EXACT SOLUTIONS OF THE DIFFUSION PROBLEM IN A RECTANGULAR CONTAINER WITH AN INTERNAL SOURCE OBTAINED BY THE FAST EXPANSIONS METHOD

Abstract

The diffusion problem in a rectangular-shaped body with the first kind boundary conditions and an internal source of substance depending on the rectangle points coordinates is solved by the method of fast expansions in general form. The exact solution containing free parameters was obtained. By changing these parameters one can get many new exact solutions. Exact solutions to the problem with a constant internal source were shown as an example. Distribution graphs of concentration and diffusion fluxes of substance for various ratios of the area size were given in the work. According to the analysis of the exact solutions it follows that the concentration and diffusion fluxes distribution will be symmetrical relative to the plane y=b/2, provided that the substance concentration in the corners of the rectangular area is equal to zero. The coordinates of the points with the highest and lowest concentrations of the symmetric distribution were determined in the article. A study of the difference in diffusion fluxes along the coordinate axes was carried out. As a result it was found that the difference in non-symmetrical diffusion fluxes is affected by a constant internal source and is not affected by the concentration of the substance in the area corners.

Keywords

• exact solution
• diffusion
• internal source
• rectangular shape
• fast expansion method