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We study a forced convective three-dimensional flow and heat transfer in the porous medium. Temperatures of the fluid and solid phases are not identical leading to local thermal non-equilibrium (LTNE) conditions, and hence we require two heat transport equations: one for each phase. This model arises when a hot (cold) fluid flows through a relatively cold (hot) porous medium. This situation usually appears in geothermal and chemical engineering applications. The flows outside the boundary layers are approximated by a linear variation along with the streamwise directions. The governing equations that model the problem are solved numerically using the Chebyshev collocation method. The chief focus is on the application of the Chebyshev collocation method for the system of ordinary differential equations that essentially governing the fluid flow and heat transfer in the boundary layers. The numerical results show that the thickness of the boundary layer is thinner in the streamwise direction, and also predicts the reverse flow in the other direction for a negative three-dimensionality parameter. The thinning of the boundary layer thickness is found for higher permeability values. In the case of forced convection regime, both interphase rate of heat transfer and porosity scaled conductivity are decreased towards zero, the temperature of the fluid phase gradually deviates from solid porous medium temperature thereby showing LTNE effects. The various results corresponding to the LTNE are shown to be a continuation of the classical heat transfer already present in the local thermal equilibrium. The temperature difference which arises out of the LTNE is suitable in most industrial applications. Further, to assess the nature of these flows for a large time, a linear stability analysis is performed to see whether or not the obtained solutions are practically realizable. It is found that all the obtained solutions are always stable, and hence are indicative of practically significant. The fluid dynamics of these mechanisms are discussed in detail.