## 6 Conclusions  
An analytical study of the natural convection flow of some water-based fractional nanofluids over an infinite vertical plate with thermal radiation and a uniform temperature on the boundary,  
Figure 4: Values of the skin friction coefficient \(C_{f}\) against the nanoparticle volume fraction \(\varphi\) for two water-based fractional nanofluids. (**a**) for \(Nr=1\); (**b**) for \(Nr=0\). Figure 3: Variation of the velocity \(u\)(\(y\),\(t\)) with the fractional parameter \(\alpha\) and the nanoparticle volume fraction \(\varphi\). (**a**) for \(\varphi=0\); (**b**) \(\varphi=0.1\); (**c**) for \(\varphi=0.2\). is developed using the Caputo time-fractional derivative. The closed forms of solutions for the dimensionless temperature and velocity fields, and the corresponding Nusselt number and skin friction coefficient, are established in equivalent forms, in terms of the Wright function and its fractional derivatives. They have been reduced to the solutions obtained in [13], corresponding to ordinary nanofluids, when the fractional parameter \(\alpha\to 1\). In order to get some physical insight into the results, which have been obtained for CuO-water and Ag-water fractional nanofluids, some numerical calculations and graphical representations have been presented in Table 2 and Figures 1-4, for different values of the fractional parameter \(\alpha\), radiative parameter \(Nr\), and the nanoparticles volume fraction \(\varphi\). The main findings are:  
The enhancement of heat transfer in the natural convection flows is lower for fractional nanofluids, in comparison to ordinary nanofluids. The thermal boundary layer is thicker for fractional nanofluids. In both cases, the fluid temperature increases and the heat transfer declines for increasing values of the nanoparticle volume fraction \(\varphi\). The flows of water-based fractional nanofluids are faster than the ordinary nanofluids. A decrease in the fractional derivative parameter increases the thickness of the velocity boundary layer. From a physical point of view, it means that the nanofluid viscosity increases with the decreasing of \(\alpha\). The dimensionless velocity of water-based fractional nanofluids, as well as that of ordinary nanofluids, is a decreasing function with respect to the nanoparticle volume fraction \(\varphi\). The skin friction coefficient in the natural convection flow is higher for fractional nanofluids, in comparison to ordinary nanofluids. The enhancement of heat transfer in natural convection is stronger for Cu-water fractional nanofluids, when compared with Ag-water fractional nanofluids. In the presence of radiation, Ag-water fractional/ordinary nanofluids achieve a lower skin friction coefficient near the plate, relative to Cu-water fractional/ordinary nanofluids. Our study shows that the nanofluids described by fractional derivatives have a significantly different behavior than ordinary nanofluids. The same conclusions were obtained in the references [33; 34]. Even if the models studied in these articles are different from the model studied here, the obtained results showed that the fractional parameter has a strong influence on the heat transfer process. **Acknowledgments:** [PERSON] is grateful to Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan for supporting this scientific research. **Conflicts of Interest:** The authors declare no conflict of interest.