Multi-scale method for the solution of the concentration distribution in a two-layered channel flow

In this work the fascinating dynamics of a two-layered channel flow characterised by the dispersion in composite media within its layers is investigated in depth. The top layer comprises of a fluid zone that allows the fluid to travel along its surface easily (with relatively higher velocity), while the bottom layer is packed with porous media. The primary objective of this research is to do an in-depth investigation of the complex two-dimensional concentration distribution of a passive solute discharged from the inflow region. A multi-scale perturbation analysis approach has been implemented to address the system’s inherent complexity. This accurate determination of the dispersion coefficient, mean concentration distribution and two-dimensional concentration distribution is accomplished deftly using Mei’s homogenisation approach up to second-order approximation, which satisfactorily capture the minor variations in the solute dynamics also. The influence of various flow and porous media elements on these basic parameters is thoroughly investigated, expanding our comprehension of the complex interaction between flow dynamics and porous media’s properties. The effect of Darcy number and the ratio of two viscosities ( $M$ ) on the dispersion coefficient depends on the height of the porous layer. As the Péclet number ratio increases, the dispersion coefficient experiences a concurrent increase, resulting in a decline in the concentration peak. The results of the analytical studies have also been compared with those results obtained using a purely computational method to establish the validity of our studies. Both the sets of results show quite good agreement with each other. In this study, alternate flow models have been used for the porous region, and the outcomes are compared to determine which approach yields more suitable results under different conditions.