Heat Transfer in MHD Flow of Maxwell Fluid via Fractional Cattaneo-Friedrich Model: A Finite Difference Approach

The idea of fractional derivatives is applied to several problems of viscoelastic fluid. However, most of these problems (fluid problems), were studied analytically using different integral transform techniques, as most of these problems are linear. The idea of the above fractional derivatives is rarely applied to fluid problems governed by nonlinear partial differential equations. Most importantly, in the nonlinear problems, either the fractional models are developed by artificial replacement of the classical derivatives with fractional derivatives or simple classical problems (without developing the fractional model even using artificial replacement) are solved. These problems were mostly solved for steady-state fluid problems. In the present article, studied unsteady nonlinear non Newtonian fluid problem (Cattaneo-Friedrich Maxwell (CFM) model) and the fractional model are developed starting from the fractional constitutive equations to the fractional governing equations; in other words, the artificial replacement of the classical derivatives with fractional derivatives is not done, but in details, the fractional problem is modeled from the fractional constitutive equations. More exactly two-dimensional magnetic resistive flow in a porous medium of fractional Maxwell fluid (FMF) over an inclined plate with variable velocity and the temperature is studied. The Caputo time-fractional derivative model (CFM) is used in the governing equations. The proposed model is numerically solved via finite difference method (FDM) along with L1-scheme for discretization. The numerical results are presented in various figures. These results indicated that the fractional parameters significantly affect the temperature and velocity fields. It is noticed that the temperature field increased with an increase in the fractional parameter. Whereas, the effect of fractional parameters is opposite on the velocity field near the plate. However, this trend became like that of the temperature profile, away from the plate. Moreover, the velocity field retarded with strengthening in the magnetic parameter due to enhancement in Lorentz force. However, this effect reverses in the case of the temperature profile.