Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 4, Problem 98P

The velocity field for an incompressible flow is given as V = 5 x 2 i 20 x y j + 100 t k . Determine if this flow is steady. Also determine the velocity and acceleration of a particle at (1, 3, 3) at: =0.2 s.

Expert Solution & Answer
Check Mark
To determine

The velocity of particle.

The acceleration of particle

Answer to Problem 98P

The velocity of particle is V=5i60j+20k_.

The acceleration of particle is a=50i+900j+100k_.

Explanation of Solution

The vector field of flow is V=5x2i20xyj+100tk, and the position of particle is (1,3,3) at time equal to 0.2s

Write the expression for the velocity field vector.

  V=ui+vj+wk

Here, the velocity in x direction is u, the velocity in y direction is v and the velocity in z direction is w.

Write the expression for the given velocity field vector.

  V=5x2i20xyj+100tk   ...... (I)

Here, the velocity in x direction is 5x2, the velocity in y direction is 20xy and the velocity in z direction is 100t.

Write the expression for the acceleration component in x direction.

  ax=ut+uux+vuy+wuz   ...... (II)

Write the expression for the acceleration component in y direction.

  ay=vt+uvx+vvy+wvz   ...... (III)

Write the expression for the acceleration component in z direction.

  az=wt+uwx+vwy+wwz   ...... (IV)

Substitute 5x2 for u, 20xy for v and 100t for w in Equation (II).

  ax=(5 x 2)t+5x2(5 x 2)x+(20xy)(5 x 2)y+100t(5 x 2)z=0+50x30+0=50x3..... (V)

Substitute 5x2 for u, 20xy for v and 100t for w in Equation (III).

  ay=(20xy)t+5x2(20xy)x+(20xy)(20xy)y+100t(20xy)z=0100x2y+400x2y+0=300x2y..... (VI)

Substitute 5x2 for u, 20xy for v and 100t for w in Equation (IV).

  az=(100t)t+5x2(100t)x+(20xy)(100t)y+100t(100t)z=100+0+0+0=100..... (VII)

Write the expression for acceleration vector.

  a=axi+ayj+azk   ...... (VIII)

Calculation:

Substitute 1 for x, 3 for y and 0.2s for t in Equation (I).

  V=5×(1)2i(20×1×2)j+(100×0.2s)kV=5i60j+20k

Substitute 1 for x in Equation (V).

  ax=50(1)3=50

Substitute 1 for x and 3 for y in Equation (VI).

  ay=300(1)2(3)=900

Substitute 900 for ay, 50 for ax and 100 for az in Equation (VIII).

  a=50i+900j+100k

Conclusion:

The answer is calculated by by the expression : ay=vt+uvx+vvy+wvz

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