Concept explainers
Calculate the nine components of the viscous stress tensor in cylindrical coordinates (see Chap. 9) for the velocity field of Prob. 10-109. Discuss.
The nine components of viscous stress tensor in cylindrical coordinate system.
Answer to Problem 110P
The nine components of viscous stress tensor in cylindrical coordinate system are
Explanation of Solution
Given information:
Velocity component for radial direction.
Velocity component for angular direction.
Here, radius for angular velocity is
Velocity component in vertical direction.
Write the expression for nine component viscous stress tensor in cylindrical co-ordinates.
Here, viscous stress tensor in purely radial direction is
viscous stress tensor in radial direction is
Write the expression for viscous stress tensor in radial direction.
Here, viscosity is
Write the expression for viscous stress tensor in angular direction.
Write the expression for viscous stress tensor in vertical direction.
Write the expression for viscous stress tensor in angular direction.
Write the expression for viscous stress tensor in vertical direction.
Write the expression for viscous stress tensor in purely vertical direction.
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the nine components of viscous stress tensor in cylindrical coordinate system are
Conclusion:
The nine components of viscous stress tensor in cylindrical coordinate system are
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Chapter 10 Solutions
Fluid Mechanics: Fundamentals and Applications
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