Heat and mass transfer through a vertical channel for the Brinkman fluid using Prabhakar fractional derivative

Abstract

In industrial and manufacturing processes like printing, coating, and painting, non-Newtonian fluid flow plays an significant role. The primary focus of the current study is to develop and advance a flow model of Brinkman flow through a channel of two parallel vertical plates. The generalization of the constitutive relations for mass and heat fluxes is done by utilizing the advanced and more generalized definition of the Prabhakar operator. The Prabhakar operator involved the 3 parameters of Mittag-Leffler mapping as a kernel of an operator, therefore, this derivative is a more general and advanced fractional derivative as compared to the other fractional derivatives. The governing equations for this flow model can be calculated by coupling the foresaid generalized constitutive relations for heat and mass fluxes. Furthermore, to have better and deeper knowledge about the behavior of flow, the developed governing equations are transformed to dimensionless form by introducing suitable relations for the involved variables. We are interested in getting the exact result for the temperature, concentration, and velocity of the flow. To achieve this goal, the Laplace transform is utilized to prescribe partial differential governing equations for temperature, velocity, and concentration. The inverse of the Laplace transform is done applying Stehfest’s and Tzou’s algorithm to transform governing equations for temperature, velocity, and concentration. The inversion results that are obtained by Stehfest’s and Tzou’s algorithm are presented in graphical form. Further, the parametric analysis of the research outcomes is also explained graphically. For this purpose, the field variables, namely temperature, velocity, and concentration, are also plotted in some graphs. From the parametric study, it is concluded that by raising the values of the fractional parameter, velocity increases, while simultaneously a decreasing trend is seen for concentration and temperature for a short time. The effects of many variables, including fraction parameters, mass Grashof number, Grashof number, Prandtl number, Schmidt number, Brinkman parameters, and magnetic field, on concentration, temperature, and velocity are depicted graphically.