Consider developing Couette flow-the same flow as Prob. 7-2 except that the flow is not yet steady-state, but is developing with time. In other words, time t is an additional parameter in the problem. Generate a dimensionless relationship between all the variables.
The functional relationship between given parameters.
Answer to Problem 53P
The functional relationship between given parameters is
Explanation of Solution
Given information:
The viscosity of the fluid is
Write the expression of function of fluid velocity.
Write the expression for the number of the pi terms.
Here, the total number of the repeating variable is
The total number of the variable are 7
Substitute
Write the expression for the first pi-terms.
Here, the constants are
Write the expression for the second pi-terms.
Write the expression for the third pi-terms.
Write the expression for the forth pi-terms.
Write the expression for the relation between the pi terms.
Write the dimensional expression for the fluid velocity.
Here, the length is
Write the dimensional expression for the viscosity of the fluid.
Here, the mass is
Write the dimensional expression for the density.
Write the dimensional expression for the length.
Write the dimensional expression for the speed of the plate.
Write the dimensional expression for the time.
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
Since, the Equation (IX) is the reciprocal of the Reynolds number hence
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
Substitute
Conclusion:
The functional relationship between given parameters is
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Chapter 7 Solutions
Fluid Mechanics: Fundamentals and Applications
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