Fractal Geometry of Mixing

The fluid flow in microchannels is laminar as a result of the size of the channel and of the fluid viscosity. In this respect the microfluidics and polymer melt flow in extruders are quite similar. Mixing of advected light particles can be achieved by using patterns of ridges on the walls. We solve numerically the Navier-Stokes equations describing flows in four patterned microchannels: (i) the staggered herring bone (SHB) which consists of periodic grooves and ridges distributed along the channel length, (ii) three fractal pattern microchannels where by employing the Weierstrass function we generate non-periodic patterns of ridges on the channel bottom. The quality of the mixing between two types of tracers is determined by using an index derived from the Shannon mixing entropy. The fractal dimension of the mixing region is determined by studying the variation of the entropy of mixing with the scale of observation. To further understand and quantify the mixing we compute fractal dimensions of Poincare plots along the channels