Impacts of the fractional derivatives on unsteady magnetohydrodynamics radiative Casson nanofluid flow combined with Joule heating

This paper presents a mathematical formulation and numerical simulations of an unsteady magnetohydrodynamic non-Newtonian nanofluid flow and entropy generation over a vertical plate using the fractional derivatives approaches. Impacts of a thermal radiation, Joule heating and convective boundary conditions on the convective flow are examined. Both of the Caputo definition and the conformable definition are used to estimate the time fractional derivatives and comparisons between these definitions are carried out for different values of the fractional derivatives order (0.85 <= delta <= 0.95 < i). The fractional partial differential system is converted to a dimensionless form then it solved numerically using an implicit finite difference method (FDM). Profiles of the velocity, temperature and nanoparticles volume fraction as well as values of the Nusselt number, Sherwood number and skin friction are surveyed using wide ranges of Casson parameter (1 <= beta <= 100 < i), the magnetic field parameter (0 <= M <= 5 < i), the radiation parameter (1 <= R-d <= 10 < i), the buoyancy ratio (0 <= Nr <= 0.5 < i), the thermophores parameter (0.1 <= Nt <= 0.5<i) and Biot number (0 <= Bi <= 0.3<i). The consequences revealed that the Nusselt number is reduced as the order of the fractional derivatives approaches to one while it causes an enhancement in the skin friction. Also, the raise in values of the buoyancy ratio and Lorentz force minimizes activity of the nanofluid fluid flow.