Numerical simulations of bubble formation from a submerged orifice and a needle: The effects of an alternating electric field

European Journal of Mechanics B/Fluids 56 (2016) 97–109

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Numerical simulations of bubble formation from a submerged orifice and a needle: The effects of an alternating electric field Shyam Sunder, Gaurav Tomar ∗ Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India

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Article history: Received 3 March 2015 Received in revised form 17 June 2015 Accepted 30 November 2015 Available online 4 December 2015 Keywords: Electrohydrodynamics Bubble formation Alternating electric field Volume-of-Fluid method

abstract In many applications, such as bubble column reactors, electric field is employed to provide a greater control on the sizes of bubbles forming at orifices and needles. In this study, we investigate the effects of an alternating electric field on the bubble dynamics. We perform numerical simulations of an alternating electric field coupled with two-phase flow using a Coupled Level-Set and Volume-of-Fluid method. We show that bubbles forming at orifices and needles decrease in size (up to 30%) only for a range of applied frequency and for other frequencies, the size of bubbles can be much bigger compared to the bubbles forming in the corresponding DC electric field case. The oscillating electric forces excite capillary waves on the bubble interface resulting in applied frequency dependent bubble oscillations. The numerically observed resonance for the needle case corresponds to 2ωτc = 0.75, where 2ω is the frequency of the oscillation of the electric field force at the interface and τc is the capillary time scale, indicating that the resonance behavior is indeed governed by the interactions between the capillary and electric field force. A decomposition of bubble profile shapes into Legendre modes shows that for orifice as well as the needle case, second mode is most dominant followed by the fourth mode. © 2015 Elsevier Masson SAS. All rights reserved.

1. Introduction In various bubble injection systems, such as those employed in aerators and bubble column reactors, it is desired to reduce the size of the detached bubbles for more efficient heat and mass transfer. In ink-jet printers, it is desired to control the size of the droplets from the nozzle for high-quality printing. Electric field has been successfully used to reduce the bubble and droplet sizes in bubble and drop injection systems, see for example, [1–5]. Electric forces have also been shown to imitate gravitational forces and cause bubble and droplet detachments from nozzles in microgravity conditions (see [6–9]). Use of alternating electric fields further enhances the electrohydrodynamic effects resulting in decreased bubble and drop sizes. Alternating electric field leads to interfacial oscillations in bubbles and drops. Sato [10,11] performed experiments to study the effects of an alternating electric field on bubble and droplet formation from needles. During the formation of distilled water (weakly conducting) droplets in air as well as in kerosene oil, a ‘synchronous region’ was observed where the number of droplets

∗

Corresponding author. Tel.: +91 80 2293 3062; fax: +91 80 2360 0648. E-mail address: [email protected] (G. Tomar).

http://dx.doi.org/10.1016/j.euromechflu.2015.11.014 0997-7546/© 2015 Elsevier Masson SAS. All rights reserved.

produced was proportional to the oscillation frequency of the externally applied electric field. Using half order Bessel and Hankel functions, Yang and Carleson [12] investigated oscillations of a conducting drop inside a dielectric fluid in the presence of an alternating electric field. They concluded that the resonant frequency for forced oscillation is equal to the natural frequency for inviscid drops and the viscous effects decrease the resonance frequency. Trinh et al. [13,14] investigated the oscillation dynamics of acoustically levitated droplets and bubbles driven by either a modulated ultrasonic field or a time varying electric field. Common conclusions from the above two studies were that the degeneracy between axisymmetric and non-axisymmetric modes is removed and due to viscous damping the resonant frequency is detuned for larger amplitudes. They observed that even numbered modes couple only weakly with the odd modes whereas the odd modes excite even modes strongly. In both the studies it was shown that the higher modes sub-harmonically excite lower modes and very less driving energy was transferred to higher order modes at their respective resonant frequencies because of high viscous damping. Kweon et al. [5] observed that an alternating electric field leads to bubble interface oscillations causing an early breakup and a substantial decrease in bubble size is observed at a certain critical applied voltage for a given frequency. Bellini et al. [15,16] observed that a uniform electric field excites only the second Legendre shape

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mode while a quadrupolar electric field excites the second as well as the fourth mode for the case of an isolated spherical bubble. Lee et al. [17] experimented with dripping pendant drops of water and DNA solution under an alternating electric field between electrodes separated by a short distance. They observed a twostage drop formation process. During the first stage, the drop oscillated and elongated forming a liquid bridge which eventually generated a drop on the lower electrode upon breakup during the second stage. Tran et al. [18] studied the axisymmetric and non-axisymmetric oscillations and jetting of liquid meniscus on a nozzle tip under ac, ac superimposed on dc, and pulsed dc voltages. They observed that the ac waveform led to jet formation at higher voltages while the meniscus oscillated axisymmetrically at lower voltages. The ac waveform along with a mean dc component generated a normal jet or a Taylor cone jet depending on the ratio of the two voltages. Tran et al. [19] performed semi-analytical and experimental studies on AC electric field driven oscillations of sessile droplet on conducting, hydrophobic and hydrophilic surfaces. They investigated the effects of liquid physical properties and the contact radius on droplet oscillations. Sommers and Foster [20] studied the effects of an electric field on the shape and volume modes of oscillating bubbles. Their experimental procedure, similar to Trinh et al. [13], consisted of trapping bubbles at the node of a standing acoustic wave and then induce oscillations using AC electric field. It was observed that the nonuniform electric field excited the first three even Legendre shape modes. They further numerically calculated the electric field inside experimentally captured bubble profiles and found increased strength of electric field inside the bubbles. Corti et al. [21] used the amplitude of the quadrupolar bubble oscillation mode to calculate the net surface charge on an oscillating bubble. By performing numerical simulations of the complete bubble formation cycle, Sunder and Tomar [22] investigated the phenomenon of bubble volume reduction under non-uniform electric field caused by a needle. They showed that non-uniform electric stresses push the bubble interface into the needle leading to a premature neck formation and bubble detachment thus resulting in smaller bubbles. Recently, Sharma et al. [23,24] studied the effects of a constant and an alternating electric field on the formation of oil droplets inside water at the exit of a ‘T’ shaped microchannel. They observed that a plug flow of the oil phase can be transformed into a flow with smaller oil droplets by the application of a constant electric field. For an alternating electric field, they observed that weakly conducting fluids are more responsive to a change in the AC frequency compared to the purely dielectric fluids and smaller droplets are formed at lower AC frequencies. In the present study, we investigate the role of alternating electric field on the bubble formation process from submerged orifices and needles. We perform numerical simulations using the Coupled Level-set and Volume-of-fluid (CLSVOF) algorithm along with an electrohydrodynamic formulation to study the effects of an alternating electric field on bubble dynamics. We consider both the gas and the liquid as perfect dielectric materials and the electrode configuration is chosen so as to obtain a non-uniform electric field (see [22]). We analyze the reduction in bubble volume in comparison to a DC electric field. We observe that for a critical frequency the volume of the detached bubbles is observed to be minimum, whereas at other frequencies an increase in the detached bubble sizes is observed. For the bubble oscillation modes, we show that the second Legendre shape mode is the most dominant oscillation mode followed by a decreasing contribution from the higher order even modes. Although the Legendre modes vary with time, the amplitude of second mode always remains positive while that of the fourth mode is always negative implying that the bubbles oscillate around an elongated, ellipsoidal mean shape. The paper is organized as follows. In Section 2, we formulate

Fig. 1. Computational domain for the case of bubble formation from an orifice, R = domain radius, H = domain height, ri = orifice radius, ρ = density, µ = viscosity, ϵ = relative permittivity, ψ = electric potential, ω = angular frequency of electric field oscillation, θ = contact angle with respect to bottom plate, g⃗ is the gravity vector, subscripts l and g denote respectively the liquid and gas phases.

the problem and present the equations governing the fluid flow and electrohydrodynamics. Section 2.4 presents the details of the numerical algorithm. Results and important conclusions are discussed in Sections 3 and 4, respectively. 2. Problem formulation 2.1. Computational domain We model bubble formation at orifices and needles using an axisymmetric formulation. Figs. 1 and 2 show the computational domain for the bubble formation from a submerged orifice and a needle, respectively. In both figures, R and H denote respectively the radius and height of an axisymmetric computational domain. For all the simulations presented in this study, we use R = 10 mm and H = 20 mm. A dielectric gas is injected through an orifice/needle, of radius ri = 1 mm and placed at the bottom, as shown in Figs. 1 and 2. In the case of gas injection through an orifice, the bottom conducting plate is given a sinusoidally varying electric potential while a zero potential is given to the top boundary. This generates an alternating but spatially uniform electric field inside the domain. In the case of bubble formation at a needle, we consider a conducting needle of internal radius ri = 1 mm, outer radius ro = 1.5 mm and height h = 6 mm with the center line of the needle coinciding with the axis of symmetry in the computational domain. A sinusoidally varying electric potential is applied to the needle and a zero potential is applied at the outer side boundary (r = R). This generates a spatially non-uniform alternating electric field. The above configurations can be realized in experiments by using a conducting wire-mesh frame which can be easily earthed or put at some reference potential. The position of the wire-frame, if it is put at a constant electric potential, would dictate the strength of the electric field in the domain. Therefore, one would have to accordingly choose the vertical location. Similarly, in the needle case, the needle is employed as an electrode with a sinusoidally varying potential and the side walls are imposed with a zero potential (see Fig. 2). In addition to the difference in the flow behavior around the growing bubble, the two configurations (Figs. 1 and 2) show different contact line dynamics. During the bubble growth at needles and orifices the contact line is expected to be pinned at the tip of the inner surface of the needle and orifice edge, respectively. Contact angle, θ , in the case of orifice is measured from the horizontal surface. Whereas, in the case of needles the contact angle should be measured from the horizontal surface or vertical inner surface depending upon the tendency of the motion

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ρQ 2

the average inlet velocity, (iii) Weber number, We = γ d3 where Q =

Vavg π d2 4

ρ

is the inlet flow rate, (iv) density ratio ρ l , (v) viscosity g ϵ ϵ E2d

µ

and (viii) perratio µ l (vii) electric Bond number, Boe = 0 γl g mittivity ratio = ϵl /ϵg . Here, subscripts l and g denote the properties of the liquid and gas, respectively. In addition to the above non-dimensional parameters bubble formation process is also a function of the contact line dynamics. We model the contact line dynamics using Gibb’s criterion with an equilibrium contact angle given by θ . Under an oscillating electric field, the frequency, ω, of the applied electric field also greatly influences the interface dynamics and thus the bubble formation process. Fig. 2. Computational domain for the case of bubble formation from a needle, R = domain radius, H = domain height, ro = needle outer radius, ri = needle inner radius, h = needle height, ρ = density, µ = viscosity, ϵ = relative permittivity, ψ = electric potential, ω = angular frequency of electric field oscillation, θ = contact angle with respect to needle inner surface, g⃗ is the gravity vector, subscripts l and g denote respectively the liquid and gas phases.

of the contact line (see Fig. 2). In our simulations, we model flow through the needle as well to capture the dynamics of the contact line accurately. In [22], we showed that the contact line can slip into the needle due to high electric stresses near the needle tip thus modifying interface profile and resulting in smaller bubbles. In the present study, we impose an oscillating electric potential at the electrodes which results in oscillating electric forces on the fluid interface that may enhance or reduce the effect of electric field on the bubble detachment process.

2.3. Electric field formulation We model the oscillating electric field under the quasi-static assumption. The electric field equations under this assumption are solved with time varying boundary conditions without considering any electromagnetic coupling. For the validity of electrostatic assumption, the maximum strength of magnetic field inside the computational domain due to the total current density must be such that cB∗ ≪ E ∗ , where B∗ and E ∗ are the characteristic scales of the magnetic and electric field respectively, and c is the speed of light. Only displacement current exists inside perfect dielectric materials under oscillating electric fields and therefore the scale of magnetic field strength is given by B∗ ∼ (l/c 2 )ωE ∗ , where l is the characteristic length scale in the domain and ω is the frequency of the electric field (see [25]). Thus we have, cB∗

2.2. Two-phase flow formulation

E∗

The fluid motion is governed by an axisymmetric, incompressible, one-fluid formulation of the Navier–Stokes equation and the continuity equation, which are given by Eqs. (1) and (2) respectively.

ρ (F )



    ∂v + v · ∇ v = −∇ p + ∇ · µ(F ) ∇ v + ∇ vT ∂t + ρ (F ) g + fγv + fEv ,

∇ · v = 0.

(1) (2)

Here v and p are the velocity and pressure, respectively. The density and viscosity are a function of the liquid fraction field, F , and are given by ρ(F ) and µ(F ), respectively. Volumetric representation γ of surface tension and electric field forces are denoted by fv γ E and fv , respectively. The surface tension force is given by fv = γ κ nδs , where γ is the surface tension coefficient, κ is the interface curvature, n is the interface normal and δs is the surface Dirac delta function. The electric forces are given by, fEv = −

ϵ0 2

E 2 ∇ϵ,

(3)

where ϵ is the relative permittivity of the fluid, E is the magnitude of the electric field and ϵ0 = 8.85 × 10−12 C/(V-m) is the permittivity of vacuum. Interface between the two phases is governed by the advection equation for the void fraction field, F ,

∂F + ∇ · (vF ) = 0. ∂t

(4)

The key non-dimensional parameters governing the bubble formation processes at orifices and needles under the electric field are: (i) Bond number, Bo =

ρ gd2 , γ

where d is the inner diame-

ter of the needle, (ii) Capillary number, Ca =

µVavg , γ

where Vavg is

∼

lω c

.

(5)

Therefore, for the validity of the electrostatic assumption, the frequency of the applied alternating electric field must be such that lω/c ≪ 1. Also, the dielectric constant of most materials depends on the frequency of applied AC electric field [26]. In the present work, the maximum values of electric field frequencies used are less than 250 Hz. Thus, the permittivity of dielectric fluids of practical interest considered here can be assumed to be constant and the assumption of electrostatics is valid. Further, the loss component in the case of purely dielectric (insulating) materials is zero and therefore no electric energy is dissipated. Such materials are often termed as lossless materials (see [26]). Due to the application of an alternating potential at the needle, the potential inside the whole domain can be written as ψ(r , z , t ) = Re[Ψ (r , z )eiωt ], where Ψ√ = ψR + iψI denotes the complex potential phasor, where i = −1 and Re[·] denotes the real part of the complex quantity. The electric field is then given by E(r , z , t ) = Re[E(r , z )eiωt ] where E = −∇ Ψ = −(∇ψR + ∇ψI ) is the electric field phasor. If the phase is constant throughout the domain (phase may be different at different locations in the domain for example in the case of traveling wave electroosmosis in [27]), the electric field phasor can be taken as real, i.e., as E = −∇ψR and the electric field becomes E = −∇ψR (r , z )Cos(ωt ). Using Gauss’ law, we get the governing equation for electric potential inside the domain in the absence of any free charges,

∇ · (ϵ∇ψ(r , z )) = 0,

(6)

where ψ ≡ ψR . The electric field alternates due to the oscillatory boundary condition for the electric potential. 2.4. Numerical implementation The governing equations, Eqs. (1) and (2), are discretized on a staggered grid using second order accurate finite difference schemes. Various scalars are located at cell centers and the vector

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components (such as velocity and electric field and surface tension forces) are defined at the corresponding cell faces. The convective fluxes in the momentum equation, Eq. (1), are calculated using second order accurate Essentially Non-Oscillatory (ENO) scheme by Shu and Osher [28]. The viscous terms are discretized using second order accurate central differences. The semi discretized equations are advanced in time using a first order explicit Euler method to give non-solenoidal auxiliary velocity field (u∗ ). Divergence correction is done using a pressure field based on a projection step which is similar to that given in [29]. The surface tension forces are modeled as volumetric body forces using the continuum surface force model of Brackbill et al. [30]. Eq. (3) when directly used to calculate electric forces at the interface yields inaccurate results because of the jump in electric field across the interface. Tomar et al. [31] modified this equation and proposed an alternative expression, fEv =

ϵ0 2



(D · n)2 ∇ ϵ0 2

  1

ϵ

(7)

∂φ + ∇ · (vφ) = 0. ∂t

(8)

The void fraction field, F , is advected geometrically using an operator split Volume-of-Fluid advection algorithm given by Pilliod and Puckett [36]. The level-set values in Eq. (8) are advected using a second order accurate ENO scheme by Shu and Osher [28]. So at each new time step, interface normal, n, is given by freshly advected values of the level-set function and the distance l of the interface from cell center is calculated from the advected void fraction, F , values. The level-sets are subsequently reinitialized to exact signed distance function from the interface at the end of each time step. The Volume-of-Fluid advection method is made second order accurate by alternating the sweep direction for the interface reconstruction every time step (see [36,33]). Physical properties, ρ(F ), µ(F ) and ϵ(F ) are computed using a smoothed Heaviside function given by,

H (φ) =

 2   0,

φ > δ; +

φ 2δ

+

1 2π

 sin



πφ δ



,

|φ| ≤ δ;

(9)

φ < −δ;

where the transition region thickness, δ = 1.51r, and 1r = 1z is the grid size in the radial and axial directions, respectively. The physical properties of the fluids, in the thin transition region of (2δ ), can be obtained as shown below:

ρ (φ) = ρl H (φ) + ρg [1 − H (φ)] µ (φ) = µl H (φ) + µg [1 − H (φ)] [1 − H (φ)] 1 H (φ) = + . ϵ (φ) ϵl ϵg

Domain boundary

(10) (11) (12)

Conditions ∂p ∂r ∂F ∂z ∂p ∂r ∂p ∂z

Axis Top Outer Bottom

∂p ∂z

Inlet

= = = = =

∂F ∂r ∂φ ∂z ∂F ∂r ∂F ∂z ∂F ∂z

= = =

∂φ ∂r ∂u ∂z ∂φ ∂r

= 0, =

∂φ ∂z

= = = ∂φ ∂z

∂v ∂r ∂v ∂z ∂v ∂r

=

∂ψ ∂r

= 0, u = 0

= 0, ψ = 0, p = 0 =

∂ψ ∂r

= 0, u = 0

= − cos(θ), ψ = ψ0 cos(2πωt ), u = v = 0   2  = 0, ψ = 0, u = 0, v (r ) = 2Vavg 1 − rri

Table 2 Boundary conditions for the needle case. Domain boundary

Conditions ∂p ∂r ∂F ∂z ∂p ∂r ∂p ∂z ∂p ∂r

Axis

 − (E · t)2 ∇ϵ ,

which gives more accurate values of electric forces at the interface in spite of the jump in physical properties. Here, D is the electric displacement vector, and n and t are the unit normal and tangent vectors, respectively. More details of the electrohydrodynamic formulation and validation are given in [31]. The interface between the injected gas phase and the ambient liquid phase is captured using the Coupled Level-set and Volumeof-fluid (CLSVOF ) method of Sussman and Puckett [32]. This algorithm conserves mass using Volume-of-Fluid method (VOF) and gives accurate values of interface normal and curvature based on the level-set function, the accuracy tests and previous applications are given in [33–35]. The interface is advected by solving the advection equations for the void fraction, F , using Eq. (4) and the level-set function, φ , using the following equation,

 1,   1

Table 1 Boundary conditions for the orifice case.

Top Outer Bottom Needle outer Needle Top Needle inner

∂p ∂z ∂p ∂r ∂p ∂z

Inlet

= = = = = = =

=

∂F ∂r ∂φ ∂z ∂F ∂r ∂F ∂z ∂F ∂r ∂F ∂z ∂F ∂r ∂F ∂z

= = = = =

∂φ ∂r ∂u ∂z ∂φ ∂r ∂φ ∂z ∂φ ∂r

= 0, = 0, =

∂φ ∂z

= = = =

∂v ∂r ∂v ∂z ∂v ∂r ∂ψ ∂z

= =

∂ψ ∂r ∂ψ ∂z

= 0, u = 0 = 0, p = 0

= 0, ψ = 0, u = 0 = 0, u = v = 0

= 0, ψ = ψ0 cos(2πωt ), u = v = 0 ∂φ ∂z ∂φ ∂r

= sin θ, ψ = ψ0 cos(2πωt ), u = v = 0 = cos θ, ψ = ψ0 cos(2πωt ), u = v = 0

  2  = 0, ψ = 0, u = 0, v (r ) = 2Vavg 1 − rri

For density and viscosity a weighted arithmetic mean (WAM) has been used, whereas, for electric permittivity a weighted harmonic mean (WHM) gives more accurate results (see [31]). Using the smoothed Heaviside function, the surface tension and electric forces are also distributed in a thin transition region around the true interface. It results in a smoother solution and better convergence properties of the pressure Poisson equation and the Laplace equation for electric potential. The surface tension body forces can be written as, fγv = γ κ (φ) ∇ H (φ)

(13)

where γ is the surface tension coefficient and κ (φ) = −∇ · |∇φ| is the interface curvature. The electric forces are given by, ∇φ

fEv = −

 ϵ0  (ϵ E · n)2 /(ϵ1 ϵ2 ) + (E · t)2 (ϵ1 − ϵ2 )∇ Hδ 2

(14)

where n and t are the unit normal and tangent vectors, respectively. The resulting pressure Poisson equation ∇ · (∇ p/ρ(F )) = ∇ · u∗ /1t, where 1t is the time step, and the Laplace equation governing the electric potential (Eq. (6)) is solved using the Linear–Algebraic system interface and the BommerAMG solver from HYPRE (see [37]). The boundary conditions for Eqs. (1), (4), (6) and (8) are shown in Tables 1 and 2 for the orifice and needle case, respectively. The contact line is initially pinned at the edge of the orifice/needle. In the case of any slippage of the interface along the surface, determined based on the Gibbs criterion, a constant equilibrium contact angle boundary condition is imposed for the interface profile through the level set function. The details of the contact line modeling are given in [22]. The inner radius of the orifice/needle is equal to 1 mm and is discretized with 16 grid points. This yields a grid cell size of 6.25 × 10−5 m. As discussed earlier, the domain size is given by R = 10 mm and H = 20 mm. With these combinations the bubble dynamics is grid and domain size independent. A variable time stepping procedure is used which automatically reduces or

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increases the time interval as required by the solution procedure. An effective CFL criterion, considering the effect of various forces on fluid acceleration, is used to set the value of the time step, 1t (see [33]). Starting values of the time steps are of the order of 10−6 s and these can go up to 10−9 s during the critical stages of bubble dynamics like detachment which are of the order of a milli second for bubble formation under uniform electric field and still lesser for the case of non-uniform electric field. Also, low values of time steps ensure that at least over a thousand time steps resolve a single oscillation of the electric field cycle. Thus, the effects of oscillating electric field on the interface dynamics are captured accurately. 3. Results and discussions Simulations have been performed for the following values of the dimensionless parameters defined in Section 2.2: We = 0.01, Bo = 0.1, Ca = 0.01 and BoE = 5 for the needle case and BoE = 8 for the orifice case unless otherwise mentioned. The ratio of permittivity is chosen to be ϵl /ϵg = 2. The liquid to gas density and viscosity ratios have been chosen as 100 and 10, respectively. The low density ratios have been chosen due to the numerical convergence issues with the multigrid based solver. We note here that at these low capillary and Weber numbers, the bubble formation process is essentially governed by the competition between the buoyancy flux, surface tension and electric field forces (that is Bond number, Bo and electric Bond number, BoE ). Therefore, the choice of low density ratio for numerical convergence has little effect on the physics discussed in this study. To study the effect of AC electric field on bubble dynamics we perform simulations for various frequencies ranging from 8 to 250 Hz. Simulations have been performed for bubble injection from an orifice and at needles with a geometry encountered in bubble column reactors. In what follows we study the effect of alternating electric field on the process of bubble formation from submerged orifices and needles. 3.1. Orifice case For the orifice case, we force the bubble interface to remain pinned at the orifice rim. As shown in our previous work, Sunder and Tomar [22], electric forces act uniformly throughout the bubble interface for the orifice case (for the configuration studied here) and for a constant phase alternating electric field the magnitude of the forces also oscillates in-phase. Eq. (3) indicates that the electric forces vary as square of the magnitude of electric field thus showing an oscillation frequency that is twice the frequency of the applied electric potential. A few bubble shapes just before the detachment are shown in Fig. 3 for the orifice case with ω = 16 Hz. Bubbles show interesting behavior with the interface of the growing bubble responding to the applied oscillating electric forces. Bubble profiles labeled as 1–30 in Fig. 3 have been shown at a time interval of 5 ms. Profile-1 shows an elongated bubble at an instance where the electric field strength is decreasing in time and subsequent bubble shapes in profiles 2–4 become more spherical. Profile-4 suggests neck initiation but it does not lead up to detachment as the bubble interface is pulled up again into a prolate shape due to the increase in the electric field strength. Such oscillations keep on occurring periodically (profile 7 is similar to profile 1) till the bubble grows big enough and the bubble necks leading to detachment. It may be noted that under a uniform electric field, the bubble shape varies from a prolate ellipsoid (when the electric forces are high) to a shape with a neck and an almost spherical top (in the absence of the electric forces). Profiles 20–25 also suggest that as the bubble grows in size (due to continuous volume flux from the orifice), it spends longer time in the necking regime with increasingly thinner necks and eventually, as shown by profiles 26–30 in Fig. 3, undergoes pinch off.

Fig. 3. Various bubble shapes just before the bubble detachment for the orifice case, BoE = 8, ω = 16 Hz. The time interval between two successive profiles is 5 ms. It may be noted that the bubble do not detach in spite of the neck initiation shown in profiles 17, 23, and 24.

Fig. 4 shows the various forces acting on the oscillating bubble interface corresponding to the case shown in Fig. 3. The electric, capillary, outer pressure forces act together to increase the gas pressure inside the bubble. It may be noted that the net force on the bubble becomes non-zero only just before the bubble detachment. This explains the non-detachment of the bubble in-spite of neck formation in the seemingly close-to-detachment profiles 17, 23, and 24 shown in Fig. 3. This suggests that the non-detachment, even when close to breakup profiles are attained, is possibly due to the insufficient growth rates of the neck instability of quasistatic shapes of smaller bubbles in comparison to the oscillation time period of the electric field. A DC electric field has been shown to have little effect on the volume of the detached bubbles from orifices (see [22,38]) although it alters the bubble dynamics substantially. However, an alternating electric field shows some variation for low oscillating frequencies. Since, for growing bubbles the natural frequency varies (decreases) with increase in their size, here we employ the time scale of the capillary wave motion (with needle diameter as the length scale) on the bubble–liquid interface, τc , for nondimensionalization of the applied electric field frequency,

 τc =

ρ d3 γ

(15)

where d is the inner surface diameter of the needle/orifice. For a 2 mm diameter orifice/needle used in this study, τc = 0.009 s.

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Fig. 4. Various forces acting on the oscillating bubble at an orifice, BoE = 8, ω = 16 Hz. The net force on the bubble becomes positive just before the bubble detachment. This may be the reason of bubble non-detachment in-spite of the neck formation shown in Fig. 3.

Fig. 5. Variation of the average bubble volume with the normalized electric field oscillation frequency for the orifice case. The bubble volumes are normalized using the bubble volume under a constant uniform electric field (4.98 × 10−8 m3 ). The AC frequencies are normalized based on the capillary time scale, τc .

Fig. 5 shows the average bubble size variation with the frequency of the electric field. With an increase in the AC frequency, the bubble volume initially decreases, reaches a minimum, and then increases to the uniform DC electric field value. This clearly indicates the possibility of resonance phenomenon between the capillary waves traveling on the growing bubble and the applied electric field frequency that leads to an early neck formation and subsequent detachment. Fig. 6 shows the bubble aspect ratio variation with time and the various electric field oscillation frequencies. The aspect ratio curve for the DC case is monotonically increasing while the curves for the AC cases show periodic variation, about a monotonically increasing aspect ratio, with time. Note that the complete growth period of the bubble is a couple of orders higher than the capillary time scale which on the other hand is comparable to the time period of the oscillating electric field. We observe that the oscillation amplitude is higher in the range 8–32 Hz and decreases substantially beyond it. It is interesting to note that near the detachment point, the aspect ratio is substantially higher in the DC case compared to the oscillating electric field case with the exception of 16 and 24 Hz cases for which the amplitude is comparable. Aspect ratio variation shows that for ω = 16 and 24 Hz, the bubble oscillation amplitude increases with the aspect ratio shooting above the DC curve thus resulting in early detachments and therefore small bubbles. Another interesting feature of the dynamics is the phenomenon of beats in the bubble oscillations indicated by the aspect ratio variation as shown in Fig. 6. As expected, the number of beats substantially increase with an increase in ω suggesting a phase lag between the electric field and the fluid response to it. Fig. 7 shows the Fourier transform of the bubble aspect ratio versus electric field oscillation frequency data (obtained after removing the aliasing error). The electric forces oscillate at twice

the frequency of the applied electric field. This is evident from Fig. 7 where the most dominant bubble oscillation frequency comes out to be equal to twice the applied AC frequency. One more observation to be noted is that at lower electric field oscillation frequencies, some minor contribution to the bubble oscillations comes from the other frequencies also. For example, at 8 Hz electric field frequency, the bubble oscillations have decreasing contributions from 16, 32, 48 Hz. These contributions from higher harmonics decrease as the applied electric field oscillation frequency increases. 3.2. Bubble formation at needles In an earlier work [22] we showed that during bubble formation at needles, the electric forces due to non-uniform electric field pull the bubble interface inside the needle. We also noted that the detached bubble volume does not vary much (6%) with the equilibrium contact angle by performing simulations in the range of 10–170°. In the present study, all the simulations have been performed using a critical contact angle of 90° with respect to needle inner surface unless mentioned otherwise. So the interface remains normal to the needle inner surface and oscillates as the bubbles form and detach. The pinning and de-pinning of the interface is modeled by employing appropriate boundary conditions for the level-set function, used for computing interface normal, as mentioned in Tables 1 and 2. The details of the numerical implementation of the Gibb’s criterion employed to model the contact line dynamics are given in [22]. The bubble profiles for the needle case are greatly influenced by the electric field oscillation frequency. At lower frequencies, the profiles are similar to the DC case except that the interface relaxes to an almost hemispherical shape when the electric forces become

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Fig. 6. Bubble aspect ratio variation with non-dimensionalized time and AC frequency for the orifice case, BoE = 8. The time has been normalized using the capillary time scale, τc = 0.009 s.

Fig. 7. Fourier transform of the aspect ratio data shown in Fig. 6.

zero during the oscillation cycle as shown in Fig. 8(a)–(h) for an AC frequency equal to 8 Hz. After the detachment of a smaller bubble (see Fig. 8(a)), the next bubble starts growing taking a conical (Fig. 8(b)) and subsequently a thin cylindrical (Fig. 8(c)) shape. This period of growth coincides with a decreasing electric field cycle thus leading to relaxation of the interface due to the capillary forces as shown in Fig. 8(d)–(f) with the electric forces equal to zero corresponding to the bubble profile shown in Fig. 8(f). The contact line during this cycle keeps rising and re-pins at the nozzle corner when the applied instantaneous oscillating electric field is zero. During these events, the bubble keeps growing in size due to the influx of gas from the needle. The subsequent growth of the bubble coincides with an increasing electric field strength and

the interface dips into the needle again ultimately leading to the neck formation (see Fig. 8(g), (h)) and consequential detachment. Due to asynchronicity between the oscillations in the electric field forces and the interface motion, we observe formation of bubbles of different sizes. For instance, if electric field would have continued increasing or was maintained constant during bubble growth shown by Fig. 8(c), the bubble would have experienced an early detachment. Fig. 9 shows the variation of bubble volume with time for an alternating electric field of frequency ω = 8 Hz and BoE = 5. The first bubble is large due to the effects of initial conditions. After the initial transients, a group of eight bubbles is repeated after every 0.176 s. The bubble volumes are normalized by the bubble volume (V0 = 4.2 × 10−10 m3 ) due to a DC non-uniform electric field with BoE = 5. Also shown at the bottom of the figure is the normalized electric force variation with time at any location on the interface. Note that although the magnitude of the electric forces at different points on the interface are different because of the nonuniform electric field, they all vary with the same phase. The value of the electric force used for normalization thus corresponds to the maximum at the respective points. Also indicated in the figure are the necking and detachment points denoted by the squares and circles respectively. The time difference between a circle and an immediate next square is the growth time for a bubble. Similarly, the time difference between a square and a next circle is the time taken by the bubble from necking till pinch-off. Since the gas flow rate is kept constant, the biggest bubbles correspond to largest growth time. The various ellipses mark the regions of electric force variation during which the sequence of bubble profile relaxation

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Fig. 8. Bubble profile relaxation when the electric forces become zero during the electric field oscillation cycle, AC frequency = 8 Hz. The various non-dimensional times t ∗ = t /τc (normalized using the capillary time scale, τc = 0.009 s) are: (a) 21.1, (b) 22.0, (c) 23.0, (d) 23.36, (e) 23.47, (f) 24.7, (g) 25.7, (h) 26.0.

Fig. 9. Normalized bubble volume versus the time period of the applied AC waveform, ω = 8 Hz, BoE = 5. The volume of the bubbles is normalized by the detached bubble volume under a constant non-uniform electric field (4.2 × 10−10 m3 ). The time has been normalized using the capillary time scale, τc = 0.009 s. It may be noted that a group of eight bubbles (shown inside the dashed quadrilateral) repeats itself.

shown in Fig. 8 takes place. After relaxation of the bubble profile, the biggest bubbles shown in Fig. 9 are produced. In between any two ellipses, there lie two instances of zero electric forces during the oscillation cycle. The bubble profile relaxes instantaneously to a hemispherical shape with a decrease in the electric field force as the bubble continues to grow due the constant influx of the gas. As is expected, no bubble forms at the instances when electric forces are near the trough and the next bubble detaches with a bigger size due to the more time available for growth when the electric forces increase after crossing the minimum. The detachment of a larger bubble is followed by the detachment of a smaller bubble corresponding to the peak in electric forces subsequent to which another bubble of similar size detaches when the electric forces are decreasing. It is interesting to note that merely applying an AC electric field does not result in more effective bubble size reduction, and the bubble formed can also be larger in size due to the asynchronicity in the bubble growth and the electric field force. At higher frequencies, the continuous low amplitude agitation of the bubble interface by the electric forces leads to capillary waves on the interface which travel from the needle towards the bubble top and are reflected back. The bubble profiles during the formation and detachment of a single bubble at various frequencies are shown in Figs. 10–12. At the bottom left in these figures is the bubble shape at the detachment of the previous bubble. Then

starting from top-left and coming down are the bubble shapes as another bubble starts growing in time. We tag some of the profiles with the corresponding times normalized by the capillary time scale. We do not plot the previously detached bubbles and the corresponding satellite bubbles formed during their detachment in these profiles. Also, the reference point (bubble contact line) has been put at the same datum irrespective of the exact location of the contact line in the needle. We discuss the variation in contact line displacement later. We observe that as the electric field oscillation frequency increases, the bubble oscillates more vigorously as is evident by the number of troughs and crests in the bubble profile series which increase with the AC frequency. Fig. 10 shows bubble formation cycle for ω = 16 Hz. The conical bubble profile at a nondimensional time t ∗ = 3.6 takes a more spherical shape at t ∗ = 5.1 which coincides with the low magnitude of electric forces and finally detaches around t ∗ = 6.5. In the absence of electric field, the bubble would have continued growing in this hemispherical shape for a longer duration. However, in the presence of nonuniform electric field imposed here, bubbles detach with much smaller volume due to high electric stresses. During the AC electric field cycle, when the electric field magnitude and thus electric forces on the bubble interface decrease to low values, especially for low frequencies, the bubble relaxes back to zero electric field case due to capillary forces.

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Fig. 11. Bubble profiles at the needle for the electric field oscillation frequency of 42 Hz (with 2ωτc = 0.75; where 2ω is the oscillation frequency of the electric forces at the bubble interface). The previously detached bubble is not shown except in the profile at the bottom left and at the top right. Two bubbles of slightly different size are produced alternately. The numerical values shown denote the non-dimensional time t ∗ (normalized using the capillary time scale, τc ).

Fig. 10. Bubble profiles at the needle for the electric field oscillation frequency of 16 Hz (with 2ωτc = 0.29; where 2ω is the oscillation frequency of the electric forces at the bubble interface). The previously detached bubble is not shown except in the profile at the bottom left and at the top right. The numerical values shown denote the non-dimensional time t ∗ (normalized using the capillary time scale, τc ).

For higher frequencies, unlike in the low frequency case of 16 Hz, the bubble interface spends less time in the hemispherical mode growth. For example, for ω = 16 Hz (with 2ωτc = 0.29) the half cycle required to increase the electric field forces (varying with twice the frequency) from zero to maximum is 1/(4ω) ∼ 15.625 ms which is comparable to the time required for detachment in the DC case (∼25.65 ms), however, for ω = 42 Hz, it is ∼5.95 ms. The total time taken by a bubble to grow and detach at 16 Hz AC frequency is 31.35 ms. Fig. 11 shows the bubble morphology for ω = 42 Hz (i.e. 2ωτc = 0.75) and in comparison to ω = 16 Hz case, now bubbles of two different sizes detach alternately. The time duration required for bubble growth and detachment is 9.1 ms for the left set and 14.6 ms for the right set of profiles in Fig. 11 and thus smaller bubbles are produced compared to ω = 16 Hz case. For higher frequencies, the interface profiles oscillate more and more vigorously, albeit with smaller and smaller amplitudes, as shown in Fig. 12 for a frequency of ω = 238 Hz (with 2ωτc = 4.26). The crests in the profile series coincide with the time instants of high electric forces while the troughs correspond to the minima of the electric forces. The time taken by a bubble to grow and detach at ω = 238 Hz AC frequency is 23.3 ms. An average size of bubbles corresponding to different values of ω can be obtained by averaging the volume of various bubbles formed. Fig. 13 shows the average bubble size variation with the frequency of electric field for the needle case. For the AC electric field case, average bubble size is substantially larger than the DC case, except for a frequency range ∈ (40, 43) Hz where a >10% decrease in bubble volume is observed. As discussed earlier, this suggests a resonance behavior between the capillary and the electric field forces. 3.3. Mode analysis We now perform the analysis of bubble oscillation modes to understand the effects of an oscillating electric field on the different modes and the role of different modes of bubble oscillation during detachment. Bubble profiles are subjected to

Fig. 12. Bubble profiles at the needle for the AC frequency of 238 Hz (with 2ωτc = 4.26; where 2ω is the oscillation frequency of the electric forces at the bubble interface). The numerical values shown denote the non-dimensional time t ∗ (normalized using the capillary time scale, τc ).

Fig. 13. Variation of the average bubble volume (normalized using the bubble volume under a DC electric field, 4.2 × 10−10 m3 ) with non-dimensional AC frequency (normalized using the capillary time scale τc ) for the needle case. Note that the minimum volume is obtained for ω = 42 Hz with 2ωτc = 0.75 where 2ω is the oscillation frequency of the electric force variation at the bubble interface.

mode decomposition using Legendre polynomials. Fig. 14(a), (b) and (c) show the shapes of a spherical bubble perturbed with the second, fourth and sixth Legendre modes, respectively. Only the top half of the profiles shown here represent the bubble shapes obtained during bubble formation with the centerline (dashed-dot line) representing the axis of symmetry. It can be observed that a positive mode 2 gives the bubble a prolate ellipsoidal shape while mode 2 with a negative amplitude makes it oblate. Similarly, a

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(a) n = 2.

(b) n = 4.

(c) n = 6.

Fig. 14. Perturbations of a sphere of unit radius using r = R ∗ (1 + 0.3Pn (cos θ)) for three different even Legendre modes.

drops. Here, as shown in Fig. 15(b), the cylindrical radial coordinate of any point on the bubble interface as a function of the axial coordinate, z, is used to define the interface. In terms of the Legendre polynomials, we can write, r (z ) = 2

R20

imax 

 ai P i

z



z0

i=0

,

(17)

where z0 is the maximum bubble height from the orifice or needle rim. For axisymmetric and mirror-symmetric drops, the odd coefficients, ai , are zero. Therefore, N 

ζ (χ ) ≡ r 2 (z )/R20 = Fig. 15. Bubble profile reconstruction using (a) the Legendre polynomials, Trinh et al. [14], (b) the method of Trentalange et al. [39].

positive mode 4 gives the bubble a shape with four lobes which are oriented along the axis while a negative mode 4 reverses the configuration yielding profiles similar to the ones shown in Fig. 12 for t /τc = 3.9. Higher order modes increase the number of lobes in the bubble shape as can be noticed in Fig. 14(c) for the sixth mode. The electric forces under a uniform electric field deform the bubble into a prolate ellipsoid while the capillary forces provide the restoring force leading to the bubble shape oscillations. The inertial forces cause the overshoot of the bubble profile from the equilibrium position while the viscous forces provide the damping. From the individual mode shapes, it can be expected that the electric forces excite the Legendre mode 2 while the interaction between the capillary forces, oscillating electric forces and the inertial forces excite the 4th and the higher order Legendre modes. In what follows, we discuss the method adopted for obtaining the coefficients corresponding to each mode and subsequently the various modes of oscillations observed during the bubble growth. The shape of an axisymmetric bubble can be described using Legendre polynomials with the help of the following equation, Trinh et al. [13],

 r (θ, t ) = R0

1+

imax 

 ci (t )Pi (cos θ ) .

(16)

i =2

Here, r (θ , t ) is the reconstructed bubble radius which is a function of the polar angle, θ , Pi (cos θ ) is the Legendre polynomial of degree i, ci (t ) are the coefficients, imax is the maximum number of modes considered for reconstruction, and R0 is the equivalent spherical radius of the bubble attached at the orifice/needle. As shown in Fig. 15(a), the distance of any point on the bubble interface with respect to the bubble center of gravity is a function of the polar angle, θ , measured from the bubble axis. Using the bubble profile from simulations and Eq. (16), we can compute the coefficients, ci . Modes of the bubble shapes can also be obtained using the formulation given in [39] for axisymmetric and mirror-symmetric

a(2·n) P(2·n) (χ )

(18)

n=0

where χ = z /z0 (hence varies from −1 to 1) and 2 N gives the maximum order of Legendre Polynomial. The condition that ζ = 0 at χ = ±1 gives a0 =

N 

a(2·n) .

(19)

n =1

Thus, Eq. (18) can be written as,

ζ (χ ) =

N 

a(2·n) P(2·n) (χ ) − 1 .





(20)

n =1

Using a bubble profile from simulation results and the equations above, we can calculate the coefficients a(2·n) . As explained in [39], the coefficients ci in Eq. (16) and the coefficients ai in Eq. (17) are related to each other as, c(2·n) = c2 = 1 +

a(2·n)

3 2

2

;

a2 +

n ≥ 2,



a(2·n) .

(21)

n≥2

From these relations, the coefficients (ci ) can be calculated for bubble profiles at various times and plotted as a function of time to describe the dominant modes during bubble formation. Fig. 15(a), (b) also show the reconstructed bubble profiles using Legendre polynomials following the method of Trinh et al. [14] and the method of Trentalange et al. [39]. We note that the direct use of Eq. (16) [13] does not describe properly the shape of a bubble attached to an orifice or a needle rim, particularly after necking, when the radius became a multi-valued function of angle, θ . The method of Trentalange et al. [39] is more accurate in describing most of the bubble shapes except at bubble tip where the bubble radius becomes a multivalued function of height. Therefore, for all the results related to Legendre modes variation in time, we have used the method of Trentalange et al. [39]. Figs. 16–19 show the bubble oscillation modes for the various orifice and needle cases. The bubble oscillation modes for the

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Fig. 16. The variation of the coefficients of the Legendre oscillation modes (C2 –C12 ) with the non-dimensional time for the orifice case, BoE = 8, ω = 8 Hz. The time has been normalized using the capillary time scale, τc .

Fig. 17. Bubble oscillation modes (C2 and C4 ) for the orifice case under AC electric field lie between the respective modes for the DC electric field and the modes for the zero electric field case, ω = 8 Hz for the AC case. The time has been normalized using the capillary time scale, τc .

orifice case are shown in Figs. 16 and 17. Fig. 16 shows oscillation modes for BoE = 8 and ω = 8 Hz. The bubble shapes at various time instants are also shown in the figure. The most dominant mode, as is clear from the coefficients, is mode-2 and the amplitude corresponding to it increases significantly as bubble grows towards detachment. Another significant mode is mode-4. Other modes have much smaller coefficients and thus have smaller contribution to the bubble shape and dynamics. This is in accordance with the experimental observations of Bellini et al. [15], Sommers and Foster [20] who observed the dominance of second and fourth Legendre modes in the electric field excited oscillations of an isolated bubble. Higher order ‘even’ modes are excited during bubble growth even under a uniform electric field where as a non-uniform electric field was required to excite these in the experimental study of Sommers and Foster [20] for an isolated, spherical bubble. We note that although the magnitude of the individual Legendre modes oscillates, the amplitude of the second mode always remains positive while that of the fourth mode is always negative. The positive second mode gives the bubble an elongated ellipsoidal shape while growth of the negative fourth mode indicates the formation of shapes similar to profile 9 in Fig. 3 marking the beginning of the neck formation. This implies that the bubbles oscillate between an elongated ellipsoidal and a nearnecking shape. Fig. 17 shows that the oscillation modes under an alternating electric field at an orifice lie between the respective modes for the DC case and the zero electric field case. This implies that the bubble profile shapes under an AC electric field lie between the corresponding shapes under the no electric field case and the DC electric field case. For the needle case, we observe in Fig. 18 that a non-uniform electric field increases the contribution of mode 2 (compare the corresponding amplitudes at similar non-dimensional times in

Fig. 17 for the orifice case). Note that for the needle case even a small BoE (∼0.25) affects the bubble formation process, whereas, for the orifice case higher values of BoE (∼8) are required to cause any significant effect. The bubble size is also much smaller for the needle case as compared to that for the orifice. This is the consequence of non-uniformity of electric forces as shown in our previous work [22]. Fig. 18 shows that for the needle case the oscillation modes (at a much lower electric Bond number of BoE (∼0.25)) also lie between the respective modes for the DC case and the mode coefficients in the absence of an electric field. It is interesting to note that where mode-2 shows large amplitude oscillations, mode-4 does not show any significant oscillations in its magnitude suggesting that for the low frequency of ω = 8 Hz (2ωτc = 0.14) essentially only mode 2 is excited. This also implies that the bubbles oscillate around an elongated, cylindrical mean shapes during oscillations at the needle. Fig. 19 shows the bubble oscillation modes at a higher AC frequency of 42 Hz with electric Bond number of 0.25. It can be observed that mode 2, 4 and mode 8 oscillate together while mode 6 is out of phase to the above mentioned modes. The amplitude of oscillation of both mode 2 and 4 for ω = 42 Hz is much higher compared to the case with ω = 8 Hz. The effect of AC field can be summarized as follows. For a frequency of oscillation ω of the electric field, the electric field force on the bubble oscillates with a frequency of 2ω. Relatively smaller sized bubbles are formed at certain frequencies for both the orifice (2ωτc = 0.43) and the needle geometries (2ωτc = 0.75). For other frequencies bubbles even of larger bubble volumes compared to the corresponding DC field case can be obtained. For needle geometries a much smaller electric field (BoE = 0.25) is required to cause a substantial decrease in the bubble volume compared to the orifice case (BoE = 8) essentially due to the non-uniformity in the electric field. The early detachments, and therefore smaller bubble

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Fig. 18. Bubble oscillation modes for the needle case also lie between the respective modes for the DC electric field and the modes for the zero electric field case, BoE = 0.25 for the AC case, ω = 8 Hz. The time has been normalized using the capillary time scale, τc .

Fig. 19. Bubble oscillation modes for the needle case at BoE = 0.25, ω = 42 Hz. The time has been normalized using the capillary time scale, τc .

Fig. 20. Contact line motion inside the needle for the case of a constant non-uniform electric field and an oscillating non-uniform electric field with an oscillation frequency of 42 and 127 Hz, BoE = 5. The time has been normalized using the capillary time scale, τc . The horizontal line at 6 mm denotes the position of the needle rim.

volumes, are due to the increase in the amplitude of oscillating modes 2 and 4. The origin of higher order modes, as discussed in some previous studies (for example see [20]), is due to the nonuniformity in the electric field. 3.4. Bubble contact line motion inside the needle for the AC case Fig. 20 shows the interface contact line motion inside the needle for the case of a DC electric field and for oscillating electric field frequencies of ω = 42 and 127 Hz. We observe that an oscillatory electric field greatly enhances the contact line motion which is a direct consequence of the capillary waves movement on the interface. Interface dips into the needle at the peak of the electric force oscillation cycle (see [22]), while it comes out and remains pinned at the needle rim when the electric forces become zero.

The troughs in the figure thus correspond to higher values of electric forces whereas reduction in the contact line displacement corresponds to lower values of electric field. For a higher frequency (ω = 127 Hz or 2ωτc = 2.26), the interface motion occurs with the corresponding higher frequency. However, the maximum contact line displacement is not affected by the AC frequency and is similar to the DC case. The interface position in the DC case, due to the continuous action of electric stresses, do not vary much with time and is affected only due to interface deformations during various bubble detachments. 4. Conclusions In this study, we investigated the effects of AC electric field frequency on the bubble formation process at orifices and needles.

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For bubble formation at orifices, we showed that an oscillating electric field reduces the size of bubbles for a critical frequency of ω = 24 Hz, whereas, any further increase in the oscillation frequency results in bubbles similar in size to those formed under a DC electric field. For bubble formation at needles with severely non-uniform electric field (also see [22]), we observe that bubbles of reduced sizes (30% smaller) are formed at frequencies around 42 Hz, whereas, bubbles even larger than the DC electric field case are formed for both higher and lower frequencies thus suggesting a resonance like phenomenon. The numerically observed resonance frequency corresponds to 2ωτc = 0.75, where 2ω isthe frequency of oscillation of the electric field force and τc = ρ d3 /γ is the capillary time scale, indicating that the observed resonance phenomenon occurs at the time scale of capillary wave motion of the bubble interface. Similar to the DC case, the phenomenon of depinning of the interface is observed, however, the interface shows large oscillations corresponding to the variations in the oscillating electric forces. Another interesting physics, that emerges from the asynchronicity in the electric field and capillary force variations, is that a group of bubbles of different sizes repeat itself after regular intervals at lower AC frequencies. For bubble oscillation modes at an orifice, we observed that the higher order ‘even’ Legendre modes are also excited in contrast to the excitation of only the second Legendre mode for an isolated spherical bubble under a uniform electric field. We observed that the Legendre modes oscillate with time and the amplitude of second Legendre mode always remains positive while that of the fourth mode is always negative implying that the bubbles oscillate between an elongated ellipsoidal and smoothed cylindrical shapes. For the bubble oscillation modes at a needle, a non-uniform electric field increases the contribution of mode 2 which oscillates while the mode 4 remains unaffected and do not oscillate. This suggests that the bubbles oscillate around a smoothed, mean cylindrical shape at the needle. We also observed that the bubble oscillation modes under an AC electric field lie between the respective modes for the DC case and the zero electric field case. So the bubbles oscillate between the corresponding shapes under the no electric field case and the DC electric field case. This study reveals interesting dynamics of bubble formation in dielectric mediums modulated by oscillating electric field. We showed that using an oscillating electric field, different bubble sizes can be obtained thus providing a better control for applications, such as controlled reactions in bubble column reactors. Also, similar electrohydrodynamic phenomenon is expected to prevail in the breakup of a pendant dielectric droplet from a faucet and drops of desired sizes can be ensued by controlling the electric field strength and frequency. These results would also be useful in the bubble formation during nucleate boiling to shift the critical heat flux of the boiling curve by causing early detachment of bubbles. We believe the present numerical analysis would also motivate experiments to study the effect of oscillation frequency of the applied electric field on bubble formation at needles and orifices. References [1] S. Ogata, K. Shigehara, T. Yoshida, H. Shinohara, Small bubble formation by using strong nonuniform electric field, IEEE Trans. Ind. Appl. 6 (1980) 766–770. [2] M. Sato, Cloudy bubble formation in a strong nonuniform electric field, J. Electrost. 8 (2) (1980) 285–287. [3] H. Cho, I. Kang, Y. Kweon, M. Kim, Study of the behavior of a bubble attached to a wall in a uniform electric field, Int. J. Multiph. Flow 22 (5) (1996) 909–922. [4] H. Cho, I. Kang, Y. Kweon, M. Kim, Numerical study of the behavior of a bubble attached to a tip in a nonuniform electric field, Int. J. Multiph. Flow 24 (3) (1998) 479–498. [5] Y. Kweon, M. Kim, H. Cho, I. Kang, Study on the deformation and departure of a bubble attached to a wall in dc/ac electric fields, Int. J. Multiph. Flow 24 (1) (1998) 145–162. [6] P. Di Marco, W. Grassi, Motivation and results of a long-term research on pool boiling heat transfer in low gravity, Int. J. Therm. Sci. 41 (7) (2002) 567–585.

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