## 2 Mathematical Modeling
### Similarity Transformation  
The following transformations were used to convert the abovementioned nonlinear partial differential equations to ordinary differential equations to ordinary differential equations,  
\[ \begin{split}& u=-\Omega yg(\eta)+\Omega xf^{\prime}(\eta)+gk( \eta)\sin\Bigl{(}\frac{\beta}{\Omega}\Bigr{)},\\ & u_{p}=-\Omega yG(\eta)+\Omega xf^{\prime}(\eta)+gK(\eta)\sin \Bigl{(}\frac{\beta}{\Omega}\Bigr{)},\\ & v=\Omega xg(\eta)+\Omega yf^{\prime}(\eta)+gs(\eta)\sin \Bigl{(}\frac{\beta}{\Omega}\Bigr{)}\\ & v_{p}=\Omega xG(\eta)+\Omega yF^{\prime}(\eta)+gS(\eta)\sin \Bigl{(}\frac{\beta}{\Omega}\Bigr{)}\\ & w=-2\sqrt{\Omega v_{nf}}f(\eta),\theta(\eta)=\ \frac{\overline{T}-\overline{T}_{w}}{\overline{T}_{0}-\overline{T}_{\infty}} \\ & w_{p}=-2\sqrt{\Omega v_{nf}}F(\eta),\theta_{p}(\eta)=\frac{ \overline{T}_{p}-\overline{T}_{w}}{\overline{T}_{0}-\overline{T}_{\infty}} \end{split}  \tag{18}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Therophysical Properties** & **Base Fluid (H\({}_{2}\)O)** & **Single-Walled Carbon Nanotubes (SWCNTs)** \\ \hline \(C\Bigl{(}\frac{I}{k\overline{g}}k\Bigr{)}\) & 4179 & 425 \\ \(\rho\Bigl{(}\frac{kg}{m^{3}}\Bigr{)}\) & 997.1 & 2600 \\ \(k\Bigl{(}\frac{\text{by}}{mk}\Bigr{)}\) & 0.613 & 6600 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermophysical characteristics of water. \[\eta=z\sqrt{\frac{\Omega}{v_{nf}}} \tag{19}\]  
The phenomenological laws for CNTs are as follows [38]:  
\[\mu_{nf}=\frac{\mu_{f}}{(1-\phi)^{2.5}}, \tag{20}\]  
\[A_{1}=\frac{\rho_{nf}}{\rho_{f}}=(1-\phi)+\frac{(\rho_{CNT})}{\rho_{f}}\phi, \tag{21}\]  
\[A_{2}=\frac{\rho_{nf}C_{nf}}{\rho_{f}C_{f}}=(1-\phi)+\frac{\rho_{CNT}C_{CNT}}{ \rho_{f}C_{f}}\phi \tag{22}\]  
\[A_{3}=\frac{K_{nf}}{K_{f}}=\frac{(1-\phi)+2\phi\frac{k_{CNT}}{k_{CNT}-k_{f}}ln \frac{k_{CNT}+k_{f}}{2k_{f}}}{(1-\phi)+2\phi\frac{k_{f}}{k_{CNT}-k_{f}}ln\frac {k_{CNT}+k_{f}}{2k_{f}}}. \tag{23}\]  
Equation (1) holds, and Equations (2)-(9) are transformed into Equations (24)-(33). \[{f^{\prime}}^{2}-2 ff^{\prime\prime}-g^{2}-\frac{\beta_{\
u}}{A_{1}}l(F^{ \prime}-f^{\prime})=f^{\prime\prime} \tag{24}\]  
\[2(f^{\prime}g-fg^{\prime})-\frac{\beta_{\
u}}{A_{1}}l(G-g)=g^{\prime\prime} \tag{25}\]  
\[kf^{\prime}-2 fk^{\prime}-1-\frac{\beta_{\
u}}{A_{1}}l(K-k)=k^{\prime\prime}, \tag{26}\]  
\[kg+sf^{\prime}-2s^{\prime}f-\frac{\beta_{\
u}}{A_{1}}l(S-s)=s^{\prime\prime}, \tag{27}\]  
\[ \begin{array}{l}(\frac{A_{1}A_{3}}{A_{2}P_{T}}-\frac{A_{1}Rd}{A_{2}P_{T}}) \theta^{\prime\prime}-(S_{1}+\theta)f^{\prime}+2f\theta^{\prime}+\frac{\delta \theta}{A_{2}}\\ -\frac{A_{1}Rd}{A_{2}P_{T}}\frac{3{g^{\prime}}^{2}}{\delta_{w}+S_{1}}(1+\frac{ 1}{\theta_{w}+S_{1}})^{2}]+\beta_{\mathcal{I}}\gamma l(\theta-\theta_{p})=0 \end{array}  \tag{28}\]  
**Dusty Fluid:**  
\[{F^{\prime}}^{2}-2F^{\prime\prime}-G^{2}-\beta_{\
u}(f^{\prime}-F^{\prime})=0, \tag{29}\]  
\[2(F^{\prime}G-FG^{\prime})-\beta_{\
u}(g-G)=0, \tag{30}\]  
\[KF^{\prime}-GS-2 FK^{\prime}-\beta_{\
u}(k-K)=0, \tag{31}\]  
\[KG+SF^{\prime}-2S^{\prime}F-\beta_{\
u}(s-S)=0, \tag{32}\]  
\[\Big{(}S_{1}+\theta_{p}\Big{)}F^{\prime}-2\theta_{p}^{\prime}-\beta_{t}\Big{(} \theta_{p}-\theta\Big{)}=0, \tag{33}\]  
The boundary conditions after transformation are  
\[ \begin{array}{l}f(0)=0,f^{\prime}(0)=0,g(0)=1,k(0)=0,\ s(0)=0,\ S(0)=0,\ \theta(0)=1-S_{1},\\ f^{\prime\prime}(\delta)=0,F^{\prime\prime}(\delta)=0,\ k^{\prime}(\delta)=0,K^{\prime}(\delta)=0,s^{\prime}(\delta)=0,\ \theta(\delta)=0,\ \theta_{p}(\delta)=0.\end{array}  \tag{34}\]  
where \(\delta\) is the normalized thickness constant.