Concept explainers
Expression showing relationship x-component of fluid velocity.
Answer to Problem 107P
Dimensionless relationship for x-component of fluid velocity is
Explanation of Solution
Given Information:
Top plate speed,
Bottom plate speed,
Steady flow, incompressible, 2 dimensional.
Concept used:
Buckinghams Pi theorem.
Calculation:
Fluid velocity − u
Distance between parallel plate − h
Distance − y
Fluid density -
Fluid viscosity -
Plate speeds − V
No. Pf parameters, n = 6
The relative velocity of the top and bottom plate.
Fluid velocity u is the function of remaining five parameters
The primary dimension of each parameter.
Velocity,
Density,
Viscosity,
Distance y,
Distance h,
Plate speeds,
No. Of primary dimensions used in the problem is 3. (M,L,T)
J=3
K=n-j=6-3=3
As per Buckinghams Pi theorem, no. of this
Since j=3, we need to select 3 repeating parameters.
According to the guideline, we should select
(1) Now creating
Dependant
The primary dimension of the above term is
Now equation becomes,
Equating each primary dimension to solve
Mass,
Length,
Time,
Putting values
(2) Independent
Primary dimensions of the above term
Now equation becomes,
Equating each primary dimension to solve
Mass,
Length,
Time,
Putting values
Equation of
(3) Independent
The primary dimension of above term.
Now equation becomes,
Equating each primary dimension to solve
Mass,
Time,
Length,
Putting all the values in the equation,
Comparing equation (1)(2)(3)
Conclusion:
Thus, we can develop a dimensionless relationship for x-component of fluid velocity which is a function of fluid viscosity
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