Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 2
Team of BrazilProblem 16: Rising bubble
Problem 16
Rising bubble
A vertical tube is filled with a viscous fluid.
On the bottom of the tube, there is a large air
bubble. Study the bubble rising from the
bottom to the surface.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 3
Team of BrazilProblem 16: Rising bubble
Contents
• Kinds of bubble
• Viscosity
• Drag forces
• Lateral Flow
• Rise velocity
Theoretical introduction
• Tests
• Comparison between theory and experiments
Experiments
Conclusion
• Smaller spherical cap
• Influence of the temperature
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 4
Team of BrazilProblem 16: Rising bubble
Kinds of bubbles
Spherical bubble
• Very viscous fluids.
• Governed by stokes low
Spherical cap bubble
• Stable to moderated viscous fluids.
Taylor’s bubble
• Height bigger than the diameter of the pipe.
• Constant velocity for all volumes.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 5
Team of BrazilProblem 16: Rising bubble
The phenomenon
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 6
Team of BrazilProblem 16: Rising bubble
Viscosity
• It’s the measurement of the resistance to a shear stress.
The viscosity is measured by a coefficient. � ∝��
��⇒� � �
��
��� is called absolute viscosity coefficient.
Kinematic viscosity �
�� is de density of the fluid
��
��
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 7
Team of BrazilProblem 16: Rising bubble
Forces acting on the bubble
Buoyancy
WeightDragforces
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 8
Team of BrazilProblem 16: Rising bubble
Forces acting on the bubble
� �1
2�����
����������� �!��
� � "bv
%��&����' (
Both are wrong, because they do not consider the boundary effects.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 9
Team of BrazilProblem 16: Rising bubble
Border of the bubble
We must have conservation of mass for
the liquid ahead and behind the
bubble.
�)��
)*���+
��� constant 2 0
� � ")+
)�
4�
2�1 "
*
4
�
*
�
"4 54
Pressure
Buoyancy
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 10
Team of BrazilProblem 16: Rising bubble
Large spherical cap or Taylor bubble velocity
�
67� "
8
3
67:5
2
31 5
32�
67:
; �⁄
� �2
367
Taylor bubble Large spherical cap
For Taylor bubble or large spherical cap bubbles, thevelocity can be described by the following equation:
For low viscosity fluids(e.g. Water), the
equation can be written as:• The velocity is constantand does not dependon the volume of thebubble.
Terminal velocity(m/s)
Diameter of the
bubble
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 11
Team of BrazilProblem 16: Rising bubble
Spherical cap bubbles
• Diameter smaller than the tube
Lateral oscillation
Wave length
=> � 2?@>
Capillary length
@> �A
Δ�6
Surface tension coefficient
Difference of density (air-liquid)
Drag coefficient
�� � 6 532
D�
D� ��7
@> �A
Δ�6
Drag force
� � ��?
2���D�
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 12
Team of BrazilProblem 16: Rising bubble
Rise velocity of spherical cap
��?
2���D� � �6E
At the equilibrium: � � B
�32
75 6� � 26E
� � ��?
2���D�
G � �6E
Finally:
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 13
Team of BrazilProblem 16: Rising bubble
Influence of temperature
• The temperature acts changing the viscosity of the fluid and the density of the air
inside the bubble.
� H I*+ �JKLMN
JLMN" 1 �K
Viscosity parameter
Reference temperature in
Kelvin
Temperature in kelvin
Accuracy of O6% for
temperature range:
0 2 J 2 100Q
Reference viscosity
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 14
Team of BrazilProblem 16: Rising bubble
Experimental description
• Vary the volume of the bubble Experiment
1
• Vary the temperatureExperiment
2
• Vary the diameter of the tubeExperiment
3
• Influence of surfactantExperiment
4
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 15
Team of BrazilProblem 16: Rising bubble
Materials
• Burette 50 ml
• Pipette 10 ml
• Glass tube 250 ml
• Syringe 10 ml
• Rubber tube
• Camera
• Computer software's
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 16
Team of BrazilProblem 16: Rising bubble
Experiment 1: Variation of the volume
Volume
Temperature
Diameter
Tube diameter: 15 mm
Syringe: 20 ml
Fluid: Glycerin
Temperature: 20Q
Source of errors:
• Measuring ruler and conversion scale: O0.5TT
• Volume of the syringe: O0.5TU
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 17
Team of BrazilProblem 16: Rising bubble
Experiment 1: Variation of the volume
0.0
5.0
10.0
15.0
20.0
25.0
0.0 5.0 10.0 15.0
Po
siti
on
(cm
)
Time (s)
9ml
0.0
5.0
10.0
15.0
20.0
25.0
0.0 5.0 10.0 15.0
Po
siti
on
(cm
)
Time (s)
8ml
ET � 0.01425 m/s ET � 0.01449 m/s
Volume
Temperature
Diameter
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 18
Team of BrazilProblem 16: Rising bubble
Experiment 1: Variation of the volume
0.0
5.0
10.0
15.0
20.0
25.0
0.0 5.0 10.0 15.0
Po
siti
on
(cm
)
Time(s)
7ml
0.0
5.0
10.0
15.0
20.0
25.0
0.0 5.0 10.0 15.0
Po
siti
on
(cm
)
Time(s)
6ml
ET � 0.01445 m/s ET � 0.01477 m/s
Volume
Temperature
Diameter
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 19
Team of BrazilProblem 16: Rising bubble
Experiment 1: Variation of the volume
0.0
5.0
10.0
15.0
20.0
25.0
0.0 5.0 10.0 15.0
Po
siti
on
(cm
)
Time(s)
5ml
0.0
5.0
10.0
15.0
20.0
25.0
0.0 5.0 10.0 15.0
Po
siti
on
(cm
)
Time(s)
4ml
ET � 0.01480 m/s ET � 0.01498 m/s
Volume
Temperature
Diameter
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 20
Team of BrazilProblem 16: Rising bubble
Experiment 1: Variation of the volume
0.0
5.0
10.0
15.0
20.0
25.0
0.0 5.0 10.0 15.0
Po
siti
on
(cm
)
Time(s)
3 ml
0.0
5.0
10.0
15.0
20.0
25.0
0.0 5.0 10.0 15.0
Po
siti
on
(cm
)
Time(s)
2 ml
ET � 0.01640 m/s ET � 0.01710 m/s
Volume
Temperature
Diameter
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 21
Team of BrazilProblem 16: Rising bubble
0
5
10
15
20
25
0 5 10 15
Po
siti
on
(cm
)
Time (s)
Position x Time
9 ml
8 ml
7 ml
6 ml
5 ml
4 ml
3 ml
2 ml
Precision: ±0.5 mm
Experiment 1: Variation of the volume
Volume
Temperature
Diameter
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 22
Team of BrazilProblem 16: Rising bubble
Experiment 1: Variation of the volume
Volume
Temperature
Diameter
Volume
(ml)
Experimental
velocity(X Y⁄ )
Theoretical
velocity (X Y⁄ )
9 0.0142 0.0146
8 0.0144 0.0146
7 0.0144 0.0146
6 0.0147 0.0146
5 0.0148 0.0146
4 0.0149 0.0146
3 0.0164 0.0146
2 0.0171 0.0146
Average 0,0151 0.0146
Experiment agrees with the theory
0.01
0.012
0.014
0.016
0.018
0.02
0 5 10
Ve
loci
ty (
m/s
)
Volume (ml)
Theory
Experiment
Out of the regime
Average error for all points: 4.62%
Average error for [4;9]: 1.37%
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 23
Team of BrazilProblem 16: Rising bubble
Experiment 2: Variation of the temperature
0
5
10
15
20
25
30
35
0 1 2 3 4 5
Po
siti
on
(cm
)
Time(s)
Position in function of time for different temperatures
55°C
45°C
35°C
25°C
Volume
Temperature
Diameter
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 24
Team of BrazilProblem 16: Rising bubble
Experiment 2: Variation of the temperature
Temperature
T (°C)
Calculated
Viscosity (m²/s)
∗ \]^_
Predicted
Velocity (m/s)
Experimental
velocity
55.0 0.409 0.0981 0.0850
45.0 0.876 0.0779 0.0694
35.0 1.973 0.0229 0.0234
25.0 4.685 0.00910 0.0103
Volume
Temperature
Diameter
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 25
Team of BrazilProblem 16: Rising bubble
Experiment 2: Variation of temperature
Volume
Temperature
Diameter0
0.02
0.04
0.06
0.08
0.1
0.12
20 30 40 50 60
Ve
loci
ty (
m/s
)
Temperature (°C)
Velocity in function of temperature
Theoretical
Experimental
Average error: 10%
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 26
Team of BrazilProblem 16: Rising bubble
Experiment 3: Variation of the diameter
-1
0
1
2
3
4
5
6
7
8
9
10
0 0.25 0.5 0.75 1 1.25 1.5
Po
siti
on
(cm
)
Time (s)
5mm
6.5mm
25mm
Fluid: Soy bean oil
Kinematic viscosity at 20Q : 7.5 ∗ 10^`
Volume
Temperature
Diameter
0,193 m/s
0,0703 m/s
0,0240 m/s
Experimental velocities
Precision: ±0.5 mm
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 27
Team of BrazilProblem 16: Rising bubble
Diameter
(mm)
Calculated
Velocity (m/s)
Experimental
Velocity (m/s)
25.0 0.2270 0.2130
6.5 0.0932 0.0790
5.0 0.0387 0.0340
0
0.05
0.1
0.15
0.2
0.25
0 10 20 30
Ve
loci
ty (
m/s
)
Diameter (mm)
Theoretical
Experimental
Volume
Temperature
Diameter
Average error: 11%
Experiment 3: Variation of the diameter
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 28
Team of BrazilProblem 16: Rising bubble
Conclusion
Diameter of the tube
Temperature
Viscosity
Diameter
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 29
Team of BrazilProblem 16: Rising bubble
References
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 30
Team of BrazilProblem 16: Rising bubble
Thank you!
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 31
Team of BrazilProblem 16: Rising bubble
Lateral Flow
•For bubble measures comparable to the tube.
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 32
Team of BrazilProblem 16: Rising bubble
Pressure and volume variation
P
0P•For the following parameters:
•Fluid: Water
•Length of beaker: 1m
0.. PhgPP += ρ
²/81.9
10,1
³/105
0
3
2
smg
PaP
mkgOH
==
=ρ
00 PPh =⇒=PahgP ³10.1.. ==∆ ρ
122
22
1
11 1.1..
VVT
VP
T
VP =⇒=
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 33
Team of BrazilProblem 16: Rising bubble
Buoyancy
Weight
Relation between them
Conclusion
The weight of the bubble
•Considering a bubble with a volume of 2x10-5m³ (20mL):
153 1021010210 −− ×=×××=
=
B
VgB lρ 4104.2 −×==⇒==
VgWVm
mgWg
g
ρρ
31
4
102.1102
104.2 −−
−
×=××=
B
W
0 so , of %6.0 ≈= WBW
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 34
Team of BrazilProblem 16: Rising bubble
• http://www.pmmh.espci.fr/~jbico/TP/taylor50bulle.pdf
Team of BrazilProblem ## Title
Team of Brazil: Bárbara Santiago, Guilherme Moreira, Ibraim Rebouças, João Gabriel Faria and Liara Guinsberg Bad Saulgau, 20th – 29th July, 2012
Reporter: Ibraim Rebouças 35
Team of BrazilProblem 16: Rising bubble
Fluid (U) C Fluid (U) C
Ammonia 1.05 Freon 12 1.76
Ethanol 5.72 Benzene 4.34
Gasoline 3.68 Mercury 1.07
Glycerin 28.0 Methanol 4.63