Self-similar solutions for the nonlinear dispersion of a chemical pollutant into a river flow

摘要

We study the nonlinear coupled boundary value problem arising from the nonlinear dispersion of a chemical pollutant initially in a river. We convert the governing system of partial differential equations into a system of ordinary differential equations through a similarity transformation, which has not previously been done for these problems. The similarity solutions can then be obtained from the resulting boundary value problem. Both problems of uniform and variable initial profiles are considered, and exact solution profiles are obtained. We next generalize these results to account for the case where a non-zero concentration of the pollutant can be added for positive values of time. Physically, this models the diffusion of a chemical pollutant into a river when a spill is in progress. Since solutions under this model are governed by more complicated equations, numerical solutions are obtained. Finally, we consider solutions which exhibit wave-like structures. Such solutions model the propagation of waves in a river under the presence of a pollutant. A number of interesting physical observations are made during the analysis of these solutions.