## 3 Numerical Solution  
By using the similarity approach, BVP (2)-(5) are transformed into ordinary differential equations (ODE). The stream functions (\(\psi\)) are introduced as:  
\[\textbf{{u}}=\mathbf{\psi}_{y},\ \textbf{{v}}=-\mathbf{\psi}_{x}, \tag{8}\]  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Property** & **Nanofluid** \\ \hline Dynamic viscosity & \(\mu_{nf}=\mu_{f}(1-\phi)^{-2.5}\) \\ Density & \(\rho_{nf}=(1-\phi)\rho_{f}+\phi\rho_{s}\) \\ Heat capacity & \((\rho C_{p})_{nf}=(1-\phi)(\rho C_{p})_{f}+\phi(\rho C_{p})_{s}\) \\ Thermal conductivity & \(\frac{k_{eff}}{k_{f}}=\Big{[}\frac{\left(k_{x}+2k_{f}\right)-2\phi\left(k_{f} -k_{x}\right)}{\left(k_{x}+2k_{f}\right)+\phi\left(k_{f}-k_{x}\right)}\Big{]}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The primary properties of the investigated third-grade nanofluid [58].  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Thermophysical Properties** & \(\rho(\textbf{kg/m}^{3})\) & \(c_{p}(\textbf{J/kgK})\) & \(k(\textbf{W/mK})\) \\ \hline SWCNT & 2600 & 425 & 6600 \\ MWCNT & 1600 & 796 & 3000 \\ CMC & 997.10 & 4179 & 0.6130 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Principal properties of the base fluid and nanoparticles at a standard temperature [59; 60].  
The similarity variables are expressed as:  
\[\mathbf{Y}(\mathbf{x},\mathbf{y})=\sqrt{\frac{\mathbf{b}}{\
u_{f}}}\mathbf{y},\ \mathbf{\psi}(\mathbf{x},\mathbf{y})= \sqrt{\
u_{f}}\mathbf{\omega}\mathbf{f}(\mathbf{Y}),\ \mathbf{\theta}(\mathbf{Y})=\frac{\mathbf{Y}-\mathbf{Y}_{ \infty}}{\mathbf{V}_{\mathbf{w}}-\mathbf{Y}_{\infty}}, \tag{9}\]  
Then, according to (2)-(4), the following equations are obtained:  
\[\mathbf{f}^{\prime\prime}+\mathbf{\phi}_{1}\Big{[}\mathbf{\phi}_{2}\big{(} \mathbf{f}^{\prime\prime}-\mathbf{f}^{2}\big{)}+\mathbf{\beta}_{1}\big{(}\mathbf{2}\mathbf{f}^{ \prime\prime}-\mathbf{f}^{\prime\prime\prime}\big{)}+\mathbf{3}(\mathbf{\beta}_{1}+\mathbf{ \beta}_{2})\mathbf{f}^{\prime}{}^{2}\big{]}\Bigg{\}} \tag{10}\] \[+\mathbf{\phi}_{1}\mathbf{\beta}_{3}\mathbf{R}_{\mathbf{c}}\mathbf{f}^{\prime}{}^{2} \mathbf{f}^{\prime\prime}-\mathbf{K}\mathbf{f}^{\prime}=\mathbf{0},\] \[\mathbf{\theta}^{\prime\prime}\left(\mathbf{1}+\mathbf{e}\mathbf{\theta}+\frac{ \mathbf{1}}{\mathbf{\phi}_{4}}\mathbf{P}_{\mathbf{r}}\mathbf{N}_{\mathbf{r}}\right)+\mathbf{e}\mathbf{\theta} ^{\prime}{}^{2}+\mathbf{P}_{\mathbf{r}}\frac{\mathbf{\phi}_{3}}{\mathbf{\phi}_{4}}\Big{[}\mathbf{ \phi}^{\prime}-\mathbf{f}\mathbf{\theta}+\mathbf{\frac{Q_{r}}{\mathbf{\phi}_{3}}}+\frac{E_{c} }{\mathbf{\phi}_{1}\mathbf{\phi}_{3}}\mathbf{f}^{\prime}{}^{2}\Big{]}=\mathbf{0}, \tag{11}\]  
with  
\[\mathbf{f}(\mathbf{0})=\mathbf{S},\ \mathbf{f}(\mathbf{0})=\mathbf{1}+\mathbf{\Lambda}\mathbf{f}^{\prime }(\mathbf{0}),\ \mathbf{\theta}^{\prime}(\mathbf{0})=-\mathbf{B}_{\zeta}(\mathbf{1}-\mathbf{\theta}(\mathbf{0})) \tag{12}\]  
where \(\mathbf{\phi}^{\prime}{}_{i}\mathbf{s}\) is \(\mathbf{1}\leq i\leq\mathbf{4}\) in Equations (10) and (11). The thermophysical characteristics of the third-grade nanofluid are shown below.  
\[\mathbf{\phi}_{1} =(\mathbf{1}-\mathbf{\phi})^{2.5},\ \mathbf{\phi}_{2}=\big{(}\mathbf{1}-\mathbf{\phi}+\mathbf{ \phi}\frac{\rho_{c}}{\rho_{f}}\big{)},\ \mathbf{\phi}_{3}=\left(\mathbf{1}-\mathbf{\phi}+\mathbf{\phi}\frac{(\rho_{c})_{x}}{(\rho_{c} )_{f}}\right) \tag{13}\] \[\mathbf{\phi}_{4} =\left(\frac{\big{(}\mathbf{k}_{x}+2\mathbf{k}_{y}\big{)}-2\mathbf{\phi} \big{(}\mathbf{k}_{y}-\mathbf{k}_{x}\big{)}}{\big{(}\mathbf{k}_{x}+2\mathbf{k}_{y}\big{)}+\bm {\phi}\big{(}\mathbf{k}_{y}-\mathbf{k}_{x}\big{)}}\right).\]