Oscillatory electro-osmotic flow through a slit channel with slipping stripes on walls

A theoretical model is presented in this paper for time-oscillating electro-osmotic flow through a plane channel bounded by two parallel plates, which are patterned with periodic stripes of distinct hydrodynamic slippage and wall potential. The flow is driven by oscillatory pressure gradient and electric field of the same frequency in the axial direction. Flows that are longitudinal or transverse to the stripes are investigated. Based on the Debye–Hückel approximation, and assuming Stokes flow, the electric potential and the velocity fields are found by the methods of eigenfunction expansion and point collocation. The phenomenological coefficients of the Onsager relations for the fluid and current fluxes are deduced as functions of the channel height, the area fraction of wall with slippage, the intrinsic slip length, the Debye parameter, the zeta potentials and the oscillation parameter. Considering several kinds of wall patterns, we extend the theoretical limits in the steady-flow regime to the oscillatory-flow regime. For a uniformly charged wall, the effective slip length obtained from the hydrodynamic problem can still be used directly in the electro-osmotic flow as if the wall were uniformly slipping. When the slipping stripes are perfectly slipping but uncharged, the presence of such stripes will always have a decreasing effect on the streaming conductance, unlike the steady case in which it gives no net effect on the flow in the limit of a very thin double layer. Furthermore, we confirm the presence of a threshold frequency, beyond which the flow will diminish significantly. The slipping fraction of the wall will always introduce a phase lag to the response and lower the threshold frequency. Increasing the wall potential in the presence of slippage can appreciably increase the streaming conductance and the phase lag.