New Analytical Solutions for Dye Diffusion Equations

The two- and three-dimensional, unsteady-state, convective-diffusive partial differential equations describing the concentration distribution of a tracer dye released as an instantaneous source (line or point, respectively) and subject to the no-flux boundary conditions at the river bottom and surface (two-dimensional) and banks (three-dimensional) have been derived analytically using integral transform methods. Previous efforts at analytical dye modeling have assumed infinite geometry in all dimensions thus avoiding the complicating second type (no-flux) boundary conditions at the river bottom, surface, and banks. In the present study, the river was considered finite in the vertical and lateral directions, the dye source location was made arbitrary, and to the writer's knowledge the results represent the first two- and three-dimensional analytical solutions of tracer modeling to appear in the literature which include the boundary effects of the river. The solutions will have particular application in the calculation of turbulent diffusion coefficients in two and three dimensions, which for the first time will analytically include any significant effects of the river boundaries.