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Numerical Study of Dielectric Fluid Bubble Behavior within Diverging External Electric Fields

Matthew R. Pearson* and Jamal S. Yagoobi.† Illinois Institute of Technology, Chicago, IL, 60616

Past numerical research in the area of pool boiling within the presence of electric fields has focused on the case of uniform field intensity. Any numerical or analytical studies of the effect of non-uniform fields have assumed that the bubbles will retain their spherical shape rather than deform. Many of these studies also ignore changes to the electric field caused by the presence of the bubbles. However, these assumptions are not necessarily accurate as, even in the case of uniform field intensity, bubbles undergo a physical deformation and the electric field can become noticeably altered in the vicinity of the bubble. This study explores the effect that an electric field can have on vapor bubbles of a dielectric fluid when the field is diverging. A three-dimensional numerical code is presented that models bubble deformation due to the presence of the diverging field. Numerical results show that the imbalance of electrical stresses at the bubble surface exerts a net dielectrophoretic force on the bubble, thereby enhancing the separation of liquid and vapor phases during pool boiling. However, there is also a downward component, not predicted by analytical approximations, serving to push the bubble against the electrode plate that arises from the distortion of the electric field around the bubble in the vicinity of a plate electrode.

Nomenclature BoE = dielectric Bond number e = bubble elongation ratio E = electric field vector E = electric field magnitude E0 = nominal electric field magnitude fe = electric body force Fe = total dielectrophoretic force exerted on bubble H = mean curvature of interface = outer-normal unit vector to interface surface ∆P = pressure difference across bubble interface (vapor pressure minus liquid pressure) r = local radius of deformed sphere R0 = undeformed bubble radius R1 = inner radial boundary of plate electrodes R2 = outer radial boundary of plate electrodes = unit vector in x-direction = unit vector in y-direction z0 = width of the plate electrodes εl = absolute electric permittivity of liquid εv = absolute electric permittivity of vapor ρ = mass density ρf = charge density Φ = potential of high-voltage electrode = potential field φ = spherical inclination angle * Graduate Student, Mechanical, Materials and Aerospace Engineering Department, 10 W. 32nd Street, AIAA Student Member. † Professor, Mechanical, Materials and Aerospace Engineering Department, 10 W. 32nd Street.

φ

n

yx

9th AIAA/ASME Joint Thermophysics and Heat Transfer Conference5 - 8 June 2006, San Francisco, California

AIAA 2006-2915

Copyright © 2006 by Matthew Pearson. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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θ = spherical azimuth angle θ0 = interior angle between two plate electrodes σ = interfacial surface tension σM = Maxwell stress intensity

Introduction S electronic chip components become faster and more compact, the magnitude of the generated heat flux is rapidly rising and it is becoming increasingly difficult to effectively remove this heat from the core of the chip

to maintain performance and reliability. As microchips continue to grow in power and capability, it is crucial to find new ways of transporting heat away from the chip. Early Central Processing Units (CPUs) in personal computers required little or no thermodynamic consideration during their design. In the latest computers, it is not uncommon to find a heat sink that dwarfs the microchip, along with a multitude of cooling fans, in order to prevent the overheat of the processor.

Micro Heat Pipes (MHPs) have shown great potential as a highly effective way of removing very high heat fluxes from sources of heat such as CPUs. Conventional heat pipes utilize the capillary force existing in wicking structures to pump liquid to an evaporator section. The liquid evaporates and returns to the condenser section. Due to the two-phase nature of the flow, very high heat fluxes can be achieved. A micro heat pipe concept, which combined phase-change heat transfer with MEMS (Micro-Electro-Mechanical Systems), was first proposed by Cotter1. Babin et al.2 further described a micro heat pipe as “a wickless, noncircular channel”. Instead of using a wicking structure, the capillary forces in micro heat pipes exist in the corners of a noncircular flow channel with micro-scale cross-sectional dimensions. Many researchers, including Mallik et al.3, Peterson and Ma4, Peterson5, Le Berre et al.6, and Lee et al.7, have used micro heat pipe designs with a triangular cross-section.

The performance of a micro heat pipe depends on many factors, including the properties of the working fluid, the magnitude of the capillary forces, the dimensions of the heat pipe, and the operating conditions. Its maximum heat transport rate is subsequently limited by various physical phenomena, which include the capillary limitation, entrainment limitation, viscous limitation, sonic limitation, and boiling limitation. The capillary limitation is reached when the capillary pressure difference generated in the corners of the non-circular MHP can no longer overcome the viscous and hydrostatic pressure losses. When this limitation is reached, the liquid in the condenser section cannot be pumped quickly enough to the evaporator section, causing the heat pipe evaporator to dry out. As a consequence, the transport of heat from the evaporator to the condenser shuts down. The entrainment limitation also results in the dry-out of the heat pipe evaporator due to the liquid droplet entrainment by the liquid/vapor interfacial shear stress. The other limitations, such as the viscous, sonic, and boiling limitations, limit the heat transport capacity of the heat pipe due to the dominant viscous forces, the choked flow, and the blocked liquid flow by bubbles, respectively. All of these limitations serve to narrow the selectable operating conditions for satisfactory MHP performance. However, Babin et al.2 suggested that the most restrictive limitation governing maximum heat transport capacity of a micro heat pipe is almost always the capillary limitation. When this limit is reached, the only way to improve the performance of the micro heat pipe is to provide an additional body force to help pump the liquid from the condenser to the evaporator.

One such body force that can be introduced is an electric body force, but there has been little research to investigate how electrohydrodynamic (EHD) phenomena may be able to help overcome this capillary limitation of micro heat pipes. However, these EHD phenomena are well researched in other areas of heat transfer, especially pool boiling. EHD phenomena involve the interaction of electric fields and flow fields in a dielectric fluid medium, and this interaction can induce a fluid motion by an electric body force. The electric body force density acting on the molecules can be expressed as8

2 21 1

2 2e f E Eερ ε ρρ

∂= − ∇ + ∇ ∂ f E (1)

The first term represents the Coulomb force, which is the force acting on the free charges in an electric field. The second and third terms, titled dielectrophoretic and electrostriction forces respectively, represent the polarization force acting on polarized charges. The third term is relevant only for compressible fluids. Thus, EHD pumps require either a free space charge or a gradient in permittivity within a liquid.

A

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In an isothermal, single-phase liquid, the permittivity gradient vanishes and the Coulomb force is the only force available for generation of a net EHD motion. In many fluids used for cooling, especially cryogenics, the Coulomb force is weak but the difference in permittivity between the liquid and vapor phases allows the dielectrophoretic force to exist and enhance separation of the two phases. In theory, this separation enhancement can be used to augment or even replace the capillary force generated in a MHP, thereby helping to overcome the capillary limitation and improve the maximum attainable heat flux. To better understand the effect of dielectrophoretic force on a micro heat pipe as a system, it is useful to first fundamentally study the behavior of a single bubble within a liquid medium under an imposed electric field.

Previous Studies Ogata and Yabe9 presented a numerical and experimental study of the deformation of vapor bubbles when

exposed to a uniform electric field. The experimental apparatus consisted of a refrigerant bath, with two horizontal plate electrodes aligned parallel to each other and 5 mm apart. The lower electrode was grounded, and a D.C. voltage of up to 30 kV was applied to the upper electrode. A 0.1 mm diameter injection hole was drilled into the lower electrode plate and connected to an air compressor, allowing the injection of air bubbles into the electric field. In the absence of electric field, this injection hole created spherical bubbles with a departure diameter of approximately 1 mm. The upper electrode was made of a bronze mesh that allowed the bubbles to pass vertically through the electrode. The experimental test fluid was Furonsorubu AE, an azeotropic mixture of Refrigerant 113 (96 percent by weight) and ethanol (4 percent by weight). A high-speed video camera recorded the bubble behavior.

The experimental findings showed that, in the presence of electric field, the bubbles generated from the injection hole drifted horizontally a small distance along the plate without migrating vertically, and became elongated. The bubbles then started to vibrate up and down with the bottoms of the bubbles maintaining contact with the plate until finally detaching and traveling upwards due to buoyancy. At 24 kV, the elongated bubbles had a maximum length of approximately 3 mm and, as expected, returned to a spherical shape upon passing through the upper electrode.

The numerical study sought to quantitatively verify the experimental findings by modeling the electric forces that affect the bubble behavior. Ogata and Yabe9 noted that, due to the homogeneity of the imposed electric field, an axis of symmetry existed along the vertical centerline of the bubble, reducing the dimensionality of the problem to two. First, the potential field was solved, recognizing that the presence of the bubble has an affect on the field. From the potential field, the electric field intensity was calculated and the electrical stresses, (i.e. Maxwell stresses), were found on the surface of the bubble interface. The bubble geometry was then found from a stress balance at the liquid-vapor interface. The potential field and bubble shape were iterated until both reached convergence. Four primary assumptions were made:

1) The effects of gravity and bubble motion were negligible. 2) The pressure inside the bubble was uniform. 3) The shape of the bubble was axisymmetric and ellipsoidal. 4) The volume of the bubble remained constant, independent of the electric field. The numerical results verified that a vertical component of the electrostatic forces (Maxwell stress and

electrostriction force) existed that pushed the bubble against the grounded plate electrode. It was also found that the horizontal electrostatic force component was approximately four times larger than the vertical component. Although this horizontal component should give a zero net horizontal force due to the axisymmetry about the vertical bubble axis, the researchers postulated that the existence of the horizontal component somehow promoted the small horizontal motion of the bubble, perhaps due to slight asymmetry of the bubble due to field and fluid inhomogeneities. However, the numerical study was designed to replicate the specific conditions and dimensions of the experimental setup, and as a result the investigators did not perform dimensional analysis to reduce the number of parameters involved in the equations.

In a separate study, Hara and Wang10 developed an analytical model to predict bubble motion in non-uniform fields. The model considered three forces acting on a vapor bubble suspended in a liquid medium – dielectrophoretic force due to electric field inhomogeneities, buoyancy force due to density differences and gravitational acceleration, and drag force due to viscosity of the liquid phase. However, the dielectrophoretic force was approximated using Eq. (2), presented by Jones and Bliss11 for very small, spherical bubbles in an external, non-uniform electric field.

( )3 2

022

l v le

v l

R Eε ε ε

πε ε

−= ∇

+F (2)

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Equation (2) is valid for bubbles that are small compared to the length scale of the electric field, do not deform from a spherical shape, and have no influence on the surrounding electric field. However, for small-scale heat transport devices such as a MHP, the size of the bubbles can be the same order of magnitude as the device dimensions. Additionally, Ogata and Yabe9 showed significant bubble deformation and distortion of the surrounding electric field for a nominally uniform electric field, so the same effects are expected to be present in equal or greater amounts for imposed electric fields that are not uniform.

Numerical Model This study investigates, for the first time ever, the deformation of a bubble when subject to a non-uniform

electric field. The fundamental challenge arises from the equations that govern the shape of a bubble that is subject to an arbitrary distribution of stresses over its surface. Previous analytical and numerical studies of bubble shape, including the study by Ogata and Yabe9, have assumed that the bubble is axisymmetric. This assumption greatly simplifies the analysis because the curvature of a line is much more easily calculated than the curvature of a surface. Due to the presence of surface tension at the interface, the curvature and shape of a bubble are directly related to the distribution of stresses over the interface. The equation giving the curvature of an axisymmetric bubble can be found in many engineering textbooks that discuss two-phase flow, but curvature of arbitrary shapes can only be found in advanced geometry textbooks.

In this work, the fundamental equations that govern the shape of a three-dimensional bubble in an arbitrary, non-uniform electric field are derived and studied from first principles. A numerical simulation predicts the bubble shape and the net dielectrophoretic force that acts on the bubble. A comprehensive non-dimensionalization procedure introduces a single dimensionless parameter that governs the bubble deformation and motion behavior. Note that the motivation for this study is significantly different from past numerical and experimental studies, including those mentioned above. Most researchers have investigated the use of dielectrophoretic force as a method of improving the heat flux of an evaporator by assisting in the removal of vapor bubbles from the heated surface (e.g., see Ref. 12). In this work, the fundamental effect of dielectrophoretic force on bubbles is investigated to examine the prospects of using this force as the underlying pumping mechanism for a two-phase heat transport device.

The non-uniform electric field is imposed using two electrodes, one under an imposed high voltage and the other grounded. The two plates are positioned a small distance apart with a fixed angle θ0 between them, as illustrated in Fig. 1. Unlike the parallel plate-electrode configuration used by Ogata and Yabe9, the electrode configuration in this study imposes a non-uniform field and thus the bubble deformation will not be axisymmetric. Note that the domain is three-dimensional and therefore there is a z-coordinate axis pointing out of the page in Fig. 1, and the computational domain extends outwards along this dimension from z = 0 to z = z0. This work models the deformation of a bubble that has already grown (either due to evaporative heat transfer or by injection of vapor) on the grounded electrode to a sphere of radius R0 in the absence of electric field. The modeling is based on assumptions that:

1) The bubble is static and the effects of gravity on the bubble shape are negligible. 2) The pressure inside the bubble is uniform and does not vary with the bubble shape.

An overview of the numerical model is provided here. A more detailed description and derivation of the governing equations involved in the numerical model can be found in Ref. 13.

The following non-dimensional variables are defined:

* φφ ≡Φ

, *

0

EE

E≡ , ( )

*20

MM

l v E

σσε ε

≡−

, *

0

PP

Rσ∆∆ ≡ , *

0H R H≡ (3)

where the nominal electric field is expressed in terms of the applied potential and the length scale as

0θ

Φ

1R2R

Liquid working fluid

x

y

Lower electrode

Upper electrode

Figure 1. Schematic of electrode configuration

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00

ER

Φ≡ (4)

Due to the assumption that pressure inside the bubble is uniform and is unaltered by the deformation of the bubble, ∆P* = 2 because the pressure difference across the interface of the undeformed (spherical) bubble is given by the Young-Laplace equation as ∆P = 2σ/R0.

In the absence of free charges in the liquid volume, the potential field in the liquid is governed by the Laplace equation,

*2 * 0φ∇ = (5)

subject to the boundary conditions listed in Table 1.

Note that, in the absence of a bubble, the seventh boundary condition is not applicable and the remaining boundary conditions and Eq. (5) are satisfied by the analytical expression14

*

0

θφθ

= (6)

The definition of electric field allows direct solution from the potential field,

* * *φ= −∇E (7)

Maxwell stresses at the liquid-vapor interface are caused by the sharp permittivity gradient that exists. By neglecting the electrostriction component of the stress, the non-dimensionalized Maxwell stresses are related to the electric field at the surface by

( )2* * * *1 12 2M Eσ = − = − ⋅E E (8)

Table 1. Potential field boundary conditions

Number Location Condition Description

1 1r R= 0rφ∂ ∂ = Electrically insulated at inner arc wall

2 2r R= 0rφ∂ ∂ = Electrically insulated at outer arc wall

3 0θ = 0φ = Electrically grounded at lower electrode

4 0θ θ= φ = Φ Fixed D.C. potential at upper electrode

5 0z = 0zφ∂ ∂ = Plane of symmetry at 0z =

6 0z z= 0zφ∂ ∂ = Electrically insulated at outer boundary

7 Bubble Surface

0nφ∂ ∂ = Electrically insulating bubble

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Electrostriction in the vapor phase is negligible for vapors that closely obey the Clausius-Mossotti relation and have a permittivity approximately equal to vacuum permittivity, and electrostriction in the liquid phase has no effect if the liquid is incompressible. Therefore, electrostriction can be suitably neglected under these circumstances. Note that although Ogata and Yabe9 neglected electrostriction in the vapor phase, the contribution of electrostriction in the liquid phase was incorrectly included.

A stress balance at the interface governs the equilibrium shape of the bubble. In the absence of additional stresses such as Maxwell stresses, the hydrostatic pressure difference between the vapor and the liquid phases is balanced by the surface tension and curvature of the interface, which gives rise to the Young-Laplace Equation. With additional stress components present, these components are introduced as additional terms in the stress balance. Accounting for hydrostatic pressure differences, surface tension effects, and Maxwell stresses, the mean curvature of the interface must everywhere satisfy Eq. (9) for equilibrium of normal stresses to exist.

( )* * *12 BoE MH P σ= − ∆ + (9)

Shear stresses due to viscosity are not considered in this analysis, due to the assumed absence of fluid motion. Note that the hydrostatic pressure difference can only exert a normal component of stress on the interface, and the Maxwell stresses will also have only a normal component of stress provided that there are no surface charges at the interface. The parameter introduced into Eq. (9) is the dielectric Bond number, representing the ratio of Maxwell stresses to surface tension stresses, and defined as

( ) 2

0 0Bo l vE

E Rε εσ

−= (10)

The mean curvature field given by Eq. (9) must be translated into a geometric surface that represents the interface between the liquid bulk and the vapor bubble. The mean curvature of an arbitrary surface defined by a vector function x(θ,φ) of two parameters θ and φ is given by15

( ) ( )( ) ( )

( )( )2 2

3 22 22

2

2H

θθ θ ϕ ϕ θϕ θ ϕ θ ϕ ϕϕ θ ϕ θ

θ ϕ θ ϕ

− ⋅ +=

− ⋅

x x x x x x x x x x x x x

x x x x (11)

where the scalar triple product is defined as

( ) ( )≡ ⋅ ×abc a b c (12)

For a spherical bubble, spherical coordinates can be used to most simply represent the interface as

0

cos cos

sin cos

sin

R

θ ϕθ ϕ

ϕ

= x (13)

where R0 is a constant equal to the bubble radius. By relaxing the requirement that the radius be constant everywhere, a more generalized interface surface can be parameterized as

( )cos cos

, sin cos

sin

r

θ ϕθ ϕ θ ϕ

ϕ

= x (14)

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where r is an unknown function of the two parameters. With this choice of parameterization vector x, Eq. (11) becomes

( ) ( ) ( )( )

( ) ( )( )

2 2 2 2

3 22 2 2

2 2 2 2 2 2

3 22 2 2

2,

2

2 3 3 2 tan

2

r r r r r r r r rH

r r r r

r r r r r r r r

r r r r

θ ϕϕ θ ϕ θϕ ϕ θθ

θ ϕ

θ ϕ ϕ θ ϕ

θ ϕ

θ ϕ

ϕ

+ − + +=

+ +

+ + + + +−

+ +

ɶ ɶ ɶ ɶɶɶ ɶɶ (15)

where

secr rθ θ ϕ=ɶ , 2secr rθθ θθ ϕ=ɶ , secr rθϕ θϕ ϕ=ɶ (16)

For a known function H(θ,φ), Eq. (15) is a second-order, non-linear partial differential equation that must be solved to give r(θ,φ), using the boundary conditions illustrated in Fig. 2. The left and right boundary conditions are periodic to reflect the periodic nature of the spherical parameterization. The top boundary condition is also periodic, while the bottom boundary condition represents the reflection of the half-bubble across the z = 0 plane. The resulting numerical solution of r(θ,φ) can be used in Eq. (14) to describe a surface of an arbitrary shape (not necessarily a sphere) that represents the interface between the vapor phase of the bubble and the liquid phase of the surrounding fluid.

It is postulated that there are multiple solutions that satisfy Eq. (15). For example, a sphere that is centered at the origin will have r(θ,φ) = R, where R is a constant, and the corresponding mean curvature will be everywhere equal to 1/R. However, the same sphere offset a given distance from the origin will have the same mean curvature, but the radius function r(θ,φ) that parameterizes the surface of the offset sphere would no longer be equal to a constant. Thus, multiple solutions of r(θ,φ) satisfy the prescribed mean curvature 1/R. It is postulated that multiple solutions of r(θ,φ) will exist for any given mean curvature field H(θ,φ). For this reason, the numerical procedure is constrained such that the geometric centroid of the resulting volume enclosed by the surface always lies at the origin of the spherical coordinate system.

The net dielectrophoretic force acting on the bubble is computed by integrating the Maxwell stresses over the surface of the interface,

* * *ˆe M

S

dAσ= − ∫∫F n� (17)

where the dimensionless force is defined as

*

0Boe

eE Rσ

= FF (18)

and

* 20A A R= (19)

2π

2π00

0, 2 ,r rϕ π ϕ=

, 2 , 2

r r

θ π θ π πϕ ϕ ±

∂ ∂= −∂ ∂

2 , 0,r rπ ϕ ϕ=

,0

0r

θϕ∂ =∂

θ

ϕ

Figure 2. Boundary conditions for geometric governing equation

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Note that the only value that must be pre-specified for the numerical simulation to operate is the value of the parameter BoE and that bubble deformation behavior depends entirely on the value of this single parameter.

Numerical Results

A. Bubble Geometry and Maxwell Stress Intensity The grid was refined until the solution was grid independent to within one percent. The three-dimensional nature

of the problem prevented grid independence from being further improved. The potential field and bubble shape were converged to a relative error of 10-5 and 10-3, respectively. Numerical data were obtained for two values of the dielectric Bond number, BoE. Figure 3 shows the numerically predicted bubble shape for BoE = 10. Coloring of the bubble surface is based on the local magnitude of the dimensionless electric field squared, (E*)2 – brighter areas denote stronger intensity, as illustrated by the color bar to the right of the figure. The coloration can also be used in Eq. (8) to determine the local Maxwell stress intensity. The axes of the figure are equally scaled to ensure that the true geometric shape of the bubble is correctly portrayed.

The figure illustrates that the bubble has departed from its original spherical shape and has become elongated. The elongation can be described quantitatively by defining an elongation ratio,

( ) ( )( ) ( )

* *

* *

max min

max min

y ye

x x

−=

− (20)

Therefore, a sphere has e = 1. A vertically elongated bubble has e > 1, and a vertically compressed (or horizontally elongated) bubble has e < 1. The elongation ratio of the bubble in Fig. 3 is e = 1.11.

Figure 4 shows a similar bubble, this time for a Dielectric Bond Number of 15. Comparison of Fig. 4 with Fig. 3 shows that the intensity and distribution of Maxwell Stresses on the bubble surface is similar. However, because of the higher value of BoE, the bubble in Fig. 4 has a larger geometric reaction to the presence of these stresses. From Eq. (20), the elongation ratio is e = 1.21. For values of BoE much above 15, the elongation of the bubble caused it to grow above the location of the upper electrode and out of the domain, which is an unacceptable condition.

B. External Potential Field Distribution Figures 5 and 6 illustrate the lines of isopotential inside the domain, at z = 0, for the two cases of BoE = 10 and

BoE = 15 respectively. Note that in the near-vicinity of the bubble, the distribution of the potential field is significantly altered due to the boundary condition that is imposed at the bubble interface. However, the bubble’s

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Electric Bond Parameter = 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Figure 3. Bubble shape in presence of electric field for BoE = 10

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

Electric Bond Parameter = 15

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Figure 4. Bubble shape in presence of electric field for BoE = 15

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effect on the potential distribution is much localized – even a short distance away from the bubble the potential field is effectively unaltered. Thus, there is a potential field boundary layer near the interface of the bubble.

C. Dielectrophoretic Force Components The most interesting aspect of the numerical results is the resulting dielectrophoretic force acting on the bubble.

The dielectrophoretic force exerted on a small, spherical, insulating bubble in the presence of an external electric field is given by Eq. (2). In general, εv < εl and the net dielectrophoretic force will act opposite to the gradient of electric field intensity (from stronger electric field to weaker electric field). However, when the bubble is attached to the lower electrode plate, the numerical data in Table 2 show that the dielectrophoretic force has a downward

component that serves to press the bubble against the plate. This component is in fact almost twice as large as the horizontal component that serves to move the bubble towards the weaker electric field (the right hand side of Figs. 5 and 6). Note that the force components listed in Table 2 are dimensionless. Therefore, the magnitudes are very similar irrespective of the Dielectric Bond Number values, but the values of the true (dimensional) forces differ after multiplication by the parameter BoE.

Discussions of Numerical Results The results of bubble deformation in non-uniform fields imposed by diverging plate electrodes appear to

conform qualitatively to the axisymmetric study of bubbles in uniform electric fields that was done by Ogata and Yabe9. The nominal electric field lines in this non-uniform electric field study, obtained by substituting Eq. (6) into Eq. (7), run in an arc between the lower electrode and the upper electrode, and bubble deformation appears to occur approximately along these arcs. The amount of deformation that the bubble exhibits in response to the electric field depends on the value of the parameter BoE. For a configuration with BoE = 0, there would be no response to the electric field and the bubble would maintain its spherical shape. Configurations with increasing values of BoE have bubbles that exhibit more significant elongation in the presence of the electric field. Due to the asymmetry of the imposed electric field, there is asymmetry in the bubble deformation. The left side of the bubble (where the nominal electric field magnitude is larger) is more deformed than the right hand side (where the nominal electric field magnitude is weaker). Note that the value of BoE depends both on fluid properties and on the potential that is applied to the high-voltage electrode, Φ. Therefore, for a given fluid, increasing the value of BoE corresponds to increasing the magnitude of this applied voltage.

0.10.10.2

0.2

0.3

0.3

0.4

0.40.5

0.5

0.6

0.6

0.7

0.7

0.70.8

0.8

0.8

0.8

0.9

0.9

0.9

1

1

1

Isopotential Lines (BoE = 10)

1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

Figure 5. Isopotential lines for BoE = 10

0.1

0.10.2

0.20.3

0.3

0.4

0.40.5

0.5

0.6

0.6

0.7

0.70.8

0.8

0.8

0.9

0.9

0.9

0.9

1

1

1

Isopotential Lines (BoE = 15)

1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

Figure 6. Isopotential lines for BoE = 15

Table 2. Dielectrophoretic force components

Force Component BoE = 10 BoE = 15

x -component ( )* ˆe ⋅F x 0.1433 0.1461

y -component ( )* ˆe ⋅F y -0.2212 -0.2335

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The direction of the dielectrophoretic force predicted by the numerical simulation differs greatly from the analytical expression given by Eq. (2). The analytical expression predicts a positive horizontal component and a small, positive upward component. Numerical results show that although the horizontal component still draws the bubble towards an area of weaker field, the vertical component is directed downwards towards the ground electrode and has almost twice the magnitude of the horizontal component. This attractive force towards the grounded electrode also exists in uniform electric field distributions, as was observed, both numerically and experimentally, by Ogata and Yabe9. The cause of this downward force component is the proximity of the bubble with the electrode. Note in Figs. 3 and 4 that the Maxwell Stress intensity at the underside of the bubble, close to the grounded plate electrode, is approximately zero. Maxwell Stresses act in the normal direction across the interface from the liquid phase to the vapor phase. Therefore, at the top of the bubble, significant Maxwell stresses are present that exert a downward force on the bubble mass. At the bottom of the bubble, there are no Maxwell stresses to balance this downward force, and the bubble is pressed against the plate. The numerical simulation does not contain the necessary governing equations to model the deformation of the bubble as it is pressed against this plate. However, concepts from two-phase flow suggest that the bubble would have a contact area with the plate (rather than a single contact point) and the liquid-vapor interface would contact the solid electrode at some contact angle relating to the surface tension properties of the liquid, vapor and solid surfaces.

The difference between the Maxwell stress intensities at the top and the bottom of the bubble can be attributed to the fact that the bubble strongly affects the potential (and therefore the electric) field in the near-vicinity. The analytical form of Eq. (2) is based on the assumption that the bubble has no influence on potential field distribution. Note that the dielectrophoretic force only depends directly on the electric field distribution at the liquid-vapor interface, where the distortion of the fields due to the bubble’s presence is at a maximum. When the length scale of the electric field is significantly larger than the length scale of the bubble, then this effect can surely be neglected. However, in this study the electric field and the bubble both have the same length scale, R0, and therefore Eq. (2) is not suitable for use in predicting the dielectrophoretic force.

Note that Maxwell Stresses depend on the squared intensity of the electric field, E2, and not on the direction of the electric field or the value of the potential field. Therefore, reversing the roles of the two electrodes (applying a high voltage to the lower electrode plate and grounding the upper electrode plate) should have no effect on the deformation or motion behavior of the bubble. It follows that when the bubble is in the vicinity of either electrode plate that it will be subject to an attractive force, regardless of the applied potential of the plate. A bubble that is exactly equidistant between the two electrodes will not be subject to an attractive force because a case of symmetry will exist. However, for a bubble that is a small distance from this centerline, a very small attractive force towards the nearest electrode can be expected. This attractive force will become larger as the bubble moves closer and closer to the electrode.

Conclusions A full non-dimensionalization procedure reveals that the static deformation of a bubble depends on a single

dimensionless parameter, the dielectric Bond number BoE. This parameter is recognized by the IEEE Dielectrics and Electrical Insulation Society as a standardized dimensionless parameter for use in electrohydrodynamics. The equations governing potential field, electric field, Maxwell stresses, and stress balance are well known. However, the concept of solving a surface from a Prescribed Mean Curvature (PMC) field is not well studied mathematically or numerically, especially in a spherical coordinate system. This work presents a finite-difference method of solving a PMC surface. The surfaces studied here are spherical in nature, but the same approach could be used for any type of surface. Although multiple solutions satisfying a PMC surface may exist, it is postulated that, because a sphere is used as the initial guess, and because the geometric centroid of the bubble is anchored to a fixed location, the final converged bubble shape is the solution that would be adopted by a physical bubble that is initially spherical.

The numerical solutions present some interesting deviations that are not predicted the analytical approximation of Eq. (2), particularly in the vicinity of one of the electrodes. The primary conclusions that can be made from the generated numerical data are as follows: • When the bubble is close to the lower electrode plate, it experiences a downward component due to the existence of downward-acting Maxwell stresses at the top of the bubble and almost non-existent Maxwell stresses at the bottom. This phenomenon arises from the distortion of the electric field in the near-vicinity of the bubble surface. • The influence of this downward dielectrophoretic force component may be significant or insignificant, depending on the magnitudes of other forces acting on the bubble. For example, in normal Earth gravity, buoyancy forces can easily overcome this attractive dielectrophoretic force. However, in the absence of gravity, this buoyancy force disappears and the dielectrophoretic force becomes dominant, greatly influencing the motion of the bubble.

American Institute of Aeronautics and Astronautics

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• The dimensions of the electrode geometry studied here are the same order of magnitude as the bubble radius. Therefore, the intensity of the non-dimensional Maxwell Stresses is relatively high and thus low values of Dielectric Bond Number are studied here. For larger plates that are positioned further apart, the non-dimensional Maxwell Stresses would be weaker and larger values of BoE could be studied without the bubble elongating beyond the boundaries of the domain.

The results show similar behavior to that observed by Ogata and Yabe9, but the behavior is now generalized to non-uniform electric fields in three dimensions. Therefore, in addition to the vertical force that attracts the bubble to the lower electrode, there is also a net force component the helps propel the bubble towards areas of lower electric field intensity. Although this work studies only the field generated between two diverging plate-electrodes, the numerical model and governing equations that have been presented are applicable to any arbitrary electrode configuration in three dimensions. This fundamental study of bubble deformation and motion in non-uniform electric fields and the numerical data generated are greatly beneficial in the design of a thermal heat transfer device powered by dielectrophoretic force.

Acknowledgments The authors thank NASA Headquarters – Microgravity Fluid Physics Program – for their financial support of

this research project. They are also grateful to the NASA Goddard Space Flight Center for their collaboration on this project.

References 1Cotter, T. P., “Principles and Prospects of Micro Heat Pipes,” Proceedings of the 5th International Heat Pipe Conference,

Tsukuba, Japan, 1984, pp. 328-335. 2Babin, B. R., Peterson, G. P., and Wu, D., “Steady-State Modeling and Testing of a Micro Heat Pipe,” ASME Journal of

Heat Transfer, Vol. 112, No. 3, 1990, pp. 595-601. 3Mallik, A. K., Peterson, G. P., and Weichold, M. H., “Fabrication of Vapor-Deposited Micro Heat Pipe Arrays as an Integral

Part of Semiconductor Devices,” Journal of Microelectromechanical Systems, Vol. 4, No. 3, 1995, pp. 119-131. 4Peterson, G. P., and Ma, H. B., “Theoretical Analysis of the Maximum Heat Transport in Triangular Grooves: A Study of

Idealized Micro Heat Pipes,” ASME Journal of Heat Transfer, Vol. 118, No. 4, 1996, pp. 734-739. 5Peterson, G. P., “Applications of Microscale Phase Change Heat Transfer: Micro Heat Pipes and Micro Heat Spreaders,”

Handbook of Microelectromechanical Systems, edited by M. Gad-el-Hak, CRC Press, New York, 2000, pp. 301-326. 6Le Berre, M., Launay, S., and Lallemand, M., “Fabrication and Experimental Investigation of Silicon Micro Heat Pipes for

Cooling Electronics,” Journal of Micromechanics and Microengineering, Vol. 13, No. 3, 2003, pp. 436-441. 7Lee, M., Wong, M., and Zohar, Y., “Integrated Micro-Heat-Pipe Fabrication Technology,” Journal of

Microelectromechanical Systems, Vol. 12, No. 2, 2003, pp. 138-146. 8Melcher, J., Continuum Electromechanics, M.I.T. Press, Cambridge, MA, 1981. 9Ogata, J., and Yabe, A., “Basic study on the Enhancement of Nucleate Boiling Heat Transfer by Applying Electric Fields,”

International Journal of Heat Mass Transfer, Vol. 36, No. 3, 1993, pp. 775-782. 10Hara, M., and Wang, Z., “An Analytical Study of Bubble Motion in Liquid Nitrogen under D.C. Non-uniform Electric

Fields,” Proceedings of the 4th International Conference on Properties and Applications of Dielectric Materials, IEEE, New York, 1994, pp. 459-462.

11Jones, T. B., and Bliss, G. W., “Bubble Dielectrophoresis,” Journal of Applied Physics, Vol. 48, No. 4, 1977, pp. 1412-1417.

12Bryan, J. E., and Seyed-Yagoobi, J., “Influence of Flow Regime, Heat Flux, and Mass Flux on Electrohydrodynamically Enhanced Convective Boiling,” ASME Journal of Heat Transfer, Vol. 123, No. 2, 2001, pp. 355-367.

13Pearson, M., “Deformation and Motion Behavior of a Bubble due to Non-Uniform Electric Fields,” M.S. Thesis, Mechanical, Materials, and Aerospace Dept., Illinois Institute of Technology, Chicago, IL, 2006.

14Crowley, J. M., Fundamentals of Applied Electrostatics, Laplacian Press, Morgan Hill, CA, 1999. 15Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., CRC Press, New York, 1998.