On Large Deformations of Oldroyd-B Drops in a Steady Electric Field

TL;DR

This work analyzes large-deformation dynamics and breakup of Oldroyd-B viscoelastic drops in a steady uniform electric field, extending classical leaky dielectric theory to non-Newtonian drops. Using axisymmetric Basilisk simulations with a log-conformation tensor formulation, the authors map deformation across six regions in the $(\sigma_r,\epsilon_r)$ phase space and quantify the roles of the electric capillary number $Ca_E$ and Deborah number $De$ in determining stable shapes, transitions, and transient behaviors. Key findings include elasticity delaying or reducing deformation in certain regions (e.g., $PR_A^+$), promoting oscillatory or pointed-end dynamics in others (e.g., $PR_B^+$), and enhancing oblate deformation with possible dimpling or breakup in the $OB^-$ regime; the results are supported by domain- and grid-convergence validation and asymptotic analyses. The work provides practical guidance for tuning drop shape and stability in microfluidic and electrohydrodynamic applications by leveraging viscoelastic effects and phase-space positioning.

Abstract

The deformation of viscoelastic drops under electric fields is central to applications in microfluidics, inkjet printing, and electrohydrodynamic manipulation of complex fluids. This study investigates the dynamics of an Oldroyd-B drop subjected to a uniform electric field using numerical simulations performed with the open-source solver Basilisk. Representative pairs of conductivity ratio ($σ_r$) and permittivity ratio ($ε_r$) are selected from six regions ($PR_A^+$, $PR_B^+$, $PR_A^-$, $PR_B^-$, $OB^+$, and $OB^-$) of the $(σ_r, ε_r)$ phase space. In regions where the first- and second-order deformation coefficients share the same sign ($PR_A^-$, $PR_B^-$, $OB^+$), deviations from Newtonian behavior are negligible. In $PR_A^+$, drops develop multi-lobed shapes above a critical electric capillary number, with elasticity reducing deformation and increasing the critical $Ca_E$ with Deborah number ($De$). In $PR_B^+$, drops form shapes with conical ends above the critical $Ca_E$, while steady-state deformation decreases with $De$ below this threshold, and critical $Ca_E$ shows non-monotonic variation. At high $Ca_E$ and $De$, transient deformation exhibits maxima and minima before reaching steady state, with occasional oscillations between spheroidal and pointed shapes. In $OB^-$, drops deform to oblate shapes and breakup above a critical $Ca_E$, with deformation magnitude increasing and critical $Ca_E$ decreasing with $De$; at low $Ca_E$ and high $De$, dimpling and positional oscillations are observed. These results elucidate viscoelastic-electric interactions and provide guidance for controlling drop behavior in practical applications.

Figures (20)

• Figure 1: Schematic of the problem setup: An Oldroyd-B drop of radius $R$ is subjected to an externally applied electric field, $\bm{E}_\infty$, aligned along the axis of symmetry marked in the schematic. The domain size is chosen to be $32R \times 32R$ to minimize the boundary effects.
• Figure 2: Variation of drop deformation with electric capillary number for $\mu_r=1$, $\rho_r=1$, $Re=1$, $\beta_i = 1$, $De = 0$, (a) $\sigma_r = 10$, $\epsilon_r=1.37$; (b) $\sigma_r = 0.1$, $\epsilon_r=2$
• Figure 3: Deformation vs. time for various sizes of the simulation domain. Simulation parameters are $\mu_r=1$, $\rho_r=1$, $Re=1$, $\beta_i = 1/9$, $\sigma_r = 10$, $\epsilon_r=1.37$, $De=5$, $Ca_E=0.4$. $R/\Delta x_{min}$ is taken as 256.
• Figure 4: (a) Deformation parameter variation with time at various refinements of the grid. (b) Steady state deformed interface of the drop for various grid refinements. Simulation parameters: $Re=1$, $Ca_E=0.4$, $De=5$, $\sigma_r=10$, $\epsilon_r=1.37$, $\beta=1/9$
• Figure 5: $\epsilon_r-\sigma_r$ phase plot on log-log scale for $\mu_r=1$. $PR$ and $OB$ indicate the prolate and oblate deformation respectively. Subscripts $A$ and $B$ for prolate deformation correspond to the flow direction from equator to poles and poles to equator respectively. Oblate deformation is always accompanied by the poles to equator flow around the drop. Superscript $+$ indicates the positive contribution to the deformation at $\order{Ca_E^2}$, whereas superscript $-$ denotes the negative contribution to the deformation at $\order{Ca_E^2}$. $(\sigma_r, \epsilon_r)$ pairs selected for the study are marked (red circle markers) on $(\sigma_r, \epsilon_r)$ phase plot on log-log scale. Expressions for the deformation coefficients at various orders are given in Appendix \ref{['appA']}.