Fox and McDonald's Introduction to Fluid Mechanics
9th Edition
ISBN: 9781118912652
Author: Philip J. Pritchard, John W. Mitchell
Publisher: WILEY

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Textbook Question
Chapter 5, Problem 1P

Which of the following sets of equations represent possible three-dimensional incompressible flow cases?

1. (a) u = 2y2 + 2xz; υ = −2xy + 6x2yz; w = 3x2z2 + x3y4
2. (b) u = xyzt; υ = −xyzt2; w = z2 (xt2yt)
3. (c) u = x2 + 2y + z2; υ = x − 2y + z; w = −2xz + y2 + 2z
To determine

The set of equation that represent three-dimensional incompressible flow.

Explanation of Solution

Given:

The given set of equations are:

1. a) u=2y2+2xz;v=2yz+6x2yz;w=3x2z2+x3y4
2. b) u=xyzt;v=xyzt2;w=z2(xt2yt)
3. c) u=x2+2y+z2;v=x2y+z;w=2xy+y2+2z

Calculation:

The criteria that must be satisfied for a flow to be incompressible is,

xu+yv+zw=0

Check the given sets to satisfy the above criteria.

For set (a),

xu+yv+zw=x(2y2+2xz)+y(2yz+6x2yz)+z(3x2z2+x3y4)=2z+6x2z2z+6x2z=12x2z0

For set (b),

xu+yv+zw=x(xyzt)+y(xyzt2)+z(z2(xt2yt))=yztxzt2+2z(xt2yt)0

For set (c),

xu+yv+zw=x(x2+2y+z2)+y(x2y+z)+z(2xy+y2+2z)=2x2+2+2x=0

Hence, of the three only set (c) satisfies the criteria for the incompressible flow.

Thus, the set of equations that represent three-dimensional incompressible flow is u=x2+2y+z2;v=x2y+z;w=2xy+y2+2z

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