4-Problems, fluid mech

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Problems
Problems
Flow Descriptions
4.1
Identify five examples of an unsteady flow and explain what features classify them as an unsteady
flow?
4.2
You are pouring a heavy syrup on your pancakes. As the syrup spreads over the pancake, would the
thin film of syrup be a laminar or turbulent flow? Explain.
4.3
Breathe in and out of your mouth. Try to sense the air flow patterns near your face while doing this.
Discuss the type of flow associated with these flow processes. If you were to blow out a candle, you would
do it while exhaling (at least most people do). Why is it easier to do this by exhaling than by inhaling?
4.4
In the system in the figure, the valve at
C
is gradually opened in such a way that a constant rate of increase
in discharge is produced. How would you classify the flow at
B
while the valve is being opened? How
would you classify the flow at
A
?
PROBLEM 4.4
Answer:
a. Unsteady, uniform;
b. nonuniform, steady
4.5
Water flows in the passage shown. If the flow rate is decreasing with time, the flow is classified as (a)
steady, (b) unsteady, (c) uniform, or (d) nonuniform.
PROBLEM 4.5
4.6
If a flow pattern has converging streamlines, how would you classify the flow?
Answer:
Nonuniform, steady or unsteady
4.7
Consider flow in a straight conduit. The conduit is circular in cross section. Part of the conduit has a
constant diameter, and part has a diameter that changes with distance. Then, relative to flow in that conduit,
correctly match the items in column A with those in column B.
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Problems
A
B
Steady flow ∂
V
s
/∂
s
= 0
Unsteady flow ∂
V
s
/∂
s
≠ 0
Uniform flow ∂
V
s
/∂
t
= 0
Nonuniform flow ∂
V
s
/∂
t
≠ 0
4.8
Classify each of the following as a onedimensional, twodimensional, or threedimensional flow.
a. Water flow over the crest of a long spillway of a dam.
b. Flow in a straight horizontal pipe.
c. Flow in a constantdiameter pipeline that follows the contour of the ground in hilly country.
d. Airflow from a slit in a plate at the end of a large rectangular duct.
e. Airflow past an automobile.
f. Air flow past a house.
g. Water flow past a pipe that is laid normal to the flow across the bottom of a wide rectangular channel.
Answer:
a. 2d,
b. 1d,
c. 1d,
d. 2d,
e. 3d,
f. 3d,
g. 2d
Pathlines, Streamlines and Streaklines
4.9
If somehow you could attach a light to a fluid particle and take a time exposure, would the image you
photographed be a pathline or streakline? Explain from definition of each.
4.10
The pattern produced by smoke rising from a chimney on a windy day is analogous to a pathline or
streakline? Explain from the definition of each.
4.11
At time
t
= 0, dye was injected at point
A
in a flow field of a liquid. When the dye had been injected for 4
s, a pathline for a particle of dye that was emitted at the 4 s instant was started. The streakline at the end of
10 s is shown below. Assume that the speed (but not the velocity) of flow is the same throughout the 10 s
period. Draw the pathline of the particle that was emitted at
t
= 4 s. Make your own assumptions for any
missing information.
PROBLEM 4.11
4.12
For a given hypothetical flow, the velocity from time
t
= 0 to
t
= 5 s was
u
= 2 m/s, ν = 0. Then, from time
t
= 5 s to
t
= 10 s, the velocity was
u
= + 3 m/s, ν = 4 m/s. A dye streak was started at a point in the flow
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Problems
field at time
t
= 0, and the path of a particle in the fluid was also traced from that same point starting at the
same time. Draw to scale the streakline, pathline of the particle, and streamlines at time
t
= 10 s.
4.13
At time
t
= 0, a dye streak was started at point
A
in a flow field of liquid. The speed of the flow is constant
over a 10 s period, but the flow direction is not necessarily constant. At any particular instant the velocity
in the entire field of flow is the same. The streakline produced by the dye is shown above. Draw (and
label) a streamline for the flow field at
t
= 8 s.
Draw (and label) a pathline that one would see at
t
= 10 s for a particle of dye that was emitted from point
A
at
t
= 2 s.
PROBLEM 4.13
Acceleration
4.14
Acceleration is the rate of change of velocity with time. Is the acceleration vector always aligned
with the velocity vector? Explain.
4.15
For a rotating body, is the acceleration toward the center of rotation a centripetal or centrifugal
acceleration? Look up word meanings and word roots.
4.16
Figure 4.24 on p. 110 shows the flow pattern for flow past a circular cylinder. Assume that the approach
velocity at
A
is constant (does not vary with time).
a. Is the flow past the cylinder steady or unsteady?
b. Is this a case of onedimensional, twodimensional, or threedimensional flow?
c. Are there any regions of the flow where local acceleration is present? If so, show where they are and
show vectors representing the local acceleration in the regions where it occurs.
d. Are there any regions of flow where convective acceleration is present? If so, show vectors
representing the convective acceleration in the regions where it occurs.
Answer:
a. Steady;
b. Twodimensional;
c. No;
d. Yes
4.17
The velocity along a pathline is given by
V
(m/s) =
s
2
t
1/2
where
s
is in meters and
t
is in seconds. The
radius of curvature is 0.5 m. Evaluate the acceleration along and normal to the path at
s
= 2 m and
t
= 0.5
seconds.
4.18
Tests on a sphere are conducted in a wind tunnel at an air speed of
U
0
. The velocity of flow toward the
sphere along the longitudinal axis is found to be , where
r
0
is the radius of the
sphere and
x
the distance from its center. Determine the acceleration of an air particle on the
x
axis
upstream of the sphere in terms of
x
,
r
0
, and
U
0
.
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Problems
PROBLEM 4.18
Answer:
4.19
In this flow passage the velocity is varying with time. The velocity varies with time at section
A-A
as
At time
t
= 0.50 s, it is known that at section
A-A
the velocity gradient in the
s
direction is + 2 m/s per
meter. Given that
t
0
is 0.5s and assuming quasi–onedimensional flow, answer the following questions for
time
t
= 0.5 s.
a. What is the local acceleration at
A-A
?
b. What is the convective acceleration at
A-A
?
PROBLEM 4.19
4.20
The nozzle in the figure is shaped such that the velocity of flow varies linearly from the base of the nozzle
to its tip. Assuming quasi–onedimensional flow, what is the convective acceleration midway between the
base and the tip if the velocity is 1 ft/s at the base and 4 ft/s at the tip? Nozzle length is 18 inches.
PROBLEMS 4.20, 4.21
Answer:
a
c
= 5 ft/s
2
4.21
In Prob. 4.20 the velocity varies linearly with time throughout the nozzle. The velocity at the base is 1
t
(ft/s) and at the tip is 4
t
(ft/s). What is the local acceleration midway along the nozzle when
t
= 2 s?
4.22
Liquid flows through this twodimensional slot with a velocity of
V
= 2(
q
0
/
b
)(
t
/
t
0
), where
q
0
and
t
0
are
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Problems
reference values. What will be the local acceleration at
x
= 2
B
and
y
= 0 in terms of
B
,
t
,
t
0
, and
q
0
?
PROBLEMS 4.22, 4.23
Answer:
a
l
= 4
q
0
/(
Bt
0
)
4.23
What will be the convective acceleration for the conditions of Prob. 4.22?
4.24
The velocity of water flow in the nozzle shown is given by the following expression:
where
V
= velocity in feet per second,
t
= time in seconds,
x
= distance along the nozzle, and
L
= length of
nozzle = 4 ft. When
x
= 0.5
L
and
t
= 3 s, what is the local acceleration along the centerline? What is the
convective acceleration? Assume quasi–onedimensional flow prevails.
PROBLEM 4.24
Answer:
a
l
= 3.56 ft/s
2
,
a
c
= 37.9 ft/s
2
Euler's Equation
4.25
State Newton's second law of motion as used in dynamics. Are there any limitations on the use of
Newton's second law? Explain.
4.26
What is the difference between a force due to weight and a force due to pressure? Explain.
4.27
A pipe slopes upward in the direction of liquid flow at an angle of 30° with the horizontal. What is the
pressure gradient in the flow direction along the pipe in terms of the specific weight of the liquid if the
liquid is decelerating (accelerating opposite to flow direction) at a rate of 0.3
g
?
4.28
What pressure gradient is required to accelerate kerosene (S = 0.81) vertically upward in a vertical pipe at a
rate of 0.3
g
?
Answer:
∂
p
/∂
z
= 65.7 lbf/ft
3
4.29
The hypothetical liquid in the tube shown in the figure has zero viscosity and a specific weight of 10
kN/m
3
. If
p
B
p
A
is equal to 12 kPa, one can conclude that the liquid in the tube is being accelerated (a)
upward, (b) downward, or (c) neither: acceleration = 0.
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