Concept explainers
(a)
Whether
Answer to Problem 4.59P
Yes,
It’s an incompressible and steady flow.
Explanation of Solution
Given information:
The two dimensional incompressible velocity potential is given as,
According to the given information, the velocity potential is,
According to the definition,
&
The Laplace equation in three dimensions can be represented as,
Calculation:
According to the explanation,
According to the given information,
By differentiating it with respect to
To find
Similarly,
We can say,
To find
Therefore, according to the equations
According to the equations,
Since,
We can say that,
Therefore, according to the above result,
As the velocity potential satisfies this equation, we can say that this is an incompressible and steady flow.
Conclusion:
Yes,
As the velocity potential satisfies this equation, we can say that this is an incompressible and steady flow.
(b)
The stream function.
Answer to Problem 4.59P
Explanation of Solution
Given information:
The two dimensional incompressible velocity potential is given as,
The incompressible flow in
For a function
According to the above equations, the new function
Calculation:
According to the subpart a,
According to the explanation given,
To find
Therefore,
To find
Similarly,
According to
We can say,
Therefore,
So, the stream function will be,
Conclusion:
The stream function is equal to
(c)
The equation of the streamline that passes through
Answer to Problem 4.59P
Explanation of Solution
Given information:
The two dimensional incompressible velocity potential is given as,
According to the subpart b, the stream function is equal to
Therefore, by substituting
Calculation:
By substituting
Conclusion:
The stream line passes through point
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Chapter 4 Solutions
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