*Engineering Quantities of Interest*

The friction factor and the temperature gradient in mathematical structure are described as:

$$\mathcal{C}\_{\text{F}} = \frac{\tau\_{\text{w}}^{\prime}}{\rho\_{\text{f}}^{\prime} \mathcal{U}\_{\text{w}}^{2}} \; \mathcal{N}u\_{\text{x}} = \frac{\mathbf{x} q\_{\text{w}}^{\prime}}{k\_{\text{f}}^{\prime} (T\_{\text{f}} - T\_{\text{\infty}})} \, \tag{17}$$

The wall shear stress and the heat-flux are expressed as:

$$\pi'\_{w} = \mu'\_{nf} \left(\frac{\partial u\_1}{\partial y}\right)\_{y=0} \rho'\_{w} = -k'\_f \left(\frac{k'\_{nf}}{k'\_f} + \frac{16\sigma'^\ast T\_1^3}{3k'^\ast k'\_f}\right) \left(\frac{\partial T\_1}{\partial y}\right)\_{y=0}.\tag{18}$$

Utilizing Equation (18) in Equation (17), the dimensionless expressions are:

$$\begin{aligned} \mathbf{C}\_{F}\mathbf{R}\mathbf{e}\_{\mathcal{X}}^{0.5} &= \mathbf{K}\_{1}F''(0) \\ \mathrm{Nu}\_{\mathcal{X}}\mathrm{Re}\_{\mathcal{X}}^{-0.5} &= -\left(\frac{1}{\mathcal{K}\_{6}} + \frac{4}{3}\mathrm{R}\_{d}(1 + (\theta\_{w} - 1)\theta(0))^{3}\right)\theta'(0) \\ &\qquad \text{(for }\gamma\,\mathrm{Al}\_{2}\mathrm{O}\_{3} - \mathrm{H}\_{2}\mathrm{O)}\end{aligned} \tag{19}$$

$$\begin{aligned} \mathbf{C}\_{F}\mathbf{R}\mathbf{e}\_{\mathcal{X}}^{0.5} &= \mathbf{K}\_{\mathsf{F}}F''(0) \\ \mathrm{Nu}\_{\mathcal{X}}\mathbf{R}\mathbf{e}\_{\mathcal{X}}^{-0.5} &= -\left(\frac{1}{\mathsf{K}\_{\mathsf{F}}} + \frac{4}{3}\mathsf{R}\_{d}(1 + (\theta\_{\mathcal{w}} - 1)\theta(0))^{3}\right)\theta'(0) \\ &\quad \text{(for } \mathsf{y}\text{ Al}\_{2}\mathsf{O}\_{3} - \mathsf{C}\_{2}\mathsf{H}\_{\mathsf{G}}\mathsf{O}\_{2}) \end{aligned} \tag{20}$$
