Let
If both integrals of the divergence theorem are equal.
Answer to Problem 16P
Both integrals of the divergence theorem are equal.
Explanation of Solution
Given information:
The vector is
The following figure shows the faces of the cube.
Figure-(1)
Consider flow is steady and incompressible.
Write the expression for the divergence theorem.
Here, unit vector is
Write the reference equation for
Here, unit vector in x-direction is
Compare the Equation (II ) with given vector.
Partially differentiate Equation (III) with respect to the
Partially differentiate Equation (IV) with respect to the
Partially differentiate Equation (V) with respect to the
Write the expression for the volume integral.
Write the expression for
Write the expression for
Write the expression for the surface integral.
Calculation:
Substitute
for
The unit vector
Substitute
The unit vector
Substitute
The unit vector
Substitute
The unit vector
Substitute
The unit vector
Substitute
The unit vector
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
From the Equation (IX) and (X) integral of the divergence theorem is equal.
Conclusion:
Both integrals of the divergence theorem are equal.
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Chapter 9 Solutions
Fluid Mechanics: Fundamentals and Applications
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