Fractal geometry of mixing

The mixing of two components in microchannel systems is quantified using both a rigorously defined mixing index, as well as fractal geometry analysis of concentration/fluorescent tracer cross-sectional images. The microchannels used in this work are equipped with slanted ridges that induce cross-sectional flows conducive of mixing. Two geometries are investigated: (i) a periodic one in which the ridges are arranged in a staggered herring bone pattern (SHB) and (ii) an aperiodic one where the position of the ridges is generated using the Weierstrass function. The quality of the mixing between two tracers distributed in the micromixers is determined by using an index derived from the Shannon mixing entropy. The fractal dimension of the mixing region is also determined by studying the variation of the entropy of mixing with the scale of observation. We show that the use of the fractal dimension analysis amplifies the differences between the two type of mixers, making it a useful measure for their optimization. The fluid flow in the microchannel mixers is laminar as a result of the size of the channel and of the fluid viscosity, making their flow characteristics similar with polymer melt flows in extruders. Thus, the methods developed have broad applicability to an array of technically important systems.