NATIONAL UNIVERSITY OF SINGAPORE
ME2134
NATIONAL UNIVERSITY OF SINGAPORE
ME2134 – FLUID MECHANICS I
(Semester I : AY2013/2014) Time Allowed : 2 Hours
INSTRUCTIONS TO CANDIDATES:
1.
This examination paper contains FOUR (4) questions and comprises SEVEN (7) printed pages.
2.
Answer ALL FOUR (4) questions.
3.
All questions carry equal marks.
4.
This is a CLOSED-BOOK EXAMINATION with authorized materials: Students are allowed to bring in TWO A4 size sheets of notes/formulae written on both sides.
5.
Programmable calculators are NOT allowed for this examination.
PAGE 2
ME2134
QUESTION 1 (a)
You saw an experiment performed on a floating steel blade, as shown in Figure 1.1.
Figure 1.1: a floating razor blade (i)
Explain briefly why the blade can float on water, despite its higher density than water. (2 marks)
(ii)
The mass of the blade is denoted by m, and the total length of its sides L, assuming that the surface tension force acts along a direction at an angle θ with a surface tension coefficient σ, show that
mg L sin Thus determine the maximum mass of the blade that water may be able to support, given that σ = 0.07 N/m, and L = 25 mm. What happens if a few drops of soap are spilled in the vicinity of the blade? (3 marks) (b)
You are tasked with measuring the viscosity of an oil emulsion, using an available viscometer as shown in Figure 1.2.
Figure 1.2: Viscometer consists of rotating cylinder of radius Ri in a holding cup of radius Ro. The liquid is filled to a level L in the annular space between the cylinder (of radius Ri) and the holding cup (of radius Ro), and the cylinder is rotated at an angular velocity of ω. In addition to filling the annular jacket, the fluid also fills the circular gap between the bottom of the cylinder and the holding cup, to a thickness h.
PAGE 3
(i)
ME2134
Ignore the fluid in the gap h, and to consider only the fluid in the annular jacket, show that the shear rate in this annular region is (8 marks)
Ri du , dr Ro Ri
and show that the torque required to rotate the cylinder is T
2 Ri3 L Ri Ro
If the measured torque is T 8.30 102 N.m at ω = 60 RPM, what is the fluid viscosity? Here Ri 20.0 mm, Ro 22.0 mm and L = 10.0 cm. (7 marks) (ii)
Using this device, you construct the shear stress versus shear rate for another fluid, as shown below in Figure 1.3.
Figure 1.3: shear stress versus shear rate of an emulsion Classify this type of fluid. (5 marks)
PAGE 4
ME2134
QUESTION 2 (a)
(b)
You are given a 2-dimensional flow u 2 x, v 2 y . (i)
What are streamlines? Find the streamline at time t = 0 passing through point (1,1). (4 marks)
(ii)
What are pathlines? Find the pathline of a particle passing through point (1,1) at time t = 0. (4 marks)
(iii)
What are streaklines? Find the streakline at time t = 0 passing through point (1,1). (2 marks)
Figure 2.1 shows a reservoir containing water with a steel rectangular gate.
y
Figure 2.1: A water reservoir with a rectangular gate. The rectangular gate is L = 2.0 m long and W = 1.0 m wide and is hinged about point C, and making an inclination tan 1 4 / 3 with the free surface. The inclination is indicated by the right triangle of sides 4-3 (the remaining side is 5) marked on the figure. On the right side, you are given the geometric properties of a rectangle. (i)
Find the depth of the centroid of the gate and calculate the magnitude of the resultant force on the gate. Find the centre of pressure where this resultant force acts through. (8 marks)
PAGE 5
(iii)
ME2134
Because of manufacturing defects, the weight of the gate is not sufficient and it opens a small gap h = 10 cm, allowing water to escape through a gap, as shown in Fig. 2.2. You are to provide an estimate of the escape velocity and flow rate through the gate by using Bernoulli Equation, stating out clearly its limitations. (7 marks)
Figure 2.2 Gate is raised by h = 10cm.
QUESTION 3 A water truck was cruising along a straight highway to a destination in the outback. The water tank measures 2.0 m (high) x 2.0 m (wide) x 8.0 m (long) and was filled with water up to a height of 1.8 m (see Figure 3.1). During the journey, a freak wind storm blew the lid off the top of the tank leaving a 2.0 m wide gap opened to the atmosphere. Sensing that something was amiss, the driver stepped on the brake paddle to decelerate the truck to a stop. When he got out of the truck to check for damages, he found that not only had the lid been blown away, 15 % of the water had also spilled on the road. Assume the dimension of the opening at the top of the tank measures 2 m x 2 m (same width as the tank).
(a)
Determine the maximum deceleration that the truck driver had applied to the brake. (8 marks)
(b)
What is the total force acting on the inner vertical surface AB of the water tank during the deceleration? Note: You need to include contribution from atmospheric pressure. (11 marks)
Without bothering to look for the missing lid, the driver went back to the truck and continued his journey by accelerating the truck at 2 m/s2 to a constant cruising speed. By his action, has the driver unknowingly spilled the water out of the tank due to his acceleration? Note: no mark will be awarded to “yes” or “no” answer, you must accompany your answer with a detailed calculation. (6 marks)
PAGE 6
ME2134
State all the assumptions made. You may use density of water = 1000 kg/m3 and atmospheric pressure = 101.32 kN/m2
8.0 m 5.0 m
2.0 m D
A
2.0 m 1.8 m
Water C
B
Figure 3.1
FLUIDS PTE LTD
Drawing not to scale
QUESTION 4 (a)
The pressure drop ( P ) along a straight circular pipe is a function of its diameter (d), length (L), surface roughness (), volume flow rate (Q), density () and dynamic viscosity () of fluid. Using , Q and d as repeating variables, express this relationship as a dimensionless function. (8 marks) d
Q L
Figure 4.1 (b)
The piping system consists of two flanged 90o elbows, “Y” junction, and a valve transports water from reservoir A to reservoir B downstream (see Figure 4.2). Water enters at sharp entrance to the pipe from tank A and sharp exits to tank B. The loss coefficient of 90o elbows (KBC and KDE) is 0.3 and that of Y-junction is 0.8. If the elbows have internal diameter of 5 cm, and all the straight pipes also have internal diameter of 5 cm and friction factor of 0.02, determine the volume flow rate to reservoir B if (i) the valve is fully closed and (ii) the valve is fully opened with loss coefficient of 0.6. If the whole pipe network is shifted bodily by 10 m vertically up the tank as shown in Figure 4.3, and if the valve is fully opened, do you expect the volume flow rate to decrease compare to case (ii) above now that the entrance and the exits of the pipes are located closer to the water surface. You must justify your answer with a detailed explanation. State all the assumptions made (17 marks)
PAGE 7
ME2134
40 m 69.5 m
Reservoir A
LAB =10 m, fAB =0.02
A
KBC = 0.3
B
5m
Reservoir B LFG =20 m, fFG = 0.02 Valve
C E
Figure 4.2
G
LCD =10 m, fCD =0.02 D
30 m
10 m
F
10 m KDE = 0.3 LEF =100 m, fEF =0.02 Y-junction
20 m
H
LFH =20 m, fFH = 0.02
Drawing not to scale
40 m
59.5 m LAB =10 m, fAB =0.02
Reservoir A 15 m
KBC = 0.3 A
B
Reservoir B 20 m
Valve
C D
LFG =20 m, fFG =0.02
LCD =10 m, fCD =0.02 E
G 10 m
F
Figure 4.3
10 m KDE = 0.3
LEF =100 m, fEF =0.02 Y-junction LFH =20 m, fFH = 0.02
- END OF PAPER -
H
30 m
NATIONAL UNIVERSITY OF SINGAPORE
ME2134 – FLUID MECHANICS I
(Semester I : AY2013/2014) Time Allowed : 2 Hours
INSTRUCTIONS TO CANDIDATES:
1.
This examination paper contains FOUR (4) questions and comprises SEVEN (7) printed pages.
2.
Answer ALL FOUR (4) questions.
3.
All questions carry equal marks.
4.
This is a CLOSED-BOOK EXAMINATION with authorized materials: Students are allowed to bring in TWO A4 size sheets of notes/formulae written on both sides.
5.
Programmable calculators are NOT allowed for this examination.
PAGE 2
ME2134
QUESTION 1 (a)
You saw an experiment performed on a floating steel blade, as shown in Figure 1.1.
Figure 1.1: a floating razor blade (i)
Explain briefly why the blade can float on water, despite its higher density than water. (2 marks)
(ii)
The mass of the blade is denoted by m, and the total length of its sides L, assuming that the surface tension force acts along a direction at an angle θ with a surface tension coefficient σ, show that
mg L sin Thus determine the maximum mass of the blade that water may be able to support, given that σ = 0.07 N/m, and L = 25 mm. What happens if a few drops of soap are spilled in the vicinity of the blade? (3 marks) (b)
You are tasked with measuring the viscosity of an oil emulsion, using an available viscometer as shown in Figure 1.2.
Figure 1.2: Viscometer consists of rotating cylinder of radius Ri in a holding cup of radius Ro. The liquid is filled to a level L in the annular space between the cylinder (of radius Ri) and the holding cup (of radius Ro), and the cylinder is rotated at an angular velocity of ω. In addition to filling the annular jacket, the fluid also fills the circular gap between the bottom of the cylinder and the holding cup, to a thickness h.
PAGE 3
(i)
ME2134
Ignore the fluid in the gap h, and to consider only the fluid in the annular jacket, show that the shear rate in this annular region is (8 marks)
Ri du , dr Ro Ri
and show that the torque required to rotate the cylinder is T
2 Ri3 L Ri Ro
If the measured torque is T 8.30 102 N.m at ω = 60 RPM, what is the fluid viscosity? Here Ri 20.0 mm, Ro 22.0 mm and L = 10.0 cm. (7 marks) (ii)
Using this device, you construct the shear stress versus shear rate for another fluid, as shown below in Figure 1.3.
Figure 1.3: shear stress versus shear rate of an emulsion Classify this type of fluid. (5 marks)
PAGE 4
ME2134
QUESTION 2 (a)
(b)
You are given a 2-dimensional flow u 2 x, v 2 y . (i)
What are streamlines? Find the streamline at time t = 0 passing through point (1,1). (4 marks)
(ii)
What are pathlines? Find the pathline of a particle passing through point (1,1) at time t = 0. (4 marks)
(iii)
What are streaklines? Find the streakline at time t = 0 passing through point (1,1). (2 marks)
Figure 2.1 shows a reservoir containing water with a steel rectangular gate.
y
Figure 2.1: A water reservoir with a rectangular gate. The rectangular gate is L = 2.0 m long and W = 1.0 m wide and is hinged about point C, and making an inclination tan 1 4 / 3 with the free surface. The inclination is indicated by the right triangle of sides 4-3 (the remaining side is 5) marked on the figure. On the right side, you are given the geometric properties of a rectangle. (i)
Find the depth of the centroid of the gate and calculate the magnitude of the resultant force on the gate. Find the centre of pressure where this resultant force acts through. (8 marks)
PAGE 5
(iii)
ME2134
Because of manufacturing defects, the weight of the gate is not sufficient and it opens a small gap h = 10 cm, allowing water to escape through a gap, as shown in Fig. 2.2. You are to provide an estimate of the escape velocity and flow rate through the gate by using Bernoulli Equation, stating out clearly its limitations. (7 marks)
Figure 2.2 Gate is raised by h = 10cm.
QUESTION 3 A water truck was cruising along a straight highway to a destination in the outback. The water tank measures 2.0 m (high) x 2.0 m (wide) x 8.0 m (long) and was filled with water up to a height of 1.8 m (see Figure 3.1). During the journey, a freak wind storm blew the lid off the top of the tank leaving a 2.0 m wide gap opened to the atmosphere. Sensing that something was amiss, the driver stepped on the brake paddle to decelerate the truck to a stop. When he got out of the truck to check for damages, he found that not only had the lid been blown away, 15 % of the water had also spilled on the road. Assume the dimension of the opening at the top of the tank measures 2 m x 2 m (same width as the tank).
(a)
Determine the maximum deceleration that the truck driver had applied to the brake. (8 marks)
(b)
What is the total force acting on the inner vertical surface AB of the water tank during the deceleration? Note: You need to include contribution from atmospheric pressure. (11 marks)
Without bothering to look for the missing lid, the driver went back to the truck and continued his journey by accelerating the truck at 2 m/s2 to a constant cruising speed. By his action, has the driver unknowingly spilled the water out of the tank due to his acceleration? Note: no mark will be awarded to “yes” or “no” answer, you must accompany your answer with a detailed calculation. (6 marks)
PAGE 6
ME2134
State all the assumptions made. You may use density of water = 1000 kg/m3 and atmospheric pressure = 101.32 kN/m2
8.0 m 5.0 m
2.0 m D
A
2.0 m 1.8 m
Water C
B
Figure 3.1
FLUIDS PTE LTD
Drawing not to scale
QUESTION 4 (a)
The pressure drop ( P ) along a straight circular pipe is a function of its diameter (d), length (L), surface roughness (), volume flow rate (Q), density () and dynamic viscosity () of fluid. Using , Q and d as repeating variables, express this relationship as a dimensionless function. (8 marks) d
Q L
Figure 4.1 (b)
The piping system consists of two flanged 90o elbows, “Y” junction, and a valve transports water from reservoir A to reservoir B downstream (see Figure 4.2). Water enters at sharp entrance to the pipe from tank A and sharp exits to tank B. The loss coefficient of 90o elbows (KBC and KDE) is 0.3 and that of Y-junction is 0.8. If the elbows have internal diameter of 5 cm, and all the straight pipes also have internal diameter of 5 cm and friction factor of 0.02, determine the volume flow rate to reservoir B if (i) the valve is fully closed and (ii) the valve is fully opened with loss coefficient of 0.6. If the whole pipe network is shifted bodily by 10 m vertically up the tank as shown in Figure 4.3, and if the valve is fully opened, do you expect the volume flow rate to decrease compare to case (ii) above now that the entrance and the exits of the pipes are located closer to the water surface. You must justify your answer with a detailed explanation. State all the assumptions made (17 marks)
PAGE 7
ME2134
40 m 69.5 m
Reservoir A
LAB =10 m, fAB =0.02
A
KBC = 0.3
B
5m
Reservoir B LFG =20 m, fFG = 0.02 Valve
C E
Figure 4.2
G
LCD =10 m, fCD =0.02 D
30 m
10 m
F
10 m KDE = 0.3 LEF =100 m, fEF =0.02 Y-junction
20 m
H
LFH =20 m, fFH = 0.02
Drawing not to scale
40 m
59.5 m LAB =10 m, fAB =0.02
Reservoir A 15 m
KBC = 0.3 A
B
Reservoir B 20 m
Valve
C D
LFG =20 m, fFG =0.02
LCD =10 m, fCD =0.02 E
G 10 m
F
Figure 4.3
10 m KDE = 0.3
LEF =100 m, fEF =0.02 Y-junction LFH =20 m, fFH = 0.02
- END OF PAPER -
H
30 m
