Electrohydrodynamic interactions of two Quincke rotors: asymptotic theory and boundary element simulations

Abstract

Weakly conducting spheres suspended in a leaky dielectric fluid can undergo spontaneous Quincke rotation when the applied electric field exceeds a critical value. We investigate how electric and hydrodynamic interactions modify this instability and the subsequent rotation of two spheres at fixed separation and orientation. Starting from electric dipole and hydrodynamic rotlet interactions, we develop a systematic far-field asymptotic theory. Away from onset, slow drift of the rotation axes makes a regular expansion fail at long times. A multiple-scale expansion gives slow phase equations that select the axes according to pair orientation and field strength. Near onset, weak interactions produce corrections comparable to the isolated rotation magnitude. A detuned multiple-scale expansion gives coupled equations for the transverse dipole amplitudes. Their linearization identifies four collective rotation modes, whose critical fields and steady branches with one active mode are determined together with their stability. At leading order near onset, electric interaction shifts all four modal thresholds equally, whereas hydrodynamic interaction makes counter-rotation appear first for pair angles below $\cos^{-1}(1/\sqrt3)\simeq54.7^\circ$ and co-rotation above it. Away from onset, the angle separating stable counter-rotation from stable co-rotation increases with field strength. Boundary element method (BEM) simulations place onset within a field interval for each configuration and resolve the growth and relaxation of rotation. Steady-speed comparisons show how pair orientation changes the balance between electric and hydrodynamic effects. We compare the unexpanded dipole-rotlet model of Das and Saintillan [Phys. Rev. E 87, 043014 (2013)] with our asymptotic theory and BEM simulations.