by Prof. Vaughan R. Voller (University of Minnesota)

Abstract

Transport in fractal media results in anomalous behaviors. Here we look at two anomalous transport examples that have an immediate physical realization.

In the first place, we consider  the effective thermal conductivity of 2-D Sierpiński Carpets and 3-D Menger Sponges. For given patterns of these fractal objects, treating the “holes” as conducting paths and the “pattern elements” as insulators, we calculate, through direct numerical simulation, effective thermal conductivities. These values have a power-law scaling with length, the exponent associated  with measures of the  fractal dimension of the conducting object. Beyond this, and in keeping with the location of this talk, we also show how this scaling relates to the classical effective conductivity treatment proposed by Maxwell. 

Secondly,  we study diffusive transport into a one-dimensional porous tube, with varying conductivity distributed as  a Cantor set; the ‘holes” in the set having a very high conductivity, in contrast to a very low value in  the “separators”. In this system, we analytically show that the advance of a wetting front follows a super-diffusive power-law behavior, with a time exponent in excess of 1/2. Further, we directly relate this anomalous exponent to the fractal dimension of the Cantor set.     

Biography

Vaughan Voller is, by academic training, an applied mathematician. His research focuses on computational and analytical modelling of transport processes; working on a wide variety of systems, from solidification of metal alloys through to the growth of sedimentary deltas. For most of his academic life he has worked in the Department of Civil, Environmental, and Geo- Engineering at the University of Minnesota where he is currently the James L. Record Professor.