Deformation of a conducting drop in a randomly fluctuating electric field

We quantify the transient deformation and breakup of a conducting drop suspended in a dielectric medium and subjected to a fluctuating electric field. Specifically, the magnitude of the field fluctuates randomly in time, while its orientation is fixed. Hence, the deformation of the drop is axisymmetric about the direction of the field. The temporal fluctuations are described by a stationary Markovian Gaussian process, characterized by a mean, variance, and correlation time. Small deformation theory predicts that the fluctuations produce a larger deformation than under a steady electric field of strength equal to the mean of the fluctuating electric field. Next, we utilize boundary integral computations to quantify the deformation and breakup of drops beyond the small deformation regime. When the mean of the fluctuating field is greater than the critical field for breakup under a steady field, we find that the average time taken to undergo breakup is less than that under an equivalent steady field. More interestingly, a certain fraction of drops are observed to undergo breakup even when the mean field is less than the critical field. The fraction of drops undergoing breakup and the range of mean electric field below the critical where breakup is observed depends on the strength of fluctuations of the electric field. An operating map is presented for the percentage of drops undergoing breakup as a function of the dimensionless mean field for different strength of field fluctuations. The present study sheds light on the response of drops in applications such as electrocoalescence and electroemulsification, where interactions with surrounding drops or disturbances in operating conditions can produce a random field around a drop, even when the applied macroscopic field is uniform.

Figure 1

Schematic of the electric field induced deformation of a conducting drop. The axis of symmetry, $\phi$, is along the direction of the electric field, $E_{\infty }^{*}$. The average magnitude of the electric field is $E_c$, and the fluctuations are denoted by ${\epsilon }^{*}$. The drop having viscosity ${\mu }_i$, permittivity ${\varepsilon }_i$ and conductivity ${\sigma }_i$ is suspended in a dielectric medium with viscosity ${\mu }_o$, permittivity ${\varepsilon }_o$ and conductivity ${\sigma }_o$. The initially undeformed state of the drop is shown by the dashed curve. The semi-major and semiminor axis of the deformed drop are denoted by $L$ and $B$, respectively. The unit normal ($\mathit{n}$) and tangential ($\mathit{s}$) vectors, the continuous tangential coordinate ($s$), the polar angle ($\theta$), and a fixed source point on the interface ($\mathit{x}$) are also shown.

Figure 2

(a) A simulated electric field with variance $G_w\lambda =0.1$. The solid line is an average over 100 realizations of the algorithm over the initial distribution of ${\epsilon }_0$. The dashed line shows the mean value of the fluctuations. (b) Probability density function of the numerically computed (circles) and theoretically predicted (solid line) electric field at $t=1$.

Figure 3

Deformation plotted as a function of the electric capillary number. For drops that deform to a steady shape, the deformation corresponds to the steady final deformation. For drops that undergo breakup, the deformation corresponds to the deformation before breakup. The insets show the final shapes of a drop that reaches a steady shape, and a drop that breaks up with pointed ends. The vertical line demarcates the transition of the system from a steady final state to breakup by pointed ends. The critical electric capillary number for the transition is ${\text{Ca}}_E\approx 0.21$.

Figure 4

Transient deformation of the drop at a mean electric capillary number $\langle {\text{Ca}}_E\rangle =0.01$ and variance in the fluctuations $G_w\lambda =0.1$. The light-gray curves correspond to the transient deformation for a given initial value of the fluctuation in electric field, ${\epsilon }_0$, calculated using boundary integral computations. The solid curve is calculated as the average of the gray curves, and represents the average transient deformation taken over 100 values of ${\epsilon }_0$. The dashed curve corresponds to the average transient deformation, $\langle \mathcal{D}\left(t\right)\rangle$, calculated using the small deformation theory Eq. (24). The dash-dotted curves predict the deformation one standard deviation from the mean, obtained from the small deformation theory Eq. (25). The dotted curve shows the transient deformation of the drop under a steady D.C. field at ${\text{Ca}}_E=0.01$, predicted by boundary integral computations.

Figure 5

(a) Transient deformation of a drop at $\langle {\text{Ca}}_E\rangle =0.1$ and $G_w\lambda =0.1$. The light-gray curves correspond to the transient deformation for a given initial value of the fluctuation in electric field, ${\epsilon }_0$, calculated using boundary integral computations. The solid curve is calculated as the average of the gray curves, and represents the average transient deformation taken over 100 values of ${\epsilon }_0$. The dotted curve shows the transient deformation of the drop under a steady D.C. field at ${\text{Ca}}_E=0.1$. (b) Probability density function of the final deformation of the drop. The dashed curve is a fit of a mixture of two normal distributions to the probability density function. The solid line shows the final deformation of the drop under a steady D.C. electric field. The top-right inset shows the mean deformed steady state of the drop at $t=25$, with the initial state shown by the dashed circle for reference.

Figure 6

(a) Transient deformation of a drop at $\langle {\text{Ca}}_E\rangle =0.28$ and $G_w\lambda =0.1$. The light-gray curves correspond to the transient deformation for a given initial value of the fluctuation in electric field, ${\epsilon }_0$, calculated using boundary integral computations. The solid curve is calculated as the average of the gray curves, and represents the average transient deformation taken over 100 values of ${\epsilon }_0$. The dotted curve shows the transient deformation of the drop under a steady D.C. field at ${\text{Ca}}_E=0.1$. (b) Probability density function of the breakup time of the drop. The dashed curve is a fit of a mixture of two normal distributions to the probability density function. The solid line shows the breakup time of the drop under a steady D.C. electric field. The top-right inset shows the mean shape of the drop before breakup at the point where the solid curve in panel (a) terminates, with the initial state shown by the dashed circle for reference.

Figure 7

(a) Transient deformation of a drop at $\langle {\text{Ca}}_E\rangle =0.19$ and $G_w\lambda =0.1$ calculated using boundary integral computations. The light-gray curves correspond to the transient deformation for initial values of the fluctuation in electric field, ${\epsilon }_0$, which predict steady deformation. The dark-gray curves correspond to the transient deformation for ${\epsilon }_0$ values that predict breakup. The solid curve is calculated as the average of the light-gray curves, and the dashed curve is calculated as the average of the dark-gray curves. The dotted curve shows the transient deformation of the drop under a steady D.C. field at ${\text{Ca}}_E=0.19$. (b) Probability density function of the final deformation (for drops reaching steady state) or final deformation before breakup (for drops that undergo breakup). The dashed curve is a fit of a mixture of two normal distributions to the probability density function. The insets show the mean drop shape at steady state ($t=45$) and just before breakup [the point where the dashed curve in panel (a) terminates], with the shape of the undeformed drop at $t=0$ shown by the dashed circles for reference.

Figure 8

Percentage of drops that undergo breakup at $t^{*}/t_c=50$ as a function of the mean electric capillary number for $G_w\lambda =0.001(\diamond ),0.01(\square ),$ and $0.1(\circ )$. The solid curve corresponds to a steady D.C. electric field. The dashed, dotted, and dash-dotted curves are drawn to guide the eye. Based on the strength of the fluctuations and $\langle {\text{Ca}}_E\rangle$, three final states are identified—all the drops reach steady deformation, some drops reach steady state and some undergo breakup, and all drops undergo breakup.

Figure 9

Convergence results for the system under a constant electric field at ${\text{Ca}}_E=0.24$. Final deformation before breakup for (a) different values of $N$ at $\mathrm{\Delta }t=0.02$ and (b) different values of $\mathrm{\Delta }t$ at $N=150$.