Universal behaviour in boundary-driven electrokinetic flows

Travelling wave charges lying on the insulating walls of an electrolyte-filled capillary give rise to oscillatory modes which vanish when averaged over the period of oscillation. They also give rise to a zero mode (a unidirectional, time-independent velocity component) which does not vanish. The latter is a nonlinear effect caused by continuous symmetry breaking due to the quadratic nonlinearity associated with the electric body force in the time-dependent Stokes equations. In this paper, we provide a unified view of the effects arising in boundary-driven electrokinetic flows (travelling wave electroosmosis) and establish the universal behaviour exhibited by the observables. We show that the incipient velocity profiles are self-similar implying that those obtained with a single experimental configuration can be employed again to attain further insights without the need of repeating the experiment. Certain results from the literature are recovered as special cases of our formulation and we resolve certain paradoxes having appeared in the past. We present simple theoretical expressions, depending on a single-fit parameter, that reproduce these profiles, which could thus provide a rapid test of consistency between our theory and future experiment. The effect becomes more pronounced when reducing the transverse dimension of the system, relative to the velocity direction, and increasing the excitation wavelength, and can therefore be employed for unidirectional transport of electrolytes in thin and long capillaries. General relations, expressing the zero mode velocity in terms of the electric potential and the geometry of the system only, can thus be easily adapted to alternative experimental settings.