Download - Rising Bubble
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