PS15
液柱内温度差マランゴニ対流における電場印加が帯電粒子挙動に与える影響
Effect of Applied Electric Field on the Behavior ofCharged Particles in Thermocapillary Convection of aLiquid Bridge
吉田優人 1, 石村美紗 2, 西野耕一 2,Yuto YOSHIDA1, Misa ISHIMURA1 and Koichi NISHINO2
12横浜国立大学大学院，Graduate School, Yokohama National University
横浜国立大学，Yokohama National University

1. Introduction
A floating zone (FZ) method1) is used for generating high-purity single crystals of silicon. In the molten
section, natural convection has an adverse effect on crystal growth. To suppress this effect, experiments of theFZ method were conducted under microgravity. However, Marangoni convection, driven by surface tension
differences, became dominant. This convection is known to adversely affect crystal growth. Therefore, it isimportant to better understand Marangoni convection and thereby manipulate it. So far, experiments havebeen conducted using a liquid bridge, where a liquid is suspended between two cylindrical disks. In previousstudies, Marangoni convection due to temperature difference in the liquid bridge was controlled by heatingor applying a magnetic field2,3). Although convection control by electric field is expected in industrial
applications due to its controllability and non-contact application, there are yet few studies4) on the effect ofelectric field on Marangoni convection in a liquid bridge. Furthermore, it is known that the application of anelectric field changes the surface tension and causes the EHD effect, but the principle of EHD generation in aliquid bridge and the dependence of the surface tension on the electric field have not yet been clarified.
In experiments on Marangoni convection, particle image velocimetry (PIV) is a powerful tool for visualizing
flow field, where we can analyze the flow field by capturing images of tracer particles injected into the fluid.
It has been demonstrated5) that the particle motion accurately represents the fluid motion in the case where asmall temperature difference is applied to the liquid bridge. On the other hand, when an electric field isapplied to the liquid bridge, the particles are observed to be driven by the electric field, suggesting that theparticles are electrically charged6-9). This effect must be clarified in order to determine whether the applicationof an electric field contributes to the convection control or simply affects the particle motion (or results in both).
Therefore, understanding the effect of charged particles is essential for evaluating convection control by anelectric field. The present study aims at conducting numerical analyses with a view to understanding how andto what extent charged particles affect the flow field and flow visualization.
1 of 9

2. Numerical analysis
Numerical simulations were performed using the commercial CFD solver STAR-CCM+. The computational
domain consists only of a liquid bridge suspended between the two concentric disks as shown in Figure 1,
where electrically charged particles exists in the fluid. A two-dimensional unsteady analysis is performed
assuming axisymmetric flow and no deformation of the liquid bridge. Flow conditions and physical propertiesof the fluid are the same as those of ground experiments conducted by Yamazaki7). He carried out a series ofPIV analysis of the motion of nylon 12 particles in 5 cSt silicone oil where temperature and potential differenceswere applied between two disks. The physical properties assumed in the present numerical simulation aresummarized in Table 1, where the one-way coupling between particles and fluid is assumed and only charged
particles are affected by the applied electric field. By performing the analysis based on this assumption, theeffect of applied electric field on the behavior of charged particles is clarified.
The governing equations for fluids are obtained from the conservation of mass, momentum and energy:𝛁 ∙ 𝒖 = 0,(1)𝜕𝒖
1+ (𝒖 ∙ 𝛁)𝒖 = − 𝛁𝑝 + 𝜈𝛁 2 𝒖 + 𝒇𝒃 ,𝜕𝑡𝜌(2)

𝜕𝑇+ (𝒖 ∙ 𝛁)𝑇 = α𝛁 2 𝑇,𝜕𝑡(3)where 𝒖 is the fluid velocity, 𝑡 is the time, 𝜌 is the density, 𝑝 is the pressure, 𝜈 is the kinematic viscosity,
𝒇𝒃 is the body force such as buoyancy and Marangoni force, 𝑇 is the temperature, and α is the thermaldiffusivity.
Marangoni force is considered using Tiwari & Nishino’s method10), which mimics a surface force as a bodyforce acting on a thin surface cell:
𝐹𝑚 = 𝜎𝑇𝜕𝑇 1,𝜕𝑧 𝛿𝑟(4)where 𝜎𝑇 is the temperature coefficient of surface tension , and 𝛿𝑟 is the surface cell thickness.
The Lagrangian method is used for the calculation of particle motion. Different from the Eulerian fluid
calculations solved by the finite volume method, the governing equations are defined for each individualparticle. The governing equation for particles is obtained from the conservation of momentum and these are
the forces acting on the particles:
𝑑𝒖𝒑= 𝑭𝒔 + 𝑭𝒃 ,𝑑𝑡

(5)

𝑭𝒔 = 𝑭𝒅 + 𝑭𝒑 + 𝑭𝑳𝑺 ,(6)

𝑭𝒃 = 𝑭𝒈 + 𝑭𝑪𝒐 ,

(7)

𝑚𝑝

where 𝑚𝑝 is the particle mass, 𝒗𝒑 is the particle velocity, 𝑭𝒔 is the surface force, 𝑭𝒃 is the body force, 𝑭𝒅is the drag force, 𝑭𝒑 is the pressure gradient force, 𝑭𝑳𝑺 is the shear lift force, 𝑭𝒈 is the gravitational force,and 𝑭𝑪𝒐 is the coulomb force.
These are the boundary conditions at the free surface, top disk, and bottom disk:
𝑢𝑟 = 0,𝜕𝑇
𝜕𝜙= 0,= 0 (𝑟 = 𝑅),𝜕𝑟𝜕𝑟(8)~(10)

2 of 9𝑢𝑟 = 0, 𝑢𝑧 = 0, T = 𝑇ℎ , 𝜙 = ∆𝐸 (z = 𝐻),(11)~(14)𝑢𝑟 = 0, 𝑢𝑧 = 0, T = 𝑇𝑐 , 𝜙 = 0 (z = 0),(15)~(18)where 𝑢𝑟 is the radial velocity, 𝑅 is the radius of the liquid bridge, 𝑢𝑧 is the axial velocity, 𝐻 is the heightof the liquid bridge, and 𝜙 is the electric potential.
In this study, the analysis is performed for 5 different particle charges of 𝑞𝑝 = 0, 1, 4.42, 10, 24.8 × 10−17 C.
The Pauthenier equation for the saturation charge due to the electric field charge is calculated as
𝑞∞ = 12π𝜀𝑝 𝑅𝑝 2 |𝑬|⁄(

𝜀𝑝+ 2),
𝜀0(19)

where 𝜀𝑝 is the particle permittivity, 𝑅𝑝 is the particle radius, 𝑬 is the electric field intensity, and 𝜀0 is thevacuum permittivity. This gives 𝑞∞ = 24.8 × 10−17 C . In addition, unpublished internal data from ourlaboratory11) shows 𝑞𝑒𝑥𝑝 = 4.42 × 10−17 C, obtained from an experiment in which an electric field was applied
to a liquid bridge without a free surface and the amount of charge was estimated from particle movement.
However, this value contains errors and should be treated with caution. Since it is not currently possible toaccurately measure the particle charge, 𝑞∞ = 0, 1, 10 × 10−17 C are added, and 5 different charge values areused to evaluate the particles under a wide range of charging conditions.

Figure 1. Geometry of the analytical liquid bridge.

Table 1. Physical properties of silicone oil and nylon 12 particles.

Silicone oil [a]Nylon 12 particles [b]Kinematic viscosity[cSt]

5Density [kg/m3]

915

Specific heat [J/(kg*K)]

Average particle
size [μm]5.0FeatureSpherical

3

Density [kg/m ]1016.94−11

Permittivity [F/m]3.72 × 10−11

Particle charge[C]0, 1, 4.42, 10, 24.8 × 10−17

1.8 × 10Permittivity [F/m]

2.6 × 10

Temperature coefficientof surface tension[N/(m*K)]−6.58 × 10−5

33 of 9

3. Results and discussion
Figure 2 shows the velocity vectors of the fluids and charged particles (𝑞𝑝 = 4.42 × 10−17 C) in the electricfield. Twenty thousand particles were initially introduced at equal intervals, and the velocity vectors of fluid
and particles at 3 [s] after the application of the electric field are indicated. The conditions are as follows:
temperature difference (∆T) of 10 [℃] and the potential difference (∆E ) of 400 [V] between the top andbottom of the liquid bridge. The temperature and electric potential distributions of the fluid are shown inFigure 3. These results are the left half of the cross section of the liquid bridge. From Figure 2, it is recognizedthat both fluid and particles show similar velocity vectors in that they flow downward near the free surface
and flow upward near the axis but that their quantitative distributions are appreciable different as discussed
in detail below.

[a] FluidFigure 2.

[b]Charged particles

Velocity vectors of fluid and charged particles, where the left and right boundaries of each figure are the freesurface and the axis of the liquid bridge, respectively.

[a] Temperature
Figure 3.

[b] Electric potential
Temperature and electric potential distributions.
4 of 9Figure 4 shows the histograms of the particle slip velocity, 𝒖𝐬𝐥𝐢𝐩 = 𝒖 − 𝒖𝐩 , where Figure 4-1 gives those forno-charged particles while Figure 4-2 gives those of charged particles (𝑞𝑝 = 4.42 × 10−17 C). Each figure shows
the particle slip velocities in axial and radial directions. In the absence of charge, the slip velocity is distributed
around 0 mm/s for both axial and radial directions. Although not shown here, the slip velocity between fluid
and ideal particles (i.e., sufficiently small particles having fluid density) is found to be quite similar to thatshown in Figure 4-1, therefore indicating that the slip velocity distribution seen here is caused by theLagrangian method used in the present numerical analysis. This must be considered in the interpretation of
the slip velocity between charged particles and fluid. Under the charged condition, the axial slip velocity isdistributed around 0.04 mm/s. This is because the charged particles are attracted in the direction of the cathodeby the applied electric field. When comparing the kurtosis 𝑘 of the 𝑢slip(𝑧) , 𝑘 = 126 for 𝑞𝑝 = 0 C and 𝑘 =46 for 𝑞𝑝 = 4.42 × 10−17 C, indicating that the distribution for the charged particles has lower peak value. In
the previous study by Kimura and Kamii12), the ratio between the moving body’s relative velocity under𝒖𝐬𝐥𝐢𝐩oscillatory flow and the terminal velocity in a stationary fluid, 𝒖 , is greater than 0.99 under the conditions
𝐭𝐞𝐫of the density ratio of 1.5 and the particle diameter of 2 mm. Their study suggests that for particles of small
diameter and near-fluid density, the oscillatory flow has no effect on the terminal velocity of the particles.
Furthermore, they showed that the particle velocity is represented by the superposition of the velocity of theoscillatory flow and the terminal velocity estimated for the stationary fluid. If this superposition is applicableto the present study, the slip velocity should be calculated using Stokes' law:𝒖𝒕𝒆𝒓 =𝑭𝒒
𝑞𝑝 𝑬== 0.041 𝑚𝑚⁄𝑠,6𝜋𝜇𝑅𝑝
6𝜋𝜇𝑅𝑝(20)where 𝜇 is the fluid viscosity. This terminal velocity can explain the peak velocity of 𝑢slip(𝑧) seen in Figure
4-2. However, it also shows that it is widely distributed around the terminal velocity. This is because the dragforce on the particles changes as the fluid velocity differs from place to place in the liquid bridge, resulting ina wide range of slip velocity. Therefore, the superposition of thermocapillary convection and particle terminal
velocity cannot represent particle behavior.

The numberof particles

𝑢slip(𝑧) [𝑚𝑚⁄𝑠]
Figure 4-1.

𝑢slip(𝑟) [𝑚𝑚⁄𝑠]

Histogram of particle slip velocity without charge.

5 of 9

The numberof particles

𝑢slip(𝑟) [𝑚𝑚⁄𝑠]𝑢slip(𝑧) [𝑚𝑚⁄𝑠]

Histogram of particle slip velocity for 𝑞𝑝 = 4.42 × 10−17 C.

Figure 4-2.

𝒖𝒑(𝒛)To evaluate particle traceability, the ratio of particle velocity to flow velocity, 𝒖𝒖𝒑(𝒛)Figure 5 shows histograms of 𝒖𝒖𝒑(𝒛)are many particles with

𝒖(𝒛)(𝒛)(𝒛), has been defined here.

at particle charges 𝑞𝑝 = 0, 1, 4.42, 10, 24.8 × 10−17 C. When 𝑞𝑝 = 0 C, there

close to 1, indicating good fidelity as fluid tracers. This indicates that the particles

follow the flow well under conditions where there is no electrical influence on the particles, such as when theparticles are not charged or when only a temperature difference is applied without applying an electric field,
due to their small particle size and the slight difference in density between particles and fluids. On the otherhand, as the charge increases, the number of particles with

𝒖𝒑(𝒛)
𝒖(𝒛)

values away from 1 increases and theirdistribution becomes wider. Additionally, for 𝑞𝑝 = 4.42 × 10−17 C or higher, there are many particles withpoor fidelity. Therefore, when selecting particles as tracer particles for PIV, it is necessary to consider theircharging characteristics. The degree of goodness as fluid tracers can be evaluated as how many particles outof all particles injected into the computational domain have a𝒖𝒑(𝒛)𝒖(𝒛)value that can be considered indicative ofgood fidelity. The ranges for good fidelity are examined as 0.9~1.1, 0.95~1.05, and 0.99~1.01, respectively, but
0.95~1.05, which is widely used as the standard, is used in this study. Figure 6 shows the particle distributionof𝒖𝒑(𝒛)
𝒖(𝒛)

𝒖𝒑(𝒛)
𝒖(𝒛)

for each charge amount. From left to right, particles satisfying 0.95 <𝒖𝒑(𝒛)< 0.95 or 1.05 < 𝒖(𝒛)

𝒖𝒑(𝒛)
𝒖(𝒛)

< 1.05 (good fidelity) and(poor fidelity) are displayed. In the case of no charge, most of the flow field isoccupied by particles with good fidelity, whereas, at 𝑞𝑝 = 4.42 × 10−17 C, particles with poor fidelity are more
widely distributed. This indicates that the applied electric field deteriorates particle fidelity. Figure 7 shows
the ratio of particles with good tracking performance on the vertical axis and the amount of charge on thehorizontal axis, and also shows a least-squares fitting curve. From this curve, it can be seen that in order for𝒖𝒑(𝒛)more than 70% of the particles to satisfy 0.95 < 𝒖(𝒛)< 1.05, the amount of charge must be kept below 𝑞𝑝 =2.03 × 10−17 C, and for more than 80% of the particles to satisfy it, the amount of charge must be kept below
𝑞𝑝 = 0.95 × 10−17 C . Therefore, when using PIV particles in liquid bridge experiments involving theapplication of an electric field, the particles should be used within a good tracking range considering theircharging characteristics.

6 of 9

The numberof particles

𝒖𝒑(𝒛)
𝒖(𝒛)

𝒖𝒑(𝒛)
𝒖(𝒛)
𝑞𝑝 = 0 C

𝑞𝑝 = 1 × 10−17 C

𝒖𝒑(𝒛)
𝒖(𝒛)

𝒖𝒑(𝒛)𝒖(𝒛)The numberof particles

𝑞𝑝 = 4.42 × 10−17 C

𝑞𝑝 = 10 × 10−17 C

The numberof particles

𝒖𝒑(𝒛)
𝒖(𝒛)
𝑞𝑝 = 24.8 × 10−17 C
Figure 5. Histograms of𝒖𝒑(𝒛)𝒖(𝒛)at 𝑞𝑝 = 0, 1, 4.42, 10, 24.8 × 10−17 C.

7 of 9[a] Good fidelity
Figure 6-1.

[b]

The distribution of[a] Good fidelity
Figure 6-2.

The distribution of𝒖𝒑(𝒛)𝒖(𝒛)for 𝑞𝑝 = 0 C (good and poor fidelity).

[b]
𝒖𝒑(𝒛)
𝒖(𝒛)

Poor fidelity

Poor fidelity

for 𝑞𝑝 = 4.42 × 10−17 C (good and poor fidelity).

Figure 7. The ratio of the good tracer particle

8 of 9

4. Conclusions
The effect of applied electric field on the charged particles in thermocapillary convection in a liquid bridge
has been investigated by numerical analysis based on the Lagrangian method for particle behavior. Bychanging the charge quantity and by examining the velocity and the number of particles, the following
findings were obtained.
1)

Slip velocities are not constant but distributed around the terminal velocity due to the electrical force.
This suggests that the particle behaviors cannot simply be represented by the superposition of fluid flow
and terminal velocity evaluated for stationary fluid.
𝒖𝒑(𝒛)

2)

Increasing particle charge shifts 𝒖3)

The choice of tracer particles considering their charging characteristics is important for the experiment of(𝒛)value away from 1 and broadens the distribution.

thermocapillary convection in a liquid bridge exposed to an electric field.

References
1)

P.H. Keck and M. J. E. Golay: Crystallization of silicon from a floating liquid zone. Phys. Rev, 89 (1953) 1297, DOI:
https://doi.org/10.1103/PhysRev.89.1297.
2) M. Kudo and H. Kawamura: Control of Oscillatory Thermocapillary Convection in a Half-Zone Liquid Bridge byCircumferential Round Heating. Thermal Science & Engineering, 14 (2006) 19, DOI: https://doi.org/10.11368/tse.14.19.
3) Y. Tsutsumida, H. Minakuchi, Y. Takagi, Y. Okano and S. Gima: Effects of magnetic fields on Marangoni convection
in a floating half-zone. The Japan Society of Mechanical Engineers, 25 (2012) 118, DOI:
https://doi.org/10.1299/jsmecmd.2012.25.118.
4) M. Haga, T. Maekawa, K. Kuwahara, A. Ohara, K. Kawasaki, T. Harada, S. Yoda and T. Nakamura: Effect of Electric
Field on Marangoni Convection under Microgravity. J. Jpn. Soc. Microgravity Appl., 12 (1995) 19, DOI:
https://doi.org/10.15011/jasma.12.1.19
5) T. Yano, K. Nishino, H. Kawamura, I Ueno, S. Matsumoto, M. Ohnishi, M. Sakurai: 3-D PTV measurement ofMarangoni convection in liquid bridge in space experiment. Experiments in fluids, 53 (2012) 9, DOI:
https://doi.org/10.1007/s00348-011-1136-9.
6) K. Yamazaki: The study of the effect of an electric field on thermocapillary flow in a hanging droplet. Master’s thesis,
Yokohama National University (2019).
7) K. Yamazaki: Behavior of liquid bridge Marangoni convection under temperature field and electric field. Graduation
thesis, Yokohama National University (2021).
8) C. Matsuno: Electrocapillary convection in the liquid film by electric dependence of surface tension. Master’s thesis,
Yokohama National University (2022).
9) K. Yamazaki: Electric field dependence of surface tension in dielectric fluids and Marangoni convection in thin liquidfilm due to external electric field. Master’s thesis, Yokohama National University (2023).
10) S. Tiwari and K. Nishino: Numerical study to investigate the effect of partition block and ambient air temperature on
interfacial heat transfer in liquid bridges of high Prandtl number fluid. J. Cryst. Growth, 300 (2007) 486, DOI:
https://doi.org/10.1016/j.jcrysgro.2006.12.049.
11) K. Yamazaki, a private letter.
12) H. Kimura and Y. Kamii: Dynamic Forces of Fluid Acting on a Moving Body in Unsteady Flow Field and Motion ofSinking Matter. Fisheries Engineering, 50 (2013) 1, DOI: https://doi.org/10.18903/fisheng.50.1_1.
© 2024 by the authors. Submitted for possible open access publication under
the terms and conditions of the Creative Commons Attribution (CC BY) license(http://creativecommons.org/li censes/by/4.0/).
9 of 9