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For the axisymmetric Poiseuille flow of Prob. 4-108.

Fluid Mechanics | 2nd Edition | ISBN: 9780071284219 | Authors: Yunus A. Cengel, John M. Cimbala ISBN: 9780071284219 39

Solution for problem 105P Chapter 4

Fluid Mechanics | 2nd Edition

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Fluid Mechanics | 2nd Edition | ISBN: 9780071284219 | Authors: Yunus A. Cengel, John M. Cimbala

Fluid Mechanics | 2nd Edition

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Problem 105P

For the axisymmetric Poiseuille flow of Prob. 4-108. calculate the linear strain rates in the x- and r-directions, and calculate the shear strain rate ɛxr. The strain rate tensor in cylindrical coordinates (r, θ. x) and (ur,uθ. ux). is

PROBLEM: Consider fully developed axisymmetric Poiseuille flow—flow in a round pipe of radius R(diameter D = 2 R), with a forced pressure gradient dP/dx driving the flow as illustrated in Fig. P11–53. (dP/dx is constant and negative.) The flow is steady, incompressible, and axisymmetric about the x-axis. The velocity components are given by

where μ is the fluid’s viscosity. Is this flow rotational or irrotational? If it is rotational, calculate the vorticity component in the circumferential (θ) direction and discuss the sign of the rotation.

FIGURE P11-53

Step-by-Step Solution:
Step 1 of 4<p>The velocity component of the flow is given. Using the velocity component calculate the linear strain in x and r direction, that is calculate  and . Also calculate the shear strain . Step 2 of 4<p>The velocity components are

,

 and

.

So the linear strain along the  direction is

Since the  component of the velocity has no  dependence.

Step 3 of 4

Chapter 4, Problem 105P is Solved
Step 4 of 4

Textbook: Fluid Mechanics
Edition: 2
Author: Yunus A. Cengel, John M. Cimbala
ISBN: 9780071284219

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For the axisymmetric Poiseuille flow of Prob. 4-108.

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