Numerical study of Rayleigh fission of a charged viscous liquid drop

It is well known that a spherical drop charged beyond its Rayleigh limit becomes unstable (when subjected to a small shape perturbation) and ejects a significant fraction of the original charge in the form of a jet. To understand the detailed mechanism of this process, we model a charged drop of a perfectly conducting viscous liquid numerically using the boundary-element method. It is found from the simulations that the drop progressively deforms into a conical shape with pointed ends just prior to breakup. For ethylene glycol droplets suspended in air, the drop attains an aspect ratio of 3.86 at its critical shape, close to the observed value of 3.85 in the experimental images available in the literature. As the numerical approach fails to predict charge ejection due to the occurrence of a singularity at this point, the charge loss fraction is estimated by determining how much minimum charge must be removed if the drop is to relax back to a spherical shape. In this manner, the charge loss due to Rayleigh fission (the difference between the original charge and the minimum charge removed) is estimated to be about 39%, which falls within the range of experimental data lying between 20% and 50%. The results on scaling laws, time scales of the process, and the role of various stresses on the dynamics of the drop are discussed.

Figure 1

Comparison of the experimental observations of evolution of drop shapes reported by Giglio et al. [25] (top) with the present simulations (bottom). The simulations are carried out starting with a prolate spheroid of eccentricity $\delta =0.1$, charge $Q=8.1\pi$, and $\lambda =1$ until a point in time when the conical tips develop. The times shown for the simulations are nondimensional reckoned with respect to the time (zero) for the occurrence of the critical shape.

Figure 2

Streamline plots of the drop near singularity. Green lines indicate the direction of fluid flow and red arrows correspond to the direction and relative magnitude of the electrical stresses acting on the drop surface along the arc length. The insets show focused tips of the corresponding shapes.

Figure 3

Variation of aspect ratio with respect to self-normalized time for the three cases $\lambda =1$, 150, and 1600 and their comparison with the experimental results reported by Giglio et al. [25]. The curves for $\lambda =150$ and $\lambda =1600$ overlap and thus cannot be distinguished.

Figure 4

(a) Profiles showing evolution of the drop surface with respect to time leading to formation of the conical tip. (b) Drop profiles rescaled to show the self-similarity of the shapes obtained with respect to nondimensional times 62.3, 62.4, 62.42, 62.45, 62.46, 62.47, and 62.48. The tip curvatures used for scaling corresponding to times are 51.494, 76.762, 88.01, 117.66, 134.78, 159.41, and 197.38, respectively.

Figure 5

Focused tip of the drop at a time just before the critical time $t_0$. It is observed that, near the tip, the drop surface is closely enveloped by a cone of semivertical angle of $24.7^{\circ }$.

Figure 6

Plots of (a) curvature $\kappa$, (b) charge density $q$, and (c) velocity $v$ of the drop tip as a function of time obtained from the simulations with their corresponding best-fitted exponents.

Figure 7

Charge at which a deformed drop relaxes back to spherical shape as a function of time at which the critical shape of the drop is frozen and charge is reduced sequentially until the drop relaxes back to a sphere. The left-hand $y$ axis indicates the scaled charge $Q_c/\pi$ and the right-hand $y$ axis shows the percentage of charge lost in the process. The numbers shown indicate the aspect ratio of the drop shape for the corresponding time.

Figure 8

Evolution of the drop in the relaxation regime from the shape with $\mathcal{A}=3.676$, when subjected to charges $Q=5.78\pi$ and $5.79\pi$. The figure gives evidence of the charge lost value corrected up to two decimal places.

Figure 9

Evolution of the drop in the relaxation regime from the critical shape, when subjected to charge $Q<4.96\pi$. The drop shapes experimentally observed by Giglio et al. [25] (top) compare well with present simulations (bottom) and relaxation to a sphere can be observed.

Figure 10

Variation of aspect ratio with respect to self-normalized time for the case $\lambda =1$ and their comparison with the experimental results reported by Giglio et al. [25] in the relaxation regime.