This study examines the flow of a Newtonian fluid enclosed between two non-Newtonian Jeffreys fluids with viscosity that varies with temperature within a composite vertical channel. Including a corotational Jeffreys liquid allows for considering stress dependence on the present deformation rate and its history. The proposed study's framework comprises three distinct regions, wherein the intermediate region governs Newtonian fluid flow under temperature-dependent viscosity. However, the outer layers oversee the flow of Jeffreys fluids within the porous medium, demonstrating temperature-dependent viscosity. The Brinkman–Forchheimer equation is employed to establish the governing equations applicable to both low and high permeabilities of the porous medium. This equation is nonlinear, making it challenging to find an analytical solution. Therefore, the regular and singular perturbation methods with matched asymptotic expansions are applied to derive asymptotic expressions for velocity profiles in various regions. The hydrodynamic quantities, such as flow rate, flow resistance, and wall shear stresses, are determined by deriving their expressions using velocities from three distinct regions. The graphical analysis explores the relationships between these hydrodynamic quantities and various parameters, including the Grashof number, Forchheimer number, viscosity parameter, Jeffreys parameter, conductivity ratio, effective viscosity ratio, absorption ratio, and the presence of varying thicknesses of different layers. An interesting finding is that a more pronounced velocity profile is noticed when the permeability is high and the viscosity parameter of the Newtonian region, denoted as α2, is lower than that of the surrounding area. This heightened effect can be linked to a relatively more significant decrease in the viscosity of the Jeffreys fluid, represented by μ1, as compared to the viscosity of the Newtonian fluid, μ2, as the temperature increases. The outcomes of this research hold special significance in situations like the extraction of oil from petroleum reserves, where the oil moves through porous layers with varying viscosities, including sand, rock, shale, and limestone.

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