Analytical Solutions for Advection and Advection-Diffusion Equations with Spatially Variable Coefficients

TLDR: In this article, the authors considered the problem of the transport of pollutant in an open channel where the flow in the channel is augmented by steady, unpolluted lateral inflow distributed along the whole length of the channel, such as a steady inflow of ground water.

Abstract: sidered here are consistent with the problem of the transport of pollutant in an open channel where the flow in the channel is augmented by steady, unpolluted lateral inflow distributed along the whole length of the channel, such as a steady inflow of ground water. Therefore the analytical solutions are solu­ tions to a practical problem. The simple expressions considered here for the spatial var­ iation of the coefficients facilitate the process of obtaining an­ alytical solutions to these equations. The spatially variable co­ efficient equations reduce to constant coefficient equations through a simple transformation. Consequently, many of the ~nown analytical solutions to the constant coefficient equa­ tions can be used to obtain analytical solutions to the spatially variable coefficient equations. Analytical solutions are provided for the advection of a sud­ den release of pollutant into the channel and for the solution of advection-diffusion equation. The advection of an initial quasi-Gaussian concentration profile in the channel is also considered. The analytical solutions are simple to evaluate and are useful for validating numerical schemes for solving the advection and advection-diffusion equation with spatially var­ iable coefficients written in either conservative or nonconser­ vative form (Zoppou and Knight 1994). The conservative and nonconservative forms of the equa­ tions are valid equations describing different physical prob­ lems. The analytical solutions to the conservative and noncon­ servative forms of the governing equations will be used to illustrate the importance of selecting the equation relevant to the physical problem, when spatially variable coefficients are involved.