We present an experimental investigation of electrohydrodynamic (EHD) flows within a neutrally buoyant drop with a radius of 2.25 mm. Utilizing particle image velocimetry and high-speed shadowgraphy, we measure the internal circulation and reported velocity profiles in the bulk and at the interface of the drop. Two leaky dielectric liquids, silicone and castor oils, are employed as the drop and as the external phase, allowing the analysis of two shape configurations: oblate and prolate. The strength of the applied uniform electric field (from 0.125 to 1.75 kV/cm) enables the analysis covering both the small-deformation limit ( $ C a E \u226a 1$) and drops with larger deformations. Our measurements show good agreement with the leaky dielectric model (LDM) for the small-deformation cases. The flows begin at the interface as a result of jump in the electric stresses, leading then to four counter-rotating vortices inside the drop. At a permanent regime, the analytical solutions adequately predict the radial and tangential velocity components within the drop. However, a nuanced behavior is noticed for larger deformations, where the LDM theory underpredicts the internal circulation. Moreover, due to the increased deformation, a non-uniform azimuthal profile is observed for the velocity at the interface, $ v \theta $. Transient measurements of this velocity component enlighten the dynamic response of the EHD flows of the drop. Following the available analytical solutions, the dynamic response is governed by the timescale of the deformation of the drop, $ \tau def = \mu a / \gamma $. We propose a critical value of $ C a E \u2248 0.1$ below which the LDM adequately describes the velocity field in both quasi steady-state and transitory regimes.

## I. INTRODUCTION

Electrohydrodynamic (EHD) flows, driven by electric stresses shearing at fluid interfaces,^{1} have garnered significant attention of the scientific community over the years. Specifically, drops dispersed in a liquid medium under a constant electric field serve as a pivotal reference case in the field, owing to their relevance in the pharmaceutical industry,^{2} drug delivery,^{3} and in the energy sector,^{4} just to mention a few. Such applications often rely on breakup and coalescence phenomena of the drops,^{5} whose dynamics is governed by EHD flows. The external circulation may either promote or obstruct the drop–drop interaction, which may lead to coalescence,^{6} while the internal circulation might affect the breakup dynamics of drop.^{1}

*a*dispersed in another liquid with the same density

_{o}*ρ*but different viscosities ( $ \mu \u0302$ and

*μ*), electrical conductivities ( $ \sigma \u0302$ and

*σ*), and permittivities ( $ \u03f5 \u0302$ and

*ϵ*). The hat symbol denotes a quantity associated with the drop phase. When an electric field with strength

*E*is applied, the discontinuity of electrical properties at the interface generates a jump in electric stresses, i.e., $ ( T e \u2212 T e \u0302 )$.

_{o}^{1}This jump is balanced by hydrodynamic and capillary stresses, as described by

Here, *γ* is the interfacial tension, and **n** is the unit vector normal to the interface. $ T i j = \u2212 p \delta i j + \mu ( \u2202 j u i + \u2202 i u j )$ are the hydrodynamic stresses, where *δ _{ij}* is the Kronecker delta function. The electric stresses are calculated by the Maxwell stress tensor $ T e i j = \u03f5 ( E i E j \u2212 E k E k \delta i j / 2 )$. Equation (1) implies that due to the discontinuities at the interface, there may be fluid motion both inside and outside of the drop and the deformation of its shape.

^{5}

^{7}conducted a comprehensive analytical work considering both the drop and the external phases as poorly conducting leaky dielectric liquids. This led to the well-known leaky dielectric model (LDM). This model assumes small currents in the system and an irrotational electric field. If the charge relaxation timescale $ \tau c = \u03f5 / \sigma $ is much smaller than other time scales, e.g., the timescale of charge convection by the flow $ \tau f = \mu / ( \u03f5 E o 2 )$, the charge convection term can be neglected. The transport equation of the charge then reduces to

^{8}

^{,}

^{7}namely,

*p*is the pressure. Equation (4) considers creeping flow inside the drop and therefore is valid for slow fluid velocities with $ R e \u226a 1$, where $ R e = \rho v \u221e a o / \mu $ is the Reynolds number. Here, $ v \u221e = \u03f5 E o 2 a o / \mu $ is the characteristic velocity of the flow. As will be shown later, all cases investigated here satisfy the creeping flow condition. Note that due to absence of charge convection, the electric forces are zero in the bulk and therefore are not present in the momentum balance. The velocity field inside the drop in permanent regime can be obtained by solving these equations in polar coordinates,

^{5}

*L*and

*W*are the drop axes parallel and perpendicular to

**E**, respectively. The relaxation of the drop toward its equilibrium spherical shape is governed by the capillary or deformation timescale $ \tau def = \mu a o / \gamma $. The relevant dimensionless number here is, therefore, the electric capillary number

*Ca*is commonly defined based on the properties of medium, not the drop phase. The electric capillary number was employed by Saville

_{E}^{8}to derive an analytical equation for the deformation parameter based on the LDM approach,

*R*and

*S*; for a horizontal

**E**, if

*R*>

*S*the internal circulation is from the poles to the equator, while the fluid motion is from the equator to the poles if

*R*<

*S*. Although additional models have been proposed to account for the transitory dynamics,

^{9–11}larger deformations

^{12}charge convection,

^{13}and drop rupture,

^{14–16}the LDM proposed by Taylor and Melcher

^{7}remains a reference case for assessing the electrohydrodynamic flows of leaky dielectric drops in the small-deformation limit ( $ C a E \u226a 1$), corroborated by diverse studies. Torza

*et al.*

^{17}were the first to report an experimental verification of the LDM obtaining good qualitative agreement, with some comments on the conditions to rupture being also reported. Further studies were performed by Vizika and Saville

^{18}and Ha and Yang

^{19}confirming the qualitative applicability of the LDM for the small-deformation limit, but commenting on the necessity of improvements to account for larger deformations. In summary, it is understood that the pioneer LDM theory is capable of describing the EHD of drops fairly well, considering the small-deformation limit and permanent regime. For a more in-depth review of this topic, readers are directed elsewhere.

^{1,5,8}From an experimental standpoint, the majority of investigations have focused on the shape of the drop,

^{20–22}on its coalescence

^{6}and on its breakup.

^{23}For the latter case, the conductivity and permittivity ratios dictate different modes of breakup. For a strong electric field and leaky dielectric fluids, if the drop is more conducting and viscous, the tangential stresses lead to the emission of jets. Otherwise, if the fluid system is inverted, a lenticular drop is formed leading to equatorial streaming.

^{24}Moreover, a non-uniform velocity field can be observed for a certain range of electric and hydrodynamic properties of the fluids, leading to the rotation of the drop.

^{25}Mikkelsen

*et al.*

^{26}reported experimental measurements of the external EHD flows of a drop; using particle image velocimetry (PIV), they analyzed the suppression of the velocity field by Marangoni flows due to surface contaminants. However, a detailed analysis of the velocity field, particularly close to the interface, nor the internal circulation were reported.

More recently, the focus has shifted toward the investigation of the transient dynamics of drops under a constant electric field.^{15,27–29} A first theoretical analysis of the dynamic response of the EHD flows was presented by Sozou.^{30} Assuming that the charge instantaneously accumulate at the interface, the velocity field was investigated with time within the creeping flow regime considering a low electric Reynolds number $ R e E = \u03f5 2 E o 2 / \sigma \mu $. Thus, once the steady state is reached the flow converged to the LDM solution, provided that $ R e E \u226a 1$; however, no closed-form solution of the transient velocity field was proposed. To fill this gap, Esmaeeli and Sharifi^{10} followed a different approach by neglecting the local fluid acceleration term $ \u2202 v / \u2202 t$ also considering creeping flow. Despite an analytical solution of the velocity field being outlined, the authors focused on the deformation–time curve of the drop with little discussion on the dynamics response of the velocity field. Moreover, a monotonic behavior was proposed, with a modified the capillary timescale governing the dynamics. Later on, it was shown by Esmaeeli and Behjatian^{11} that the monotonic behavior adequately describes the dynamics of a slightly deformed drop with large Ohnesorge numbers $ O h = ( \mu a o / \gamma ) / ( a o 2 \rho / \mu )$, whereas an oscillatory time response is observed when $ O h \u226a 1$. Lanauze *et al.*^{31} extended the solution of Sozou^{30} by considering charge relaxation under creeping flow using a boundary integral method. A similar approach has been performed by Das and Saintillan^{9} who developed a second-order perturbation solution to include both charge relaxation and charge convection in the creeping flow regime.

Based on the literature review outlined above, experimental inquiries into the internal circulation of a leaky dielectric droplet are to best of our knowledge not available. Even within the small-deformation limit at permanent regime, where the LDM theory applies fairly, the experimental validation of the LDM model is thus far confined to the deformation of the drop. As for the dynamic response of the drop, existing experimental studies have primarily focused on the shape of the drop, while a proper quantification of fluid circulation is, to the best of our knowledge, restricted to numerical and theoretical investigations. Moreover, the definition of a “small-deformation limit” remains vague to date. A criterion that quantitatively determines the extent of the applicability of the LDM approach is yet to be reported.

Therefore, experimental measurements of the EHD circulation within a neutrally buoyant drop are still necessary for a full comprehension of the phenomenon, particularly regarding the temporal evolution of the velocity field outside the small-deformation limit for which no investigations are reported. In this study, we propose a novel methodology based on particle image velocimetry (PIV) to quantify the internal circulation of neutrally buoyant drop under a constant electric field. The methodology provides accurate measurements of the velocity field inside the drop for a wide range of *Ca _{E}* and

*Re*.

_{E}## II. EXPERIMENTAL METHODOLOGY

### A. Setup and test fluids

Figure 2(a) shows the experimental setup used in this work. The experiments are conducted in a plexiglass cuvette with a base of 50 × 40 mm^{2} and 40 mm of height, as shown in Fig. 2(b). The cuvette is completely filled with a stagnant leaky dielectric liquid, where a drop of $ a o = 2.25$ mm is inserted using a micropipette [also shown in Fig. 2(b)]. The size of the drop is kept fixed for all measurements. Silicone oil (8453.290, VWR) and castor oil (24667.290, VWR) are either the continuous medium or the drop, depending on the case analyzed. Table I summarizes the properties of the fluids. The electrical permittivities and conductivities are measured using broadband dielectric spectroscopy (Novocontrol-concept 80, Germany) following the parallel electrode method.^{32} The density, dynamic viscosity, refractive indexes, and the interfacial tension are obtained from the supplier at 20 °C (the temperature control is ensured by using a ventilation system). The electric field is generated using two vertical copper electrodes with a thickness of 1 mm spaced 20 mm apart. A high-voltage supply system (Hartlauer Elektronik CC400-32, Germany) is used to apply the desired voltage. The right electrode shown in Fig. 2(b) is connected to the high-voltage supply, while the left electrode is grounded, thus generating an electric field in the horizontal direction.

. | Density, ρ (kg/m^{3})
. | Viscosity, μ (Pa s)
. | Electrical permittivity, ε (F/m) . | Electrical conductivity, σ (S/m)
. | Refractive index, n
. |
---|---|---|---|---|---|

Silicone oil | 970 | 0.50 | 2.83 × 10^{−11} | 8.71 × 10^{−13} | 1.41 |

Castor oil | 961 | 0.78 | 4.16 × 10^{−11} | 3.01 × 10^{−11} | 1.47 |

. | Density, ρ (kg/m^{3})
. | Viscosity, μ (Pa s)
. | Electrical permittivity, ε (F/m) . | Electrical conductivity, σ (S/m)
. | Refractive index, n
. |
---|---|---|---|---|---|

Silicone oil | 970 | 0.50 | 2.83 × 10^{−11} | 8.71 × 10^{−13} | 1.41 |

Castor oil | 961 | 0.78 | 4.16 × 10^{−11} | 3.01 × 10^{−11} | 1.47 |

Table II presents the cases examined in this study. By utilizing the same fluids and modifying the composition of both the continuous phase and the drop, we achieve two distinct final shapes: the oblate (cases I–IV) and the prolate (cases V–VIII) configurations. A drop is considered oblate or prolate when its major axis is perpendicular or parallel to the direction of the electric field, **E**, respectively. The strength of the electric field is $ E o = \Delta V / \Delta x$ where $ \Delta V$ is the voltage and $ \Delta x$ is the distance between the electrodes. *E _{o}* spans from 0.25 to 1.75 kV/cm and from 0.125 to 1.10 kV/cm for the oblate and prolate configurations, respectively. First, the creeping flow condition assumed in Eq. (4) is verified by calculating the Reynolds number. In the cases investigated here, $ v \u221e$ spans from $ 8 \xd7 10 \u2212 5$ to $ 4 \xd7 10 \u2212 3$ m/s, leading to Reynolds numbers in the range of $ 10 \u2212 5 < R e < 10 \u2212 3$. Therefore, the creeping flow assumption can be considered here. The electric dimensionless numbers, namely,

*Re*and

_{E}*Ca*, are also presented in Table II.

_{E}*E*has been adapted for each case to facilitate the comparison between them. Due to the changes in the electrical properties of the fluids, particularly the conductivity, it is possible to keep

_{o}*D*nearly constant and the

*Ca*in the same order of magnitude when inverting the system of fluids. In this work, care has been taken to avoid working in conditions close to the rupture of the drop, that might happen when $ C a E \u2248 1$.

_{E}^{1}The LDM theory has two main assumptions: (i) small drop deformations ( $ C a E \u226a 1$) and (ii) no charge convection ( $ R e E \u226a 1$). Thus, from an inspection of Table II, it is natural to expect the LDM to be applicable for cases I, II, V, and VI, for which both of the aforementioned conditions are largely satisfied. However, the limit conditions for hypothesis (i) and (ii) are not clear to date, nor the relative importance of deformation over charge convection. It will be shown in this study that the applicability of the LDM theory relies more on

*Ca*than on

_{E}*Re*.

_{E}Case . | Oil . | Drop initial radius, a_{o} (mm)
. | Strength of the electric field, E, (kV/cm)
. _{o} | $ S = \u03f5 \u2041 \u03f5$ . | $ R = \sigma \u0302 \sigma $ . | $ M = \mu \u0302 \mu $ . | $ C a E = \u03f5 E o 2 a o \gamma $ . | $ R e E = \u03f5 2 E o 2 \sigma \mu $ . | D
. | D
. _{T} | Shape . | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Drop . | Medium . | |||||||||||

I | Silicone | Castor | 2.25 | 0.25 | 0.68 | 0.029 | 0.735 | 0.018 | 0.046 | −0.0030 | −0.0033 | Oblate |

II | 0.50 | 0.060 | 0.184 | −0.0115 | −0.0108 | |||||||

III | 0.85 | 0.175 | 0.533 | −0.0625 | −0.0432 | |||||||

IV | 1.75 | 0.740 | 2.257 | −0.2016 | −0.1382 | |||||||

V | Castor | Silicone | 2.25 | 0.125 | 1.47 | 34.48 | 1.36 | 0.003 | 0.287 | 0.0019 | 0.0014 | Prolate |

VI | 0.30 | 0.012 | 1.655 | 0.0122 | 0.0110 | |||||||

VII | 0.60 | 0.071 | 6.620 | 0.0493 | 0.0320 | |||||||

VIII | 1.10 | 0.240 | 22.25 | 0.196 | 0.1077 |

Case . | Oil . | Drop initial radius, a_{o} (mm)
. | Strength of the electric field, E, (kV/cm)
. _{o} | $ S = \u03f5 \u2041 \u03f5$ . | $ R = \sigma \u0302 \sigma $ . | $ M = \mu \u0302 \mu $ . | $ C a E = \u03f5 E o 2 a o \gamma $ . | $ R e E = \u03f5 2 E o 2 \sigma \mu $ . | D
. | D
. _{T} | Shape . | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Drop . | Medium . | |||||||||||

I | Silicone | Castor | 2.25 | 0.25 | 0.68 | 0.029 | 0.735 | 0.018 | 0.046 | −0.0030 | −0.0033 | Oblate |

II | 0.50 | 0.060 | 0.184 | −0.0115 | −0.0108 | |||||||

III | 0.85 | 0.175 | 0.533 | −0.0625 | −0.0432 | |||||||

IV | 1.75 | 0.740 | 2.257 | −0.2016 | −0.1382 | |||||||

V | Castor | Silicone | 2.25 | 0.125 | 1.47 | 34.48 | 1.36 | 0.003 | 0.287 | 0.0019 | 0.0014 | Prolate |

VI | 0.30 | 0.012 | 1.655 | 0.0122 | 0.0110 | |||||||

VII | 0.60 | 0.071 | 6.620 | 0.0493 | 0.0320 | |||||||

VIII | 1.10 | 0.240 | 22.25 | 0.196 | 0.1077 |

### B. High speed shadowgraphy

High speed shadowgraphy measurements are conducted according to the experimental setup shown in Fig. 2. The shadowgraph images are obtained in order to quantify the contour of the drop using a single CCD camera (CX3-25, LaVision, Germany) with a spatial resolution of 5296 × 4584 pixels with 12 bits of digital output, coupled to a 105 mm lens (Nikkon, Japan). A LED system is used as a background source illumination. The shadowgraph images of the drop are then recorded with a frequency of 20 Hz. The acquired images have a scale factor of 4.18 *μ*m per pixel. Figure 3 shows sample images of the drop, at steady-state condition, for the cases shown in Table II. Clearly, the drop evolves from a perfectly spherical shape ( $ D \u2192 0$) when no electric field is applied [Figs. 3(a) and 3(f)] to either an oblate (top row) or a prolate (bottom row) shape, depending on the fluid composition of the system. An image binarization process is conducted to calculate *D* from the raw images as presented in Table I for each case investigated. For cases I and V, *D* assumes a value of −0.0030 and 0.0019, respectively, as shown in Figs. 3(b) and 3(g). Note that *D* is negative for the oblate shape. Good agreement with Eq. (9) is observed since *D _{T}* is −0.0033 and 0.0014 for cases I and V, respectively. A similar observation is made when

*E*is increased in cases II and VII, as shown in Figs. 3(c) and 3(h), respectively. As those cases typically fall within the small-deformation limit ( $ C a E \u226a 1$), good agreement with the LDM theory is expected.

_{o}^{8}However, a large deviation is noticed when we consider drops with larger deformations, e.g., cases III and VII with

*D*being −0.0625 and 0.0493, respectively. For such cases, Eq. (9) under estimates our experimental measurements by a factor of 30% approximately. When

*E*is increased further, such deviations tend to augment, up to roughly 45% for case VIII. It is evident that the LDM theory applies for cases with small deformation only, in accordance with previous studies.

_{o}^{1,5}Finally, we stress that buoyancy effects are negligible based on good agreement with the analytical solution.

### C. Particle image velocimetry

The internal circulation of the drop is quantified by adding polyamide particles coated with RhodamineB dye (20 *μ*m) and illuminating the drop at its plane of symmetry with a laser sheet [Fig. 2(a)]. A monocavity Nd:YAG laser (Quantel BRIO SP23, France) with a wavelength of 532 nm and an energy of 64 mJ served as the laser source. The reflections of the laser at the interface of the drop were reduced by using a bandpass filter, only allowing the passage of the light emitted by the fluorescent particles. Slow recirculation within the drop allowed image acquisition at a relatively low frame rate (20 Hz), enabling the use of a monocavity laser instead of a double-cavity one. Mirrors, spherical lenses, and cylindrical lenses are arranged to generate a laser sheet approximately 150 *μ*m thick. The image acquisition system for the PIV measurement is the same used from the shadowgraph experiments, as shown previously. When a dielectric particle is placed within an electric field, it experiences a dipole momentum due to its polarization that may induce the migration of the particles in the direction of **E**. Therefore, preliminary experiments in a single-phase environment were performed that showed that the electric field did not promote any appreciable motion of the tracers. The polarization force experienced by the particles $ F p$ is proportional to the difference in electrical permitivities: $ F p \u221d ( \u03f5 p \u2212 \u03f5 \u0302 )$, where the index *p* refers to the particle. Therefore, $ F p$ is small since in our case $ \u03f5 p \u2248 \u03f5 \u0302$. Considering the weight of a particle $ F g = m p g$ one can estimate that *F _{g}* is order $O$ (6) when compared to

*F*, explaining the reason why the particles are not affected by

_{p}**E**.

The vector fields shown in Fig. 4 (right half of each subfigure) for cases II and V are obtained using the Davis software (version 10, LaVision, Germany) with a cross correlation iterative multi-pass routine of round interrogation windows of 64 × 64 pixels with 50% overlap in both directions. A geometric mask is applied to remove false vectors outside the drop. Additional remarks of the procedure used in the PIV calculation are shown in Appendix A. The velocity vectors are then corrected based on the refraction of the light emitted by the particles as it crosses the drop interface. Details of this simplified procedure are shown in Appendix B. Figure 4 provides a visual comparison between the PIV velocity fields and the analytical velocity vectors (left half of each subfigure) given in Eqs. (5) and (6), showing good qualitative agreement.

## III. RESULTS AND DISCUSSION

Figures 5 and 6 show the measured velocity vectors for the cases shown in Table II, considering the permanent regime. For the sake of clarity, the vectors are shown for the upper left quadrant only that corresponds to $ \u2212 1 < x / a o < 0$ and $ 0 < y / a o < 1$. By making the positions dimensionless as a function of the initial radius *a _{o}*, a visual inspection of the shape of the drop can also be performed. Note that the dashed line shown in the figures qualitatively represent the interface of the drop. An observation is possible from an inspection of cases I and II: the velocity field remains somewhat symmetrical with respect to the diagonal, i.e., $ x / a o = y / a o$. The circulation is bounded at $ x / a o = y / a o = 0$, while the tangential component $ v \theta $ is maximum at the diagonal crossing the quadrant, close to the interface. A stagnation zone is also identified at $ x / a o = \u2212 0.2$ and $ y / a o = 0.3$ for case I and at $ x / a o = \u2212 0.2$ and $ y / a o = 0.4$ for case II. The slight deviation regarding the horizontal prediction is consistent with the oblate shape, as shown previously in Table II. As will be discussed later in this paper, the center of the stagnation zone is indeed shifted even for the small-deformation limit as a consequence of the stronger $ v r$ component along the direction

*θ*= 0. Additionally, a remark can be drawn for the prolate configuration [cases V and VI shown in Figs. 6(a) and 6(b), for instance], which seem to have stronger asymmetries than for the oblate configuration, as discussed later in Appendix A.

However, the larger deformations observed for cases III and IV lead to an increase in the non-uniformity of the velocity field, thus shifting the center of the circulation zone further to $ x / a o = \u2212 0.2$ and $ y / a o = 0.5$ and $ x / a o = \u2212 0.2$ and $ y / a o = 0.6$, respectively. Clearly, the symmetry axis along the diagonal line of the quadrant is completely lost for these two cases. When the composition of the fluids is reversed, i.e., cases V to VIII, a similar observation is made regarding the evolution of the velocity vectors with *E _{o}*: a non-uniform distribution is noticed when

*E*is larger, particularly cases VII and VIII. Under those circumstances, it is expected that the analytical velocity field [Eqs. (5) and (6)] and shown in Fig. 4 no longer matches our measurements. Finally, we stress that some radial components of the velocity are identified close to interface for some cases that arise from the geometrical placement of the interrogation window relative to the interface of the drop, as further discussed in Appendix A.

_{o}### A. Radial velocity profiles

Figure 7 shows radial profiles of $ v r$. The spatial coordinates are now converted from the Cartesian (*x*, *y*) to the polar ( $ r , \theta $) system to improve the clarity of the results. The radial profiles are shown along the *r* distance for three different values of *θ*, namely, 0, $ \pi / 4$, and $ \pi / 2$. Let us consider cases II and VI shown in Figs. 7(a) and 7(b), respectively. Clearly, the figure suggests a very good agreement between the experimental measurements (shown in the figure as the markers) and the analytical profiles obtained (lines in Fig. 7). Considering the profiles at the axes of symmetry of the drop, i.e., *θ* = 0 and $ \theta = \pi / 2$, a parabolic behavior (slightly inclined toward the region where $ r / a o \u2192 1$) is present for both cases analyzed. At $ \theta = \pi / 4 , \u2009 v r$ oscillates due to the presence of an additional stagnation point, located at the center of the vortex, roughly at $ r / a o = 0.7$. According to Eqs. (5) and (6), the internal EHD circulation of the drop is not perfectly symmetrical along the diagonal line for the region that corresponds to $ 0 < r / a o < 1$ and $ 0 < \theta < \pi / 2$ as a consequence of the non-linearity of the equations. For instance, $ | | v r | |$ is larger in the direction of *θ* = 0 when compared to the direction of $ \theta = \pi / 2$. Consequently, considering case II, for instance, the center of the stagnation zone is shifted toward the upper pole of the drop.

A different $ v r$ profile is observed when we shift our analysis to more deformed drops, e.g., cases III and VII. According to Eq. (7), the contribution of *E _{o}* is solely regarding the magnitude of the velocity components, as indicated in Figs. 7(c) and 7(d) that show the exact same profile of $ v r$ from the LDM theory. On the other hand, this clearly deviates from our measurements, as the profiles of $ v r$ no longer match the analytical prediction. Although for case III a similar profile is still observable, the theory under predicts the magnitude of $ v r$. The deviation is particularly significant for case VII [shown in Fig. 7(d)], where the profile obtained experimentally is completely asymmetric. Strictly speaking, the LDM applies only $ C a E \u226a 1$, which was the case shown in Figs. 7(a) and 7(b). It is evident here that Eqs. (5) and (6) are applicable only for small deformations, as only the initial radius

*a*is considered. However, charge convection and charge relaxation could also play a significant role for stronger electric fields when

_{o}*Re*is closer to the unity.

_{E}^{9,12}However, extensions of the LDM theory to account for such effect usually focus on the deformation parameter

*D*and not on the velocity field inside the drop, which is the focus of the present study. Our measurements suggest that deviations in the internal circulation are already evident when $ C a E \u2248 0.1$.

### B. Tangential velocity at the interface

The tangential component of the velocity at the interface of the drop, $ v \theta i$, is shown in Figs. 8 and 9 for the cases shown in Table II. The measurements correspond to all the velocity fields shown in Figs. 5 and 6. The experimental values (indicated as markers) closely align with the analytical solution predicted in Eq. (6) for the cases shown in Figs. 8(a), 8(b), 9(a), and 9(b). However, it is noteworthy that there is a slight non-uniform trend of the azimuthal profile of $ v \theta i$ even within the small-deformation limit. Mainly, the azimuthal profiles tend to be thinner and slightly inclined toward $ \theta \u2192 0$; a probable consequence of the slight deformation observed even for those cases, which is not all taken into account by the LDM model. However, such deviation is small and a good agreement can be considered here. Again, the measurements of $ v \theta i$ evolve to a non-uniform azimuthal profile when *E _{o}* is further increased, as shown in Figs. 8(c), 8(d), 9(c), and 9(d). In addition to the significant underprediction of our measurements by the LDM model, the polar coordinate

*θ*when $ v \theta i$ is maximum, i.e., $ v \theta i m$ moves toward the upper ( $ \theta \u2192 \pi / 4$) and lower ( $ \theta \u2192 0$) regions of the quadrant, respectively, for cases III to IV, and VII to VIII.

### C. Dynamic response of the velocity at the interface

We now shift our focus to the dynamic response of the velocity at the interface of the drop. Figure 10 shows the velocity vectors at different instants of time. We focus here on case II, as this case still corresponds to a nearly spherical shape but with slightly larger velocity magnitudes. The time instant *t* is made dimensionless based on the timescale of the deformation of the drop *τ _{def}*. Clearly, the internal structures of the velocity field evolve with time. EHD flows originate at the interface due to a discontinuity in electric stresses [see the stress balance given in Eq. (1)]. The velocity first appears as a tangential component at the interface ( $ t / \tau def = 0$), followed by an increase in the radial component within the drop. The magnitude of the vectors in the bulk of the drop increases with time for later values of time when $ t / \tau def = 0.8$ and $ t / \tau def = 2.6$ until a clear vortical structure is identified on the right top corner of the drop at $ t / \tau def = 5.3$ (d) after which the permanent regime is established. Although we show the transient velocity vector only for case II, a similar qualitative behavior has been observed for all other cases, regardless the magnitude of

*E*. Note that when $ t / \tau def = 0$, shown in Fig. 10(a), the vectors close to the interface are not tangent to it given that, considering a transitory regime, the drop is experiencing shape deformation, inducing a horizontal motion of the tracers. This is in accordance with Eq. (11) that lead to streamlines that cross the interface of the drop before the permanent regime is achieved.

_{o}^{10}We focus here only on the internal EHD circulation of the drop, for which the following streamfunction is proposed:

*A*and

*B*carry the time-dependency of the velocity field,

*τ*is a capillary timescale that governs the dynamics

*M*into account. Note that $ v r$ and $ v \theta $ obtained from this theory eventually converge to the LDM prediction at steady state when $ t \u2192 \u221e$.

The dynamic response of the $ v \theta i m$ is shown in Fig. 11 (markers) for all the cases investigated. Conveying the point made from the investigation of the transient velocity vectors shown in Fig. 10, that the EHD circulation begins first as a tangential component at the interface when the electric field is applied, it is foreseeable to have an exponential growth of $ v \theta i m$ over time until the steady state condition is reached. By a direct comparison of our measurements to the transient analytical solution of $ v r$ and $ v \theta $, shown as the lines in Fig. 11, it is observable that the streamfunction given in Eq. (11) is capable of properly describing the dynamic response of the tangential velocity at the interface of the drop, at least when regarding its maximal value $ v \theta i m$. However, we use here the usual definition of the capillary timescale as the governing parameter of the dynamic response so that $ \tau = \tau def = \mu a o / \gamma $, which seemed to be a better fit to our measurements. Apparently, the viscosity ratio *M*, although important in the dynamic response of the shape of the drop, has minor effect on the velocity at the interface. As the transient solution proposed by Esmaeeli and Sharifi^{10} is based on the same assumptions than the LDM theory regarding the steady-state condition, it is expected to see an under prediction of our measurements for the cases that correspond to larger deformations. We confirm here the critical value of $ C a E \u2248 0.1$, below which the analytical solutions for both the transitory and permanent regimes are applicable. It is also noteworthy that a monotonic growth is observed for all cases investigated. This is in corroboration with other remarks from Esmaelli and Behjatian^{11} who argue that an oscillatory dynamic response can be expected for less viscous fluids, which is not the case of the current measurements.

## IV. CONCLUSIONS

Experimental measurements of the electrohydrodynamic (EHD) flows inside a neutrally buoyant leaky dielectric drop have been reported for the first time. Special focus is given for the tangential component of the velocity at the interface $ v \theta i$, for which the dynamic response is also investigated. By using two leaky dielectric oils, namely, silicone and castor oils, both the oblate and prolate configurations were investigated. In addition, the strength of the constant electric field *E _{o}* spanned from 0.125 to 1.75 kV/cm, thus covering both the small-deformation limit ( $ C a E \u226a 1$) and also larger deformations. The internal circulation at the plane of symmetry of the drop was measured in an Eulerian referential using particle image velocimetry (PIV) and high-speed shadowgraph.

While a similar overall behavior of the tangential velocity $ v \theta $ and $ v r$ in the permanent regime was observed compared to the analytical prediction, the LDM theory under predicts our measurements when the drop adopts an ellipsoidal shape. However, even within the small-deformation limit, some deviations in $ v \theta $ are observed due to the deformation of the drop, which is not accounted for by the LDM model. We confirmed that the electrohydrodynamic flows are indeed driven by the electric stress jump at the interface, as described by the balance equation $ n \xb7 [ ( T \u2212 T \u0302 ) + ( T e \u2212 T \u0302 e ) ] = \gamma n ( \u2207 \xb7 n )$. The velocity starts as a tangential component at the interface, followed by an increase in the radial component within the drop, leading to four counter-rotating vortices. The dynamic response of the velocity field is then described by the capillary timescale $ \tau def = \mu a / \gamma $, for which current theories available in the literature have a fair agreement. We propose a critical electric capillary number of $ C a E \u2248 0.1$, below which the LDM approach adequately represents the both the transitory and quasi-steady dynamics of the velocity field. However, we stress that no analytical solution is present for the case of larger deformations, thus the novel methodology presented here offers significant improvements toward the capability to understand the internal EHD flows of a neutrally buoyant drop, as it offers less restrictions to the experimental conditions employed when compared to the analytical solutions that are available in the literature.

## ACKNOWLEDGMENTS

This research was funded by the Normandie Region, under Grant No. 22E05147. We also acknowledge the technical support of Franck Lefebvre, Gilles Godard, and Said Idlahcen from CORIA during the implementation of the experimental setup. The measurements of the dielectric properties were conducted by Nicolas Delpouve and Laurent Delbreilh from the GPM (Groupe de Physique des Matériaux) of the UFR Sciences et Techniques de Rouen, whose contribution is also hereby acknowledged.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Joel R. Karp:** Data curation (equal); Formal analysis (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). **Bertrand Lecordier:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). **Mostafa S. Shadloo:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – original draft (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: UNCERTAINTY OF THE PIV MEASUREMENTS

A sample image of the tracers particles used in the PIV measurements is shown in Fig. 12(a). To provide a qualitative analysis of the EHD flows inside the drop, streaklines are shown for Cases II and VI, as displayed in Figs. 12(b) and 12(c), respectively. The streaklines are obtained by adding the intensity of each pixel for a series of 20 images that corresponds to 2 s of motion. Thus, the path of the tracers can be identified, leading to the same structure shown in Fig. 4. The streaklines shown here are similar to the ones reported by Taylor^{33} where four counter-rotating vortices are identified.

In order to verify the accuracy of the PIV measurements, Fig. 13 shows a scalar field of the correlation value (left half of each subfigure) for the same cases shown in Figs. 12 and 4. The correlation value, provided by Davis, is the parameter used for post-processing procedures of vector validation. Figure 13 shows that the correlation value approaches the unity within the better part of the drop. A smaller correlation value is observed for some regions close to the interface of the drop, when the interface crosses the interrogation window. Similarly, in the poles of the drop, the correlation value is smaller due to the accumulation of particles. As a consequence of such conditions, there may be a small radial component of the velocity even at the interface of the drop, which may lead to vectors slightly crossing the interface, as shown in Fig. 6(b), and at the poles of the drop (Fig. 4).

To verify the accuracy of the PIV measurements further, Fig. 13 also shows (right half of the sub-figures) the magnitude of the uncertainty of the velocity measurements, $ \delta V * / V m *$ (also provided by Davis), where $ V m *$ is the maximum value of the velocity magnitude inside the drop. By doing so, it is clear that $ \delta V *$ is small when compared to the calculated velocity. For both cases, $ \delta V * / V m *$ is highest at the interface of the drop and although higher uncertainties are observed for Case VII [shown in Fig. 13(b)], $ \delta V * / V m * < 0.01$ everywhere within the drop, highlighting the accuracy of our measurements. The typically higher uncertainties for the prolate configuration (cases V to VIII) are basically associated with increased experimental difficulties when the drop deforms along **E**, therefore reducing its distance to the electrodes. However, this effect seems to be small and does not affect the discussion.

### APPENDIX B: CORRECTION OF IMAGE DISTORTION

*n*

_{1}), the light will be refracted once it crosses the interface of the drop with the medium (refractive index

*n*

_{2}), as shown in Fig. 14(a), according to Snell's law of refraction

*α*is the angle formed between the light beam emitted from the particles and the vector normal at the interface of the drop. Since $ n 1 \u2260 n 2$, the image captured by the CCD camera will be distorted. Considering case I, where $ n 2 > n 1$, it follows that $ \alpha 1 > \alpha 2$, thus the particles will appear to be closer to the interface than they actually are with $ r * > r$, where the $*$ symbol denoted the virtual position, obtained from the distorted image.

*f*is simply the ratio between the virtual and real positions of the interface and can be obtained directly from the following equation:

_{i}