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Importance of studying Brinkman–Navier–Stokes flow under the influence of electric and magnetic fields lies in its relevance to fundamental physical phenomena, its applications in various fields of science and engineering and its potential for technological advancements. The study of Brinkman–Navier–Stokes flow under the influence of electric and magnetic fields, along with additional factors such as Joule heating, fractional derivatives and convection, represents a multifaceted and challenging problem in fluid dynamics and applied mathematics. In this study, we explore the combined effects of Joule heating due to electric current passing through the fluid and fractional derivatives that describe nonlocal behaviors. The fractional formulation leads to a relaxation mechanism that exhibits delay and recollection of the fluid motion. The organization of direct temperature and concentration changes in MHD flow is made possible by this formalization. A finite difference/finite element method is used to calculate the flow dynamics issue and fractionally linked fields. The physical factors explain how the topic under examination is relevant.

The enhancement of the working ability of the industrial fluid is the need of the present era; nanofluid is an emerging field in science and technology. In this study, the Brinkman-type fluid model is used and is generalized using the Fourier’s and Fick’s laws. The graphene oxide nanoparticles are dispersed in the base fluid water. The fractional partial differential equations are then solved via the Laplace and Fourier transform method. The obtained solutions for velocity, heat transfer, and mass transfer are plotted in graphs. The results show that velocity profile decreases for Brinkman-type fluid parameter and volume fraction of the nanoparticles. The plot for the fractional parameter shows that different plots can be drawn for a fixed time and other physical parameters, which is the memory effect.

This paper proposes a new method for the development of the Caputo time fractional model. The method relies on generalized Fourier’s and Fick’ laws to describe the flow behavior of Brinkman-type fluids. An analysis of the free convection flow through a channel is carried out using a new transformation method. This transformation affects fluid energy and concentration equations. The specific governing equations are solved using a Laplace transform and Fourier sine transform. We obtain the solutions of the governing partial differential equations (PDEs) in terms of the Mittag–Leffler function. Mathematical software has been used for both graphical and numerical computation in order to examine the effects of embedded parameters. From graphical and tabular analysis, fractional-order solution provides more than one layer for fluid behavior, thermal, and concentration distribution in the channel. Experimentalists and engineers can choose from many best-fitted layers to compare their data and results. A deviation in the velocity profile’s behavior is also seen for larger values of the Brinkman parameter.