The interaction between electric currents and a magnetic field is used to produce body (Lorentz) forces in electrolyte solutions. By appropriate patterning of the electrodes, one can conveniently control the direction and magnitude of the electric currents and induce spatially and temporally complicated flow patterns. This capability is useful, not only for fundamental flow studies, but also for inducing fluid flow and stirring in minute devices in which the incorporation of moving components may be difficult. This paper focuses on a theoretical and experimental study of magnetohydrodynamic flows in a conduit with a rectangular cross section. The conduit is equipped with individually controlled electrodes uniformly spaced at a pitch L. The electrodes are aligned transversely to the conduit's axis. The entire device is subjected to a uniform magnetic field. The electrodes are divided into two groups A and C in such a way that there is an electrode of group C between any two electrodes of group A. We denote the various A and C electrodes with subscripts, i.e., A(i) and C(i), where i=0,+/-1,+/-2, .... When positive and negative potentials are, respectively, applied to the even and odd numbered A electrodes, opposing electric currents are induced on the right and left hand sides of each A electrode. These currents generate transverse forces that drive cellular convection in the conduit. We refer to the resulting flow pattern as A. When electrodes of group C are activated, a similar flow pattern results, albeit shifted in space. We refer to this flow pattern as C. By alternating periodically between patterns A and C, one induces Lagrangian chaos. Such chaotic advection may be beneficial for stirring fluids, particularly in microfluidic devices. Since the flow patterns A and C are shifted in space, they also provide a mechanism for Lagrangian drift that allows net migration of passive tracers along the conduit's length.