A study of the generalized nonlinear advection-diffusion equation arising in engineering sciences

Abstract

In this work, we examine a nonlinear partial differential equation of fluid mechanics, namely, the generalized nonlinear advection–diffusion equation, which portrays the motion of buoyancy driven plume in a bent-on porous medium. Firstly, we classify all (point) symmetries of the equation, which prompt three cases of n. Next, for each case, we construct an optimal system of one-dimensional subalgebras and use them to perform symmetry reductions and symmetry invariant solutions. In a bid to explain the physical significance of some invariant solutions secured, we present a graphic display of some solutions in 3D, 2D as well as density plots via the exploitation of numerical simulations. Besides, we categorically state here that the results obtained in this study are new when compared with the outcomes previously achieved by Loubens et al., 2011 Quart. Appl. Math. 69 389–401. Interestingly, kink shape soliton, dark soliton, singular soliton together with exponential function solution wave profiles are displayed to make this work more valuable. Furthermore, we determine the conserved vectors in two different ways: engaging the general multiplier approach and Ibragimov’s conservation law theorem. Finally, we provide the physical meaning of these conservation laws.