Investigation of the influence of the parameters of the mathematical model of two-dimensional stationary convective diffusion on the calculations of the concentration of depositing solid particles in the channel

Dmitry A. Tukmakov

Federal Research Center Kazan Scientific Center of the Russian Academy of Sciences

The paper investigates the influence of various parameters of the mathematical model of diffusion on the results of calculations of the convective diffusion of a dispersed impurity in a liquid. The need to study and model the diffusion processes of dispersed impurities is associated with the environmental problems of deepening the riverbed. Natural conditions in fishery reservoirs are adversely affected as a result of hydraulic engineering works. The extraction of sand and gravel mixtures in riverbeds leads to pollution of watercourses with suspended solids and, accordingly, to an increase in turbidity, which has a negative impact on aquatic ecosystems. This paper presents the results of a theoretical study of the diffusion of a solid impurity and evaluates the influence of both different approaches in modeling and the parameters of disperse media. The equations of the mathematical model are derived from the equation of convective diffusion in a two-dimensional non-stationary form. The watercourse is assumed to be rectilinear, of constant depth, with a constant average longitudinal velocity, while the transverse and vertical averaged velocities of the watercourse are assumed to be equal to zero. Let us direct the Ox axis along the coast towards the current, the Oz axis vertically upwards, and the Oy axis transverse to the flow. It is assumed that the point source maintains its intensity long enough to be able to solve the problem in the stationary approximation. After applying simplifications, a flat mathematical model of stationary diffusion is obtained. By the method of separation of variables in the form of a Fourier series, a solution was obtained for the set of equations of the mathematical model of diffusion of several fractions of particles. The solution is implemented in the form of a computer program. The influence of various model parameters (particle settling rate, accounting for particle sedimentation, and the linear size of particles of a dispersed impurity) on the results of calculations of the diffusion of a dispersed impurity in the longitudinal and transverse directions was determined using the method.

mathematical modeling, diffusion, inhomogeneous media, dispersed suspension