Electrohydrodynamics of a Viscoelastic Drop

Abstract

The behavior of a drop in electric fields is critical for diverse natural and industrial applications, including inkjet printing and electrohydrodynamic atomization. This complex phenomenon is governed by the balance of normal stresses at the deformable interface, dictating the drop's deformation, stability, and eventual breakup. While Newtonian drop deformation is well studied, research on viscoelastic drops remains limited.

This study investigates the deformation and breakup dynamics of a viscoelastic drop subjected to an electric field, employing both analytical and numerical methodologies. Analytically, an asymptotic solution is derived for an Oldroyd-B drop suspended within an Oldroyd-B ambient fluid under the limiting conditions of small deformation (characterized by a small electric capillary number) and weak viscoelasticity (quantified by a small Deborah number), considering a spherical axisymmetric coordinate system and the Stokes flow assumption. At a viscosity ratio of unity, the contribution to deformation due to viscoelasticity is always negative, indicating that prolate deformation diminishes with increasing Deborah number, whereas the magnitude of oblate deformation increases with Deborah number. Additionally, the viscoelasticity of the drop has a more pronounced influence on deformation than that of the surrounding ambient fluid.

Asymptotic analysis shows that drop behavior largely depends on the conductivity ratio (drop to ambient) and permittivity ratio (drop to ambient). Based on the first-order deformation coefficient, drops exhibit either prolate or oblate deformation. Oblate deformation always features flow from poles to the equator. Prolate deformation, however, can have flow either from pole to equator or from equator to pole. Thus, the phase plot of conductivity ratio and permittivity ratio is divided into six distinct regions, depending on the type of deformation (prolate or oblate), the flow direction, and the sign of the second-order deformation coefficient. If the first- and second-order deformation coefficients share the same sign, there exists a critical electric capillary number beyond which the drop breaks up or deforms into multilobed or pointed shapes. Conversely, if these coefficients have opposite signs, the drop can maintain stable spheroidal shapes even at large electric capillary numbers.

In the numerical part of the study, we investigate the effect of a steady electric field on the deformation of an LPTT drop. We find that the deviation from Newtonian behavior is negligible when the first- and second-order deformation coefficients have opposite signs. Therefore, a detailed numerical study, including a comparison between LPTT and Oldroyd-B drops, focuses on regions where the signs of the first- and second-order deformation coefficients are the same. When both coefficients are positive and the flow direction is from equator to pole, the drop deforms into a multilobed shape or undergoes multilobed breakup when the capillary number exceeds a critical value. In this scenario, the steady-state deformation decreases with increasing Deborah number for both LPTT and Oldroyd-B drops, indicating that elasticity resists deformation. For the case where both deformation coefficients are positive and the flow is from pole to equator, the drop deforms into a steady spheroidal shape below the critical electric capillary number and forms pointed shapes above it. The drop deformation decreases monotonically with Deborah number for the Oldroyd-B drop; however, it exhibits a non-monotonic trend for the LPTT drop. When both first- and second-order deformation coefficients are negative, the drop deforms into a stable oblate shape below the critical electric capillary number and breaks up above it. While the deformation magnitude increases monotonically with Deborah number for the Oldroyd-B fluid, its dependence on Deborah number is non-monotonic for the LPTT drop.

Finally, this study investigates the behavior of an Oldroyd-B drop in an alternating electric field for regions showing the same signs of the first- and second-order deformation coefficients, considering three frequencies of the applied alternating electric field: 100 Hz, 10 Hz, and 1 Hz.