Hydraulics II
Problem Set No. 1:
Fundamentals of Fluid Flow
Problem 1
Tests on a sphere are conducted in a wind tunnel at an air speed of U0 . The velocity of
flow toward the sphere along the longitudinal axis is found to be
(︃
u = −U0
r3
1 − 03
x
)︃
where r0 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, r0 , and
U0 .
Problem 2
A fluid has a velocity components of vx = (3x2 + 1) m/s and vy = (4txy) m/s, where x
and y are in meters and t is in seconds. Determine the streamlines that pass through
point (1 m, 3 m) at times t = 1 s and t = 1.5 s. Plot these streamlines for 0 ≤ x ≤ 5 m.
Problem 3
[︃
]︃
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A fluid has velocity components of vx =
m/s and vy = (2ty) m/s, where x and
2x + 1
y are in meters and t is in seconds. Determine the pathline that passes through point
(2 m, 6 m) at time t = 2 s. Plot this pathline for 0 ≤ x ≤ 4 m.
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Hydraulics II
Problem 4
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 specific
weight of the liquid, γ, if the liquid is decelerating at a rate of 0.3g?
Problem 5
The closed tank shown, which is full of liquid, is accelerated downward at 1.5g and to
the right at 0.9g. Here L = 3 ft, H = 4 ft, and the specific gravity of the liquid is 1.2.
Determine PC − PA and PB − PA .
Problem 6
Pressurized air in a flask causes water to flow out a tube with a speed of V = 3 m/s.
Model the flow by assuming that the Bernoulli equation applies. What is the absolute
pressure of the gas?
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Hydraulics II
Problem 7
Water flows through a vertical contraction (venturi) section. Piezometers are attached
to the upstream pipe and minimum area section as shown. The mean velocity in the pipe
is v = 5 ft/s. The difference in elevation between the two water levels in the piezometers
is ∆z = 6 in. What is the velocity at the minimum area?
Problem 8
The
“spherical”)︁ Pitot probe shown is used to measure the flow velocity in hot water
(︁
ρ = 965 kg/m3 Pressure taps are located at the forward stagnation point and at 90◦
from the forward stagnation point. The speed of fluid next to the surface of the sphere
varies as 1.5v0 sin θ, where v0 is the free-stream velocity and θ is measured from the
forward stagnation point. The pressure taps are at the same level; that is, they are in
the same horizontal plane. The piezometric pressure difference between the two taps is 3
kPa. What is the free-stream velocity v0 ?
Problem 9
The vx and vy velocity components of a flow field are given by vx = −ωy and vy = ωx.
Determine the vorticity and the rate of rotation of flow field.
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Hydraulics II
Problem 10
The velocity components for a two-dimensional flow are
vx =
vy =
Cx
x2 + y 2
Cy
x2 + y 2
where C is a constant. Is the flow irrotational?
Problem 11
The velocity in the outlet pipe from this reservoir is 30 ft/s and h = 18 ft. Because of
the rounded entrance to the pipe, the flow is assumed to be irrotational. Under these
conditions, what is the pressure at A?
Problem 12
A closed tank of liquid (SG = 1.2) is rotated about a vertical axis (see figure), and at
the same time the entire tank is accelerated upward at 4 m/s2 . If the rate of rotation is
10 rad/s, what is the difference in pressure between points A and B (PB − PA )? Point B
is at the bottom of the tank at a radius of 0.5 m from the axis of rotation and point A
is at the top on the axis of rotation.
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