Derive a differential equation for velocity
Answer to Problem 7.66P
The differential equation is,
Therefore, we can say
Explanation of Solution
Given information:
Diameter of the sphere is
Density of the sphere
The density of the fluid is
Drag co-efficient is
The specific gravity S of the sphere material is defined as,
The Newton’s law of downward motion is defined as,
Where,
The buoyancy force is defined as,
The drag force is defined as,
In above equation,
Calculation:
Apply Newton’s law of downward motion,
The weight of the sphere
Therefore,
Calculate the weight of the sphere,
Calculate the Buoyancy force exerted on sphere,
Substitute in equation (1),
But, according to explanation we can rewrite above equation as,
Replace buoyancy force in terms of
Rearrange,
Assume,
Therefore,
Integrate above equation to find
Therefore, we get,
Substitute,
Where,
Conclusion:
The differential equation is,
Therefore, we can say
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Chapter 7 Solutions
Fluid Mechanics
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