Solute Transport along Temporally and Spatially Dependent Flows through Horizontal Semi-Infinite Media: Dispersion Proportional to Square of Velocity

According to the hydrodynamic dispersion theories, the dispersion parameter is proportional to a power n of the velocity; the power ranges between 1 and 2. Based on the value n=1, analytical solutions of the dispersion problems along temporally dependent flow domains were obtained in previous works. In the present work, two dispersion problems are addressed for n=2. Using the Laplace transform technique, analytical solutions are obtained for two-dimensional advection-diffusion equations describing the dispersion of pulse-type point source along temporally and spatially dependent flow domains, respectively, through a semi-infinite horizontal isotropic medium. Point sources of a uniform and varying nature are considered. The inhomogeneity of the medium is demonstrated by the linearly interpolated velocity in the space variable. Introduction of new space variables enable one to reduce the advection-diffusion equation in both problems to a one-dimensional equation with constant coefficients. The solutions are graphically shown.