Confinement effect on electrically induced dynamics of a droplet in shear flow

Deformation and breakup of droplets in confined shear flows have been attracting increasing attention from the research community over the past few years, as attributable to their implications in microfluidics and emulsion processing. Reported results in this regard have demonstrated that the primary effect of confinement happens to be the inception of complex oscillating transients, monotonic variation of droplet deformation, and droplet stabilization against breakup, as attributable to wall-induced distortion of the flow field. In sharp contrast to these reported findings, here, we show that a nonintuitive nonmonotonic droplet deformation may occur in a confined shear flow, under the influence of an external electric field. In addition, we demonstrate that the orientation angle of a droplet may either increase or decrease with the domain confinement under the influence of an electric field, whereas the same trivially decreases with the increase in degree of confinement in the absence of any electrical effects. Unlike the typical oscillatory transients observed in microconfined shear flows, we further bring out the possibility of an electrohydrodynamically induced dampening effect in the oscillation characteristics, as governed by a specific regime of the relevant dimensionless electrical parameters. Our results reveal that instead of arresting droplet deformation, the unique hydrodynamics of microconfined shear flow may augment the tendency of droplet breakup, and is likely to alter the droplet breakup mode from midpoint pinching to edge pinching at high electric field strength. These results may bear far reaching implications in a wide variety of applications ranging from the processing of emulsions to droplet based microfluidic technology.

Figure 1

Schematic illustration of the present problem setup, where a liquid droplet having radius $\overline{a}$ is placed in another liquid medium in a confined domain experiencing both the uniform electric field and simple shear flow. The upper and lower walls move in opposite directions with speed $\overline{U}$. Positive and negative electrodes are attached to the upper and lower walls, respectively. The walls are separated by a distance $2\overline{H}$ and ${\overline{E}}_{\infty }$ denotes the magnitude of the imposed electric field. The orientation angle of the droplet between the major semiaxis and the horizontal axis is denoted by ${\theta }_d$.

Figure 2

Variation of $D_{\infty }$ with Ca for (a) a PD-PD system with $S=15$, (b) a LD-LD system with (R, S) = (2, 0.5). Others parameters are $M=1$, $\lambda =1$, Wc = 0.2, and Re = 0.01.

Figure 3

Effect of confinement on the (a) $D_{\infty }$, (b) ${\theta }_{d,\infty }$ of a PD-PD system with $S=2$. Others parameters are $\mathrm{Ca}=0.1$, $\lambda =1$, and $\mathrm{Re}=0.01$.

Figure 4

Variation of (a) shear rate, (b) $E^2$ with the vertical distance at the tip of the droplet ($x=0.85$) for a PD-PD system with $S=2$. Other parameters are $\mathrm{Re}=0.01$, $\lambda =1$, and $M=10$.

Figure 5

Effect of confinement on the (a) $D_{\infty }$, (b) ${\theta }_{d,\infty }$ of a PD-PD system with $S=0.1$. Others parameters are $\mathrm{Ca}=0.1$, $\lambda =1$, and $\mathrm{Re}=0.01$.

Figure 6

Effect of confinement on the (a) $D_{\infty }$, (b) ${\theta }_{d,\infty }$ of a LD-LD system with (R, S) = (2, 0.5). Other parameters are $\mathrm{Ca}=0.1$, $\lambda =1$, and $\mathrm{Re}=0.01$.

Figure 7

Effect of confinement on the (a) $D_{\infty }$, (b) ${\theta }_{d,\infty }$ of a LD-LD system with (R, S) = (10, 1). Other parameters are $\mathrm{Ca}=0.1$, $\lambda =1$, and $\mathrm{Re}=0.01$.

Figure 8

Effect of confinement on the (a) $D_{\infty }$, (b) ${\theta }_{d,\infty }$ of a LD-LD system with (R, S) = (0.033, 0.4397). Others parameters are $\mathrm{Ca}=0.1$, $\lambda =1$, and $\mathrm{Re}=0.01$.

Figure 9

Transient variation of deformation parameter (a) a PD-PD system with $S$ = 15, (b) a LD-LD system with (R, S) = (2, 0.5). Others parameters are $M=1$, $\lambda =1$, Wc = 0.2, and Re = 0.01.

Figure 10

Variation of (a) $L$ and (b) ${\theta }_d$ with $t$ for a PD-PD system with $S=2$. Other parameters are $\mathrm{Ca}=0.4$, $\lambda =1$, $\mathrm{Re}=0.01$, and $\mathrm{Wc}=0.909$. The inset shows the contours of the droplet at steady state for $M=0$ and $M=15$.

Figure 11

Variation of (a) $L$ and (b) ${\theta }_d$ with $t$ for a PD-PD system with $S=0.1$. Other parameters are $\mathrm{Ca}=0.4$, $\lambda =1$, $\mathrm{Re}=0.01$, and $\mathrm{Wc}=0.909$. The inset shows the contour of the droplet at steady state for $M=0$ and $M=7.5$

Figure 12

Variation of (a) $L$ and (b) ${\theta }_d$ with $t$ for a LD-LD system with (R, S) = (0.5, 2). Other parameters are $\mathrm{Ca}=0.4$, $\lambda =1$, $\mathrm{Re}=0.01$, and $\mathrm{Wc}=0.909$. The inset shows the contour of the droplet at steady state for $M=0$ and $M=0.75$.

Figure 13

Variation of (a) $L$ and (b) ${\theta }_d$ with $t$ for LD-LD system with (R, S) = (2, 0.5). Other parameters are $\mathrm{Ca}=0.4$, $\lambda =1$, $\mathrm{Re}=0.01$, and $\mathrm{Wc}=0.909$. The inset shows the contour of the droplet at steady state for $M=0$ and $M=7.5$.

Figure 14

Effect of confinement on droplet breakup for (a) a PD-PD system with $S=15$ and (b) a LD-LD system with (R, S) = (0.5, 2). Other parameters are $\mathrm{Re}=0.01$, $\lambda =1$, and $M=1$.

Figure 15

Variation of (a) $E^2$ and (b) shear rate with the vertical distance at the tip of the droplet ($x=2.11$) for a LD-LD system with (R, S) = (0.5, 2). Other parameters are $\mathrm{Re}=0.01$, $\lambda =1$, and $M=1$.

Figure 16

Transient evolution of droplet shape for (i) $\mathrm{C}{\mathrm{a}}_E=0$, (ii) $\mathrm{C}{\mathrm{a}}_E=1.6$, and (iii) $\mathrm{C}{\mathrm{a}}_E=16$ for a LD-LD system with (R, S) = (2, 0.5). Other parameters are $\mathrm{Re}=0.01$, $\lambda =1$, $\mathrm{Ca}=1.6$, and $\mathrm{Wc}=0.55$.

Figure 17

Regime diagram of breakup and nonbreakup mode in ${\mathrm{Ca}}_E$ versus Wc space. Other parameters are $\mathrm{Re}=0.01$, $\lambda =1$, $\mathrm{Ca}=0.6$, $S=2$, and $R=0.5$.

Figure 18

(a) Variation of steady state deformation parameter with domain confinement for different values of Ca. Other parameters are values are λ = 1 and $\mathrm{Re}=0.05$. (b) Variation of steady state deformation parameter with electric capillary number. Other properties are $R=10$, $S=1.37$, λ = 0.874, and $\mathrm{Re}=0.01$.

Figure 19

Grid and Cn independence study. Deformation of the droplet interface in (a) simple shear flow with Ca = 0.1, $\mathrm{Wc}=0.8$; (b) shear flow with electric field having $S$ = 15, Ca = 0.14, $M$ = 1, Wc = 0.2. (c) Breakup dynamics of the droplet in the combined presence of shear flow and electric field. Other parameters are $\mathrm{Re}=0.01$, $\mathrm{Wc}=0.9$, $\lambda =1$, $\mathrm{Ca}=0.6$, $S=2$, $R=0.5$, and $\mathrm{C}{\mathrm{a}}_E=0.4$.

Figure 20

Comparison of present simulation result and the numerical results of Halim and Esmaeeli [66].The parameters considered are $S=0.5$, λ = 1, $\mathrm{Re}=1,\overline{\rho }=0.5,M=1$, and $\mathrm{C}{\mathrm{a}}_E=0.25$.

Figure 21

Transient variation of deformation parameter for different values of Pe. Other parameters are $\mathrm{Ca}=0.1$, $M=10$, $R=2$, $S=0.5,\mathrm{Cn}=0.01$, λ = 1, and $\mathrm{Re}=0.01$.