Substance advection by a steady 2D stream of the viscous fluid in a lengthy free-surfaced canal

Spreading of a low-concentrated admixture in the 2D (length - depth) stream of the viscous fluid in an open lengthy canal is considered; the admixture's dissipation and diffusion are taken into account. Apart from being long, the canal is assumed to be low-sloping, with a given shape of the bottom-line. A mathematical model under consideration is derived by the small parameter technique, starting from the 2D steady Navier-Stokes equations for the incompressible fluid and the unsteady diffusion equation for the moving medium. The main feature to this model is taking account of the deep--cross structure of the stream and it lets us investigate peculiarities of the substance transfer. An interesting particular case is then a rise of the near-surface opposite flow which may be caused e.g. by the wind action. The wide range of the main parameters to the problem does not allow to point the only one particular discretization scheme which would be superior. To our mind, in most cases some refined upwinding technique should be used to approximate the convective term. As to the time-stepping process, partially implicit (e.g., implicit with respect to the convective term) integration schemes occured to be most efficient because of an easy solvability of the corresponding equation (usually it is a tridiagonal linear system)