fluid_mechanics

Problems 185
4.62 In the region just downstream of a sluice gate, the water
may develop a reverse flow region as is indicated in Fig. P4.62
and Video V10.9. The velocity profile is assumed to consist of
two uniform regions, one with velocity V a 10 fps and the other
with V b 3 fps. Determine the net flowrate of water across the
portion of the control surface at section 122 if the channel is 20 ft
wide.
Plate
v(x)
x
Sluice gate
Control surface
A
B
V b = 3 ft/s
(1) (2)
F I G U R E P4.62
4.63 At time t 0 the valve on an initially empty 1perfect vacuum,
r 02 tank is opened and air rushes in. If the tank has a volume
of V 0 and the density of air within the tank increases as
r r 11 e bt 2, where b is a constant, determine the time rate of
change of mass within the tank.
†4.64 From calculus, one obtains the following formula 1Leibnitz
rule2 for the time derivative of an integral that contains time in both
the integrand and the limits of the integration:
d
dt x 21t2
x 1 1t2
f 1x, t2dx
x 2
x 1
V a = 10 ft/s
0f
0t dx f 1x 2, t2 dx 2
dt
1.8 ft
1.2 ft
f 1x 1 , t2 dx 1
dt
Discuss how this formula is related to the time derivative of the
total amount of a property in a system and to the Reynolds transport
theorem.
4.65 Water enters the bend of a river with the uniform velocity
profile shown in Fig. P4.65. At the end of the bend there is a region
of separation or reverse flow. The fixed control volume ABCD
coincides with the system at time t 0. Make a sketch to indicate
(a) the system at time t 5 s and (b) the fluid that has entered and
exited the control volume in that time period.
V = 1 m/s
B
10 m
A
Control volume
F I G U R E P4.65
0.5 m/s
D
C
1.2 m
4.66 A layer of oil flows down a vertical plate as shown in
Fig. P4.66 with a velocity of V 1V 0h 2 2 12hx x 2 2 ĵ where V 0
and h are constants. (a) Show that the fluid sticks to the plate and
that the shear stress at the edge of the layer 1x h2 is zero. (b) Determine
the flowrate across surface AB. Assume the width of the
plate is b. (Note: The velocity profile for laminar flow in a pipe has
a similar shape. See Video V6.13.)
y
Oil
F I G U R E P4.66
h
4.67 Water flows in the branching pipe shown in Fig. P4.67 with
uniform velocity at each inlet and outlet. The fixed control volume
indicated coincides with the system at time t 20 s. Make a sketch
to indicate (a) the boundary of the system at time t 20.1 s, (b) the
fluid that left the control volume during that 0.1-s interval, and (c)
the fluid that entered the control volume during that time interval.
V 3 = 2.5 m/s
(3)
Control volume
0.8 m
0.6 m
F I G U R E P4.67
0.5 m
4.68 Two plates are pulled in opposite directions with speeds of
1.0 ft/s as shown in Fig. P4.68. The oil between the plates moves
with a velocity given by V 10 yî ft/s, where y is in feet. The fixed
control volume ABCD coincides with the system at time t 0. Make
a sketch to indicate (a) the system at time t 0.2 s and (b) the fluid
that has entered and exited the control volume in that time period.
Control
volume
1 ft/s
B
A
0.1 ft
0.1 ft
y
0.2 ft 0.2 ft
F I G U R E P4.68
V 1 = 2 m/s
4.69 Water is squirted from a syringe with a speed of V 5 ms by
pushing in the plunger with a speed of V p 0.03 ms as shown in
Fig. P4.69. The surface of the deforming control volume consists of
the sides and end of the cylinder and the end of the plunger. The system
consists of the water in the syringe at t 0 when the plunger
is at section 112 as shown. Make a sketch to indicate the control surface
and the system when t 0.5 s.
D
C
(1)
(2)
u(y) = 10y ft/s
V 2 = 1 m/s
x
1 ft/s