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ISBN: 9780071284219 39

## Solution for problem 98P Chapter 4

Fluid Mechanics | 2nd Edition

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Fluid Mechanics | 2nd Edition

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Problem 98P

Combine your results from Problem given below  to form the two-dimensional strain rate tensor in the ɛij in the xy-plane,

Are the x- and y-axes principal axes?

PROBLEM: For the two-dimensional Poiseuille flow of Prob. 4-100, calculate the linear strain rates in the x- and y-directions, and calculate the shear strain rate ɛxy.

PROBLEM 4 - 100: Consider fully developed two-dimensional Poiseuille flow—flow between two infinite parallel plates separated by distance h,with both the top plate and bottom plate stationary, and a forced pressure gradient dP/dxdriving the flow as illustrated in Fig. P11–48. (dP/dxis constant and negative.)

FIGURE P11-48

The flow is steady, incompressible, and two-dimensional in the xy-plane. 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 z-direction. Do fluid particles in this flow rotate clockwise or counterclockwise?

Step-by-Step Solution:

Solution to 98P

Step 1</p>

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

Where, u is the component of velocity in x direction, v is the component of velocity in y direction,  is the pressure gradient, μ is the fluid viscosity, h is the height.

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>

εxx=εyy=0 since,

And,

Now,

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

since v=0

Step 3 of 3

##### ISBN: 9780071284219

Fluid Mechanics was written by and is associated to the ISBN: 9780071284219. The full step-by-step solution to problem: 98P from chapter: 4 was answered by , our top Engineering and Tech solution expert on 07/03/17, 04:51AM. The answer to “Combine your results from below to form the two-dimensional strain rate tensor in the ?ij in the xy-plane, Are the x- and y-axes principal axes?PROBLEM: For the two-dimensional Poiseuille flow of Prob. 4-100, calculate the linear strain rates in the x- and y-directions, and calculate the shear strain rate ?xy.PROBLEM 4 - 100: Consider fully developed two-dimensional Poiseuille flow—flow between two infinite parallel plates separated by distance h,with both the top plate and bottom plate stationary, and a forced pressure gradient dP/dxdriving the flow as illustrated in Fig. P11–48. (dP/dxis constant and negative.) FIGURE P11-48The flow is steady, incompressible, and two-dimensional in the xy-plane. The velocity components are given by where ? is the fluid’s viscosity. Is this flow rotational or irrotational? If it is rotational, calculate the vorticity component in the z-direction. Do fluid particles in this flow rotate clockwise or counterclockwise?” is broken down into a number of easy to follow steps, and 143 words. Since the solution to 98P from 4 chapter was answered, more than 266 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Fluid Mechanics, edition: 2. This full solution covers the following key subjects: flow, Dimensional, strain, calculate, plate. This expansive textbook survival guide covers 15 chapters, and 1547 solutions.

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