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

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 ($\sigma_r$) and permittivity ratio ($\epsilon_r$) are selected from six regions ($PR_A^+$, $PR_B^+$, $PR_A^-$, $PR_B^-$, $OB^+$, and $OB^-$) of the $(\sigma_r, \epsilon_r)$ 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.