*4.1. Case I: Prescribed Surface Temperature*

For this case, the thermal boundary conditions are:

$$T = T\_w = T\_\infty + b\mathbf{x}^{2n-1} \qquad \text{at } y = 0$$

and:

$$T \to \infty \qquad \text{ as} \quad y \to \infty \tag{27}$$

where *n* stands for wall temperature parameter, and when *n* = <sup>1</sup> <sup>2</sup> , we have the isothermal boundary condition. The dimensionless temperature θ(η) is assumed to be of the form:

$$\Theta(\eta) = \frac{T - T\_{\infty}}{T\_{\text{av}} - T\_{\infty}} \tag{28}$$

Substituting Equations (19), (20), (27), and (28) into Equation (26) gives:

$$\Theta'' + Pr\left\{ f\Theta' - \left(\frac{2(2n-1)}{n+1}\right) f'\Theta + E\_c(f'')^2 \right\} = 0 \tag{29}$$

also, the boundary conditions (27) become:

$$
\theta(\mathfrak{n}) = 1 \qquad \text{at} \qquad \mathfrak{n} = 0
$$

*Coatings* **2019**, *9*, 490

$$\theta(\eta) \to 0 \qquad\quad\text{as}\qquad\eta \to \infty \tag{30}$$

Here, *Pr* and *Ec* represent the Prandtl and Eckert numbers, respectively, defined as:

$$Pr = \frac{\upsilon}{\alpha} \qquad \text{and} \qquad Ec = \frac{\mathcal{U}^2 w}{c\_p (T - T\_\infty)} \tag{31}$$

The physical quantity of interests here is the local Nusselt number *Nux*, which is defined by:

$$N\mu\_x = \frac{\mathfrak{x}q\_{\text{uv}}}{\mathfrak{x}}(T\_{\text{uv}} - T\_{\text{\textinfty}}) \tag{32}$$

where *qw* is the wall heat flux from the plate and is expressed as:

$$q\_w = -k \left(\frac{\partial T}{\partial y}\right)\_{y=0} \tag{33}$$
