22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Deformation of air and water bubbles in mineral oil under electric field
H. Du1,2, S. Pancheshnyi1 and A. Krivda1
1
ABB Corporate Research, Segelhofstrasse 1K, 5405, Baden-Dättwil, Switzerland
2
EPFL, Lausanne, Switzerland
Abstract: High electric field strength is applied to air and water bubbles in mineral oil to
investigate how the bubble deforms.
Three kinds of behavior were observed
experimentally: elongation along field lines, Coulombic repulsion and break-up. The effect
of the type of the power source (AC or DC), the conductivity of the bubble, the size of the
bubble as well as gravity and viscosity of liquid dielectric was investigated numerically and
a good qualitative agreement of the simulation and experiment is obtained.
Keywords: bubble, liquid dielectrics, electric field, deformation, conductivity
1. Measurements
Previous studies [1-5] show that a bubble in liquid
dielectric could be elongated in the direction parallel to
the electric field, detached from the electrode because of
accumulation of free charges or broken up into smaller
bubbles due to instabilities.
The experimental setup used in this work is shown in
Fig. 1. Two parallel electrodes were immersed in the
liquid dielectric and a bubble was injected into it and
attached to the top (air) or bottom (water) electrode
initially. Although the size of each electrodes is different,
this doesn’t have influence on the experiments since the
bubble is typically smaller compared to the electrodes that
the electric field could be regarded as uniform. The
parameters of our experiments are summarized below:
• Voltage sources: AC 0-14 kV r.m.s. (20-50Hz) or DC
0-20 kV.
• Dielectric liquid: Nytro 10XN mineral oil (ε ≈ 2).
• Type of bubbles: air (ε ≈ 1) or water (ε ≈ 80).
• Gap distances: L = 5-22 mm.
• Initial diameter of bubbles: D 0 = 1-6 mm.
Fig. 1. Schematic diagram of the experimental setup.
1.1 Nonconductive air bubble in mineral oil
Air bubbles were injected into the mineral oil and
attached on the top electrode initially. The experimental
results are shown in Fig. 2. In (a), positive DC voltages
were applied on the top electrode. The size of air bubble
is quite bigger than ones in other two cases. The voltages
increased from 0 to 20 kV with time. With increasing
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voltages, the air bubble elongated along the direction of
electric field. In (b), the only difference from the
experiments of (a) is the size of air bubble. For the small
bubble in (b), it first elongated and began to detach from
the top electrode at 20 kV. In (c), AC was used. The air
bubble elongated, as well as oscillated, as voltage
increased. At certain point the air bubble broke up into
one big and one small bubble. And then they immerged
and broke up into two big bubbles again. The two
bubbles repelled each other and one bubble moved
outside and broke up during the movement. What should
be mentioned here is that instability of shape of the air
bubble could be observed after the applied voltages
reached a certain value.
Fig. 2.
Air bubble in mineral oil under E.
a) D_0 = 5.6 mm, L = 7 mm, DC +20 kV (top);
b) D_0 = 2.6 mm, L = 7mm, DC +20 kV (top);
c) D_0 = 4.5 mm, L = 7 mm, AC 14 kV r.m.s. 50 Hz.
Voltages increased from 0 kV to maximum at the
timescale of 1 minute.
For air and mineral oil, the permittivity ratio
ε air ⁄ε liquid ≈ 0.5, and therefore the bubble can be
considered non-conductive.
Hence, the air bubble
elongated in liquid under electric field due to the
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dielectrophoretic and the electrostrictive forces.
However, for high E, the assumption of no free charges
may not be valid. The nonconductive air bubble could
convert into conductive one due to partial discharges.
Detachment from the electrode could be observed in the
cases of (b) and (c), which possibly results from
Coulombic repulsion between charges with same sign.
The reason for the absence of detachment in (a) is
possibly because of the size of the air bubble. If we
increase the voltage, the repulsion could be observed even
for (a). As for the influence of the voltage shape, air
bubble would oscillate sinusoidally in AC so that it’s not
easy to observe the elongation by eyes, while the
elongation behavior is pronounced in DC. In contrast, air
bubble becomes easier to break up in AC due to
oscillation and instability. In addition, the polarity of DC
didn’t play a role on the deformation.
1.2 Conductive water bubble in mineral oil
Considering the big difference of permittivity ratio
between air and water, we could expect more pronounced
deformations for a water bubble. Here, the water bubble
was injected into mineral oil and attached to the bottom
electrode. After we applied AC voltages, the water
bubble firstly elongated, which is similar to behavior of
the air bubble. The water bubble then became unstable
with higher voltages and finally ejected clusters of small
bubbles at the tip. The small bubbles oscillated in the
medium. The experimental results are shown in Fig. 3.
2. Simulation with COMSOL Multiphysics
Fig. 4 shows the two classes of geometry considered in
our simulation. Both cases are 2D-axisymmetric in a 1-cm
gap.
Fig. 4. An air bubble is placed in the mineral oil under
uniform E. Left: the gravity is neglected and therefore the
air bubble could stay in the middle when no voltage is
applied. Right: the gravity is considered and the air
bubble is attached to the top electrode with certain contact
angle initially.
We used the laminar two-phase flow level set method
for tracking interface movement. The equations for fluid
motion are according to incompressible Navier-Stokes
equations similar to [6]:
𝜌
𝜕𝜕
𝜕𝜕
+ 𝜌(𝑢 ∙ 𝛻)𝑢 = 𝛻[−𝑝𝑝 + 𝜇(𝛻𝛻 + (𝛻𝛻)𝑇 )] +
𝐹𝑠𝑠 + 𝜌𝜌 + 𝐹𝑒 , 𝛻 ∙ 𝑢 = 0
The electrostatics interface sets up the following
equations for the electric potential V:
−∇ ∙ (𝜀0 𝜀𝑟 ∇V) = 0
here, ε 0 is the permittivity of vacuum, and ε r is the
relative permittivity. The electric volume force (F e ) is
given by the divergence of the Maxwell stress tensor (T):
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𝐹𝑒 = 𝛻 ∙ 𝑇, 𝑇 = 𝐸𝐷𝑇 − (𝐸 ∙ 𝑇)𝐼,
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Fig. 3. Water bubble in mineral oil. D_0 = 3 mm; L = 22
mm; AC 14 kV rms 50 Hz. Voltage increased smoothly
from zero to maximum at the timescale of 1 minute.
As we can see from the simulations section, the
moderately or perfectly conductive bubbles would
elongate at very low electric field and then become
unstable and break up into small bubbles above certain
critical E, while only elongation took place for
nonconductive air bubble. Here, as the water bubble is
conductive, the behavior of instability and break-up is
more pronounced than that of a non-conductive air
bubble.
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where electric field is the field and D is the displacement
field:
𝐸 = −∇V, 𝐷 = 𝜀0 𝜀𝑟 𝐸
2.1 Air bubble in mineral oil without gravity
The simulated results of non-conductive air bubble
deformation at 8 kV and 20 kV is presented in Fig. 5.
The final shape of the air bubble for different voltages is
shown in Fig. 6a. As we can see, there is a critical
voltage for instability and break-up between 14 kV/cm
and 20 kV/cm in our simulation. Below the critical
voltage, only elongation takes place; above it, elongation
and break-up could be obtained Fig. 6b is the plot of
elongation ratio (γ=a/b) with the electric field. The slope
of the curve increases with electric field suggesting that
higher electric fields lead to more pronounced
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Fig. 5. Simulation of deformation process for non-conductive air bubble in mineral oil at 8 (a) and 20 kV/cm (b).
Initial diameter of the bubble is 2 mm (black circle).
Fig. 6. a) Simulation of final shape of air bubble for different voltages; b) Elongation ratio (γ = b/a) as function of
electric field in kV/mm for nonconductive air bubble. Initial diameter of the bubble is 2 mm (black circle).
deformation. For example, at 2.5 kV/mm the elongation
ratio is already about 16, which is unstable state for the air
bubble.
For simulation of conductive air bubble, we define the
permittivity of air as large as 10 000 to represent the
conductive case in electrostatics. Air bubble has same
elongation phenomenon but reaches instability
remarkably faster. The elongation ratio of transition
shape for higher voltage is bigger. In addition, less time
is needed to reach the instability for higher voltages.
What should be mentioned here is that the form of
instability of conductive bubble is different from that of
nonconductive bubble. For the nonconductive one, the
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instability takes place with formation slender waist; while
for conductive one the instability takes place at the tips of
bubble.
2.2 Air bubble in mineral oil with gravity
In most previous publications related to numeric analysis
of deformation behavior of bubbles in an electric field, the
term of gravity was usually neglected. In order to check if
gravity has a limited influence on the deformation, the
gravity term is added in this part of the simulation. When
no electric field exists in the liquid, the injected air bubble
should rise until touching the top electrode because of the
gravity term and finally the air bubble attached on the top
electrode with a certain contact angle.
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Fig. 7 shows simulation results of deformation process
for a non-conductive air bubble at 2 kV/mm. The air
bubble firstly elongates along E. After about 0.1 s, the
bubble becomes unstable and finally ejects small bubbles.
Fig. 7.
Simulation of deformation process for
non-conductive air bubble at 20 kV with gravity term.
Fig. 8 shows simulation results of deformation process
for a conductive air bubble at 5 kV. The air bubble firstly
elongates along E. After short time, the tip of the air
bubble become unstable and form a sharp pin. Up to this
point, the Comsol stops calculating. Compared to
nonconductive bubble, the instability takes place even for
low electric field (in this case of 0.5 kV/mm).
3. Conclusions
In this study, a high electric field strength is applied to
both air and water bubbles in mineral oil to investigate
bubble deformations.
Three kinds of deformation behaviors of bubbles are
observed experimentally: elongation along field lines,
Coulombic repulsion and break-up.
The effect of the type of the power source (AC or DC),
the conductivity of the bubble, the size of the bubble as
well as gravity and viscosity of liquid dielectrics are
investigated numerically. A good qualitative agreement
of the simulated results and the experiments is obtained.
The simulation results indicate that a conductive bubble
breaks up faster and at lower fields compared to a nonconductive one: 25 ms and 0.5 kV/mm vs. 88 ms and
2 kV/mm for a 2-mm air bubble in mineral oil,
respectively.
4. References
[1] L. Rayleigh. “On the Capillary Phenomena of Jets”.
Proc. Royal Soc. London, 29, 71 (1879)
[2] C. Garton and Z. Krasucki. “Bubbles in insulating
liquids: stability in an electric field”. Proc. R. Soc.
Lond. A. Math. Phys. Sci., 280, 211 (1964)
[3] Y. Kweon, M. Kim, H. Cho and I. Kang. “Study on
the deformation and departure of a bubble attached
to a wall in dc/ac electric fields”. Int. J. Multiphase
Flow, 24, 145 (1998)
[4] A. Nosseir, I. Hashad, E. Taha and A. El-Zein.
“Electrically induced pressure in mineral oil under
external bubble injection”. J. Electrostat., 12, 511
(1982)
[5] M. Talaat and A. El-Zein. “Analysis of air bubble
deformation subjected to uniform electric field in
liquid dielectric”. Int. J. Electromagn., 2, 4 (2012)
[6] C. Dan C and L. Lie. “Impact of air bubble
deformation on dielectric liquid subjected to strong
electric field”. High Power Laser and Particle
Beams, 11, 29 (2011)
[7] T. Elperin, A. Fominykh and Z. Orenbakh.
“Simultaneous Convective Heat and Mass Transfer
During Gas Bubble Dissolution in an Alternating
Electric Field”. Chem. Engng. Res. Design, 83,
1237 (2005)
Fig. 8. Simulation of deformation process for conductive
air bubble at 5 kV with gravity term.
We note that the direct simulation of nonisothermal
absorption of a solvable dielectric gas from a stagnant
bubble by a surrounding dielectric liquid under the
influence of the alternating electric field requires special
care in order to resolve the enhancement of mass or heat
transfer between a bubble or droplet and a surrounding
fluid in the presence of electric field [7].
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