Deformation and stability of a viscous electrolyte drop in a uniform electric field

We study the deformation and breakup of an axisymmetric electrolyte drop which is freely suspended in an infinite dielectric medium and subjected to an imposed electric field. The electric potential in the drop phase is assumed to be small, so that its governing equation is approximated by a linearized Poisson-Boltzmann or modified Helmholtz equation (the Debye-Hückel regime). An accurate and efficient boundary integral method is developed to solve the low-Reynolds-number flow problem for the time-dependent drop deformation, in the case of arbitrary Debye layer thickness. Extensive numerical results are presented for the case when the viscosity of the drop and surrounding medium are comparable. Qualitative similarities are found between the evolution of a drop with a thick Debye layer (characterized by the parameter $\chi \ll 1$, which is an inverse dimensionless Debye layer thickness) and a perfect dielectric drop in an insulating medium. In this limit, a highly elongated steady state is obtained for sufficiently large imposed electric field, and the field inside the drop is found to be well approximated using slender-body theory. In the opposite limit $\chi \gg 1$, when the Debye layer is thin, the drop behaves as a highly conducting drop, even for moderate permittivity ratio $Q={\epsilon }_1/{\epsilon }_2$, where ${\epsilon }_1,{\epsilon }_2$ is the dielectric permittivity of drop interior and exterior, respectively. For parameter values at which steady solutions no longer exist, we find three distinct types of unsteady solution or breakup modes. These are termed conical end formation, end splashing, and open end stretching. The second breakup mode, end splashing, resembles the breakup solution presented in a recent paper [R. B. Karyappa et al., J. Fluid Mech. 754, 550 (2014)]. We compute a phase diagram which illustrates the regions in parameter space in which the different breakup modes occur.

Figure 1

An electrolyte fluid drop with viscosity $\lambda \mu$ is surrounded by a nonionic fluid with viscosity $\mu$. A constant electric field, directed along the $z$ axis, is imposed in the far field.

Figure 2

Comparison of numerically computed results (solid lines) for the surface potential ${\varphi }_s$ and analytical solutions (cross symbols) for a spherical particle, from Ref. [49]. Here $Q=10$ and $\chi$ is indicated in the figure.

Figure 3

Comparison between computed results (solid lines with filled symbols) and small deformation theory from Ref. [49] and Appendix pp2 (dashed lines) with $Q$ and $\chi$ indicated in the figure. Insets: Magnification of results near $E_b=0$.

Figure 4

Steady-state deformation curves for various $Q$ and $\chi$, corresponding to a highly conducting drop, compared with the small deformation result $D_f=(9/16)E_b$.

Figure 5

Comparison of drop deformation for various $\chi$ and $Q=5$ together with the analytical results based on a spheroidal approximation for $\chi =0$.

Figure 6

Steady drop deformation for $Q=50$ and various $\chi$. Theoretical steady-state response curve using a spheroidal approximation for $\chi =0$ [14, 37] is shown by a dashed curve. The subplot provides an enlarged view of the numerical data near the first turning point. The drop profile for $E_b$ greater than the turning point on the lower branch exhibits unsteady pointed ends.

Figure 7

Comparison of drop shape with a spheroid that has the same aspect ratio. Top left: aspect ratio $l/b\approx 10$ for a slightly conducting drop with $\chi =0.1$, $E_b=6.5$, $Q=5$. Bottom left: aspect ratio $l/b\approx 4.4$ with $\chi =0.25$, $E_b=1.6$, $Q=5$. Right: comparisons between calculated surface potentials and slender-body approximation, for same parameter values as in left panels.

Figure 8

Upper panel: Evolution of drop with $Q=50$, $\chi =10$, and $E_b=0.26$. Bottom panel: Drop shapes at breakdown of the numerical scheme for $Q=50$ and $E_b=0.26$.

Figure 9

Evolution of interface tip curvature ${\kappa }_{\mathrm{tip}}$ vs $1/\left({\kappa }_{\mathrm{tip}}{U}_{n,\mathrm{tip}}\right)$ for the three cases in Fig. 8. Here, $U_{n,\mathrm{tip}}$ is the normal velocity at the tip. Following the scaling law ${\kappa }_{\mathrm{tip}}=O\left({\tau }^{-\delta }\right)$ and $U_{n,\mathrm{tip}}=O\left({\tau }^{\delta -1}\right)$ from Fontelos et al. [17], we plot ${\kappa }_{\mathrm{tip}}$ vs the time to singularity $\tau \sim 1/\left({\kappa }_{\mathrm{tip}}{U}_{n,\mathrm{tip}}\right)$. The estimated or average slope $\delta \approx 0.71$ is shown as a black solid line.

Figure 10

Breakup of a viscous drop for $Q=10$ and $E_b=0.26$ with $\chi =10$ in upper panel ($t=0,0.25,0.49,0.74,0.98,1.23,1.47,1.66,1.77,1.83,1.86,1.88,1.90,1.92,1.94$) and $\chi =30$ in the middle panel ($t=0,0.30,0.60,0.90,1.20,1.35,1.50,1.56,1.58,1.59,1.61$). The inset of the middle panel show local finger formation before breakup for $\chi =30$. The lower two panels show tip profiles at different resolution $N$ for $\chi =30$ at times before (left) and after (right) the snail head is formed. The profiles are well resolved, at least up to the onset of snail formation.

Figure 11

Breakup of a viscous drop for $Q=5$ and $E_b=1.65$ with $\chi =0.25$. Top: drop profiles at times $t=0,1.73,29.31,32.14,33.56,34.41$. Middle: interfacial potential ${\varphi }_s$ vs $s$, at the same $t$ as top. Bottom: maximum normal velocity vs $t$.

Figure 12

Boundaries separating steady ($S$-region) and unsteady ($U$-region) solutions in the $E_b-\chi$ plane for various $Q$.

Figure 13

Phase diagram of steady shapes and breakup modes in $Q-\chi$ space with $E_b=0.5$. Interface shapes at markers (a)–(c) on the phase diagram are shown in panels at right and below. (a) End splashing ($Q=5,\chi =10$), (b) conical end formation ($Q=20,\chi =2$), and (c) open end stretching ($Q=10,\chi =0.5$).