A computational “toolbox” can be used for the a priori design of optimized fluidic components. These components include 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. Applying the advection maps in sequence generates a representation of the outflow distribution for complex designs that combine one or more features. Therefore, the effect of the complex three-dimensional flow field can 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 toolbox is applied to determine optimal combinations of features for specified mixer sizes and mixing metrics.