## 2 Formulation of the Governing Equations
The following set of scale variables has been used for converting the velocity and energy equations (Equations (4) and (12)) into a dimensionless form as:  
\[\mathcal{U}=\overline{\mathcal{U}}\dot{\pi}^{-1},\quad\quad\mathcal{V}= \overline{\mathcal{V}}\dot{\pi}^{-1},\quad\quad x=\overline{x}\dot{\pi}^{-1},\quad\quad\mathcal{F}=\overline{\mathcal{F}}\mathcal{R}^{-1},\quad\ \dot{\pi}(\zeta)=\frac{\overline{\mathcal{T}}-\overline{\mathcal{T}}_{u}}{ \overline{\mathcal{T}}_{l}-\overline{\mathcal{T}}_{u}}, \tag{14}\]  
which in result, gives us the following set of ordinary differential equations:  
\[\mathcal{F}^{iv}+\dot{\Lambda}_{1}\big{[}\mathcal{R}\big{(}\mathcal{F} \mathcal{F}^{\prime\prime\prime}-\mathcal{F}^{\prime}\mathcal{F}^{\prime \prime}\big{)}+\alpha(3\mathcal{F}^{\prime\prime}+\zeta\mathcal{F}^{\prime \prime\prime})\big{]}=0, \tag{15}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Title** & **H\({}_{2}\)O (f)** & **Al\({}_{2}\)O\({}_{3}\) (\(\Phi_{1}\))** & **Cu (\(\Phi_{2}\))** \\ \hline \(\overline{\mathcal{p}}\)\(\left(\text{kg m}^{-3}\right)\) & 997.0 & 3970 & 8933 \\ \(\mathcal{C}_{p}\)\(\left(\text{J kg}^{-1}\text{K}^{-1}\right)\) & 4180 & 765 & 385 \\ \(\mathcal{K}\)\(\left(\text{Wm}^{-1}\text{K}^{-1}\right)\) & 0.6071 & 40 & 400 \\ \hline \end{tabular}
\end{table} 
Table 1: Thermo-physical properties of base fluid (H\({}_{2}\)O) and nanoparticles Al\({}_{2}\)O\({}_{3}\) and Cu [17; 18]. \[\frac{\bar{\kappa}_{Inf}}{\bar{\kappa}_{f}}\vartheta^{\prime\prime}+\Pr\hat{\Lambda }_{2}(\alpha\bar{\zeta}+\mathcal{R}\mathcal{F})\vartheta^{\prime}=0, \tag{16}\]  
where,  
\[\left. \begin{array}{l}\hat{\Lambda}_{1}=\frac{v_{f}}{v_{Inf}}=\frac{(1-\[PERSON]_{ 2})^{5/2}}{(1-\[PERSON]_{1})^{-5/2}}\left[\left\{1-\[PERSON]_{1}\left(1-\frac{\bar{ \rho}_{1}}{\bar{\rho}_{f}}\right)\right\}(1-\[PERSON]_{2})+\frac{\bar{\rho}_{2}}{ \bar{\rho}_{f}}\[PERSON]_{2}\right],\\ \hat{\Lambda}_{2}=\left[1-\[PERSON]_{1}\left(1-\frac{\left(\bar{\rho}_{C}\right)_{ \mu_{1}}}{\left(\bar{\rho}_{C}\right)_{f}}\right)\right](1-\[PERSON]_{2})+\frac{ \left(\bar{\rho}_{C}\right)_{\mu_{2}}}{\left(\bar{\rho}_{C}\right)_{f}}\[PERSON]_{ 2}.\end{array} \right\} \tag{17}\]  
The reduced dimensionless form of boundary conditions is defined as:  
\[\mathcal{A}-\mathcal{F}(\overline{\chi}/\bar{\kappa}(\overline{\ell}))=0, \quad\mathcal{F}^{\prime}(\overline{\chi}/\bar{\kappa}(\overline{\ell}))=0, \quad\vartheta(\overline{\chi}/\bar{\kappa}(\overline{\ell}))=1,\quad\text{ when }\overline{\chi}=-\bar{\kappa}(\overline{\ell})\]  
\[1-\mathcal{F}(\overline{\chi}/\bar{\kappa}(\overline{\ell}))=0,\quad\mathcal{ F}^{\prime}(\overline{\chi}/\bar{\kappa}(\overline{\ell}))=0,\quad\vartheta( \overline{\chi}/\bar{\kappa}(\overline{\ell}))=0,\quad\text{when }\overline{\chi}=-\bar{\kappa}( \overline{\ell}) \tag{18}\]  
where, \(\mathcal{A}=\overline{\mathcal{V}}_{l}/\overline{\mathcal{V}}_{u}\) is the symbol for permeability parameter. Moreover, \(Pr=\varepsilon_{f}/v_{f}\) denotes Prandtl number. The local heat transfer rate (i.e., local Nusselt number) is defined by:  
\[Nu=-\frac{\bar{\kappa}/\bar{\kappa}_{f}}{(\overline{\mathcal{T}}_{l}- \overline{\mathcal{T}}_{u})}\left(\bar{\kappa}_{Inf}\frac{\partial\overline{ \mathcal{T}}}{\partial\overline{\chi}}\right)_{\overline{\chi}=0}, \tag{19}\]  
Subsequently, the set of scale variables converts the above equation into dimensionless rate of heat transfer both at the bottom and top wall, which is then expressed by the following relations:  
\[Nu_{l}=-\frac{\bar{\kappa}/\bar{\kappa}_{f}/\bar{\kappa}_{Inf}}{\bar{\kappa} _{Inf}}. \tag{20}\]  
\[Nu_{u}=-\frac{\bar{\kappa}/1}{\bar{\kappa}_{f}/\bar{\kappa}_{Inf}} \tag{21}\]  
It is pertinent to mention here that, if \(\Phi_{1}=\Phi_{2}=0\), then hybrid nanofluid is no longer be preserved and converted to the regular fluid (water). When \(\Phi_{1}\
eq 0,\ \Phi_{2}=0\), then water-based Al\({}_{2}\)O\({}_{3}\) nanofluid has been achieved. For Cu-Al\({}_{2}\)O\({}_{3}\)/H\({}_{2}\)O-based hybrid nanofluid, the relation \(\Phi_{1}\
eq 0,\ \Phi_{2}\
eq 0\) must be satisfied.