The kinematics of oscillatory cross flow has been studied numerically as a means for generating chaotic mixing in microfluidic devices for both confined and continuous throughput flow configurations. The flow is analyzed using numerical simulation of the unsteady Navier–Stokes equations combined with tracking of single and multispecies passive tracer particles. Two characteristics of chaotic flow are demonstrated: the stretching and folding of material lines leading to particle dispersion and a positive “effective” Lyapunov exponent. The primary mechanism for the generation of chaotic flow is a periodic combination of stretching (which occurs via shear in the channels) and rotation (which occurs via the timing of the oscillations), making these systems effective tendril-whorl type flows. First, the case of confined mixing is studied. It is shown that chaotic flow is generated in a cross-cell device when sinusoidally driven, out-of-phase, perpendicular fluid streams intersect in the flow domain. Calculations indicate that the flow becomes chaotic in the center region starting at a Strouhal number on the order of 1. A degree of mixing based on a relative mixing entropy as high as 91% is obtained. Approximately 10–15 sinusoidal cycles are needed in order to effectively mix different groups of passive tracer particles. In the second phase of the analysis, the cross flow mixing mechanism is utilized in a continuous operation by combining a throughput channel flow with an oscillatory cross flow in a configuration called the star-cell geometry. It is shown that the oscillatory flow remains chaotic even in combination with the throughput flow, and a degree of mixing in the 80%–90% range is obtained for the range of parameters studied here.