Modelling substance spreading in 2d steady free-surfaced stream

Problem about spreading of admixture having low concentration in 2D steady stream of viscous fluid is considered with taking into account dissipation and diffusion processes. The area of stream is an open low-sloping channel with the given shape of the bottom-line. The channel is assumed to be long in the one direction, i.e. ratio of average depth of the channel to its length may be considered as a small parameter. Proposed mathematical model of the process is derived by the small parameter method starting from 2D steady Navier–Stokes equations for incompressible fluid and unsteady diffusion equation in the moving medium. The main character feature of this model is taking into account deep-cross structure of the stream, and it makes us to be able to investigate the pecularities of the substance transfer in case of onset of an opposite flow in surface layer caused, for instance, by wind action. For numerical modelling explicit and economizing convectionally implicit difference schemes are submitted, so is their stability analysis. Numerical results are shown in case when at initial moment admixture is lumped in given cross-section.