## 2 Mathematical Modeling
\begin{split}&(1+\xi^{2})F^{\prime\prime}+3\zeta F^{\prime}+(1- \phi_{Al_{2}O_{3}})^{2.5}(1-\phi_{Cu})^{2.5}\times\bigg{[}(1-\phi_{Cu})\big{(}1- \big{(}1-\frac{\rho_{Al_{2}O_{3}}}{\rho f}\big{)}\phi_{Al_{2}O_{3}}\big{)}\\ +&\phi_{Cu}\big{(}\frac{\rho_{Cu}}{\rho_{f}}\big{)} \bigg{]}\big{[}\zeta FF^{\prime}-HF^{\prime}+F^{2}-G^{2}\big{]}+(1-\phi_{Al_{2}O _{3}})^{2.5}(1-\phi_{Cu})^{2.5}\big{[}\zeta p^{\prime}+2p-MF\big{]}=0\\ \\ &(1+\xi^{2})G^{\prime\prime}+3\zeta G^{\prime}-(1-\phi_{Al_{2}O _{3}})^{2.5}(1-\phi_{Cu})^{2.5}\times\bigg{[}(1-\phi_{Cu})\big{(}1-\big{(}1- \frac{\rho_{Al_{2}O_{3}}}{\rho f}\big{)}\phi_{Al_{2}O_{3}}\big{)}\\ +&\phi_{Cu}\big{(}\frac{\rho_{Cu}}{\rho_{f}}\big{)} \bigg{]}\big{[}\zeta FG^{\prime}-HG^{\prime}\big{]}-(1-\phi_{Cu})^{2.5}(1-\phi _{Al_{2}O_{3}})^{2.5}MG=0\\ \\ &(1+\xi^{2})H^{\prime\prime}+3\zeta H^{\prime}+(1-\phi_{Al_{2}O _{3}})^{2.5}(1-\phi_{Cu})^{2.5}\times\bigg{[}(1-\phi_{Cu})\big{(}1-\big{(}1- \frac{\rho_{Al_{2}O_{3}}}{\rho f}\big{)}\phi_{Al_{2}O_{3}}\big{)}\\ +&\phi_{Cu}\big{(}\frac{\rho_{Cu}}{\rho_{f}}\big{)} \bigg{]}\big{[}\zeta FH^{\prime}-HH^{\prime}+H+FH\big{]}-(1-\phi_{Cu})^{2.5}(1- \phi_{Al_{2}O_{3}})^{2.5}p^{\prime}=0,\\ \\ &\frac{k_{Inf}}{k_{nf}}\big{[}(1+\xi^{2})\theta^{\prime\prime}+ \zeta(1-2n)\theta^{\prime}+n^{2}\theta\big{]}+Pr\bigg{[}(1-\phi_{Cu})\big{(}1 -\big{(}1-\frac{(\rho_{Cu})_{Al_{2}O_{3}}}{(\rho_{Cu})f}\big{)}\phi_{Al_{2}O_ {3}}\big{)}\\ +&\phi_{Cu}\big{(}\frac{(\rho{C}_{p})_{Cu}}{\rho_{C _{pf}}}\big{)}\bigg{]}\big{[}\zeta F\theta^{\prime}-nF\theta-H\theta^{\prime} \big{]}+\frac{M}{(1-\phi_{Cu})^{2.5}}(1-\phi_{Al_{2}O_{3}})^{2.5}\big{[}f^{2}+ G^{2}\big{]}=0\\ \end{split}  \tag{12}\]  
The altered terms are as follows:  
\[ \begin{split}& H(0)=F(0)=0,G(0)=r\varepsilon_{ur},\theta(0)=1,\\ & F(\zeta_{0})=H(\zeta_{0})=0,G(\zeta_{0})=Re\varepsilon_{\Omega },\theta(\zeta_{0})=0.\\ \end{split}  \tag{13}\]  
The volumetric proportions of \(Al_{2}O_{3}\) and \(Cu\) are represented by \(\phi_{Al_{2}O_{3}}\) and \(\phi_{Cu}\), respectively. Meanwhile, \(k_{Inf}\) and \(k_{f}\) denote the thermal conductivity of the hybrid nanoliquid and water, respectively. The unique thermal properties of the hybrid nanoliquid and water are delineated as follows [43]:  
\[v_{Inf}=\frac{\mu_{Inf}}{\rho_{Inf}},\mu_{Inf}=\frac{\mu_{f}}{(1-\phi_{Al_{2} O_{3}})^{2.5}(1-\phi_{Cu})^{2.5}} \tag{14}\]  
\[\frac{\rho_{Inf}}{\rho_{f}}=(1-\phi_{Cu})\big{(}1-\big{(}1-\frac{\rho_{Al_{2}O _{3}}}{\rho f}\big{)}\phi_{Al_{2}O_{3}}\big{)}+\phi_{Cu}\big{(}\frac{\rho_{Cu}} {\rho_{f}}\big{)}, \tag{15}\]  
\[\frac{(\rho{C}_{p})_{Inf}}{(\rho{C}_{p})_{f}}=(1-\phi_{Cu})\big{(}1-\big{(}1- \frac{(\rho{C}_{p})_{Al_{2}O_{3}}}{(\rho{C}_{p})f}\big{)}\phi_{Al_{2}O_{3}}\big{)} +\phi_{Cu}\big{(}\frac{(\rho{C}_{p})_{Cu}}{(\rho{C}_{p})_{f}}\big{)} \tag{16}\]  
\[\frac{\sigma_{Inf}}{\sigma_{bf}}=\bigg{[}\frac{(\sigma_{Cu}-\sigma_{bf})3\phi_ {Cu}}{(\sigma_{Cu}+2\sigma_{bf})+(\sigma_{bf}-\sigma_{Cu})\phi_{Cu}}+1\bigg{]},\frac{\sigma_{bf}}{\sigma_{f}}=\bigg{[}\frac{(\sigma_{f}-\sigma_{Al_{2}O_{3}})3 \phi_{Al_{2}O_{3}}}{(\sigma_{Al_{2}O_{3}}-\sigma_{f})-(\sigma_{Al_{2}O_{3}}+2 \sigma_{f})\phi_{Cu}}+1\bigg{]}, \tag{17}\]  
\[k_{Inf}=\bigg{(}\frac{2k_{nf}+k_{Cu}+2\phi_{Cu}(k_{Cu}-k_{nf})}{2k_{nf}+k_{Cu}- \phi_{Cu}(k_{Cu}-k_{nf})}\bigg{)}k_{nf}.\frac{k_{nf}}{k_{f}}=\frac{k_{Al_{2}O_{3 }}+2k_{f}-2\phi_{Al_{2}O_{3}}(k_{f}-k_{Al_{2}O_{3}})}{2k_{f}+k_{Al_{2}O_{3}}+ \phi_{Al_{2}O_{3}}(k_{f}-k_{Al_{2}O_{3}})}. \tag{18}\]