Analytic Solutions for Asymmetric Model of a Rod in a Lattice Fluid

We consider the problem of determining analytically some exact solutions of the concentration u(x, y, t) of particles moving by diffusion and advection or drift. It is assumed that the advection is nonlinear. The driven diffusive flow is impeded by an impenetrable obstacle (rod) of length L. The exact solutions for u are evaluated for small and big values of vL/D, where v is the drift velocity and D is the diffusion coefficient. The results show that in some regions in the (x, y) plane the concentration first increases (or decreases) monotonically and then is nearly constant after some critical length L. The location at which u is nearly constant depends on the nature of the driving field v/D. This problem has relevance for the size segregation of particulate matter which results from the relative motion of different-size particles induced by shaking. Methods of symmetry reduction are used in solving the nonlinear advection-diffusion equation in (2+1) dimensions.