Electrohydrodynamic interaction, deformation, and coalescence of suspended drop pairs at varied angle of incidence

The electrohydrodynamic interaction of liquid drop pairs suspended in another immiscible liquid and subjected to uniform electric field is examined using the leaky dielectric model and the explicit forcing lattice Boltzmann method, by taking into account the nonlinear inertia effects. This facilitates explorations of wider parametric contrast, precise electrohydrodynamic interaction, and postcoalescence breakup phenomena that remained unknown. The influence of dielectrophoretic and electrohydrodynamic forces in prolate- and oblate-shaped deformations, coalescences, and repulsive motion of leaky drop pairs appearing at widely varied incidence angle $\left(\alpha \ge 0^{\circ }\right)$ to the applied electric field is studied. The electrically driven flow at $\alpha =0^{\circ }$ evolves in the form of decisively important outflow- and inflow-natured counterrotating vortex pairs in and around the drops. With suitably tuned conductivity $\left({\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}\right)$ and permittivity $\left({\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\right)$ ratios of drop fluid to surrounding outer medium, the relative impacts of attractive electric force versus outflow- or inflow-natured vortex pair-induced hydrodynamic force was optimized to distinctly facilitate the prolate- and the oblate-type deformations of a drop pair, their coalescence, departure, and the postcoalescence breakup. For varied ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}$ over a range $0.25\le {\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\le 20.0$, and using fixed conductivity ratio ${\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=5.0$ and electric capillary number $\mathrm{C}{\mathrm{a}}_E=0.46$ (the ratio of electric force and surface tension), the dipolar electric force is appropriately set with respect to the surrounding four outflow-type outer vortex pair-induced hydrodynamic force to enforce two prolately deformed drops move apart for ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}<{\left({\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\right)}_{\mathrm{crit}1}\approx 2.4$ and coalesce for $2.4<{\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}<9.57$. The low pressure that grew at the neck of near-contact or coalescing drops facilitated the transport of inner fluid into neck region and helped the drop pair's coalescence. For ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}>{\left({\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\right)}_{\mathrm{crit}3}\approx 9.57$, exceeding a critical value, the deformed oblate drop pair moved closer and coalesced; and depending on ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}$ the coalesced drop subsequently broke into a number of satellite or daughter drops spread in a direction perpendicular to the electric field; and such a phenomenon is newly identified. For the nonaligned drop pairs with $\alpha <54.7^{\circ }$ or $\alpha >125.3^{\circ }$, the attractive radial $\left({\mathit{F}}_{\mathit{r}}\right)$ component of dielectrophoretic force drove two drops closer, while the torque produced by tangential $\left({\mathit{F}}_{\mathit{t}}\right)$ components of the electric force made them rotate and align to the electric field upon coalescence. For $54.7^{\circ }<\alpha <125.3^{\circ }$ the dipolar radial force $\left({\mathit{F}}_{\mathit{r}}\right)$ appeared repulsive and drop pair moved apart. Importantly for nonaligned drop pairs both oblate and prolate deformations are noticed, for suitably selected dielectric properties.

Figure 1

Sketch of two identical drops appearing at an angle α to the electric field $E$ and placed at a distance $d$ apart. The radial $\left(F_r\right)$ and tangential $\left(F_t\right)$ components of the induced electric force are shown to act at respective drop centers.

Figure 2

Computation of surface tension for a single drop through Laplace's law. The slope (surface tension) is 0.268.

Figure 3

(a) Computed effect of the electric capillary number $\left(\mathrm{C}{\mathrm{a}}_E\right)$ on the deformation rate $D$ of a drop and comparison between present result and available numerical predictions [12, 50] for ${\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=0.1,{\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=2.0$, 0.54 ≤ Re ≤ 4.09; (b1) simulated variation of $D$ with ${\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}$ and comparison with existing numerical results [14] at ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=10.0,\mathrm{C}{\mathrm{a}}_E=0.18$; (b2) the computed prolate drop deformation (red solid curve is the interface) and electrically induced velocity field at ${\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=14.5,{\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=10.0,\mathrm{C}{\mathrm{a}}_E=0.18$; (b3) computed oblate drop deformation and induced velocity at ${\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=1.81,{\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=10.0,\mathrm{C}{\mathrm{a}}_E=0.18$; (c) sketches of locally grown inflow- and outflow-type vortex pairs and induced influences.

Figure 4

(a) A sketch of two identical liquid drops aligned with the direction of the applied electric field. (b) The computed tangential electric stress $\left(\tau \right)$ along the interface of the left-side droplet for $d_0/R=2.2,{\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=1.0,{\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=10$, and comparison with Stokes solution of Baygents et al. [26] and Sozou [51]. (c) The computed drop velocity $\left(u\right)$ with varied permittivity ratio $\left({\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\right)$ at fixed ${\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=5.0,\mathrm{C}{\mathrm{a}}_E=0.46$. The positive $u$ values represent divergence of two drops, while negative $u$ shows two drops move closer. PR denotes prolate, OB denotes oblate shape; exponents +/− denote two outflow-type outer vortex pairs induced equator to pole and pole to equator flow directions, as shown in insets; div denotes divergence of two drops, and coa denotes coalescence of two drops.

Figure 5

Simulated prolate deformation and relative motion of a drop pair for increased permittivity ratio over $0.5<{\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\le 9.57$, at fixed ${\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=5.0,\mathrm{C}{\mathrm{a}}_E=0.46,\alpha =0^{\circ }$. (a1) The transient divergence of a drop pair at ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=1.0$ [within stretch $A$, ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}<{\left({\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\right)}_{\mathrm{crit}1}=2.4]$, and (a2) induced velocity field at $t=1.8T$; $\mathrm{Re}=1.14$. (b1) The transient electrocoalescence of a drop pair at ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=2.5$, detected in the stretch $B\left[{\left({\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\right)}_{\mathrm{crit}1}=2.4<{\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}<5.0={\left({\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\right)}_{\mathrm{crit}2}\right]$, and (b2) induced velocity field at $t=2.6T$; $\mathrm{Re}=0.71$. (c1) The transitional coalescence of a drop pair at ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=7.0$ [within stretch $C$, ${\left({\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\right)}_{\mathrm{crit}2}=5.0<{\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}<9.57={\left({\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\right)}_{\mathrm{crit}3}]$, and (c2) the evolution of the surrounding velocity field showing the 90° switching of outflow-natured two outer vortex pairs; $\mathrm{Re}=0.57$.

Figure 6

The detected oblate deformation of a drop pair, transient coalescence, and the postcoalescence break up to different modes for increased ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}$ over the stretch D $[{\left({\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\right)}_{\mathrm{crit}3}=9.57<{\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\le 20.0]$, at fixed ${\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=5.0,\mathrm{C}{\mathrm{a}}_E=0.46,\alpha =0^{\circ }$. (d1) The coalescence and subsequent two-component break up at ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=11,\mathrm{Re}=1.71$; (d2) the streamlines at $t=4.2T$ reveal the development of the outflow-natured vertically upward moving outer vortex pair around the upper half of the drop that drives it to elongate in a direction perpendicular to the electric field and facilitates breakup. Similarly, another outflow-natured downward moving near-interfacial outer vortex pair dominates around the bottom half of the drop. (d3) The detected oblate coalescence and three component breakup at ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=15,\mathrm{Re}=2.85$. (d4) The transient coalescence and five component postcoalescing breakup at ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=20,\mathrm{Re}=4.27$; and (d5) surrounding streamline pattern at $t=3.48T$ reveal the active dominance or role of the developed vertically upward moving outflow-natured near-interface outer vortex pair that enforces the droplet to elongate in a direction orthogonal to the electric field due to the paired vortex-induced thrust. (d6) The corresponding velocity field at $t=0.6T$ that facilitates drop pair to move closer and elongate in the vertical direction.

Figure 7

(a1) The simulated pressure $\left(p\right)$ contours; (b1) pressure variation along the horizontal line of symmetry $y=0.5$; (c1) streamline pattern, and near-contact dimple-shaped interface deformation (red solid curve is the interface), as two drops begin to coalesce at ${\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=10.0,{\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=2.5,\mathrm{C}{\mathrm{a}}_E=0.46$. (a2) The simulated pressure $\left(p\right)$ contours; (b2) extracted pressure variation along the line $y=0.5$; (c2) streamline patterns and near-contact Taylor cone formation at the interface (red solid curve is the interface), as two drops just begin to coalesce at a reduced ${\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=5.0$; ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=2.5,\mathrm{C}{\mathrm{a}}_E=0.46$.

Figure 8

Mutual responses of a drop pair aligned at different angles α to the applied electric field: attraction $(•)$; attraction and rotation $(▴)$; repulsion $(\blacksquare )$; repulsion and rotation $(▰)$.

Figure 9

(a) Sketch of the attractive electric force ${\mathit{F}}_{\mathit{e}}$ acting on two drops placed at an angle $\alpha ={35}^{\circ }$ with the direction of electric field; ${\mathit{F}}_{\mathit{e}}$ is the total force, and ${\mathit{F}}_{\mathit{r}},{\mathit{F}}_{\mathit{t}}$ correspond to radial and tangential force components shown to act at drop centers. (b) Simulated transient alignment and coalescence of a drop pair to an oblate shape. (c1) The induced vortical velocity field around the left drop at $t=8.9T$ (red solid curve is the drop interface). (c2) The combined streamline plot at $t=8.9T$ around two drops revealing the physical process of rotating coalescence at $\alpha ={35}^{\circ }$. (d) The velocity vector plot at $t=10.7T$ shows the role of the developed two outflow-natured vortex pairs in the interface expansion process upon merging. ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=0.9,{\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=0.5$ and $\mathrm{C}{\mathrm{a}}_E=0.76,\mathrm{Re}=1.39,d=5R$.

Figure 10

(a) Sketch of a repulsive electric force acting on the drop pair at $\alpha ={60}^{\circ }$; ${\mathit{F}}_{\mathit{e}}$ is the total force, and ${\mathit{F}}_{\mathit{r}},{\mathit{F}}_{\mathit{t}}$ correspond to radial and tangential components shown to act at drop centers. (b) Simulated temporal divergence process of deformed oblate drop pair. (c) The developed velocity field around the left drop at $t=7.9T$ (red solid curve is the drop interface). (d) The combined streamlines at $t=7.9T$, revealing the clockwise rotating divergence of the oblate drop pair at $\alpha ={60}^{\circ }$. ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=0.9,{\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=0.5,\mathrm{C}{\mathrm{a}}_E=0.76,\mathrm{Re}=1.39,d=5R$.

Figure 11

(a) Simulated transient alignment and coalescence of the deformed prolate drop pair at $\alpha ={35}^{\circ }$; (b) simulated divergence of the deformed prolate drop pair at $\alpha ={60}^{\circ }$. ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=0.9,{\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=2.0,\mathrm{Re}=1.48,\mathrm{C}{\mathrm{a}}_E=1.0$. Note that, the physical domain size is much larger, however, for clarity we capture a small part of it.

Figure 12

Using interpolation 1 [Eq. (29)], the computed (a) density, and (b) permittivity (ε) profiles across the drop interface and along the horizontal line of symmetry $y=0.5$. (c) Simulated drop deformation and velocity field obtained by adopting two different interpolation schemes: for the top half the interpolation 1 is used, and for the bottom half the interpolation 2 [Eq. (A1)] is used to calculate the electric properties; (d) computed tangential electric stress obtained by adopting interpolation 1 [Eq. (29)] and interpolation 2 [Eq. (A1)].

Figure 13

Comparison of spurious currents [39] generated by (a) the present explicit forcing model, and (b) the Shan and Chen [33] model, for a static droplet.

Figure 14

The simulated (c1) oblate deformation of a drop pair $\left(\alpha =0^{\circ }\right)$ and the clear nonbreaking electrocoalescence that occurred at a reduced $\mathrm{C}{\mathrm{a}}_E=0.33$, and for ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=15.0,{\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=5.0,\mathrm{Re}=1.94$ [when ${\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}>{\left({\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}\right)}_{\mathrm{crit}3}=9.57]$; and (c2) the induced surrounding velocity field at $t=T$ that facilitate the drop pair to move closer, while the surrounding vortical flow behavior unfolds supporting role of the developed outflow-natured upward and downward moving two dominant outer vortex pairs that dictate a drop to elongate vertically. Note that, at this lower $\mathrm{C}{\mathrm{a}}_E=0.33$ the coalesced drop do not break, as the outflow-type two outer vortex pair could not create sufficient vertically opposite thrust.

Figure 15

Simulated Taylor cone formation for a single high-conductivity leaky drop. ${\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}=30.0,{\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}=1.37$, and $\mathrm{C}{\mathrm{a}}_E=0.68$.