MAE 101B
Quiz #1 - Solution
Problem 1
1. For fully developed laminar flow in a pipe, (a) determine the radial distance from the
pipe axis at which the velocity equals the average velocity. (b) At this position find an
expression for the shear stress in terms of the shear stress at the wall.
After you have finished this on scrap please:
(a&b) Clearly list assumptions (4 pts)
(1)
(2)
(3)
(4)
Laminar flow
Fully developed flow
Steady flow
Incompressible flow
(a) Identify and number relevant equations i.e. the ones you use directly (6 pts)
2
R 2  p    r  
1. u  
  1    
4  x    R  
2. V  
R 2  p 
 
8  x 
(a) Describe clearly and succinctly your strategy for getting the answer (6 pts)
1. Set eqn. 2 equal to eqn. 1 and solve for r.
(a) Write down your answer (6 pts)
R
r
2
(b) Identify and number relevant equations i.e. the ones you use directly (6 pts)
1.  rx  
u r  p 
  
y 2  x 
(b) Describe clearly and succinctly your strategy for getting the answer (6 pts)
1. Find wall shear by plugging in r = R to eqn. 1.
R
2. Find shear at r 
by plugging it into eqn. 1 as well.
2
3. Divide shear at average velocity location by the shear at the wall.
(b) Write down your answer (6 pts)
r
1

w
2
Problem 2
2. Water flows through a 25 mm diameter tube that suddenly enlarges to a diameter of
50mm. The flow rate through the enlargement is 1.25 Liters / s. (a) Calculate the
pressure rise across the enlargement. (b) Compare with the value for frictionless flow.
*Note: 1 m3 = 1000 L; when using Figure 8.14 pick one of the following K’s: 0.0, 0.19,
0.25, 0.56, 0.87 or 1
After you have finished this on scrap please:
d = 25mm
D = 50mm
(a&b) Clearly list assumptions (4 pts)
(1)
(2)
(3)
(4)
Incompressible flow
Horizontal section
Steady flow
Uniform flow at each section: 1   2  1
Q = 1.25 liters per minute
(a) Identify and number relevant equations i.e. the ones you use directly (6 pts)
p
 p

V2
V2
1. hlT   1  1 1  gz1    2   2 2  gz2 
2
2

  

2
Q
V
2. hlT  K 1
3. V1 A1  V2 A2
4. V1 
A
2
(a) Describe clearly and succinctly your strategy for getting the answer (6 pts)
2
1
d
1. Use Fig. 8.14 to find AR     and K=0.56 for this expansion.
4
D
2. Use eqn. 3 to find V2  f V1 
3. Solve eqn. 4 using given Q and A in the smaller pipe.
4. Plug eqn. 2 into eqn. 1.
4. Using assumption 2 simplify eqn. 1.
5. Plug in known and acquired values and rearrange to find the pressure rise, p2  p1 .
(a) Write down your answer (6 pts)
p2 – p1 = 1.22 kPa
(b) Identify and number relevant equations i.e. the ones you use directly (6 pts)
Use simplified eqn. 1 from part a.
(b) Describe clearly and succinctly your strategy for getting the answer (6 pts)
1. For frictionless flow K=0, so plug that into above acquired eqn. and solve for p2  p1 .
2. Use
p actual
and find the ratio
p frictionless
(b) Write down your answer (6 pts)
p actual
= 40.3 %
p frictionless
Problem 3
From the graph, and/or what you’ve learned over the last two weeks, which of the
following statements are most true. (d)
a.
b.
c.
d.
Turbulent shear stress is 0 at the wall and 1 at the pipe centerline
Turbulent shear stress is 1 at the wall and 0 at pipe centerline
Turbulent shear stress is independent of Re number
None of the above
Problem 4
Which of the statements below, relating to the no-slip condition, are most correct: (d)
a. No slip occurs because of the large number of collisions between the fluid molecules
and the wall
b. No slip condition is independent of the material that the wall is made of
c. Although the velocity of the fluid at the wall is the same as that of the wall, it is not
comprised of the same molecules, i.e. molecules come and go within the fluid layer at the
wall
d. all of the above
Problem 5
Which of the statements below, relating to laminar flow in a duct with a pressure gradient
and a moving wall, are most correct: (b)
a. It is possible to have 0 shear stress at the upper wall if the pressure gradient and wall
velocity have opposite signs
b. it is possible to have 0 shear stress at the upper wall if the pressure gradient and wall
velocity have the same sign
c. it is not possible to have 0 shear stress at the upper wall
Problem 6
Which of the statements below, relating to turbulent shear stress are most correct: (e)
a. The expression for turbulent shear stress can be developed from the governing
equations
b. Because of the no slip condition the turbulent shear stress is 0 at the wall
c. Throughout most of the pipe, if the flow is turbulent, then turbulent shear stress is
greater than the laminar shear stress
d. Turbulent shear stress is 0 at the center of the pipe because the mean velocity gradient
is 0
e. All of the above
Problem 7
From the graph below and/or what you have learned the last two weeks, the following
statement(s) are most true: (e)
a. The relationship between f and Re depends on the fluid, i.e. air or water
b. Throughout laminar and turbulent flow the pressure drop decreases with increasing
velocity
c. The formulas relating f to Re for laminar and turbulent flow are both empirically
derived
d. Transition between laminar and turbulent flow always occurs at Te = 2300
e. None of the above