The dimensionless relationship between given parameters.
Answer to Problem 73P
The functional relationship between given parameters is
Explanation of Solution
The angular velocity is
Write the expression for the elevation difference.
Write the expression for the number of the pi terms.
Here, the total number of the repeating variable is
Substitute
Write the expression for the first pi term.
Write the expression for the independent pi term.
Write the expression for third pi term.
Write the expression for fourth pi term.
Write the expression for Reynold's number.
Write the dimensional formula for
Here, the dimension of mass is
Write the dimensional formula for elevation difference.
Write the dimensional formula for density.
Write the dimensional formula for radius.
Write the dimensional formula for angular velocity.
Write the dimensional formula for acceleration due to gravity.
Write the dimensional formula for time.
Write the dimensional formula for viscosity.
Calculation:
Substitute
Compare the power of the dimensional terms of
Compare the power of the dimensional terms of
Compare the power of the dimensional terms of
Substitute
Substitute
Compare the power of the dimensional terms of
Compare the power of the dimensional terms of
Compare the power of the dimensional terms of
Substitute
Write the relation between the two pi terms.
Write the relation between Froude Number and second pi term.
Compare Equation (X) and Equation (XI).
Substitute
Substitute
Compare the power of the dimensional terms of
Compare the power of the dimensional terms of
Compare the power of the dimensional terms of
Substitute
Substitute
Compare the power of the dimensional terms of
Compare the power of the dimensional terms of
Compare the power of the dimensional terms of
Substitute
Equation (XI) is the reciprocal of Reynold's Number.
Modify the Equation (XI).
Substitute
Write the expression for relation between pi terms.
Substitute
Conclusion:
The functional relationship between given parameters is
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