Prediction and measurement of leaky dielectric drop interactions

Application of an electric field across the curved interface of two fluids of low but nonzero conductivities, or “leaky dielectrics,” can give rise to electric stresses that drive sustained fluid flow. In a uniform dc electric field of sufficiently weak magnitude, the electric and velocity fields around an isolated, neutrally buoyant leaky dielectric drop at zero Reynolds number are fore-aft and azimuthally symmetric about the applied field axis. Consequently, the drop remains stationary. The presence of a second drop breaks these symmetries, resulting in the relative motion of the drop pair. Recently, Sorgentone et al. derived an analytical expression for the relative velocity of a pair of widely separated drops of identical constitution, asymptotic in the inverse separation distance between the drop centroids [C. Sorgentone, J. I. Kach, A. S. Khair, L. M. Walker, and P. M. Vlahovska, Numerical and asymptotic analysis of the three-dimensional electrohydrodynamic interactions of drop pairs, J. Fluid Mech. 914, A24 (2021)]. In the present work, we generalize the theory of Sorgentone et al. to interactions of dissimilar drops (of different size or constitution), and the pairwise additive interactions of three or more drops. We perform experiments on silicone oil drops suspended in castor oil, and we compare to asymptotic predictions of the drop pair trajectories. Experimental trajectories of drops with their line of centers initially at an arbitrary angle to the field direction are shown to be qualitatively predicted by our theory. We show results of experiments of dissimilar drops and of three and four drops, again observing qualitative agreement with our theoretical predictions.

Figure 1

Example of steady electric and velocity fields around a leaky dielectric drop in a uniform electric field, where $M=2,R=1/5,S=1/3$. In this example, the drop is slightly deformed to a prolate spheroid with undeformed radius $a$, major axis $L_{\parallel }$, and minor axis $L_{\perp }$ with respect to the applied field direction. The viscosity $\mu$, conductivity $\sigma$, and permittivity $\varepsilon$ of the drop phase are denoted with subscript $d$, and the suspending phase properties are denoted with subscript $s$. Both the electric and flow fields have fore-aft symmetry about the drop equator perpendicular to the applied field direction and are axially symmetric about the poles in parallel to the applied field direction. Internal electric field lines are omitted, however the internal field is uniform and parallel to the applied field direction.

Figure 2

Depiction of a coordinate system for two spherical drops of radius $a$. The centers of the drops are separated by a distance $d$, with the angle between the line of centers and the applied field denoted as $\mathrm{\Theta }$.

Figure 3

Schematic of the experimental setup. Drops are initially placed in the same focal plane, where they tend to remain throughout the experiment.

Figure 4

Phase diagram for the behavior of a pair of equiviscous ($M=1$), identical, leaky dielectric drops with permittivity ratio $S$ and conductivity ratio $R$, based on Eq. (4.8) from Sorgentone et al. [42]. The solid black line denotes combinations of $S$ and $R$ for which $\mathrm{\Phi }=0$ and the orientation of the drops is steady in time. In the (blue) region above the black line, $\mathrm{\Phi }<0$, such that material systems in that region will show drop pair rotation toward $\mathrm{\Theta }=\pi /2$. Below the black line, $\mathrm{\Phi }>0$. In the (orange) region between the lines of $\mathrm{\Phi }=0$ and $RS=1$, drop pairs will rotate toward parallel and attract when $\mathrm{\Theta }<54.7^{\circ }$. In the green region below $RS=1$, drop pairs will rotate toward parallel, however whether they attract or repel at $\mathrm{\Theta }<54.7^{\circ }$ depends on the separation distance relative to $d_c$ for the given $M, S$, and $R$.

Figure 5

(a) Centroid separation between two 350 cSt silicone oil drops in castor oil aligned parallel to the field. (b) Centroid separation between two drops aligned perpendicular to the field. Error bars represent a standard deviation of interpolated and time-averaged experiments. In both cases, $sin\left(2\mathrm{\Theta }\right)=0$ and minimal rotation of the drop pair is observed. $E_{\infty }=1.79\mathrm{kV}/\mathrm{cm}$ and $a=620\mu \text{m}$.

Figure 6

Parallel and perpendicular drop trajectories for various applied voltages, with ${\text{Re}}_E={\varepsilon }_s^2{E}_{\infty }^2/{\sigma }_s{\mu }_s$. (a) Drops are aligned parallel to the field direction. (b) The same trajectories as (a), with time normalized by ${\tau }_c$. (c) Drops are aligned perpendicular to the field. (d) The same trajectories as (c), with time normalized by ${\tau }_c$. Trajectories collapse upon normalization of time by ${\tau }_c={\mu }_s/{\varepsilon }_s{E}_{\infty }^2$, indicating that the maximum voltage tested of 5 $\mathrm{kV}$ will not appreciably alter the dynamics of the drop interactions compared to lower voltages.

Figure 7

(a) Separation distance between the centers of drops unaligned with the field direction, and (b) angle between the line of centers of the drops and the field direction, for drops initially placed with $d_0/a=4.7$ and ${\mathrm{\Theta }}_0={21}^{\circ }$. (c) Separation distance and (d) angle between line of centers and field, for drops initially placed with $d_0/a=4.4$ and ${\mathrm{\Theta }}_0={39}^{\circ }$. The vector between the drops points from the drop on the left to the one on the right, and angles plotted are $\left|\mathrm{\Theta }\right|$ (therefore always between $0^{\circ }$ and ${90}^{\circ }$).

Figure 8

(a) Separation distance between the centers of drops unaligned with the field direction, and (b) angle between the drops' line of centers and the field direction. (c) Photos taken from the experiment showing the progression of drop positions in time. At the initial time, ${\mathrm{\Theta }}_0>54.7^{\circ }$, and drops initially repel. The vector between the drops points from the drop on the left to the one on the right, and angles plotted are $\left|\mathrm{\Theta }\right|$.

Figure 9

(a) Normalized separation distance $d/a_{\text{avg}}$ for drops of dissimilar size. The solid black line indicates theoretical prediction of the separation distance from Eq. (11); the dotted black line indicates the prediction for identical drops with Eq. (13) using an average radius of $505.5\mu \text{m}$. The inset shows the initial position of drops when electric field is applied. (b) The angle between line of centers and the field direction shown with corresponding values of the second Legendre polynomial. The largest angle observed between the line of centers and the applied field is $\mathrm{\Theta }\approx 0.11$, which gives a value of 0.98 for the second Legendre polynomial.

Figure 10

Trajectories of drops in the plane made by the line connecting the centers of the drops and the field direction. In all cases, $S_{\text{blue}}=S_{\text{red}}=1, R_{\text{blue}}=10, R_{\text{red}}=0.1$. (a) $M=M_{\text{blue}}=M_{\text{red}}={10}^6, d_0/a=4.1, {\mathrm{\Theta }}_0={29}^{\circ }$. (b) $M=1,d_0/a=4.1, {\mathrm{\Theta }}_0={29}^{\circ }$. (c) $M=1, d_0=5.4, {\mathrm{\Theta }}_0={22}^{\circ }$. (d) Plots of separation distance vs time for the cases (a) (solid line), (b) (dashed line), and (c) (dot-dashed line). For cases (b) and (c), $d_c/a=5.1$.

Figure 11

(a) Separation distance between drop pairs for the three-drop system. (b) Rotational dynamics between drop pairs for the three-drop system. Angles plotted are $\left|\mathrm{\Theta }\right|$. The vertical dotted line indicates the moment drops 2 and 3 form a doublet, after which the theory is formally inapplicable. The kinks in the theoretical trajectories occur upon predicted formation of the doublet. (c) Photos of the initial and final configuration of drops, with a schematic identifying the vectors plotted in (a) and (b).

Figure 12

(a) Separation distance between drop pairs within the four-drop system. (b) Rotational dynamics between drop pairs within the four-drop system. Angles plotted are $\left|\mathrm{\Theta }\right|$. Here the experiment is stopped upon formation of the doublet between drops 2 and 3, however the theoretical prediction of doublet formation occurs at about 48 s. (c) Photos of the initial and final configuration of drops, with a schematic identifying the vectors plotted in (a) and (b).