Bistability (and singularity) in the onset of drop Quincke rotation*

Particles in a sufficiently strong electric field spontaneously rotate, provided that charge relaxation is slower in the particle than in the suspending fluid. It has long been known that drops also exhibit such "Quincke rotation," with the electrohydrodynamic flow engendered by electrical shear stresses at the interface resulting in an increased critical field; however, the hysteretic onset of the instability observed for sufficiently low-viscosity drops has so far defied theory—including direct numerical simulations which have hinted to the formation of interfacial charge-density "shocks" in this regime. We shall illuminate the emergence of bistability and singularity in the onset of drop Quincke rotation, by studying the Taylor–Melcher leaky-dielectric model in a simplified two-dimensional setting involving a circular (non-deformable) drop, allowing for arbitrary viscosity ratios and field strengths.