Abstract
In this study, two different fractional models are developed for a second grade fluid coupled with the energy equation. These two fractional models are based on the definitions of Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC) fractional derivatives. The fluid transport is considered over an infinite upright plate, that is nested in a permeable medium under the influence of an imposed magnetic field. The variable wall velocity and Newtonian heating boundary conditions are simultaneously applied for the very first time. Based on two different fractional derivatives, double fractional analysis is accomplished to evaluate the resulting two models. The momentum and energy solutions are separately established for each model by employing Laplace transform and Durbin's numerical Laplace inversion. The verification analysis of these obtained solutions is performed with the aid of Zakian's and Stehfest's algorithms. The rise and fall in momentum and energy profiles due to variation of incipient parametric values are elucidated graphically and physical arguments behind these behaviors are interpreted. The velocity and temperature gradients are evaluated at the boundary to estimate the wall shear stress and heat transfer rate in terms of the skin friction coefficient and Nusselt number. The noteworthy physical impacts of incipient parameters on shear stress and heat transfer rate are analyzed through tabular study. In comparison, it is found that the energy profile for the ABC model is higher than CF model and ordinary model, respectively. Similarly, for velocity distribution, the ABC model exhibits the highest profile as compared to CF and ordinary models in the case of ramped velocity. However, an exactly opposite pattern of velocity profiles is observed for an isothermal velocity case. On the basis of this comparative analysis, it can be stated that the ABC model is the best choice to appropriately explain the memory effect of energy and momentum distributions.