Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 9, Problem 104P
To determine
The mathematical equation for each boundary condition.
The pressure boundary condition is appropriate for this flow field.
The velocity boundary conditions in terms of cylindrical coordinates and velocity component at free surface.
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Consider liquid in a cylindrical tank. Both the tank and the liquid rotate as a rigid
body. The free surface of the liquid is exposed to room air. Surface tension effects
are negligible. Present the boundary conditions required to solve this problem.
Specifically, what are the velocity boundary conditions in terms of cylindrical
coordinates (r, 0, z) and velocity components (ur, uo, U:) at all surfaces, including the
tank walls and the free surface? What pressure boundary conditions are appropriate
for this flow field? Explain.
Free
surface
Ps Paa
Liquid
2.0 m
7:
10.0 m
= 2²-²²
Us
B
10.0 m
Figure Q1-2
Question 2
Air flow at a constant speed (Us = 10 m/s) is forming a two-dimensional incompressible
laminar boundary layer along a flat plate The velocity profile inside the boundary layer is
given by:
2.0 m
(Equation 1)
At x = 1.00 m, the boundary layer thickness is given as 6.6094 mm. At this location:
a) Determine the shear stress at the wall, at y = 3 mm and y = 10 mm.
b) Calculate the boundary layer displacement thickness.
c) Calculate the mass flow rate through the boundary layer per unit width.
d) Calculate the mass flow rate per unit width of an ideal flow going through the same
height as the boundary layer thickness.
e) Through calculation relate the difference between the mass flow rates in parts (c)
and (d) to the local boundary layer displacement thickness. In not more than 60
word justify your answer. Use sketch(s) to illustrate your justification.
f) Does the assumed velocity profile satisfy the pressure boundary condition? In…
An incompressible Newtonian liquid is confined between two concentric circular cylinders of infinite length— a solid inner cylinder of radius Ri and a hollow, stationary outer cylinder of radius Ro. The inner cylinder rotates at angular velocity ?i. The flow is steady, laminar, and two-dimensional in the r?-plane. The flow is also rotationally symmetric, meaning that nothing is a function of coordinate ? (u? and P are functions of radius r only). The flow is also circular, meaning that velocity component ur = 0 everywhere. Generate an exact expression for velocity component u? as a function of radius r and the other parameters in the problem. You may ignore gravity.
Chapter 9 Solutions
Fluid Mechanics: Fundamentals and Applications
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