Shape evolution of compound droplet in combined presence of electric field and extensional flow

Studies on compound droplet are of emerging importance in biology and engineering. Here, we bring out unique morphodynamics of a compound droplet as a consequence of interplay between an imposed electric field and an extensional flow. As compared to the deformation characteristics of compound droplet reported in the sole presence of extensional flow [Stone and Leal, J. Fluid Mech. 211, 123 (1990)], our results implicate several nonintuitive findings on the dynamical evolution of the droplet. These exclusive new features include the interconversion of shape-evolution patterns of the compound droplet system contingent on the strength of background flow, electric field strength, electrophysical properties, and hitherto-unveiled post-breakup dynamics. Depending on these key parameters, in addition to three steady-state configurations, two new modes of droplet pinch-off are observed: Mode I: polar pinch-off; Mode II: equatorial pinch-off. Interestingly, the pinch-off time is found to vary nonmonotonically with the strength of the electric field. In sharp contrast to the extensional viscosity of a compound droplet in absence of other external fields [Stone and Leal, J. Fluid Mech. 211, 123 (1990)], we show that the additional presence of electric field brings in an intricate dependence on the electrical properties of the inner droplet; a paradigm that is not prevalent in pure extensional flow.

Figure 1

Schematic representation of the compound droplet subjected to combined presence of uniform electric field and linear uniaxial extensional flow. The radius of the inner droplet and outer droplets are ${\overline{a}}_1$ and ${\overline{a}}_2$.

Figure 2

Schematic representation of the problem set up. The computational domain is axisymmetric with rectangular size $\overline{L}\times \overline{L}$. A cylindrical coordinate system $\left({\overline{r}}_c,\overline{z}\right)$ is considered and attached to the droplet center.

Figure 3

(a) Alteration of $D_{\infty }$ (steady-state deformation) with Ca (capillary number) for compound droplet when Re (Reynolds number) $=0.01$, $K$ (radius ratio) $=0.5$ and ${\lambda }_{12}\left(\mathrm{viscosity}\mathrm{ratio}\right)={\lambda }_{32}$ (viscosity ratio) $=1$. (b) Alteration of $D_{\infty }$ with $\mathrm{C}{\mathrm{a}}_E$ (electric capillary number) for a single droplet at $\mathrm{Re}=0.01$, $R$ (conductivity ratio) = 10, $S$ (permittivity ratio) $=1.37$, λ (viscosity ratio) $=0.874$. (c) Variation of $D_{\infty }$ with $\mathrm{C}{\mathrm{a}}_E$ for a single droplet having $R=2$, $S=1$, $\lambda =10$, $\mathrm{Re}=0.01$. (d) Variation of effective extensional viscosity of dilute emulsion of single droplets with λ for a droplet having $\mathrm{Ca}=0.1$. For single droplet ${\lambda }_{ij}\left[\right(ij)\in \{12,23\left\}\right]$ is expressed as λ. 1, 2, and 3 are used to denote the inner fluid, the outer fluid, and the suspending fluid, respectively.

Figure 4

Alteration of $D_{\infty }$ (steady-state deformation) with Ca (capillary number) for (a) inner droplet (b) outer droplet of a system with $\left(S_{23},R_{23}\right)=\left(2,0.5\right)$ and $\left(S_{12},R_{12}\right)=\left(0.5,2\right)$. Other used parameters are $\mathrm{Re}\left(\mathrm{Reynolds}\mathrm{number}\right)=0.01$, ${\lambda }_{12}\left(\mathrm{viscosity}\mathrm{ratio}\right)={\lambda }_{32}\left(\mathrm{viscosity}\mathrm{ratio}\right)=1$, $M\left(\mathrm{Mason}\mathrm{number}\right)=1$, $K$ (radius ratio) = 0.5. Here, $R_{ij}\left(\mathrm{conductivity}\mathrm{ratio}\right)={\sigma }_i/{\sigma }_j$, $S_{ij}\left(\mathrm{permittivity}\mathrm{ratio}\right)={\varepsilon }_i/{\varepsilon }_j$, and ${\lambda }_{ij}\left(\mathrm{viscosity}\mathrm{ratio}\right)={\mu }_i/{\mu }_j$, where $\left(ij\right)\in \left\{12,23\right\}$. 1, 2, and 3 are used to denote the inner fluid, outer fluid, and suspending fluid, respectively. The viscosity, electrical conductivity, and electrical permittivity are denoted by μ, σ, and ε, respectively.

Figure 5

Variation of $D_{\infty }$ (steady-state deformation) with $M$ (Mason number) for (a) inner (b) outer droplet of a leaky dielectric system when Ca (capillary number) = 0.1, $\mathrm{Re}\left(\mathrm{Reynolds}\mathrm{number}\right)=0.01$, ${\lambda }_{12}\left(\mathrm{viscosity}\mathrm{ratio}\right)={\lambda }_{32}\left(\mathrm{viscosity}\mathrm{ratio}\right)=1$, $K\left(\mathrm{radius}\mathrm{ratio}\right)=0.5$, $R_{12}\left(\mathrm{conductivity}\mathrm{ratio}\right)={\left(R_{23}\right)}^{-1}$, and $S_{12}\left(\mathrm{permittivity}\mathrm{ratio}\right)={\left(S_{23}\right)}^{-1}$.  Here, $R_{ij}={\sigma }_i/{\sigma }_j$, $S_{ij}={\varepsilon }_i/{\varepsilon }_j$, and ${\lambda }_{ij}={\mu }_i/{\mu }_j$, where $\left(ij\right)\in \left\{12,23\right\}$. 1, 2, and 3 are used to denote the inner fluid, outer fluid, and suspending fluid, respectively. The viscosity, electrical conductivity, and electrical permittivity are denoted by μ, σ, and ε, respectively.

Figure 6

Alteration of $D_{\infty }$ (steady-state deformation) with $M$ (Mason number) for different leaky dielectric systems having (a) $\left(R_{23},S_{23}\right)=\left(0.5,2\right)$, $\left(R_{12},S_{12}\right)=\left(2,0.5\right)$, where $R_{23}<S_{23}$ and $R_{12}>S_{12}$ and (b) $\left(R_{23},S_{23}\right)=\left(2,0.5\right)$, $\left(R_{12},S_{12}\right)=\left(0.5,2\right)$, where $R_{23}>S_{23}$ and $R_{12}<S_{12}$. The droplet shapes for different values of $M$ are shown in the inset of Figs. 6 and 6. Important simulation parameters are capillary number $\left(\mathrm{Ca}\right)=0.05$, Reynolds number $\left(\mathrm{Re}\right)=0.01$, radius ratio $\left(K\right)=0.5$, and viscosity ratio $\left({\lambda }_{12}\right)=\left({\lambda }_{32}\right)=1$. Here, $R_{ij}$ (conductivity ratio) $={\sigma }_i/{\sigma }_j$, $S_{ij}$ (permittivity ratio) $={\varepsilon }_i/{\varepsilon }_j$ and ${\lambda }_{ij}$ (viscosity ratio) $={\mu }_i/{\mu }_j$, where $\left(ij\right)\in \left\{12,23\right\}$. 1, 2, and 3 are used to denote the inner fluid, outer fluid, and suspending fluid. The viscosity, electrical conductivity, and electrical permittivity are denoted by μ, σ, and ε, respectively.

Figure 7

Regime plot shows different patterns of droplet shape for different values of Ca (capillary number) and $\mathrm{C}{\mathrm{a}}_E$ (electric capillary number). Mode I: polar pinch-off; Mode II: equatorial pinch-off; Steady state I: Outer droplet is oblate shaped and inner droplet is prolate shaped; Steady state II: Both the droplets are prolate shape. Others parameters are $\left(R_{23},S_{23}\right)=\left(0.5,2\right)$, $\left(R_{12},S_{12}\right)=\left(2,0.5\right)$, $K$ (radius ratio) $=0.5$, Re (Reynolds number) = 0.01 and ${\lambda }_{12}$ (viscosity ratio) $={\lambda }_{32}$ (viscosity ratio) = 1. Here, $R_{ij}$ (conductivity ratio) $={\sigma }_i/{\sigma }_j$, $S_{ij}$ (permittivity ratio) $={\varepsilon }_i/{\varepsilon }_j$, and ${\lambda }_{ij}$ (viscosity ratio) $={\mu }_i/{\mu }_j$, where $\left(ij\right)\in \left\{12,23\right\}$. The viscosity, electrical conductivity, and electrical permittivity are denoted by μ, σ, and ε, respectively. 1, 2, and 3 are used to denote the inner fluid, outer fluid, and suspending fluid, respectively.

Figure 8

Variation (a) $E^2$ (electric field squared) at Ca (capillary number) = 0.1 and (b) ${\dot{S}}_R$ (shear rate) along a probe drawn from the center of the droplet to the upper electrode. Other parameters are $\left(R_{23},S_{23}\right)=\left(0.5,2\right)$, $\left(R_{12},S_{12}\right)=\left(2,0.5\right)$, $K$ (radius ratio) = 0.5, Re (Reynolds number) = 0.01, and ${\lambda }_{12}$ (viscosity ratio) $={\lambda }_{32}=1$. Here, $R_{ij}s$ (conductivity ratio) $={\sigma }_i/{\sigma }_j$, $S_{ij}$ (permittivity ratio) $={\varepsilon }_i/{\varepsilon }_j$, and ${\lambda }_{ij}$ (viscosity ratio) $={\mu }_i/{\mu }_j$, where $\left(ij\right)\in \left\{12,23\right\}$. 1, 2, and 3 are used to denote the inner fluid, outer fluid, and suspending fluid, respectively. The viscosity, electrical conductivity, and electrical permittivity are denoted by μ, σ, and ε, respectively.

Figure 9

(a) Regime plot shows different patterns of droplet breakup for different values of $\mathrm{C}{\mathrm{a}}_E$ (electric capillary number) and $R$ (conductivity ratio). Mode I: polar pinch-off; Mode II: equatorial pinch-off; Steady state I: Outer droplet is oblate shaped and inner droplet is prolate shaped; Steady state II: Both the droplets are prolate shape; Steady state III: Outer droplet is prolate shaped and the inner droplet is oblate shaped. (b) Variation $E^2$ (electric field squared) along a probe drawn from the center of the droplet to the upper electrode. Others parameters are $S_{23}$ (permittivity ratio) = 2, Ca (capillary number) = 0.35, λ (viscosity ratio) = 1, $\mathrm{C}{\mathrm{a}}_E$ (electric capillary number) = 0.15, $K$ (radius ratio) = 0.5, Re (Reynolds number) = 0.01, $\left(R\right)=R_{23}={\left(R_{12}\right)}^{-1}$ and $S_{23}={\left(S_{12}\right)}^{-1}$. Here, $R_{ij}$ (conductivity ratio) $={\sigma }_i/{\sigma }_j$, $S_{ij}$ (permittivity ratio) $={\varepsilon }_i/{\varepsilon }_j$ and ${\lambda }_{ij}$ (viscosity ratio) $={\mu }_i/{\mu }_j$, where $\left(ij\right)\in \left\{12,23\right\}$. 1, 2, and 3 are used to denote the inner fluid, outer fluid, and suspending fluid, respectively. The viscosity, electrical conductivity, and electrical permittivity are denoted by μ, σ, and ε, respectively.

Figure 10

Influence of electric field on the pinch-off time of the droplet. Others parameters are $R_{23}\left(\mathrm{conductivity}\mathrm{ratio}\right)={\left(R_{12}\right)}^{-1}$ and $S_{23}\left(\mathrm{permittivity}\mathrm{ratio}\right)={\left(S_{12}\right)}^{-1}$, ${\lambda }_{12}\left(\mathrm{viscosity}\mathrm{ratio}\right)={\lambda }_{32}=1$, $K$ (radius ratio) = 0.5, Ca (capillary number) = 0.35, and Re (Reynolds number) = 0.01. Here, $R_{ij}={\sigma }_i/{\sigma }_j$, $S_{ij}={\varepsilon }_i/{\varepsilon }_j$, and ${\lambda }_{ij}={\mu }_i/{\mu }_j$, where $\left(ij\right)\in \left\{12,23\right\}$. 1, 2, and 3 are used to denote the inner fluid, outer fluid, and suspending fluid. The viscosity, electrical conductivity, and electrical permittivity are denoted by μ, σ, and ε, respectively.

Figure 11

Influence of electric field on the post pinch-off behavior of the droplet. Others parameters are $\left(R_{23},S_{23}\right)=\left(0.1,2\right)$, $\left(R_{12},S_{12}\right)=\left(10,0.5\right)$, Ca (capillary number) = 0.35, $K$ (radius ratio) = 0.5, Re (Reynolds number) = 0.01, and viscosity ratio $\left({\lambda }_{12}\right)={\lambda }_{32}=1$ . $\mathrm{C}{\mathrm{a}}_E$ is the electric capillary number. Here, $R_{ij}$ (conductivity ratio) $={\sigma }_i/{\sigma }_j$, $S_{ij}$ (permittivity ratio) $={\varepsilon }_i/{\varepsilon }_j$, and ${\lambda }_{ij}$ (viscosity ratio) $={\mu }_i/{\mu }_j$, where $\left(ij\right)\in \left\{12,23\right\}$. 1, 2, and 3 are used to denote the inner fluid, outer fluid, and suspending fluid. The viscosity, electrical conductivity, and electrical permittivity are denoted by μ, σ, and ε, respectively.

Figure 12

Distribution of (a) uniaxial extensional flow at $\mathrm{C}{\mathrm{a}}_E$ (electric capillary number) = 0, (b) EHD flow at $\mathrm{C}{\mathrm{a}}_E=0.4$, (c) electric force distribution at $\mathrm{C}{\mathrm{a}}_E=0.4$. The arrow size denotes the strength of the respective variable. Other parameters are $\left(R_{23},S_{23}\right)=\left(0.1,2\right)$, $\left(R_{12},S_{12}\right)=\left(10,0.5\right)$, Ca (capillary number) = 0.35, $K$ (radius ratio) = 0.5, Re (Reynolds number) = 0.01, and ${\lambda }_{12}$ (viscosity ratio)$={\lambda }_{32}=1$. Here, $R_{ij}$ (conductivity ratio) $={\sigma }_i/{\sigma }_j$, $S_{ij}$ (permittivity ratio) $={\varepsilon }_i/{\varepsilon }_j$, and ${\lambda }_{ij}$ (viscosity ratio) $={\mu }_i/{\mu }_j$, where $\left(ij\right)\in \left\{12,23\right\}$. The viscosity, electrical conductivity, and electrical permittivity are denoted by μ, σ, and ε, respectively. 1, 2, and 3 are used to denote the inner fluid, the outer fluid, and the suspending fluid.

Figure 13

Variation of extensional viscosity with viscosity ratio for leaky dielectric having Ca (capillary number) $=0.1$. (b) Deformation of the outer interface $\left(D_{\infty ,23}\right)$ vs. viscosity ratio (λ) for the leaky dielectric system at $M$ (Mason number) $=3$. Other parameters are $\left(R_{23},S_{23}\right)=\left(0.1,2\right)$, $K$ (radius ratio) = 0.5, Ca (capillary number) = 0.1, and Re (Reynolds number) = 0.01, λ (viscosity ratio) $={\lambda }_{12}={\lambda }_{23}$. Here, $R_{ij}$ (conductivity ratio) $={\sigma }_i/{\sigma }_j$, $S_{ij}$ (permittivity ratio) $={\varepsilon }_i/{\varepsilon }_j$, and ${\lambda }_{ij}$ (viscosity ratio) $={\mu }_i/{\mu }_j$, where $\left(ij\right)\in \left\{12,23\right\}$. 1, 2, and 3 are used to denote the inner fluid, outer fluid, and suspending fluid. The viscosity, electrical conductivity, and electrical permittivity are denoted by μ, σ, and ε, respectively.