Group classification and symmetry reduction of a (2+1) dimensional diffusion-advection equation

Based on classical Lie group method, we consider the continuum problem of the driven diffusive flow of particles past an impenetrable obstacle (rod) of length L. The infinitesimals of the diffusion-advection equation in (2+1) dimensions were found for an arbitrary nonlinear advection. The symmetries corresponding to different forms of the nonlinear advection are obtained. Three models are studied in details. The results show that the presence of an obstacle, whether stationary or moving, in a driven diffusive flow with nonlinear drift will distort the local concentration profile to a state which divided the (x, y)-plane into two regions. The concentration is relatively higher in one side than the other side, apart from the value of D/νL where D is the diffusion coefficient and υ is the drift velocity. This problem has relevance for the size segregation of particulate matter which results from the relative motion of different-size particles induced by shaking. Also, the obtained solutions include soliton, periodical, rational and singular solutions.