Concept explainers
An incompressible fluid of density
Answer: Eu = f (Re,
The non -dimensional relationship parameters.
The non-dimensional for first pi terms.
The non-dimensional for second pi terms.
The non- dimensional for third pi terms.
The non-dimensional for fourth pi terms.
Answer to Problem 62P
The non -dimensional parameter for first pi terms is Euler number.
The non- dimensional parameter for second pi terms is Reynolds number.
The non -dimensional parameter for third pi terms is aspect ratio.
The non -dimensional parameter for fourth pi terms is roughness ratio.
The non-dimensional relationship is
Explanation of Solution
Given information:
A homogenous wire with a mass per unit length is
Write the expression for the moment of inertia of the link 3.
Here, the moment of inertia of the link 3 is
Write the expression for the moment of inertia of the link 4.
Here, the moment of inertia of the link 4 is
Write the expression for the centroidal component.
Write the expression for the moment of inertia of the link 5.
Here, the moment of inertia of the link 5 is
Write the dimension of the diameter of the pipe in
Here, the dimension for diameter of the pipe is
Write the dimension of the length of pipe in
Here, the dimensions for length of the pipe is
Write the dimension of the height of pipe in
Here, the dimension for the height of the pipe is
Write the expressions for the density.
Here, the mass is
Substitute
Write the expression for the pressure.
Here, the pressure is
Substitute
Write the dimension for the viscosity.
Write the dimension for the velocity.
Write the expression for the number of pi-terms.
Here, the number of variable is
Write the expression for first pi terms.
Here, the constant are
Write the dimension for pi term.
Write the expression for second pi terms.
Write the expression for third pi terms.
Write the expression for fourth pi terms.
Write the expression for relation between the pi terms.
Calculation:
The number of variables are
Substitute
Substitute
Compare the coefficients of
Compare the coefficients of
Compare the coefficients of
Substitute
Substitute
The non-dimensional for first pi terms is Euler number.
Substitute
Compare the coefficients of
Compare the coefficients of
Compare the coefficients of
Substitute
Substitute
The non-dimensional for second pi terms is Reynolds number.
Substitute
Compare the coefficients of
Compare the coefficients of
Compare the coefficients of
Substitute
Substitute
The non-dimensional for third pi terms is aspect ratio.
Substitute
Compare the coefficients of
Compare the coefficients of
Compare the coefficients of
Substitute
Substitute
The non-dimensional for fourth pi terms is roughness ratio.
Substitute
Conclusion:
The non -dimensional parameter for first pi terms is Euler number.
The non- dimensional parameter for second pi terms is Reynolds number.
The non -dimensional parameter for third pi terms is aspect ratio.
The non -dimensional parameter for fourth pi terms is roughness ratio.
The non-dimensional relationship is
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Chapter 7 Solutions
Fluid Mechanics: Fundamentals and Applications
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- In deriving the vorticity equation, we have used the identity divergence x (divergence P) = 0 Show that this identity is valid for any scalar lamda by checking it in Cartesian and cylindrical coordinates.arrow_forwardWhen fluid in a pipe is accelerated linearly from rest, itbegins as laminar flow and then undergoes transition toturbulence at a time t tr that depends on the pipe diameterD , fluid acceleration a , density ρ , and viscosity μ .Arrange this into a dimensionless relation between t trand D .arrow_forwardA smooth 12-cm-diameter sphere is immersed in a streamof 20°C water moving at 6 m/s. The appropriate Reynoldsnumber of this sphere is approximately(a) 2.3 E5, (b) 7.2 E5, (c) 2.3 E6, (d) 7.2 E6, (e) 7.2 E7arrow_forward
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