On the anisotropic advection-diffusion equation with time dependent coefficients

Abstract

The advection-diffusion equation with time dependent velocity and anisotropic time dependent diffusion tensor is examined in regard to its non-classical transport features and to the use of a non-orthogonal coordinate system. Although this equation appears in diverse physical problems, particularly in particle transport in stochastic velocity fields and in underground porous media, a detailed analysis of its solutions is lacking. In order to study the effects of the time-dependent coefficients and the anisotropic diffusion on transport, we solve analytically the equation for an initial Dirac delta pulse. We discuss the solutions to three cases: one based on power-law correlation functions where the pulse diffuses faster than the classical rate $\sim t$, a second case specifically designed to display slower rate of diffusion than the classical one, and a third case to describe hydrodynamic dispersion in porous media.