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Combine your results from Probs. 4?65 and 4?66 to form the

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

Solution for problem 63P 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 63P

Combine your results from Probs. 4‒65 and 4‒66 to form the two-dimensional strain rate tensor Ɛij in the xy-plane,

Under what conditions would the x- and y-axes be principal axes?

PROBLEM 4 -65:  For the velocity field of Prob. 4‒63, calculate the linear strain rates in the x- and y-directions.

PROBLEM 4-66: For the velocity field of Prob. 4‒63, calculate the shear strain rate in the xy-plane.

PROBLEM 4-63: A general equation for a steady, two-dimensional velocity field that is linear in both spatial directions (x and y) is

Where U and V and the coefficients are constants. Their dimensions are assumed to be appropriately defined. Calculate the x- and y-components of the acceleration field.

Step-by-Step Solution:

Solution to 63P

Step 1</p>

We need to find the two dimensional strain rate tensor for a flow with the velocity field equation as given below,

The fluid is assumed to be incompressible, steady and two dimensional.

The strain rate tensor εij in the x and y plane is given by,

 Where, εxx and εyy are the linear strain rates in x and y directions respectively and εxy and εyx are the shear strain rates in x and y directions respectively.

By symmetry,

εxy=εyx

Step 2</p>

Linear strain rate in x direction:

Linear strain rate in y direction:

Thus the linear strain rates in x and y directions are as follows

εxx=a1

εyy=b2

Shear strain rate is given  by,

u and v are velocity components in x and y directions.

Step 3 of 4

Chapter 4, Problem 63P is Solved
Step 4 of 4

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

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Combine your results from Probs. 4?65 and 4?66 to form the

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