**3. Formulation of Entropy**

The volumetric EG (entropy generation) for γAl2O<sup>3</sup> nanoparticles is expressed as:

$$H\_G = \frac{k\_f'}{T\_\infty^2} \left[ \frac{k\_{nf}'}{k\_f'} + \frac{16\sigma'^\ast T\_1^3}{3k'^\ast k\_f'} \right] \left(\frac{\partial T\_1}{\partial y}\right)^2 + \frac{\mu\_{nf}'}{T\_\infty} \left(\frac{\partial u\_1}{\partial y}\right)^2 + \frac{\sigma\_{nf}' \mathcal{B}^2}{T\_\infty} u\_1^2 \tag{21}$$

The characteristic EG rate can be written as:

$$H\_{\mathcal{S}0} = \frac{k\_f'(\Delta T)^2}{L^2 T\_{\infty}^2} \tag{22}$$

By using the ratio of Equations (21) and (22), the EG number is described as:

$$H\_{\mathcal{S}} = \frac{H\_{\mathcal{G}}}{H\_{\mathcal{S}0}} \tag{23}$$

Implementing Equation (8) in Equations (21) and (22), we obtain:

$$H\_{\mathcal{S}} = \text{Re}\_{L} \left( \frac{1}{\mathbf{K}\_{6}} + \frac{4}{3} \text{R}\_{d} (1 + (\theta\_{w} - 1)\theta)^{3} \right) \theta'^{2} + \frac{\text{Re}\_{L} \text{Br}}{\Omega} \text{K}\_{1} \mathbf{F}''^{2} + \text{K}\_{3} \frac{\text{MBr} \text{Re}\_{L}}{\Omega} \mathbf{F}'^{2} \Big) \tag{24}$$
 
$$\text{(for } \gamma \text{Al}\_{2}\text{O}\_{3} - \text{H}\_{2}\text{O)}$$

$$H\_{\mathcal{S}} = \text{Re}\_{L} \left( \frac{1}{\text{K}\_{8}} + \frac{4}{3} \text{R}\_{d} (1 + (\theta\_{w} - 1)\theta)^{3} \right) \theta'^{2} + \frac{\text{Re}\_{L} \text{Br}}{\Omega} \text{K}\_{5} \text{F}''^{2} + \text{K}\_{3} \frac{\text{MBr} \text{Re}\_{L}}{\Omega} \text{F}'^{2} \Bigg\} \tag{25}$$
 
$$\text{(for } \gamma \text{Al}\_{2}\text{O}\_{3} - \text{C}\_{2}\text{H}\_{6}\text{O}\_{2})$$

where the parameters Ω = ∆*T*/*T*∞, *Br* = *µ* 0 *f* (*Uw*) 2 /*k* 0 *<sup>f</sup>*∆*T*, Re*<sup>L</sup>* = *aL*2/*ν* 0 *<sup>f</sup>*(1 − *Ct*) are described as the temperature difference and the Brinkman and Reynolds numbers, respectively.

The assessment of the Bejan *Be* number is vital in sequence to investigate the heat transfer irreversibility, and range of values is between 0 and 1. The *Be* number in dimensionless form is described as:

$$Be = \frac{\text{Re}\_{L}\left(\frac{1}{\mathbf{K}\_{6}} + \frac{4}{3}\mathcal{R}\_{d}(1 + (\theta\_{w} - 1)\theta)^{3}\right)\theta^{2}}{\text{Re}\_{L}\left(\frac{1}{\mathbf{K}\_{6}} + \frac{4}{3}\mathcal{R}\_{d}(1 + (\theta\_{w} - 1)\theta)^{3}\right)\theta^{2} + \frac{\text{Re}\_{L}\mathcal{R}r}{\text{I}\Omega}\mathcal{K}\_{1}\mathcal{F}^{\prime\prime} + \mathcal{K}\_{3}\frac{\text{MBr}\text{Ra}\_{L}}{\text{I}\Omega}F^{2}}}\right) \tag{26}$$

$$(\text{for }\chi\text{Al}\_{2}\mathcal{O}\_{3} - \text{H}\_{2}\mathcal{O})$$

$$Be = \frac{\text{Re}\_L\left(\frac{1}{\text{K}\_8} + \frac{4}{3}\text{R}\_d(1 + (\theta\_w - 1)\theta)^3\right)\theta^2}{\text{Re}\_L\left(\frac{1}{\text{K}\_8} + \frac{4}{3}\text{R}\_d(1 + (\theta\_w - 1)\theta)^3\right)\theta^{\prime 2} + \frac{\text{Re}\_L\text{Br}}{\Omega}\text{K}\_5\text{F}^{\prime \prime 2} + \text{K}\_3\frac{\text{MRr\text{Re}\_L}}{\Omega}F^{\prime 2}}} \left\{ \begin{array}{l} \\ \text{(for } \gamma\text{Al}\_2\text{O}\_3 - \text{C}\_2\text{H}\_6\text{O}\_2) \end{array} \right\} \tag{27}$$

It is concluded from the expressions mentioned above that the irreversibility of fluid friction dominates when *Be* differs from 0–0.5, while the heat transport irreversibility dominates when *Be* differs from 0.5–1. The value of *Be* demonstrates that the irreversibility of fluid friction and heat transfer equally contribute to EG.
