Electric field induced droplet deformation and breakup in confined shear flows

A numerical investigation of electrohydrodynamics of an initially spherical droplet suspended in a continuous fluid subjected to shear flow in the presence of an electric field is presented. The numerical framework is based on coupling of a multicomponent lattice Boltzmann method with a leaky dielectric model. Simulations reveal distinct deformation and breakup behavior of the droplet for a fixed channel confinement and for widely different viscosity ratio $\lambda$. For each $\lambda$, defined as ratio of droplet to outer fluid viscosity ($\lambda ={\mu }_d/{\mu }_c$), computations are performed for two specific combinations of electrical properties given by ratios of conductivity $R(={\sigma }_d/{\sigma }_c)$ and permittivity $S(={\varepsilon }_d/{\varepsilon }_c)$. Simulations show that the droplet orients toward the direction of shearing motion for $R<S$, whereas it orients along the direction of applied electric field when $R>S$. For $R>S$, the droplet elongation increases with an increase in electric field and breakup of the droplet into smaller droplets is observed beyond a threshold value. The application of electric field also results in the breakup of highly viscous droplets which otherwise are very difficult to break in shear flows. In contrast, the droplet elongation for $R<S$ is observed to be dependent upon the competing interplay of electric and shear stresses acting at the droplet interface. The cumulative effect of electric field and shear flow alters the shear stress acting at the droplet interface, thereby leading to a deviation in the droplet dynamics when $R<S$.

Figure 1

Schematic of a droplet of initial radius $a$ suspended in a confined Couette flow configuration under the influence of an electric field. The initial droplet profile is represented by a dashed line and the elongated profile is denoted by a solid line. The deformed droplet is oriented at an angle $\theta$ with the horizontal direction. The major and minor axes of the droplet are denoted by $L$ and $B$. The droplet and the outer fluid are assumed to be immiscible and the density of both the fluids is considered to be the same. The top and bottom walls of the domain are separated from each other by a distance $H$ and are moving in the opposite direction. An electric field $\mathbf{E}$ is acting along the vertical direction.

Figure 2

Comparison of droplet deformation $D$ predicted by the analytical solution of Taylor [14] and our numerical simulations.

Figure 3

Comparison of radial and angular velocity components obtained from the present mathematical model and analytical solution for $\theta =\pi /4$.

Figure 4

Shape evolution of a droplet immersed in a confined Couette flow configuration with confinement ratio $\zeta =0.5$ and $\text{Re}=0.1$ in a computational domain of size $25a\times 4a\times 4a$. The droplet and the surrounding fluid were assumed to have equal viscosities. For $\text{Ca}=0.35,$ the droplet deforms into a steady ellipsoidal shape with its axis inclined at an angle with the flow direction. The droplet elongates indefinitely in the case of $\text{Ca}=0.4,$ leading to the binary breakup of the droplet.

Figure 5

Evolution of elongation and orientation angle of the droplet corresponding to case A with $R=10$ and $S=2$, and $\lambda =1$ for [(a) and (b)] $\text{Ca}=0.2$ and [(c) and (d)] $\text{Ca}=0.3$ as ${\text{Ca}}_E$ is varied in the range of 0, 0.1, 0.2, 0.4, and 0.6, respectively. Time $t$ has been normalized using $T=a/u_o$, where $u_o$ is the wall velocity. The confinement ratio is considered to be fixed $\zeta =0.5$. The orientation angle initially increases and after attaining a peak value it decreases. For a fixed Ca, the droplet elongation increases within ${\text{Ca}}_E$ and beyond a critical ${\text{Ca}}_E$ droplet breakup occurs.

Figure 6

A sequence of shape evolution of a leaky dielectric droplet at Ca = 0.3 and $\zeta =0.5$ for (a) ${\text{Ca}}_E$=0, (b) ${\text{Ca}}_E=0.2$, (c) ${\text{Ca}}_E=0.4,$ and (d) ${\text{Ca}}_E=0.6$. The fluids represent case $A$ ($R$ = 10 and $S$ = 2) and are assumed to have equal viscosities. Figure 6 shows the evolution of the droplet into a prolate ellipsoid in the absence of electric field, whereas Fig. 6 depicts the breakup of the droplet into smaller droplets at ${\text{Ca}}_E=0.2$. This indicates that the critical capillary number required for the droplet breakup in the absence of an electric field varies with the application of electric field. With increase in ${\text{Ca}}_E$, the deformation induced in the droplet increases and the droplet breakup transitions from binary to multiple breakup.

Figure 7

Evolution of elongation and orientation angle of the droplet corresponding to case B ($R=0.1$ and $S=0.5$) for [(a) and (b)] $\text{Ca}=0.2$ and [(c) and (d)] 0.3, for ${\text{Ca}}_E$ 0, 0.1, 0.2, 0.4, and 0.6, respectively. The fluids are considered to have equal viscosity and the confinement ratio is considered to be fixed, $\zeta =0.5$. The orientation angle decreases continuously as ${\text{Ca}}_E$ increases. For low values of ${\text{Ca}}_E$, the droplet elongation decreases with increase in ${\text{Ca}}_E$. The steady-state droplet profiles for $\text{Ca}=0.2$ and 0.3 obtained for ${\text{Ca}}_E=0$ and 0.6 are illustrated in the inset images of Figs. 7 and 7.

Figure 8

Effect of viscosity ratio on the (a) deformation and (b) orientation angle of the droplet corresponding to case A ($R>S$) at $\text{Ca}=0.3$. Highly viscous droplets are denoted by filled symbols, whereas less viscous droplets are represented by unfilled symbols. For $\lambda =0.1$ and 10, the droplet elongation and rotation toward the transverse direction increase with increase in ${\text{Ca}}_E$. For $\lambda =0.1$ and 10, the droplet breakup occurs at ${\text{Ca}}_E$ = 0.5 and 0.425, respectively. The inset image of Fig. 8 depicts the droplet profile at steady state and close to droplet breakup for $\lambda =0.1$ and 10.

Figure 9

A sequence of shape evolution of a leaky dielectric droplet at Ca = 0.3, $\lambda =0.1,$ and $\zeta =0.5$ for ${\text{Ca}}_E$ = (a) 0.2, (b) 0.425, and (c) 0.50. The fluids represent case A where $R>S$ ($R$ = 10 and $S$ = 2). Figure 9 shows the evolution of the droplet shape into a prolate ellipsoid with its major axis inclined at an angle with the flow field direction. At ${\text{Ca}}_E=0.425$, the droplet displays a damped oscillatory behavior before attaining a steady shape. Further increase in ${\text{Ca}}_E$ results in the binary breakup of the droplet and indicates a shift in the critical capillary number required for the droplet breakup in the absence of an electric field.

Figure 10

Interface evolution of a leaky dielectric droplet at Ca = 0.3, $\lambda$ = 10, and $\zeta$ = 0.5 for ${\text{Ca}}_E$ = (a) 0.2 and (b) 0.425. The fluids corresponds to case $A$ where $R>S$ ($R$ = 10 and $S$ = 2). Figure 10 shows the transformation of the droplet shape into a prolate ellipsoid with rounded ends. As ${\text{Ca}}_E$ is increased, the droplet elongation and binary breakup of the droplet occurs at ${\text{Ca}}_E=0.425$. The breakup of highly viscous droplet at lower value of ${\text{Ca}}_E$ as compared to less viscous droplet (Fig. 9) indicates that it is easier to break highly viscous droplet in a confined geometry under the combined influence of electric field and shear flow than a less viscous droplet.

Figure 11

Effect of viscosity ratio on the (a) deformation and (b) orientation angle of the droplet corresponding to system $B$ at $\text{Ca}=0.3$. Highly viscous droplets are denoted by filled symbols, whereas less viscous droplet is represented by unfilled symbols. For $\lambda =0.1$ and 10, the droplet alignment toward the horizontal direction continuously increases with increase in ${\text{Ca}}_E$. For $\lambda =0.1$ and low values of ${\text{Ca}}_E$, the droplet elongation decreases with increase in ${\text{Ca}}_E$, whereas for $\lambda =10$ the droplet elongation decreases continuously as ${\text{Ca}}_E$ is increased from 0 to 0.6. The inset image of Fig. 11 depicts the steady-state profiles of the droplet at ${\text{Ca}}_E$ for $\lambda =0.1$ and 10, respectively.

Figure 12

Effect of curvature on the binary breakup of a droplet along the plane normal to the direction of flow field. Figure 12 illustrates the schematic of droplet in $xy$ plane at $z$ = 0 and Fig. 12 shows the variation in droplets minor axis in $z$ = 0 plane with time for $\lambda =0.1$, 1, and 10.

Figure 13

Charge distribution and the resulting electric force distribution acting on the droplet interface in the presence of an electric field. (a) For case A, the upper (lower) half of the droplet is negatively (positively) charged, causing the electric field lines to converge toward (move away) from the upper (lower) half of the droplet interface. (b) For case B, positive (negative) charge resides on the upper (lower) half of the droplet interface, resulting in the divergence (convergence) of the field lines toward the droplet. [(c) and (d)] The resultant electric force ${\mathbf{F}}_E$ for case A (B) acts away (toward) the droplet. The inset image highlights that the resultant electric force can be resolved along the vertical $F_{E,y}$ and horizontal component $F_{E,x}$.

Figure 14

Velocity vectors and contours of pressure developed in fluids corresponding to system $A$ with the variation of ${\text{Ca}}_E$ at $\text{Ca}=0.3$. For all viscosity ratios, with increase in ${\text{Ca}}_E$ the droplet stretches toward the electrode, leading to an increase of pressure near the droplet ends.

Figure 15

Variation of shear rate near the droplet poles with increase in ${\text{Ca}}_E$ for $R>S$ at $\text{Ca}=0.3$ and $\lambda =1$.

Figure 16

Evolution of (A) pressure and velocity vectors and (B) variation of shear rate near the droplet poles at $\text{Ca}=0.3$ for fluids corresponding to case B ($R<S$) with ${\text{Ca}}_E$. For all viscosity ratios, with increase in ${\text{Ca}}_E$ the droplet poles orient toward the horizontal direction, thereby leading to an increase in the gap size between the droplet poles and electrode. As a result, negligible pressure build up occurs near the droplet poles. Further, an increase in ${\text{Ca}}_E$ leads to a decrease in shear rate exerted by continuous fluid on the droplet poles with increase in ${\text{Ca}}_E$.

Figure 17

Schematic illustration of the effective deformation induced in the droplet under the combined influence of electric field and shear flow for (a) $R>S$ and (b) $R<S$.

Figure 18

Effect of confinement on the deformation and orientation induced in a less viscous droplet under the cumulative effect of electric field and shear flow for Ca = 0.3, ${\text{Ca}}_E$ = 0.3, and $\lambda$ = 0.1. The unconfined domain is denoted by unfilled symbols, whereas the filled symbols represent a confined domain. The inset image depict the steady droplet shape obtained in case of confined geometry and the droplet profile prior to breakup in case of the unconfined domain.

Figure 19

Effect of confinement on the deformation and orientation induced in a highly viscous droplet under the cumulative effect of electric field and shear flow for Ca = 0.3, ${\text{Ca}}_E$ = 0.425, and $\lambda$ = 10. The unconfined domain is denoted by unfilled symbols, whereas the filled symbols represent a confined domain. The inset images depict the droplet profile prior to binary breakup obtained in the case of confined geometry and the steady droplet profile obtained in the case of the unconfined domain.