Magnetohydrodynamics may potentially provide a convenient means for controlling fluid flow and stirring fluids in minute fluidic networks. The branches of such fluidic networks consist of conduits with rectangular cross sections. Each conduit has two individually controlled electrodes positioned along opposing walls and additional disk-shaped electrodes deposited in the conduit's interior away from its sidewalls. The network is positioned in a uniform magnetic field. When one applies a potential difference between a disk-shaped electrode and two wall electrodes acting in tandem, circulatory motion is induced in the conduit. When the potential difference alternates periodically across two or more such configurations, complicated (chaotic) motions evolve. As the period of alternation increases, so does the complexity of the flow. We derive a two-dimensional, time-independent expression for the magnetohydrodynamic creeping flow around a centrally positioned disk-shaped electrode in the limit of zero radius. With the aid of this expression, the trajectories of passive tracers are computed as functions of the alternations protocol and the electrodes' locations. The theoretical results are qualitatively compared with flow visualization experiments.