## 3 Derivation of Fractional Model for Blood-Gold Nanofluid  
Firstly, Equations (8)-(11) will be transformed to dimensionless form. For this purpose, the following dimensionless variables:  
\[f=\frac{u}{u_{0}},\xi=\frac{u_{0}}{v_{f}}y,\tau=\frac{t}{t_{0}},t_{0}=\frac{v_{f} }{u_{0}^{2}},\Phi=\frac{T-T_{\infty}}{T_{W}-T_{\infty}}\]  
are incorporated into Equations (8)-(11) to yield:  
\[ \begin{array}{l}\phi_{0}\Big{\{}\frac{\partial f(\xi,\tau)}{\partial\tau}+ \lambda_{1}\frac{\partial^{2}f(\xi,\tau)}{\partial\tau^{2}}\Big{\}}=\phi_{1} \Big{\{}\frac{\partial^{2}f(\xi,\tau)}{\partial\xi^{2}}+\lambda_{2}\frac{ \partial^{2}f(\xi,\tau)}{\partial\xi\partial^{2}}\Big{\}}\\ -\phi_{2}M\Big{\{}f(\xi,\tau)+\lambda_{1}\frac{\partial f(\xi,\tau)}{\partial \tau}\Big{\}}-\frac{\phi_{1}}{K}\Big{\{}f(\xi,\tau)+\lambda_{2}\frac{\partial f (\xi,\tau)}{\partial\tau}\Big{\}}+\phi_{3}Gr\theta(y,t)\end{array}  \tag{13}\]  
\[\phi_{4}\frac{\partial\Phi(\xi,\tau)}{\partial\tau}=\frac{\phi_{5}}{\Pr}\frac{ \partial^{2}\Phi(\xi,\tau)}{\partial\xi^{2}} \tag{14}\]  
and  
\[f(\xi,0)=0,\Phi(\xi,0)=0,\,\forall\xi\geq 0 \tag{15}\]  
\[f(0,\tau)=\left\{ \begin{array}{ll}\tau;&\text{if}\,0<\tau<1\\ 1;&\text{if}\,\tau>1\\ \Phi(0,\tau)=\left\{\begin{array}{ll}\tau;&\text{if}\,0<\tau<1\\ 1;&\text{if}\,\tau>1\end{array}\right\}\end{array} \right\} \tag{16}\]  
where:  
\[\lambda_{1}=\frac{\lambda}{t_{0}},\,\,\lambda_{2}=\frac{\lambda_{\tau}}{t_{0}},\,\,M=\frac{t_{0}\sigma_{f}B_{0}^{2}}{\rho_{f}},\,\,K=\frac{k}{t_{0}v_{f} \phi}\,\,Gr=\frac{g(\beta_{7}v)_{f}(T_{W}-T_{\infty})}{u_{0}^{2}},\,\,\Pr= \Big{(}\frac{uC_{p}}{k}\Big{)}_{f},\phi_{0}=(1-\phi)+\phi\frac{\rho_{5}}{ \rho_{f}},\]  
\[\phi_{1}=\frac{1}{(1-\phi)^{2S}},\,\phi_{2}=\frac{\sigma_{nf}}{\sigma_{f}},\, \,\phi_{3}=(1-\phi)+\phi\frac{(\rho\beta_{7})_{s}}{(\rho\beta_{7})_{f}},\,\, \phi_{4}=(1-\phi)+\phi\frac{(\rho C_{p})_{s}}{(\rho C_{p})_{f}},\,\,\phi_{5}= \frac{k_{nf}}{k_{f}}\]  
are the dimensionless relaxation time, retardation time, magnetic number, permeability of the porous medium, thermal Grashof number, and Prandtl number, respectively. In addition to this, \(\phi_{0}\), \(\phi_{1}\), \(\phi_{2}\), \(\phi_{3}\), \(\phi_{4}\) and \(\phi_{5}\) are constant terms that appeared during calculations. Now, the Caputo fractional derivative with the Laplace transform (LT) is given by [18; 34]. \[{}^{C}D_{\tau}^{\delta}g(\xi,\tau)=\frac{1}{\Gamma(n-\delta)}\int\limits_{0}^{ \tau}\frac{g^{(n)}(\xi,\tau)}{(\tau-s)^{\delta+1-n}}ds, \tag{17}\]  
\[\mathcal{L}\big{[}{}^{C}D_{\tau}^{\delta}g(\xi,\tau)\big{]}(q)=\overline{g}( \xi,q)=q^{\delta}\mathcal{L}[g(\xi,\tau)]-q^{\delta-1}g(\xi,0), \tag{18}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Material**} & **Base Fluid** & **Nanoparticles** \\ \cline{2-3}  & **Blood** & **Gold** \\ \hline \(\rho(\text{kg}/\text{m}^{3})\) & 1053 & 1250 \\ \(C_{p}(\text{J}/\text{kg}\,\text{K})\) & 3594 & 129 \\ \(k(\text{W}/\text{m}\,\text{K})\) & 0.492 & 318 \\ \(\beta_{7}\times 10^{-5}(\text{K}^{-1})\) & 0.8 & 1.41 \\ \(\sigma\) & 0.18 & 4.45\(\times 10^{7}\) \\ \(\Pr\) & 21 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Numerical_ values of base fluid and solid material nanoparticles [31; 32; 33]. where \(D_{\tau}^{\delta}g(\xi,\tau)\) is the Caputo fractional derivative, \(\delta\) is the fractional-order, and \(\Gamma(. )\) is the gamma function. The kernel of Caputo fractional derivatives is singular for \(\tau=s\). The Caputo--Fabrizio (CF) fractional derivative without singular kernel and its LT is given by [19]. \[C\!F\!D_{\tau}^{\delta}g(\xi,\tau)=\frac{1}{n-\delta}\!\int\limits_{0}^{\tau} \exp\!\left(-\frac{\delta(\delta-\tau)}{1-\delta}\right)\!\frac{\partial g( \xi,s)}{\partial s}ds, \tag{19}\]  
\[\mathcal{L}\!\left\{C\!F\!D_{\tau}^{\delta}g(\xi,\tau)\right\}\!(q)=\overline {g}(\xi,q)=\frac{q\mathcal{L}\!\left\{g(\xi,\tau)\right\}-g(\xi,\tau)}{(1- \delta)q+\delta}, \tag{20}\]  
with local exponential kernel and order \(\delta\).