Analytical solutions are obtained for one-dimensional advection-diffusion equation with variable coefficients in a longitudinal finite initially solute free domain, for two dispersion problems. In the first one, temporally dependent solute dispersion along uniform flow in homogeneous domain is studied. In the second problem the velocity is considered spatially dependent due to the inhomogeneity of the domain and the dispersion is considered proportional to the square of the velocity. The velocity is linearly interpolated to represent small increase in it along the finite domain. This analytical solution is compared with the numerical solution in case the dispersion is proportional to the same linearly interpolated velocity. The input condition is considered continuous of uniform and of increasing nature both. The analytical solutions are obtained by using Laplace transformation technique. In that process new independent space and time variables have been introduced. The effects of the dependency of dispersion with time and the inhomogeneity of the domain on the solute transport are studied separately with the help of graphs.
Abstract:In this article, a mathematical model is presented for the dispersion problem in finite porous media in which the flow is twodimensional, the seepage flow velocity is periodic, and dispersion parameter is proportional to the flow velocity. In addition to these, first-order decay and zero-order production parameters have also been considered directly proportional to the velocity. Retardation factor is taken into account in the present problem. First-type boundary condition of periodic nature is considered at the extreme end of the boundary. Mixed-type boundary condition is assumed at the origin of the domain. A classical mathematical substitution transforms the original advection-dispersion equation into diffusion equation in terms of other dependent and independent variables, with constant coefficients. Laplace transform technique is used to obtain the analytical solution.
Analytical solutions are obtained for a one-dimensional advection-dispersion equation with variable coefficients in a longitudinal domain. Two cases are considered. In the first one the solute dispersion is time dependent along a uniform flow in a semi-infinite domain while in the second case the dispersion and the velocity both have spatially dependent expressions. Analytical solutions are obtained by introducing new independent variables with the help of certain transformations. Result and discussions are given by different graphs.
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