English abstract
The goal of this project is to study the mathematical model for the diffusion of dust particles emitted from a fixed source. The study of this model is relevant to some industrial, and environmental applications. Most industrial establishments have factories with chimneys through which the fumes escape into the atmosphere outside the factory, These fumes diffuse into the surroundings causing pollution and forming a health hazard. In arid lands, like Oman, strong winds can cause the dust from the ground to travel with wind and can cause danger to tarmac roads and also dirt in houses.
In this project we study the solution of the time-dependent diffusion equation in the presence of a point source whose strength is dependent on time. This poses an initial boundary value problem for a second order linear partial differential equation. The steady state case of the same problem was studied by Sharan et al.(1996) when the uniform source is situated at ground level. The differential equation is either of the elliptic or parabolic type. We use the method of Fourier and Laplace transforms to obtain the solution in closed form for a source of general time dependence. A number of special cases, in which the source function of time is explicitly given and special values of the diffusion parameters are taken, are examined in detail. The explicit solutions obtained sometimes depend on standard mathematical functions. The computations of these solutions are sometimes achieved by using standard NAG library routines. In other cases, special codes were written for that purpose. The solutions are represented graphically using MATLAB. The profiles show the strong dependence of the concentration of dust on the speed of the wind, the source, and its position in the vertical direction. It is also found that the diffusion parameters play an important role in the spread of the dust particles in the atmosphere. In the case when diffusion is present only in the vertical direction, we find that for small times the pollution spreads with a front that travels with the speed of the wind. The front is a representation of a characteristic curve of the equation. When diffusion is present only in the direction of the wind, there is no discontinuity front and the pollution spreads slowly into the direction of the wind.
The solutions for all the special cases considered are examined for large values of the time. It is found that in all cases, the solution approaches that of the corresponding steady state solution of the equation.
