Scaling Analysis and Self-Similarity of One-Dimensional Transport Process

Convection-diffusion equation has been widely used to model a variety of flow and transport processes in earth sciences, including spread of pollutants in rivers, dispersion of dissolved material in estuaries and coastal waters, flow and transport in porous media, and transport of pollutants in the atmosphere. In this study, the conditions under which one-dimensional convection-diffusion equation becomes self-similar are investigated by utilizing one-parameter Lie group of point scaling transformations. By the numerical simulations, it is shown that the one-dimensional point source transport process in an original domain can be self-similar with that of a scaled domain. In fact, by changing the scaling parameter or the scaling exponents of the length dimension, one can obtain several different down-scaled or up-scaled self-similar domains. The derived scaling relations obtained by the Lie group scaling approach may provide additional understanding of transport phenomena at different space and time scales and may provide additional flexibility in setting up physical models in which one dimensional transport is significant