Electrohydrodynamics of a viscous drop with inertia

Most of the existing numerical and theoretical investigations on the electrohydrodynamics of a viscous drop have focused on the creeping Stokes flow regime, where nonlinear inertia effects are neglected. In this work we study the inertia effects on the electrodeformation of a viscous drop under a DC electric field using a novel second-order immersed interface method. The inertia effects are quantified by the Ohnesorge number $\text{Oh}$, and the electric field is characterized by an electric capillary number ${\text{Ca}}_E$. Below the critical ${\text{Ca}}_E$, small to moderate electric field strength gives rise to steady equilibrium drop shapes. We found that, at a fixed ${\text{Ca}}_E$, inertia effects induce larger deformation for an oblate drop than a prolate drop, consistent with previous results in the literature. Moreover, our simulations results indicate that inertia effects on the equilibrium drop deformation are dictated by the direction of normal electric stress on the drop interface: Larger drop deformation is found when the normal electric stress points outward, and smaller drop deformation is found otherwise. To our knowledge, such inertia effects on the equilibrium drop deformation has not been reported in the literature. Above the critical ${\text{Ca}}_E$, no steady equilibrium drop deformation can be found, and often the drop breaks up into a number of daughter droplets. In particular, our Navier-Stokes simulations show that, for the parameters we use, (1) daughter droplets are larger in the presence of inertia, (2) the drop deformation evolves more rapidly compared to creeping flow, and (3) complex distribution of electric stresses for drops with inertia effects. Our results suggest that normal electric pressure may be a useful tool in predicting drop pinch-off in oblate deformations.

Figure 1

Sketch of the problem: A leaky dielectric viscous drop (in domain ${\mathrm{\Omega }}^{-}$) immersed in another leaky dielectric fluid (in domain ${\mathrm{\Omega }}^{+}$), with an external electric field $E_0$ in the $z$ direction. Subscript “+” and “−” denote the exterior and interior fluids, respectively.

Figure 2

Convergence results for the proposed IIM. ($•$) are simulations using the FEM implementation for the base state from Supeene et al. [8]. The ratio is shown as negative values, with the $(-)$ sign denoting oblate deformation.

Figure 3

Drop deformation $D$ as a function of electric capillary number ${\text{Ca}}_E$ for prolate ($D>0$) and oblate ($D<0$) drops. For the prolate drop conductivity and permittivity ratios are ${\sigma }_r=10,{\varepsilon }_r=0.1$. For the oblate drop ${\sigma }_r=0.1,{\varepsilon }_r=2$. The comparisons show results from the IIM and BIM [7] ($\diamond$) simulations, as well as the prediction from the spheroidal model [5, 15] (dash-dotted). The Ohnesorge numbers are $\text{Oh}=0.459$ (solid) and $\text{Oh}=0.1836$ (dashed).

Figure 4

Comparison between creeping flow (left panel) and Navier-Stokes flow (right panel) for steady-state shape of upper half of drop. The red arrows represent the electric force acting on the drop interface. (a, b) Prolate drop: ${\sigma }_r=10,{\varepsilon }_r=0.1$. The Ohnesorge and electric capillary numbers are (a) $\text{Oh}=0.1836,{\text{Ca}}_E=0.32$, and (b) $\text{Oh}=0.459,{\text{Ca}}_E=0.32$. (c) Oblate drop: ${\sigma }_r=0.1,{\varepsilon }_r=2$. The Ohnesorge and electric capillary numbers are $\text{Oh}=0.459,{\text{Ca}}_E=0.253$.

Figure 5

Normal (a) and tangential (b) electric stresses for Fig. 4. (prolate drop). ${\text{Ca}}_E=0.32,\text{Oh}=0.1836$ (solid) and $\text{Oh}\to \infty$ (dashed, limit of creeping flow). In the legend, $\mathit{St}$ denotes Stokes, and NSt denotes Navier-Stokes.

Figure 6

Normal (a) and tangential (b) electric stresses for Fig. 4 (oblate drop). ${\text{Ca}}_E=0.253,\text{Oh}=0.459$ (solid) and $\text{Oh}\to \infty$ (dashed, limit of creeping flow). In the legend, $\mathit{St}$ denotes Stokes, and NSt denotes Navier-Stokes.

Figure 7

Comparison between creeping flow (left panel) and Navier-Stokes flow (right panel) for steady-state shape of upper half of drop. (a) Prolate “A”: ${\sigma }_r=0.1,{\varepsilon }_r=0.04$; the electric capillary and Ohnesorge numbers are ${\text{Ca}}_E=15,\text{Oh}=0.459$. (b) Prolate “B”: ${\sigma }_r=0.01,{\varepsilon }_r=0.1$; the electric capillary and Ohnesorge numbers are ${\text{Ca}}_E=15,\text{Oh}=0.6426$. (c) Oblate: ${\sigma }_r=2,{\varepsilon }_r=20$; the electric capillary and Ohnesorge numbers are ${\text{Ca}}_E=0.27,\text{Oh}=0.6426$. The red arrows represent the electric force acting on the drop interface.

Figure 8

Electric stresses for drop deformations shown in fig. 7. (a)-(b): ${\sigma }_r=0.1,{\varepsilon }_r=0.04,{\text{Ca}}_E=15,\text{Oh}=0.459$. (c)-(d): ${\sigma }_r=0.01,{\varepsilon }_r=0.1,{\text{Ca}}_E=15,\text{Oh}=0.6426$. (e)-(f): ${\sigma }_r=2,{\varepsilon }_r=20,{\text{Ca}}_E=0.27,\text{Oh}=0.6426$.

Figure 9

Onset of drop pinch-off for the oblate shape. Conductivity and permittivity ratios are ${\sigma }_r=0.1,{\varepsilon }_r=2$. $\text{Oh}=0.6426$ and ${\text{Ca}}_E=0.415,0.52,0.72$ at times $T=5.78,4.24,2.98$, from top to bottom. The arrow represents the direction of the electric field.

Figure 10

Comparison of onset of drop pinch-off between the Stokes (dashed) and Navier-Stokes (solid) simulations with $\text{Oh}=0.6426$ and ${\text{Ca}}_E=0.52$. The inset shows the detailed drop shape at the center. This pinch-off dynamic is more apparent at higher electric capillary number, as shown in Fig. 11.

Figure 11

Comparison of extreme drop deformation, between the Stokes (dashed) and Navier-Stokes (solid) simulations. $\text{Oh}=0.6426,{\text{Ca}}_E=0.52,$ (top) and ${\text{Ca}}_E=0.72$ (bottom). For ${\text{Ca}}_E=0.52,T=6.15$ (Stokes) and $T=4.24$ (Navier-Stokes). For ${\text{Ca}}_E=0.72,T=3.96$ (Stokes) and $T=2.98$ (Navier-Stokes). The arrow represents the direction of the electric field.

Figure 12

Flow field at the onset of drop pinch-off for the oblate shape. Conductivity and permittivity ratios are ${\sigma }_r=0.1,{\varepsilon }_r=2$. $\text{Oh}=0.6426$ and ${\text{Ca}}_E=0.415,0.52,0.72$ at times $T=5.37,3.9,2.93$, from top to bottom. The arrow represents the direction of the electric field.

Figure 13

Normal (a) and tangential (b) electric stresses for the oblate deformation with ${\text{Ca}}_E=0.72$. Other parameters are $\text{Oh}=0.6426$ (solid) and $\text{Oh}\to \infty$ (dashed, limit of creeping flow). The symbols in (a) show the value of the stress at various positions on the drop interface (inset).