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.

在这项工作中，我们研究了流体力学的非线性偏微分方程，即广义非线性对流-扩散方程，它描绘了浮力驱动的羽流在弯曲多孔介质中的运动。首先，我们对方程的所有（点）对称性进行分类，这提示n 的三种情况. 接下来，对于每种情况，我们构建一维子代数的最优系统，并使用它们来执行对称约简和对称不变解。为了解释一些固定解决方案的物理意义，我们通过数值模拟的利用，在 3D、2D 和密度图中展示了一些解决方案的图形显示。此外，我们在此明确声明，与 Loubens 等人，2011 Quart 先前取得的结果相比，本研究中获得的结果是新的。应用程序 数学。69 389–401。有趣的是，显示了扭结形状孤子、暗孤子、奇异孤子以及指数函数解波剖面，使这项工作更有价值。此外，我们以两种不同的方式确定保守向量：使用一般乘法器方法和 Ibragimov 守恒定律定理。最后，我们提供了这些守恒定律的物理意义。