Flow and Diffusion: Exact Solutions of Different Models Obtained with Traveling Wave and Self-Similar Transformations

In this research work, we study different mathematical models that are capable of describing the particle diffusion phenomena in fluid flows. One-dimensional Cartesian models are defined to analyze the diffusion phenomena while they occur in a background stream. We derive exact solutions of the convection–diffusion equation with traveling wave and self-similar transformations in the case where the flow velocity is constant. The countable set of solutions presented is derived from the Gaussian distribution and from further solutions of the diffusion equation, which have even symmetry. In the case where the speed of the stream may vary, models for incompressible fluid flows are also considered. The investigated partial differential equation (PDE) systems are reduced to ordinary differential equation (ODE) systems with the self-similar Ansatz. The derived ODE systems for certain incompressible flows can be solved analytically, resulting in formulas containing Kummer’s M and Kummer’s U functions.