A described exemplary embodiment of the present invention includes a computational "toolbox" for the a priori design of optimized fluidic components. These components consist of a channel under low-Reynolds number, pressure-driven flow, with an arrangement of grooves cut into the top and/or bottom to generate a tailored cross-channel flow. An advection map for each feature (i.e., groove of a particular shape and orientation) predicts the lateral transport of fluid within the channel due to that feature. We show that applying these maps in sequence generates an excellent representation of the outflow distribution for complex designs that combine these basic features. The effect of the complex three-dimensional flow field can therefore be predicted without solving the governing flow equations through the composite geometry, and the resulting distribution of fluids in the channel is used to evaluate how well a component performs a specified task. The generation and use of advection maps is described, and the toolbox is applied to determine optimal combinations of features for specified mixer sizes and mixing metrics.