## 5 Numerical Results and Discussion  
The natural convection flow of water-based fractional nanofluids over an infinite vertical plate with thermal radiation and a uniform temperature on the boundary, is analytically studied. Closed form solutions for the dimensionless temperature and velocity fields, and the two entities of physical interest, the Nusselt number and skin friction coefficient, are determined in equivalent forms, in terms of the Wright function or its fractional derivatives. It is worth pointing out that all of these solutions have been immediately reduced to the known solutions, based on the literature for ordinary nanofluids. A table containing the thermo-physical properties of copper oxide (CuO) and silver (Ag) is also included for later use. In order to bring to light the influence of the fractional parameter on the heat and mass transfer in the natural convection flow of the above-mentioned fractional nanofluids, and therefore to obtain some physical insight into the present results, some numerical calculations have been carried out for different values of the fractional parameter \(\alpha\), radiation parameter \(Nr\), and the nanoparticle value fraction \(\varphi\). For comparison, the diagrams of dimensionless temperature and velocity fields, and the Nusselt number and skin friction coefficient corresponding to fractional nanofluids (for different values of the fractional parameter \(\alpha\in(0,1)\)) and those of ordinary nanofluids (when \(\alpha=1\)), are depicted in Figures 1-4. As was expected, in all cases, the diagrams corresponding to the fractional nanofluids tend to superpose over those of ordinary nanofluids, when \(\alpha\to 1\). the ordinary ones. An opposite trend appears at a later point in time. With respect to the volume fraction \(\varphi\), the temperature is an increasing function for both ordinary and fractional nanofluids, and it smoothly decreases from the maximum value of one on the boundary, to the zero value far away from the plate. In Table 2, the temperature at different values of \(y\) and of the fractional parameter \(\alpha\), is given for nanofluids containing nanoparticles of CuO or Ag, and these values are in full accordance with those resulting from Figure 1. Furthermore, as results from this table, the Ag nanoparticles induce larger temperature values and a lower heat transfer to ordinary or fractional nanofluids, in comparison to CuO nanoparticles. The effect of enhancing the heat transfer rate against \(\varphi\) with the fractional parameter \(\alpha\), for CuO-water and Ag-water nanofluids, is presented in Figure 2, in the absence or presence of thermal radiation. For both fractional nanofluids, the Nusselt number Nu is an increasing function with respect to \(\alpha\). It is also an almost linearly increasing function of \(\varphi\). Further, the heat transfer rate for the CuO-water fractional nanofluid is always a little higher than that corresponding to Ag-water fractional nanofluid. However, the difference between them increases with increasing values of \(\varphi\). The influence of the fractional parameter \(\alpha\) and of the nanoparticle volume fraction \(\varphi\), on the dimensionless velocity \(u(y,t)\) against \(y\), is brought to light by Figure 3a-c; which is also seen for a nanofluid with CuO nanoparticles. Near the plate, the nanofluid velocity increases up to a maximum value and then asymptotically decreases to the zero value for \(y\) values greater than 2.5, but it is a decreasing function with respect to the two parameters \(\alpha\) and \(\varphi\). Consequently, the boundary layer thickness is lower for ordinary nanofluids, in comparison to fractional nanofluids. From a physical point of view, it means that the nanofluid viscosity decreases for increasing values of \(\alpha\) or \(\varphi\), and the ordinary nanofluids exhibit a stronger capacity in flow. This implies that the viscoelasticity strengthens the flow resistance with a decrease in the fractional parameter [33].