Ronny Berndtsson, Lund University
WRITTEN TEST 1 (22 POINTS)
FLUID MECHANICS FOR W, VVR120
30 SEPTEMBER 2013, 08:00-10:00
Open book: Text book, lecture notes, collection of equations, exercise problems, dictionary,
and pocket calculator.
Remember: Answers should be short and concise, structured and in punctual form. Just one
solved question on each paper. Calculations should be clear with assumptions and
units. Write your name on each paper.
General:
Use (if no other information given) for water: 20oC) =1.0103 kg/m3,  = 1.010-3
Pas, and for g = 9.81 m/s2, patm = 100 kPa.
Test results: Results will be displayed 10 October, 2013.
Grading: Given on basis of exam or 2 written tests. More information is given in course
program.
PROBLEM 1 (2 points)
A thin wire is set to float on top of a water surface. If the density of the wire material is 1.4 · 103
kg/m3 what is the maximum radius of the wire in order for it to stay floating?
PROBLEM 2 (3 points)
A ship of mass 1200 t floats in sea-water. What volume of sea-water does it displace? If the ship
enters fresh water, what weight of cargo must be unloaded so that the same volume of water is
displaced as before (density of fresh water = 1000 kg/m³, relative density of sea-water = 1.03; 1 t
= 1000 kg)?
PROBLEM 3 (8 points)
The figure below shows an L-shaped gate that operates by pivoting about hinge O when the
water level in the channel rises to a certain height H above O. A counterweight W attached to the
gate provides closure of the gate at low water levels. With the channel empty the force of the
gate at S is 1.635 kN. The distance l is 0.5 m and the gate is 2 m wide in to the paper.
Determine the distance H when:
a) the gate begins to open due to the hydrostatic force.
b) the force acting on S from the gate is at maximum. What is the magnitude of this force?
Assume effects of friction are negligible.
W
air
H
S
water
O
Channel bottom
l
PROBLEM 4 (6 points)
Water flows through a sudden pipe contraction according to below figure at a flow rate Q = 3
dm3/s. Upstream the contraction at section 1 the pipe diameter is 40 mm and downstream at
section 2 the diameter is 25 mm. At section 2 the gauge pressure is 30 kPa. Calculate assuming
no energy losses
a) the gauge pressure at section 1.
b) direction and total force exerted by the fluid on the contraction.
1
2
Q
PROBLEM 5 (3 points)
A jet of water with a velocity of 20 m/s is directed upwards at an angle of 45o to the horizontal.
If the air resistance is neglected, what horizontal distance x from the nozzle will it reach for a
maximum elevation?
Solutions:
1. A section of the wire floating in water looks like:
σ
d
Water
W
where d is diameter = 2 r. Assume surface tension σ = 0.073 N/m (from table) and contact angle
between wire and water equal to 0.
A vertical force balance per unit length of wire (1 m) gives: W= 2 σ <=> V w = 2 σ <=>
π r2 g 1.4 · 103 = 2 σ <=> r = √(2 σ / (π g 1.4· 103)) = 0.0018 m.
Answer: radius = 1.8 mm.
2.
Mass of sea water displaced = 1200 t
Density of sea-water
=1000×1.03 kg/m³
Volume displaced = mass/density = 1200×1000/1000×1.03 m³ = 1165 m³ of sea-water
The same volume of fresh water weights 1165×1000 kg = 1165 t
Therefore, weight of cargo to be unloaded = 1200 - 1165 = 35 t
Answer: Volume displaced sea-water = 1165 m³ and 35 ·103 kg.
3. a)
Answer: a) H= 1.0 m b) H = 0.5 m and 3270 kN (acting downward along the vertical).
4) Continuity gives
V1 · A1 = V2 · A2 = 0.003 m3/s <=> V1 = 2.39 m/s; V2 = 6.11 m/s
Bernoulli gives (z1 = z2):
p1/w + V21/2g = p2/w + V22/2g <=>
p1/w + 2.392/2g = 30·103/w + 6.112/2g <=>
p1 = 45.8 kPa
Using the momentum eqn for the control volume between section 1 and 2 and x-axis along the
flow direction gives:
ΣFx = ρ·Q(V2 – V1) <=>
F1
F
F2
F1 – F2 - F = ρ·Q(V2 – V1) <=>
45.8· 103 · 0.042· π/4 - 30.0· 103 · 0.0252· π/4 – F = 1000·0.003 (6.11 – 2.39) <=>
F = 31.6 N
Answer: a) Pressure at 1 = 45.8 kPa; b) Force is 31.6 N directed to the right in the horizontal.
5.
Distance x for zmax of a free jet trajectory is
Vx0·Vz0/g = V0·cos45o ·V0·cos45o = V0/2g = 202/2g = 20.38 m.
Answer: 20.38 m.