*4.2. Energy Equations*

The temperature Equation (3) regulate the boundary layer into non-dimensional ordinary differential equations which can be explained as,

$$\begin{aligned} \theta'' \left( \frac{\overline{k}\_{nf}}{\overline{k}\_f} + \frac{4}{3}R \right) - p\_r \left( (1-\rho)^{2.5} + \varrho \left( \frac{(\rho \mathbb{C}\_p)\_s}{(\rho \mathbb{C}\_p)\_f} \right) \right) (f\theta' - 2\theta f') &= 0 \text{ for } Al\_2O\_3 - H\_2O, \\ \theta'' \left( \frac{\overline{k}\_{nf}}{\overline{k}\_f} + \frac{4}{3}R \right) - p\_r \left( (1-\rho)^{2.5} + \varrho \left( \frac{(\rho \mathbb{C}\_p)\_s}{(\rho \mathbb{C}\_p)\_f} \right) \right) (f\theta' - 2\theta f') &= 0 \text{ for } \gamma - Al\_2O\_3 - water, \\ \theta'' \left( \frac{\overline{k}\_{nf}}{\overline{k}\_f} + \frac{4}{3}R \right) - p\_r \left( (1-\rho)^{2.5} + \varrho \left( \frac{(\rho \mathbb{C}\_p)\_s}{(\rho \mathbb{C}\_p)\_f} \right) \right) (f\theta' - 2\theta f') &= 0 \text{ for } \gamma - Al\_2O\_3 - \mathbb{C}\_2H\_6O\_2. \end{aligned} \tag{14}$$

#### *4.3. Boundary Conditions*

The related boundary conditions are as follows:

For Momentum equation

$$f(0) = 0, \ f'(0) = 1 \text{ \& } f'(\infty) = 0. \tag{15}$$

For Energy equation

$$
\theta(0) = 1 \quad \& \quad \theta(\infty) = 0. \tag{16}
$$
