Exact Solutions for Relative Magnetic Field Effects on Casson Hybrid Nanofluid Flow with Exponential thermal Memory

<title>Abstract</title> <p> <bold>Accurate modeling of transient heat transfer in conducting hybrid nanofluids (HNF) is essential for the scheme of effective heat transfer (HT) management system, especially in situations where the thermal memory effect cannot be accounted for by using the classical Fourier law. In light of this, this study seeks to examine the unsteady magnetohydrodynamic (MHD) flow and HT of a Casson HNF, made up of a 50:50 base fluid of water-ethylene glycol with particles of magnetite ferrite (Fe3O4) and cobalt ferrite (CoFe2 O4) in a vertical porous channel. The classical Fourier law for heat conduction has been modified through the use of the Caputo-Fabrizio fractional integral (CFFI) operator. The exact analytical solution of the governing equations is achieved using the combination of Laplace transform (LT) along with finite Fourier sine transform (FFST). Analytical expressions for the local Nusselt number and skin friction coefficient have also been derived. It has been revealed through graphical illustrations that the fractional memory effect plays an important role in determining the thermal as well as hydrodynamic properties of the flow. In addition to this, a rise in the hybrid nanoparticle (NP) volume fraction (</bold> <italic>ϕ</italic> <bold>) consequences in the enhancement in the temperature profile whereas heat generation/absorption leads to diminish the thermal field. It is observed that the velocity profile is greatly impacted by the Casson fluid parameter ( β</bold> _{<bold>0</bold>} <bold>), magnetic field (M ), and permeability of porous medium ( K ).</bold> </p>