## 2 Statement of the Problem  
Let us consider the unsteady free convection flow and heat transfer of a nanofluid, modeled by the Caputo time-fractional derivative, past an infinite vertical plate situated in the \((x_{1},z_{1})\)-plane of a fixed Cartesian coordinate system \(Ox_{1}y_{1}z_{1}\). At the initial moment \(t_{1}=0\), the fluid and the plate are at rest, with a constant ambient temperature \(T_{\infty}\). We also consider the radiation effect and assume the radiative heat flux to be applied, perpendicular to the plate. Since the plate is infinite, all of the physical quantities describing the fluid motion are functions of \(y_{1}\) and \(t_{1}\). The fluid is a water-based nanofluid containing nano particles of CuO or Ag, whose thermo-physical properties are given in Table 1[13; 28]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Basic Fluid/Nanoparticles** & \(\rho\) **(Kg/m3)** & \(C_{p}\)**(J/Kg K)** & \(k\) **(W/m-K)** & \(\beta\) **\(\times\) 10\({}^{5}\) (1/K)** \\ \hline Pure water & 997.1 & 4179 & 0.613 & 21 \\ Copper oxide (CuO) & 6320 & 531.8 & 76.5 & 1.80 \\ Silver (Ag) & 10,500 & 235 & 429 & 1.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermo-physical properties of water and nanoparticles. Assuming a small difference between the fluid temperature \(T(y_{1},t_{1})\) and the stream temperature \(T_{\infty}\), and adopting the Rosseland approximation [5], the radiative heat flux \(q_{r}(y_{1},t_{1})\) can be linearized to:  
\[q_{r}(y_{1},t_{1})=-\frac{16\sigma^{*}T_{\infty}{}^{3}}{3k^{*}}\frac{\partial T (y_{1},t_{1})}{\partial y_{1}}, \tag{1}\]  
where \(\sigma^{*}\) is the Stefan-Boltzman constant and \(k^{*}\) is the mean absorption coefficient. In the following, we consider the nanofluid model proposed by [PERSON] and [PERSON] [29], and take into consideration the usual [PERSON]'s approximation. In this case, the governing equations can be written as [13]:  
\[\rho_{nf}\frac{\partial u_{1}(y_{1},t_{1})}{\partial t_{1}}=\mu_{nf}\frac{ \partial^{2}u_{1}(y_{1},t_{1})}{\partial y_{1}{}^{2}}+g(\rho\beta)_{nf}[T(y_{ 1},t_{1})-T_{\infty}];y_{1},t_{1}>0, \tag{2}\]  
\[(\rho c_{p})_{nf}\frac{\partial T(y_{1},t_{1})}{\partial t_{1}}=k_{nf}(1+ \frac{16\sigma^{*}T_{\infty}{}^{3}}{3k_{nf}k^{*}})\frac{\partial^{2}T(y_{1},t _{1})}{\partial y_{1}{}^{2}};y_{1},t_{1}>0. \tag{3}\]  
If no slipping exists between the fluid and the plate, the appropriate initial and boundary conditions are:  
\[u_{1}(y_{1},0)=0,\,T(y_{1},0)=T_{\infty};\,y_{1}\geq 0, \tag{4}\]  
\[u_{1}(0,t_{1})=0,\,T(0,t_{1})=T_{w};\,\,t_{1}>0, \tag{5}\]  
\[u_{1}(y_{1},t_{1})\to 0,\,T(y_{1},t_{1})\to 0\,\,\text{as}\,\,y_{1}\to\infty. \tag{6}\]  
In the above relations, \(u_{1}(y_{1},t_{1})\) is the fluid velocity in the \(x_{1}\)-vertical direction, \(T_{w}\) is the constant plate temperature (\(T_{w}>T_{\infty}\) or \(T_{w}<T_{\infty}\) corresponds to the heated or cooled plate, respectively), \(g\) is the acceleration due to gravity, \(\rho_{nf}\) is the density of the nanofluid, \(\mu_{nf}\) is the dynamic viscosity of the nanofluid, and \(\beta_{nf}\) is the thermal expansion coefficient of the nanofluid. Their expressions, as well as the expression of \((\rho c_{p})_{nf}\), are given by:  
\[\rho_{nf}=(1-\varphi)\rho_{f}+\varphi\rho_{s},\,\,\mu_{nf}=\frac{\mu_{f}}{(1- \varphi)^{2.5}}, \tag{7}\]  
\[(\rho\beta)_{nf}=(1-\varphi)(\rho\beta)_{f}+\varphi(\rho\beta)_{s},\,\,\,( \rho c_{p})_{nf}=(1-\varphi)(\rho c_{p})_{f}+\varphi(\rho c_{p})_{s} \tag{8}\]  
where \(\varphi\) is the nanoparticle volume fraction, \(\rho_{f}\) is the density of the base fluid, \(\rho_{s}\) is the density of the solid particle, and \(c_{p}\) is the specific heat at constant pressure.