Rising Bubble

Kamila Součková

16

2

Task

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.

3

Understanding the Task

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.large = length > width

4

Understanding the TaskA 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.

large = length > width

= stable flow

5

Understanding the TaskA 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.

large = radius comparable to radius of tube(length > width)

= stable flow

6

Understanding the Task

What to Study:

• motion• shape of bubble

Depending on:

• volume of bubble V• diameter of tube D• properties of liquid

“Study the bubble”

ANALYSIS OF THE SYSTEM

8

Properties of Liquiddensity viscosity surface

tensionρ = resistance to

flow• dynamic µ• kinematic

result of cohesive intermolecular forces

minimizes surface areaσ

water 1000 kg/m3 1.0 kg/(s·m) (at 20°C) 71.97 mN/m

oil 890 kg/m3 43.7 kg/(s·m) 32 mN/m

ethanol 789 kg/m3 0.9 kg/(s·m) 22.27 mN/m

soap 1020 kg/m3 5 kg/(s·m) 3 mN/m

9

Forces in the Systemgravity acting on liquid

gravity acting on bubble

liquid will flow down,

pushing bubble up

airliq

water needs to flow around

10

Forces in the Systemgravity acting on liquid

gravity acting on bubble

airliq

resistance due to viscosity

pressure inside bubble

stresses due to surface tensionand pressure in the liquid

water needs to flow around

liquid will flow down,

pushing bubble up

MOTION OF THE BUBBLE

Speed of rising• stabilized soon due to viscous forces

0 5 10 15 20 25 300

0.020.040.060.080.1

0.120.140.160.180.2

time / s

velo

city

/ m

·s-1

water,D = 1.4cm

0 2 4 6 8 10 12 14 16 180

0.05

0.1

0.15

volume of bubble / ml

velo

city

/ m

·s-

1Speed of rising• does not depend on bubble volume:

oil,D = 1.4cm

0 2 4 6 8 10 12 14 160

0.05

0.1

0.15

volume of bubble / mlvelo

city

/ m

·s-

1

water,D = 0.9cm

14

Speed of Rising: Different Liquids, Tubes

water oil

ethan

olso

ap0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16 D = 0.94cmD = 1.40cmD = 1.56cm

velo

city

/ m

·s-1

density ρ

dynamic

viscosity µ

surface

tension σ

water 1000 kg/m3

1.0 kg/(s·m)

71.97 mN/m

oil 890 kg/m3

43.7 kg/(s·m)

32 mN/m

ethanol 789 kg/m3

0.9 kg/(s·m)

22.27 mN/m

soap 1020 kg/m3

5kg/(s·m)

3mN/m

Description of Bubble Rising

v

u(z)

w

z

16

Starting with…

• unknowns:– velocity profile of the thin layer flowing around

the bubble u(z)

– velocity of bubble v– width of layer flowing around bubble w

vu(z)

w

17

Starting with…

• unknowns:– velocity profile of the thin layer flowing around

the bubble u(z)

– velocity of bubble v– width of layer flowing around bubble w

gravity + shear force → 𝑢 (𝑧 )=− 𝜌𝑔2𝜇 (𝑤¿¿2−2𝑤𝑧)¿

vu(z)

w

w

18

• unknowns:– velocity profile of the thin layer flowing around

the bubble u(z)gravity + shear force →

– velocity of bubble v– width of layer flowing around bubble w

𝑢 (𝑧 )=− 𝜌𝑔2𝜇 (𝑤¿¿2−2𝑤𝑧)¿

Starting with…

continuity (liquid ↓ = air ↑ ) → 2𝜋 𝑅𝜌𝑔𝑤3

3𝜇 =𝜋 𝑅2𝑣

vu(z)

w

19

Starting with…

• unknowns:– velocity profile of the thin layer flowing around

the bubble u(z)gravity + shear force →

– velocity of bubble v– width of layer flowing around bubble w

continuity (liquid ↓ = air ↑ ) →

𝑢 (𝑧 )=− 𝜌𝑔2𝜇 (𝑤¿¿2−2h𝑧 )¿

2𝜋 𝑅𝜌𝑔𝑤3

3𝜇 =𝜋 𝑅2𝑣

vu(z)

h

20

Combine the equations

Rg

hv 1

32

3

velocity profile continuity

𝑢 (𝑧 )=− 𝜌𝑔2𝜇 (𝑤¿¿2−2h𝑧 )¿

2𝜋 𝑅𝜌𝑔𝑤3

3𝜇 =𝜋 𝑅2𝑣tried to find 3rd equation → no analytical solution; very difficult to solve numerically

Bubble Rising:Experimentally Prove Expected v/h3

• need to find h → measure change in bubble length

21

motionin bubble oflength :restat bubble oflength :0

LL

0LcHL expectation:

“head”

Lcylindrical “body”

0 10 20 30 400

10

20

30

40

length of bubble at rest L0 / cm

leng

th o

f bub

ble

in m

otio

n L

/ cm

Bubble Rising:Experimentally Prove Expected v/h3

22

motionin bubble oflength :restat bubble oflength :0

LL

0LcHL expectation:

02.015.1

cm52.006.1

cH

L = 1.15L0 + 1.06

23

Bubble Rising:Experimentally Prove Expected v/h3

• Prolonging of the cylindrical part:0LcHL

mm06.052.012

0

220

LLRw

wRLRL

exp

02.015.1

cm62.006.1

cH

24

Bubble Rising:Experimentally Prove Expected v/h3

• Pluding experimental velocity: sm/13.0expv

2193

exp

exp ms1031.001.1 wv

25

Bubble Rising:Experimentally Prove Expected v/h3

Theory correlates with experiment✓mm8

msmN002.1

sm81.9

mkg1000

2

2

3

R

g

2193

th

th ms1005.082.032 Rg

wv

2193

exp

exp ms1031.001.1 wv

Speed of Rising: More Experiments

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.400

0.02

0.04

0.06

0.08

0.1

0.12

dynamic viscosity / kg/(s·m)

velo

city

/ m

·s-1

Isolate Parameter: Viscosity• hot water: viscosity changes with temperature

much more than density, surface tension

00.20.40.60.81

1.21.41.61.82

f(x) = 4.86472162721733 x^-0.581128821406901Table valuesPower (Table values)

temperature / °C

dyna

mic

vis

cosi

ty /

kg/(s

·m)

27

http://www.engineeringtoolbox.com/water-dynamic-kinematic-viscosity-d_596.html

SHAPE OF BUBBLES

28

29

Characterizing the System

• Reynolds number: viscous vs inertial forces

• Eötvös (Bond) number: surface tension vs gravitational forces

2

Eo gL

tensionsurface :bubble theoflength typical:

gas and liquid of densitiesin difference :

L

vL

Re

30

Shapes of Bubble

ReEo 5 10 20

5

10

20

hone

y

sham

poo

wat

er

oil

?pe

trol

31

Shapes of Bubble: Comparison with Literature (no tubes)

source: Jinsong Hua, Jing Lou, 2007, Numerical simulation of bubble rising in viscous liquid

32

Mystery: Tail• shampoo has a clearly

visible tail• sometimes a small tail

can be observed in honey too (esp. when not exactly vertical)

• not mentioned in literature• our explanation: very

viscous → does not “fill” the bottom fast enough

CONCLUSION

34

gravity acting on liquid

gravity acting on bubble

pressure inside bubblestresses due to surface tension

liquid will flow down, pushing bubble up

airliq

resistance due to viscosity

Conclusion

force analysis

35

w

Conclusion

force analysis (gravity, viscosity, pressure, surface tension)

flow in the tube

)2(2

)( 2 hzzgzu

36

Conclusion

force analysis (gravity, viscosity, pressure, surface tension)

flow in the tube (velocity profile)

speed of rising

Rg

hv 1

32

3

theory correlates with experiment✓

37

Conclusion

force analysis (gravity, viscosity, pressure, surface tension)

flow in the tube (velocity profile)

speed of rising (theory + correlating exp for various parameters)

shapes of bubbles

38

Conclusion

force analysis (gravity, viscosity, pressure, surface tension)

flow in the tube (velocity profile)

speed of rising (theory+correlating exp for various parameters)

shapes of bubbles (depending on Re, Eo)

“study the bubble” ✓

39

Thank you for your attention!

force analysis (gravity, viscosity, pressure, surface tension)

flow in the tube (velocity profile)

speed of rising (theory+correlating exp for various parameters)

shapes of bubbles (depending on Re, Eo)

“study the bubble” ✓

APPENDIX

42

Characteristics of the System: Flow

• Reynolds number: defines relative importance of viscous and inertial forces

– in our case always flow is ⇒ laminar

vL

Re

bubble of velocity :liquid ofdensity :

v

bubble oflength :liquid of viscositydynamic :

L

210

• Refound in articles =

• Reussualy expected =

Estimations from continuity equation:

Reynolds number

43

=

𝑢𝑤 ρµ

2

2 2

vRuw

vRuRw

v – velocity of bubbleu – velocity of liquidR- radius of bubbleρ- density of liquidµ - viscosity of liquidw- thickness of layer

Rwv

u

12= Refound in articles

44

Velocity Profile

• velocity profile of the thin layer flowing around the bubble u(z)

gravity + shear force

)2(2

)( 2 hzzgzu

⇒

(holds for laminar flow and sufficiently long bubbles)

w

source: Zbierka FX (collection of solved physics problems)

45

Velocity of Rising, Width of Layer

• velocity of bubble v• width of layer flowing around bubble h

continuityamount of liquid going down = amount of air going up

32 3πRρgh

integrating + simplifying *velocity profile

= vR2

2DR

* simplification: thin layer

(approximation:cylinder)

46

Description of Bubble Rising

vRπRρgh 23

32

)2(

2)( 2 hzzgzu

Rg

hv 1

32

3

velocity profile continuity

(const for a given liquid & tube)

47

Beyond the Ratio

•

→ experiment: v depending on R

13

Rhv RhRv , ??

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

radius /cm

velo

city

/ m

·s-1

48

Beyond the Ratio

•

→ experiment: v depending on R

13

Rhv RhRv , ??

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

∅ of 3 valuesquadratic fit

radius /cm

velo

city

/ m

·s-1

2 1