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# Solved: Consider the steady, incompressible, ISBN: 9780071284219 39

## Solution for problem 67P Chapter 4

Fluid Mechanics | 2nd Edition

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Problem 67P

Consider the steady, incompressible, two-dimensional flow field of  Prob. 4‒69. Using the results of Prob. 4‒69(a). do the following:

(a) From the fundamental definition of   shear strain rate (half of the rate of decrease of the angle between two initially perpendicular lines that intersect at a point), calculate shear strain rate εxy  in the xy-plane.(Hint: Use the lower edge and the left edge of the fluid particle, which intersect at 90° at the lower-left corner of the particle at the initial time.)

(b) Compare your results with those obtained from the equation for Ɛxy  in Cartesian coordinates, i.e., PROBLEM: Consider steady, incompressible, two-dimensional shear flow for which the velocity field is where a and b are constants. Sketched in Fig. P4‒69 is a small rectangular fluid particle of dimensions dx and dy at time t. The fluid particle moves and deforms with the flow such that at a later time (t + dt). the particle is no longer rectangular, as also shown in the figure. The initial location of each corner of the fluid particle is labeled in Fig. P4‒69. The lower-left corner is at (x. y) at time t. where the x-component of  velocity is u = a + by. At the later time, this corner moves to (x + u dt, y), or (a) In similar fashion, calculate the location of each of the other three corners of the fluid particle at time t ‒ dt.

(b) From the fundamental definition of   linear strain rate (the rate of increase in length per unit length). calculate linear strain rates Ɛxx and Ɛyy.

(c) Compare your results with those obtained from the equations for Ɛxx and Ɛyy in Cartesian coordinates, i.e..  FIGURE P4‒69

Step-by-Step Solution:
Step 1<p>We have to first calculate the shear strain using the fundamental definition of shear strain rate from the fundamental definition.

Then we have to compare the result we obtained using the equation Part (a)

Step 2</p>

The fundamental definition of the shear strain given by the half of the rate of decrease of angle between the two initially perpendicular line in the given object.

The following figure shows a rectangular particle in the fluid and how it deforms due to the velocity gradient in the fluid. The initial position of the lower left corner is given by and the upper left corner was . Now after time , the positions are

The position of the lower left corner  And the position of upper left corner is  Hence, the change in the position of the top left corner is  Hence, the angle is given by (since is very small)

So the angle is given by Hence, the rate of change of the angel is given by So the shear strain is given by So the calculated shear strain from the fundamental definition is .

Part (b)

Step 3 of 3

##### ISBN: 9780071284219

The answer to “Consider the steady, incompressible, two-dimensional flow field of Prob. 4?69. Using the results of Prob. 4?69(a). do the following:(a) From the fundamental definition of shear strain rate (half of the rate of decrease of the angle between two initially perpendicular lines that intersect at a point), calculate shear strain rate ?xy in the xy-plane.(Hint: Use the lower edge and the left edge of the fluid particle, which intersect at 90° at the lower-left corner of the particle at the initial time.)(b) Compare your results with those obtained from the equation for ?xy in Cartesian coordinates, i.e., PROBLEM: Consider steady, incompressible, two-dimensional shear flow for which the velocity field is where a and b are constants. Sketched in Fig. P4?69 is a small rectangular fluid particle of dimensions dx and dy at time t. The fluid particle moves and deforms with the flow such that at a later time (t + dt). the particle is no longer rectangular, as also shown in the figure. The initial location of each corner of the fluid particle is labeled in Fig. P4?69. The lower-left corner is at (x. y) at time t. where the x-component of velocity is u = a + by. At the later time, this corner moves to (x + u dt, y), or (a) In similar fashion, calculate the location of each of the other three corners of the fluid particle at time t ? dt.(b) From the fundamental definition of linear strain rate (the rate of increase in length per unit length). calculate linear strain rates ?xx and ?yy.(c) Compare your results with those obtained from the equations for ?xx and ?yy in Cartesian coordinates, i.e.. FIGURE P4?69” is broken down into a number of easy to follow steps, and 279 words. The full step-by-step solution to problem: 67P from chapter: 4 was answered by , our top Engineering and Tech solution expert on 07/03/17, 04:51AM. Fluid Mechanics was written by and is associated to the ISBN: 9780071284219. This textbook survival guide was created for the textbook: Fluid Mechanics, edition: 2. This full solution covers the following key subjects: particle, fluid, rate, corner, strain. This expansive textbook survival guide covers 15 chapters, and 1547 solutions. Since the solution to 67P from 4 chapter was answered, more than 399 students have viewed the full step-by-step answer.

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