for the x- component of velocity u as a function of radius r and the other parameters of the problem. Calculate the volume flow rate and average axial velocity through the pipe. 102 α Pipe wall Fluid: p. R X

Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter D or radius R = D/2 inclined at angle a. There is no applied pressure gradient (@P/x = 0). Instead, the fluid flows down the pipe due to gravity alone. We adopt the coordinate system shown, with x down the axis of the pipe. Derive an expression for the x- component of velocity u as a function of radius r and the other parameters of the problem. Calculate the volume flow rate and average axial velocity through the pipe. 102 α Pipe wall Fluid: p. R X

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Description: Consider steady, incompressible, laminar flow of a Newtonian fluid in aninfinitely long round pipe of diameter D or radius R = D/2 inclined at anglea. There is no applied pressure gradient (@P/x = 0). Instead, the fluidflows down the pipe due to gra

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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Question

Consider steady, incompressible, laminar flow of a Newtonian fluid in an
infinitely long round pipe of diameter D or radius R = D/2 inclined at angle
a. There is no applied pressure gradient (@P/x = 0). Instead, the fluid
flows down the pipe due to gravity alone. We adopt the coordinate system
shown, with x down the axis of the pipe. Derive an expression for the x-
component of velocity u as a function of radius r and the other parameters
of the problem. Calculate the volume flow rate and average axial velocity
through the pipe.
10₂
α
Pipe wall
Fluid: p. p
R

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Step 1: Determine the given variables:

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Step 2: Write the incompressible continuity equation in cylindrical coordinates:

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Step 3: Take the x-component of incompressible Navier-stokes equation:

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Step 4: Find the volume flow rate through the pipe:

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Step 5: Find the average axial velocity through the pipe:

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