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Nanofluids are gaining extensive attention due to their thermo-physical properties in technological and industrial fields for controlling the effects of heat transfer. Classical nanofluid studies are generally confined to models described by partial differential equations of an integer order where the memory effect and hereditary properties of materials are neglected. In order to overcome these downsides, the present work focuses on studying nanofluids with fractional derivative formed by differential equations with Caputo time derivatives that provides memory effect on nanofluid characteristics. Also, investigation on natural convective flow, heat, and mass transfer of nanofluids formed by different base fluids with different shapes of copper nanoparticles past an infinite vertical plate with radiation effect is carried out. The governing fractional differential equations are solved by employing Laplace transform technique with suitable boundary conditions. The different base fluids-water (H 2 O), SA:sodium alginate (C 6 H 9 Na O 7 ), and EG:ethylene glycol (C 2 H 6 O 2 ) and different shapes of nanoparticles-blade, brick, platelet, and cylinder are considered for the study.The exact solutions are obtained for the temperature, concentration, and velocity distributions and the respective Nusselt number, Sherwood number, and skin-friction coefficient. The influence of non-dimensional parameters provides physical interpretations of temperature, concentration, and velocity fields, Nusselt number, Sherwood number, and skin friction in detail with the help of graphical representations. From the results, it is found that nanofluid with water-based blade-shaped nanoparticle exhibits more velocity and temperature distributions. Also, strengthen of fluid flow, temperature, and concentration of nanofluids are inversely correlate with fractional order derivatives.

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The present numerical study reports the thermal performance of the straight porous fin with temperature-dependent thermal conductivity, radiation, and magnetic field effects. The heat transfer model comprising the Darcy's law for simulating flow with solid-fluid interactions in porous medium, Rosseland approximation for heat transfer through radiation, Maxwell equations for magnetic field effect and linearly varying temperature dependent thermal conductivity, results into highly nonlinear ordinary differential equation. The governing equation is solved using a finite difference scheme with suitable boundary conditions. The obtained solutions are physically interpreted by considering the impact of different nondimensional parameters on thermal performance, efficiency, and effectiveness of the system through plotted graphs. A detailed result with regard to the Nusselt number at the fin base is calculated. The results obtained are observed to be in excellent agreement with previous studies. From the study, it is observed that there is a significant effect on the thermal performance of the fin in the presence of porous constraints; also, results reveal that the nonlinear thermal conductivity parameter strengthens the thermal performance, efficiency, and effectiveness of the fin. Furthermore, the results of the study reveal that the rate of heat transfer of the fin increases with the increase in the magnetic parameter and radiation parameter.