A unified approach to two-dimensional linear advection-dispersion equation in cylindrical coordinates on a finite domain

Abstract

A new analytical solution of the two-dimensional linear advection-dispersion equation (2-D LAD) in cylindrical coordinates on a finite domain is developed for solute transport in a radial porous medium and a circular source subject to steady and uniform groundwater. The analytical solutions of the 2-D LAD equation with the first- and third-type inlet boundary conditions are obtained by utilizing the finite Hankel transform and the unified transform method, also known as the Fokas method. More precisely, after employing the second kind finite Hankel transform, the 2-D LAD equation is converted to the 1-D LAD equation, which invites the Fokas method, to undertake a refined analysis. As a consequence, we derive a new exact analytical solution of the 2-D LAD equation in cylindrical coordinates with more general boundary conditions, and its physical applications are notable in several physical circumstances. In particular, it can be shown that for large Péclet number, the type of the inlet boundary conditions has no significant discrepancy between the solutions. Furthermore, it is also shown that for large Péclet number, the exit boundary of finite or infinite domain has no significant influence on the solutions. More importantly, we show that if the inlet boundary value is asymptotically t-periodic for large t, the solution of the 2-D LAD in a radial geometry is also asymptotically t-periodic for large t with the same period, which can be used to understand the effect of periodic boundary conditions for solute transport in porous media. All these analytical predictions are consistent with numerical results.