Multidimensional transport process of solute in a hydromagnetic third-grade fluid flowing through a channel: An approach of method of moments

The current research investigates, for the first time, the multidimensional transport process of solute in a hydromagnetic, viscous, incompressible, unidirectional, steady, fully developed, third-grade fluid flowing through a channel saturated by a porous medium under the influence of a constant pressure gradient. First-order heterogeneous boundary reactions are applied at both channel walls. A regular perturbation method is applied to derive an approximate steady velocity profile for the third-grade fluid. Aris's method of moments is employed on the governing time-dependent advection–diffusion equation, followed by an implicit finite-difference scheme to study the dispersion process of solute through the channel. The first four central moments are used on the Hermite polynomial representation to ascertain the axial distribution of the solute's mean concentration. The concentration profiles of the solute in both longitudinal and transverse directions are obtained using Aris's method of moments, extending beyond one-dimensional axial dispersion. The study demonstrates how solute dispersion in a third-grade fluid is influenced by Darcy number, Hartmann number, magnetic field inclination, and absorption parameter. It is observed that the flow velocity significantly drops across the channel as the third-grade parameter, Hartmann number, and angle of inclination of the magnetic field increase. However, the fluid velocity increases as the Darcy number rises. The dispersion coefficient decreases with lower values of the third-grade parameter and Darcy number, but the opposite scenario is observed for the increment of the Hartmann number, inclination angle of the magnetic field, and reaction parameter. It is evident that the dispersion of solute enhances by 177.89% when the Darcy number rises from 0.1 to 0.2, and by 54.93% when it increases from 0.2 to 0.3. The dispersion of solute decreases by 1.15% when the third-grade parameter increases from 0.1 to 0.5. As Darcy number and absorption parameter raise, the amplitude of the mean concentration distribution of the solute sharply declines. On the other hand, when Hartmann number, angle of inclination of the magnetic field, and the third-grade parameter enhance, the peak of the tracer's mean concentration distribution increases. It is also seen that the mean concentration distribution of solute decreases by 63.91% when the Darcy number rises from 0.1 to 0.2, and by 24.44% when it increases from 0.2 to 0.3. Also, the mean concentration distribution of solute increases by 0.019% when third-grade parameter advances from 0.1 to 0.5, and by 0.19% when it improves from 0.5 to 1. The results provide insights into complex transport mechanisms relevant to petrochemical engineering, pharmaceutical processes, lubricant manufacturing, and food industries.