**3. Model of Generalized Hybrid Nanoliquid**

In numerical and experimental investigations on the behaviors of nanofluid, modeling their physical quantities using condensed mathematical relationships between solid particles and regular liquid is a common procedure. Numerous experiments have been performed to validate such terms for nanofluids dilute in scattering of a single sort of solid material [31] and mixtures of two kinds of particles (Suresh et al. [32]). Devi and Devi [33] recommended a collection of correlation for hybrid nanofluid physical-quantities. They approached the liquid involving a single sort of nanoparticle as the regular liquid and the other sort of nanoparticle as the individual particle. The relationship of thermal conductivity and viscosity matched Suresh et al.'s [32] experimental outcomes. In the approach of Devi and Devi [33], there exists the non-linear terms owing to the communication of two sorts of distinct nanoparticles. However, in dilute mixtures where the volumetric fractions of nanoparticle are generally tiny, the impacts of these non-linear conditions may not be important. Thus, it realistically ignores the non-linear terms in the model of Devi and Devi. The hybrid nanofluid model in simplified form and the models of normal nanofluid and Devi and Devi are scheduled in Table 1, where Type A signifies the conventional nanoliquid model, Types B and C, respectively; indicate the hybrid nanoliquid model of Devi and Devi and the model of hybrid nanoliquid in simplified form. Devi and Devi [33] used the approach of the recurrence formulae to signify the density, viscosity, thermal conductivity, and specific heat of the hybrid nanoliquid related generally to the N-th type of nanoparticles as

$$
\rho\_{\rm HNF} \equiv \rho\_{\rm HNF\_N} = (1 - \phi\_N)\rho\_{\rm HNF\_{N-1}} + \phi\_N \rho\_{\rm S\_N} \tag{19}
$$

$$
\mu\_{\rm HNF} \equiv \mu\_{\rm HNF\_N} = \frac{\rho\_{\rm HNF\_{N-1}}}{(1 - \phi\_N)^{2.5}} \tag{20}
$$

$$k\_{\rm HNF} \equiv k\_{\rm HNF\_N} = \frac{k\_{\rm S\_N} + k\_{\rm HNF\_{N-1}}(M-1) - \phi\_N(M-1)\left(k\_{\rm HNF\_{N-1}} - k\_{\rm S\_N}\right)}{k\_{\rm S\_N} + k\_{\rm HNF\_{N-1}}(M-1) + \phi\_N\left(k\_{\rm HNF\_{N-1}} - k\_{\rm S\_N}\right)}\tag{21}$$

$$\left(\rho c\_p\right)\_{\rm HNF} \equiv \left(\rho c\_p\right)\_{\rm HNF\_N} = \phi\_N \left(\rho c\_p\right)\_{S\_N} + (1 - \phi\_N) \left(\rho c\_p\right)\_{\rm HNF\_{N-1}} \tag{22}$$

$$
\beta\_{\rm HNF} \equiv \beta\_{\rm HNF\_N} = (1 - \phi\_N)\beta\_{\rm HNF\_{N-1}} + \phi\_N \beta\_{\rm S\_N} \tag{23}
$$

From the above correlations, ignoring the non-linear terms, one obtains

$$
\rho\_{\rm HNF} = 1 - \rho\_F \sum\_{a=1}^{N} \phi\_a + \sum\_{a=1}^{N} \phi\_a \rho\_{S\_a} \tag{24}
$$

$$
\mu\_{\rm HNF} = \frac{\mu\_F}{\left(1 - \sum\_{a=1}^{N} \phi\_a\right)^{2.5}} \tag{25}
$$

$$\left(\rho c\_p\right)\_{HNF} = 1 - \left(\rho c\_p\right)\_F \sum\_{a=1}^N \phi\_a + \sum\_{a=1}^N \phi\_a \left(\rho c\_p\right)\_{S\_a} \tag{26}$$

$$\beta\_{\rm HNF} = 1 - \beta\_F \sum\_{a=1}^{N} \phi\_a + \sum\_{a=1}^{N} \phi\_a \beta\_{S\_a} \tag{27}$$

It is worth mentioning that for *kHNF*, we remain the recurrence Formula (21) because of the connections amid dissimilar particles can barely be uttered by the Maxwell equations. Additionally, throughout the research the values of *M* = 3 is taken which implies that the shape of the particle is spherical.


**Table 1.** Thermo-physical models of nanofluid and hybrid nanofluid.
