We consider the effects that step changes in zeta potential and cross section have on electroosmosis in long-and-narrow channels with arbitrary cross-sectional shapes. The Stokes equation of flow is solved analytically utilizing the thin Debye layer approximation to provide effective slip velocities on the channel walls. The effects of channel dimensions, surface potentials, applied pressure drop, and applied voltage are discussed. One anecdotal case, a two-region rectangular channel, is presented to illustrate the solution. The flow in each region is a combination of a uniform electroosmotic flow and a nonuniform pressure-driven flow. The electroosmotic pumping causes the pressure gradient in each region to adjust so that the flow rate is the same in each region and the overall applied pressure drop is met, resulting in convex velocity profiles in some regions and concave velocity profiles in other regions. By appropriate choice of the applied pressure drop, flat velocity profiles may be achieved in one or more regions. (C) 2003 Elsevier Inc. All rights reserved.