Abstract

Microchannel Mixing and Multifractal Dimensions

Miron Kaufman, Professor of Physics, Cleveland State University

Since the fluid flow is microchannels is laminar, mixing of advected light particles is achieved by using patterns on the walls. We solve numerically the Navier-Stokes equations describing flows in patterned microchannel: the staggered herring bone which consists of periodic grooves and ridges distributed along the channel length and a couple of fractal microchannels where by employing a 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 Shannon-Renyi entropic measures and fractal dimensions of Poincare plots along the channels taken for multiple mixing cycles. We perform a multi-scale analysis of the images. For each image we considered 30 versions, starting with the original image, smallest scale of observation (largest number of pixels) 157,094 and ending at the largest scale of observation with only 180 pixels. A renormalization group process of local averaging of pixels transforms the n’th version into n+1’th version. The entropy is shown to grow logarithmically with the number of pixels. The slope of entropy vs. ln(number bins) determines the fractal dimension. This method can provide a rigorous assessment of the quality of mixing and the means of comparison between different micromixer designs.We calculate the fractal dimensions at different cross sections along the three microchannels. Initially the two fractal mixers show little difference in the structure dimensions from the staggered herring bone channel. As the fluid flow progresses the difference between the dimensions exhibited by the three mixers increases. The insights gained by considering the geometry of the Poincare sections will be discussed.