Breakup of a leaky dielectric drop in a uniform electric field

Electrohydrodynamic (EHD) breakup phenomena for a leaky dielectric drop suspended in another immiscible viscous dielectric and subjected to a uniform electric field are examined using the leaky dielectric theory and the explicit forcing lattice Boltzmann method, by taking into account full nonlinear inertia effects. The breakup modes are first computed for varied conductivity of the drop fluid, as the viscosity ratio $\lambda$ $\left(={\mu }_{\mathrm{in}}/{\mu }_{\mathrm{out}}\right)$ is momentarily set to unity, that is, for the slightly conducting $\left(R={\sigma }_{\mathrm{in}}/{\sigma }_{\mathrm{out}}<10\right)$, moderately conducting $\left(10\le R\le 20\right)$, and highly conducting $\left(R>20\right)$ cases. For slightly conducting drops $\left(R=5\right)$ only one breakup mode via two symmetrical necks persists for permittivity ratios $0.05<Q={\varepsilon }_{\mathrm{in}}/{\varepsilon }_{\mathrm{out}}<3.0$ and electric capillary number ${\mathrm{Ca}}_E>\mathrm{C}{\mathrm{a}}_{E,\mathrm{critical}}$ (ratio of electric and surface tension forces), despite significant length-scale variation of mother and daughter drops. At higher $Q$ (for increased drop permittivity) two necks move closer to the bulbous midpart of the extended droplet, which helps enlarge two daughter drops. However, in the case of moderately conducting drops $\left(10\le R\le 20\right)$ the number of necks increased to four for increased ${\mathrm{Ca}}_E$. Accordingly two pairs of symmetrical daughter drops are created because of recurrent fluctuations of the electrical shear stress and centerline momentum flux. For highly conducting cases of $R>20$, depending on ${\mathrm{Ca}}_E$, three distinctly elongated droplet states are formed prior to breakup, which results in the onset of three different breakup modes, namely, via formations of lobed ends $\left({\mathrm{Ca}}_E\le 0.264\right)$, pointed ends $\left({\mathrm{Ca}}_E\le 0.68\right)$, and nonpointed ends $\left({\mathrm{Ca}}_E>0.83\right)$. While being consistent with past measurements, here we precisely characterize the associated breakup mechanisms and physics in terms of the interactive electric pressure, electric shear stress, and hydrodynamic pressure plus velocity gradients. Since the EHD drop breakup is a dynamic process, on an elongated slender drop the activated locally distinct driving forces, i.e., electric pressure at the end regions and tangential electric stress in the midsection, effectively lead to neck formations by virtue of the created high centerline velocity gradient. Accordingly, resulting variations of local extension rate and net mass flux toward drop ends or into intermediate bulbous regions facilitate the multiple-mode drop breakup via the inertia effect, whereas the developed negative curvature around a neck encourages capillary breakup. We also explicitly reveal the effect of the viscosity contrast $\lambda$, which particularly influences the breakup characteristics over a broader range of conductivity ratios.

Figure 1

Schematic of a leaky dielectric drop of radius $a$ suspended in another immiscible leaky dielectric fluid and subjected to a uniform electric field $\vec{E}$.

Figure 2

Computed surface tension for a single drop by Laplace's law. The slope (surface tension) is 0.27.

Figure 3

Comparison of the present results, creeping flow solution of Lac and Homsy [23], and Navier-Stokes solution of Nganguia et al. [31] for the ${\mathrm{Ca}}_E$-dependent stable drop deformation ($D$), as permittivity ratio $Q$ is varied at fixed $R=10$, $\lambda =1$. Computed different cases are $Q=0.1$ ($0.135\le \mathrm{Re}\le 0.46$), $Q=1.37$ ($0.058\le \mathrm{Re}\le 0.468$), and $Q=5.0$ ($0.0684\le \mathrm{Re}\le 0.411$).

Figure 4

Phase diagram for the EHD drop deformation and breakup in ($R$, ${\mathrm{Ca}}_E$) space; $\lambda =1$, $Q=1.37$. The dashed line is the boundary between $stable$ (lower region) and $unstable$ deformation regimes. SPN and MPN denote single and multiple pairs of pinching neck. LF, PE, and NPE correspond to varying breakup modes via lobe formation, pointed end, and nonpointed end, respectively.

Figure 5

(a) The time-dependent evolution and breakup process of the slightly conducting drop at $R=5.0$, $Q=0.05$, $\lambda =1.0$, ${\mathrm{Ca}}_E=1.3$, and $\mathrm{Re}=2.58$. (b1) Near-field instantaneous flow behavior around an elongated droplet during neck formation; $t=4.5T$; (b2), (b3) extracted pressure and velocity distribution along the drop centerline at $t=4.5T$; (b4), (b5) profiles of cross-sectional $u$-velocity magnitude at $x=1.45$ (representative of drop-midsection region) and $x=0.98$ (represents drop-end region) at $t=4.5T$. (c1), (c2) The nondimensional tangential electric stress and electric pressure distribution on drop surface at $t=4.5T$. The tangential electric stress directed toward the left is taken as positive.

Figure 6

Influence of $Q(>1)$ on the breakup of a slightly conducting drop. (a) The time-dependent evolution of the drop shape in the breakup process at $R=5$, $Q=1.37$, $\lambda =1$, ${\mathrm{Ca}}_E=1.3$, and $\mathrm{Re}=1.9$. (b1) Near-field instantaneous flow behavior around an elongated droplet during neck formation, $t=5T$; (b2), (b3) extracted instantaneous pressure and velocity distribution along the drop centerline at $t=5T$. (c1), (c2) The nondimensional tangential electric stress and electric pressure distribution on drop surface at $t=5T$.

Figure 7

Temporal evolution and breakup of an intermediately conducting drop at $R=10$, $Q=1.37$, $\lambda =1$, ${\mathrm{Ca}}_E=0.55$, and $\mathrm{Re}=0.65$.

Figure 8

Impact of increased ${\mathrm{Ca}}_E$ and resulting multicomponent breakup of an intermediately conducting drop. (a) Time-dependent drop shape evolution in the process of breakup at $R=10$, $Q=1.37$, $\lambda =1$, ${\mathrm{Ca}}_E=0.83$, and $\mathrm{Re}=0.98$. (b1) The near-field flow behavior around an elongated drop during neck formation at $t=4T$; (b2), (b3) extracted pressure and velocity distribution along the drop centerline at $t=4T$. (b4)–(b6) Profiles of cross-sectional $u$-velocity magnitude at $x=1.46$, $x=1.04$, and $x=0.78$; $t=4T$. (c1), (c2) Distributions of nondimensional tangential electric stress and electric pressure on drop surface at $t=4T$.

Figure 9

(a) Transient evolution and breakup of a highly conducting drop at $R=30$, $Q=1.37$, $\lambda =1$, ${\mathrm{Ca}}_E=0.264$, and $\mathrm{Re}=0.145$. (b1) Near-field flow behavior around an elongated drop during neck formation at $t=3.5T$; (b2), (b3) extracted pressure and velocity distribution along the drop centerline at $t=3.5T$. (c1), (c2) Distributions of nondimensional tangential electric stress and electric pressure on the drop surface at $t=3.5T$.

Figure 10

Impact of varied ${\mathrm{Ca}}_E$ on modifying edge shape of a highly conducting drop. Deformed drop shape prior to onset of breakup, near-field flow behavior, and distribution of imposed electric stress along the right upper-half surface at $R=30$, $Q=1.37$, $\lambda =1$. (a1), (a2) ${\mathrm{Ca}}_E=0.68$, $\mathrm{Re}=0.374$; (b1), (b2) ${\mathrm{Ca}}_E=0.83$, $\mathrm{Re}=0.457$.

Figure 11

Impact of reduced drop viscosity on breakup mode of an intermediately conducting drop. (a) Transient evolution of a drop in the process of breakup at $R=10$, $Q=1.37$, $\lambda =0.087$, ${\mathrm{Ca}}_E=0.89$, and $\mathrm{Re}=0.015$. Close-up views of the neck region and induced flow: (b1) profile of the neck and generated velocity field prior to pinch-off, $t=14T$; (b2) shapes of a daughter drop and curved edge of the mother drop following pinch-off, and near-field streamline pattern; $t=14.5T$.

Figure 12

Multilobed evolution and breakup of an intermediately conducting highly viscous leaky drop. (a) Transient evolution of drop shape in the process of breakup at $R=10$, $Q=1.37$, $\lambda =5$, ${\mathrm{Ca}}_E=0.67$, and $\mathrm{Re}=0.73$. Close-up views of the neck region and induced flow: (b1) profile of a longer neck and generated velocity field prior to pinch-off; $t=4.5T$; (b2) developed pointed edge shape of the mother drop following pinch-off, while the streamline pattern reveals dominance of an $inflow$ type near-edge vortex pair in the mother drop upon breakup; $t=5T$.

Figure 13

Impact of reduced viscosity on the breakup pattern of a highly conducting drop. Time-dependent droplet evolution in the process of breakup at $R=30$, $Q=1.37$, $\lambda =0.087$, ${\mathrm{Ca}}_E=0.61$, and $\mathrm{Re}=0.0047$.

Figure 14

Impact of increased drop viscosity on the breakup pattern of a highly conducting drop. (a) Transient drop evolution in the process of breakup at $R=30$, $Q=1.37$, $\lambda =3$, ${\mathrm{Ca}}_E=0.4$, and $\mathrm{Re}=0.11$; (b) deformed drop shape with conical ends that occurred prior to breakup at higher ${\mathrm{Ca}}_E=0.6$ and $R=30$, $Q=1.37$, $\lambda =3$, $\mathrm{Re}=0.165$.

Figure 15

(a) Invariance of breakup sequence at $R=30.0$, $Q=1.37$, $\lambda =1.0$, and ${\mathrm{Ca}}_E=0.264$ (with respect to Fig. 9) on a refined lattice spacing, where, $T^{\prime }=100\mathrm{\Delta }{t}^{\prime }=0.75T$. (b1), (b2) Close-ups of grid resolution in the neck region for two different adopted lattices. (c) The neck interface for two different lattices, as defined by $C_{\mathrm{in}}=0.5$ curves. (d) Coincidence of neck shape for two different lattices, at a near threshold $C_{\mathrm{in}}=0.85$.