## 2 Formulation of the Governing Equations
Furthermore, \(\mu_{lnf}\) represents the viscosity of the hybrid nanofluid. In literature, several theoretical models including the effects of nanoparticle material [32], particle size [33] and liquid layering [34; 35] have been available in order to estimate the viscosity of nanofluids. In 1952, [PERSON] [36] proposed a correlation for the effective viscosity of the nanofluid which typically involves the viscosity of the host fluid and the nanoparticles concentration. Besides, this expression is reliable for dilute solutions. In case of hybrid nanofluids, this expression would take the following mathematical form [37]:  
\[\mu_{lnf}=\frac{\mu_{f}}{\left(1-\Phi_{1}\right)^{5/2}\left(1-\Phi_{2}\right) ^{5/2}}. \tag{5}\]  
Similarly, by taking the mixture principle into account, the effective density \(\left(\overline{\rho}_{lnf}\right)\)[37] proposed by [PERSON] and [PERSON] [38] and the volumetric heat capacity \(\left(\overline{\rho}\mathcal{C}_{p}\right)_{lnf}\)[37; 39] for \(\left(\mathrm{Cu-Al_{2}O_{3}/H_{2}O}\right)\) hybrid nanofluid can be estimated via Equations (6) and (7), which is given as under:  
\[\frac{\overline{\rho}_{lnf}}{\overline{\rho}_{f}}=(1-\[PERSON]_{2})\Bigg{[}1- \Bigg{(}1-\frac{\overline{\rho}_{s1}}{\overline{\rho}_{f}}\Bigg{)}\[PERSON]_{1} \Bigg{]}+\Phi_{2}\frac{\overline{\rho}_{s2}}{\overline{\rho}_{f}}, \tag{6}\]  
\[\frac{\left(\overline{\rho}\mathcal{C}_{p}\right)_{lnf}}{\left(\overline{\rho }\mathcal{C}_{p}\right)_{f}}=(1-\Phi_{2})\Bigg{[}1-\Bigg{(}1-\frac{\left( \overline{\rho}\mathcal{C}_{p}\right)_{s1}}{\left(\overline{\rho}\mathcal{C}_ {p}\right)_{f}}\Bigg{)}\Phi_{1}\Bigg{]}+\Phi_{2}\frac{\left(\overline{\rho} \mathcal{C}_{p}\right)_{s2}}{\left(\overline{\rho}\mathcal{C}_{p}\right)_{f}}. \tag{7}\]  
Furthermore, among various thermo-mechanical properties, the thermal conductivity is the one that plays a key role in transferring the heat. [PERSON] and [PERSON] [5] proposed an efficient model for thermal conductivity which is applicable for different shapes of nanocomposites. In context to hybrid nanofluids, the expression is defined as [37]:  
\[\hat{\kappa}_{lnf}=\hat{\kappa}_{bf}\frac{\hat{\kappa}_{s2}+(m\!-\!1)\hat{ \kappa}_{bf}-(m\!-\!1)\Phi_{2}\Big{(}\hat{\kappa}_{bf}-\hat{\kappa}_{s2}\Big{)} }{\hat{\kappa}_{s2}+(m\!-\!1)\hat{\kappa}_{bf}+\Phi_{2}\Big{(}\hat{\kappa}_{bf }-\hat{\kappa}_{s2}\Big{)}}, \tag{8}\]  
where,  
\[\frac{\partial\overline{\mathcal{V}}}{\partial\overline{\mathcal{V}}}+\frac{ \partial\overline{\mathcal{U}}}{\partial\overline{x}}=0, \tag{1}\]0  
In the above relations, \(m\) and \(n\) represents the shape factor of \(\mathrm{Cu}\) and \(\mathrm{Al_{2}O_{3}}\) nano-sized particles respectively. \(\hat{\kappa}_{bf}\) is the thermal conductivity of the \(\mathrm{Al_{2}O_{3}}\)-nanofluid, while the thermal conductivity of base fluid \(\mathrm{H_{2}O}\) is expressed by \(\hat{\kappa}_{f}\). The thermal conductivity of \(\mathrm{Al_{2}O_{3}}\) and \(\mathrm{Cu}\) are respectively denoted by \(\hat{\kappa}_{s1}\) and \(\hat{\kappa}_{s2}\). Furthermore, \(\Phi_{1}\) stands for the volume fraction of \(\mathrm{Al_{2}O_{3}}\) nanoparticles while the solid volume fraction for \(\mathrm{Cu}\) nanoparticles are represented by \(\Phi_{2}\). The base fluid viscosity, density and specific heat are simultaneously symbolized by \(\mu_{f}\), \(\overline{\rho}_{f}\) and \(\left(\mathcal{C}_{p}\right)_{f}\). \(\overline{\rho}_{s1}\) and \(\overline{\rho}_{s2}\) denotes the densities of \(\mathrm{Al_{2}O_{3}}\) and \(\mathrm{Cu}\) respectively. At constant pressure, \(\left(\mathcal{C}_{p}\right)_{s1}\) and \(\left(\mathcal{C}_{p}\right)_{s2}\) respectively indicates the specific heat of \(\mathrm{Al_{2}O_{3}}\) and \(\mathrm{Cu}\) nanoparticles. Table 1 has been decorated for the thermo-physical properties of base fluid \(\mathrm{H_{2}O}\) and nanoparticles \(\mathrm{Al_{2}O_{3}}\) and \(\mathrm{Cu}\). The suitable boundary conditions for the present flow problem are specified as:  
\[