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Team of Brazil Problem 16 Rising bubble reporter: Ibraim Rebouças

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Team of Brazil

Problem 16

Rising bubble

reporter:

Ibraim Rebouças

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

==

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