Concept explainers
Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter D or radium R = D/2 inclined at angle a. (Fig- 9-103). There is no applied pressure gradient
Answer:
FIGURE P9-103
The expression for average velocity through the pipe.
The expression for volume flow rate through the pipe.
The expression for
Answer to Problem 103P
The expression for average velocity through the pipe is
The expression for volume flow rate through the pipe is
The expression for
Explanation of Solution
Given information:
The diameter of the pipe is
pipe is
The flow is assumed to be Newtonian flow. The velocity field is assumed to be axial symmetric no swirl
The no slip condition at the pipe wall implies that when
The axis of the pipe is symmetry when
Write the expression for continuity equation for incompressible fluid.
Here, the change in distance along
Write the expression for the velocity component along
Here, the function of the radius is
Write the expression for angle
Write the expression for
Here, the density is
Write the expression for volume flow rate through pipe.
Here, the volume flow rate is
Write the expression for change in the area of the pipe.
Here, the change in the distance along
Write the expression for average velocity through the pipe.
Here, the velocity through the pipe is
Write the expression for area of pipe.
Here, the radius of the pipe is
Calculation:
Substitute
Integrate Equation (X) with respect to
Here, the constant is
Substitute
Double integration of Equation (X) with respect to
Here, the constant is
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Conclusion:
The expression for average velocity through the pipe is
The expression for volume flow rate through the pipe is
The expression for
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Chapter 9 Solutions
Fluid Mechanics: Fundamentals and Applications
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