*2.2. Entropy Generation*

$$\begin{split} S\_G &= \frac{k\_f}{T\_{vo}^2} \Big[ 1 + \frac{16\mu^\* T^3}{3k^\* k\_f} \Big] \left(\frac{\partial T}{\partial z}\right)^2 + \frac{1}{T\_{vo}} \Big[ \mu\_f \Big( \left(\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2 \Big) + \beta\_1 \Big[ 2\frac{\partial u}{\partial r} \Big(\frac{\partial u}{\partial z}\Big)^2 \\ & - \left(\frac{\partial u}{\partial z}\frac{\partial v}{\partial z}\frac{\mathbf{v}}{r} + \frac{\partial v}{\partial r}\frac{\partial u}{\partial z}\frac{\partial v}{\partial z}\right) + u\frac{\partial u}{\partial z}\frac{\partial^2 u}{\partial r \partial z} + w\frac{\partial u}{\partial z}\frac{\partial^2 v}{\partial z^2} + \frac{\partial u}{\partial z}\left(\frac{\partial u}{\partial z}\right)^2 - \frac{\partial v}{\partial r} \left(\frac{\partial u}{\partial z}\right)^2 \\ & + \frac{\upsilon}{r} \Big(\frac{\partial u}{\partial z}\Big)^2 \Big] + \frac{\sigma\_f B\_0^2}{T\_{vo} (1 + m^2)} \Big(\mu^2 + \upsilon^2\Big) + \frac{R\_D}{C\_{vo}} \left(\frac{\partial C}{\partial z}\right)^2 + \frac{R\_D}{T\_{vo}} \left(\frac{\partial C}{\partial z}\frac{\partial T}{\partial z}\right) \end{split} \tag{21}$$

Equation (21) is reduced as

$$\begin{split} N\_G &= \gamma\_1 \Big[ 1 + R d \big( 1 + \theta \left( \Theta\_w - 1 \right) \big)^3 \Big] \theta'^2 + Br \big( f''^2 + \mathfrak{g}'^2 \big) + L \theta' \Phi' + \\\ MeBr &\overset{\text{(}f' \text{ } f'' \text{ }}{\end{split}}{\sim} \left( f' \big)^2 - 2f'' \big\{ \text{g} \text{ } g' - 2f \big\} \mathfrak{g}' \big) + \frac{R \text{r} \text{M}}{\left( 1 + \text{m}^2 \right)} \Big( f'^2 + \mathfrak{g}^2 \big) + L \frac{\chi\_2}{\wp\_1} \phi'^2 \end{split} \tag{22}$$

The Bejan number is defined as

$$Be = \frac{\gamma\_1 \left[1 + Rd(1 + \theta(\theta\_w - 1))^3\right] \theta'^2 + L\theta'\phi' + L\frac{\gamma\_2}{\gamma\_1}\phi'^2}{\int \gamma\_1 \left[1 + Rd(1 + \theta(\theta\_w - 1))^3\right] \theta'^2 + Br\left(f''^2 + g'^2\right) + L\theta'\phi' + } \tag{23}$$

$$WeBr\text{Re}\left(\begin{array}{c} f'f''^2 - 2f''\text{ }\mathcal{S}' \\ -2fg'\text{ }\mathcal{S}'' \end{array}\right) + \frac{BrM}{\left(1 + m^2\right)}\left(f'^2 + g^2\right) + L\frac{\gamma\_2}{\gamma\_1}\phi'^2\end{array} \tag{23}$$

where *NG* = *T*∞*h*<sup>2</sup>*SG*/*kf*(*Tw*−*T*∞) is the entropy generation, *Br* = <sup>μ</sup>*f*(*r*α1) <sup>2</sup>/*kf*(*Tw*−*T*∞) is the Brinkman number, *L* = *RD*(*Cw*−*C*∞)/*kf* is the diffusion parameter, γ<sup>1</sup> = (*Tw*−*T*∞)/*T*<sup>∞</sup> and γ<sup>2</sup> = (*Cw*−*C*∞)/*C*<sup>∞</sup> are the temperature and concentration ratios, respectively.

*Coatings* **2020**, *10*, 610
