Numerical study on electrohydrodynamic multiple droplet interactions

We present a numerical study of inviscid multiple droplet coalescence and break-up under the action of electric forces. Using an embedded potential flow model for the droplet hydrodynamics, coupled with an unbounded exterior electrostatic problem, we are able to perform computations through various singular events and analyze the effects of the electrical field intensity on droplet interactions. Laboratory experiments on the electrodynamics of droplet pairs show a much richer, and sometimes unexpected, behavior than that of isolated droplets. For example, it has been found that opposite charged droplets tend to repel each other when the electric field intensity is above a certain critical value. Although the mathematical model employed in this work incorporates very simple flow and electric assumptions, many of the droplet coalescence patterns seen in laboratory experiments can be reproduced. In this model, the interaction pattern of two droplets of radii $R_0$ separated a distance $D_0$, depends on the ratio $X_0=D_0/R_0$ and the applied uniform electric field intensity, $E_{\infty }$. By performing a vast number of numerical simulations we are able to characterize the coalescence modes before and after drop merging as a function of these two parameters. The simulations predict that droplet repulsion occurs within a narrow interval of $E_{\infty }$ values, different for each $X_0$. Surprisingly, in this $E_{\infty }$ interval, a sharp transition between two power-law precoalescence flow regimes is seen. The evolution of several flow characteristics before and after coalescence, and the shape of the deformed droplets at coalescing time and the double cone angle, are also addressed and analyzed in detail. Cone angles below ${35}^{\circ }$ lead to droplet coalescence for any $X_0$ value, which is in accordance with previously reported studies. Finally, it is shown that the model and algorithm can handle multiple droplet interactions. The simulations qualitatively match results from water in oil experiments in microchannels, despite the fact that the exterior fluid is not considered in the mathematical model.

Figure 1

Physical sketch of the droplet pair geometry. Left: Two spherical droplets of radius $R_0$ at initial time. Right: The deformed droplets with aspect ratio $a/b$. Points on the right droplet, $(0,z_C)$ the droplet center, and $(0,z_A)$ the droplet left end point, are depicted.

Figure 2

Coalescence modes on ($X_0,E_{\infty }$) space. Each point on the graph corresponds to a numerical simulation. For $X_0=0,0.4,1,1.67$ the postcoalescence sequence for increasing values of $E_{\infty }$ is oscillation, satellites, and jetting. The repulsion mode appears for $X_0=1$ and $X_0=1.67$ at different $E_{\infty }$ intervals.

Figure 3

Evolution of the droplets aspect ratio $a/b$ for various electric fields intensities $E_{\infty }$. From top to bottom the graphs correspond to $X_0=0,1,1.67$, respectively. For $X_0=0.4$ the precoalesced droplets do not oscillate. The repulsion mode appears between precoalescence oscillation and precoalescence pure deformation. The curve colors and line styles represent the corresponding postcoalescence modes.

Figure 4

Left: Evolution of the droplets center axial coordinate $z_C$ for various electric field intensities and indicated $X_0$ values. Right: Same for the evolution of the drop left point $z_A$. The curves color and line styles represent the corresponding postcoalescence modes.

Figure 5

Left: Coalescence times $t_c$ versus $E_{\infty }$. Right: Corresponding log-log plot with the exponents of the linear fittings. The log is natural logarithm. For $X_0=1,1.67$ there is a curve jump between the two distinct flow regimes.

Figure 6

Double cone angle $\beta$ versus $E_{\infty }$, for $X_0=0.4,1,1.67.$

Figure 7

Average droplet velocity $V_M=D_0/t_c$ versus $E_{\infty }$ for $X_0=0.4,1,1.67$. For $X_0=1,1.67$ the average velocity is $V_M\approx 0.5$ at the curve jumps.

Figure 8

Calculated cone angles at contact points for $X_0=1.67$ and $E_{\infty }=0.2,0.4,0.6.$

Figure 9

Zoomed drops facing shapes for $E_{\infty }=0.7$. The individual droplets are about to emit jets and an air bubble is trapped inside the fluid.

Figure 10

Aspect ratio $a/b$ evolution after contact for $X_0=0$ at indicated $E_{\infty }$ values. The postcoalescence sequence is oscillation, satellites, and jetting.

Figure 11

Evolution of the postcoalesced droplets aspect ratio $a/b$ for various electric fields intensities. Each graph specifies the corresponding $X_0$ value.

Figure 12

Front profiles at various times for $X_0=1.67$ and $E_{\infty }=0.29.$ At $t=2.2144$ two opposite charged satellite droplets pinch-off.

Figure 13

Aspect ratio evolution $a/b$ for $E_{\infty }=0.4$ before and after coalescence time (black) and drop volume evolution (blue). Jet ejection frequencies can be seen in the inset figure. At $t=1.0409$ coalescence takes place and conservation of volume holds after passing through the singularity.

Figure 14

Front profiles at various times for $X_0=1.67$ and $E_{\infty }=0.4$. At $t=1.3746$ the protruding cylinders develop Taylor cones from which tip streaming occurs.

Figure 15

Front profiles at various times for $X_0=1.67$ and $E_{\infty }=0.45$. After first contact at $t=0.3654$ the droplets recoil and fail to coalesce. The droplets carry opposite charges and at $t=0.5673$ jet emission takes place.

Figure 16

Comparison between flow variables for repulsion, $E_{\infty }=0.385$, versus coalescence, $E_{\infty }=0.3$ at contact time. The rescaled separation is $X_0=1$. Panel (a) shows that the repulsion front profile is more deformed than the coalescence profile and panel (b) shows velocity potentials. In panel (c) the opposite axial velocity behavior is depicted and panel (d) shows the opposite curvature signs at contact point.

Figure 17

Coalescence times $t_c$ versus $E_{\infty }$ for $X_0=0.8$. The curves corresponding to $X_0=0.4,1,1,67$ has been included as a reference. The transition zone between regimes occurs in the interval $E_{\infty }\in [0.3,0.32]$ where recoiling takes place.

Figure 18

Six droplets aspect ratio $a/b$ evolution for $E_{\infty }=0.4$ before first coalescence. Droplets 1 and 6 undergo same deformation. Same behavior for droplets 2 and 5 and droplets 3 and 4.

Figure 19

Three-dimensional renderings of six droplets front profiles for $E_{\infty }=0.4$ at the following times: (a) $t=0$, (b) $t=1$, (c) $t=1.2$, (d) $t=1.2858$, (e) $t=1.3001$, (f) $t=1.3648$, (g) $t=1.3985$, and (h) $t=1.4099$. The horizontal axis is $z\in [-4,4]$ and the vertical axis is $r\in [-0.5,0.5]$. Note that the coalescence front propagates inward.

Figure 20

Nine droplets aspect ratio $a/b$ evolution for $E_{\infty }=0.4$ before first coalescence. All droplets deform differently.

Figure 21

Three-dimensional renderings of nine droplets front profiles for $E_{\infty }=0.4$ at the following times: (a) $t=0$, (b) $t=0.0281$, (c) $t=0.0488$, (d) $t=0.0785$, (e) $t=0.0850$, (f) $t=0.0910$, (g) $t=0.1176$, and (h) $t=0.1213$. The horizontal axis is $z\in [-5,4]$ and the vertical axis is $r\in [-0.5,0.5]$. Coalescence fronts start propagating where the droplets are closer.