Concept explainers
(a)
The velocity profile approach from the outer cylinder wall to the inner cylinder wall.
Answer to Problem 98P
The Velocity profile is
Explanation of Solution
Given information:
Write the expression for Navier-Stokes equation for cylindrical coordinates along
Here, the dynamic viscosity is
Write the expression for gap between the outer and inner radius.
Here, the gap between the outer and inner radius
Write the expression for distance from one end of the wall.
Write the expression for speed of the upper plate.
Here, the angular velocity at inner radius is
Calculation:
Integrate Equation (I).
Here, the integration constant is
Integrate Equation (II).
Here, the integration constant is
Apply boundary condition in Equation (III).
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Therefore, the final velocity profile is linearly varying from the outer radius.
Conclusion:
The Velocity profile is
(b)
The type of flow when the outer wall approaches to infinity and the inner cylinder radius is very small.
Answer to Problem 98P
This type of flow when the outer wall approaches to infinity and the inner cylinder radius is very small is called as linear vortex flow in a cylinder.
Explanation of Solution
Given information:
The outer cylinder wall approaches to infinity and the inner cylinder radius is very small, so
Calculation:
Applying boundary condition in Equation (V).
Substitution
Substitute
Therefore, the final velocity profile represents a linear vortex flow in a cylinder.
Conclusion:
This type of flow when the outer wall approaches to infinity and the inner cylinder radius is very small is called as linear vortex flow in a cylinder.
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Chapter 9 Solutions
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