**4. Heat Transfer**

Following the authors [30,31], we will solve the system (37) and (38) using power-law equations for the scaling velocity *U* and the boundary layer thickness δ:

$$\mathcal{U}(\mathbf{x}) = \Psi \mathbf{x}^{\mathcal{W}},\tag{39}$$

$$
\delta(\mathbf{x}) = \Phi \mathbf{x}^{\mathrm{n}},\tag{40}
$$

where *m*, *n*, Ψ, and Φ are yet unknown constants. Solutions (39) and (40) satisfy the requirement that the scaling velocity *U* and the boundary layer thickness δ are equal to zero at the leading edge of the plate at *x* = 0. To find these constants, we substitute Equations (39) and (40) into Equations (37) and (38). As a result, we have:

$$(2m+n)s\Phi\Psi^2\mathbf{x}^{2m+n-1} = -\mathbf{v}\frac{\Psi^2}{\Phi}\mathbf{x}^{m-n} + gF\Phi\mathbf{x}^n,\tag{41}$$

$$(m+n)t\Phi\Psi\mathbf{x}^{m+n-1} = a\frac{z}{\Phi}\mathbf{x}^{-n}.\tag{42}$$

Let us equate the exponents at the coordinate *x*. As a result, we have a system of equations, whose solution is:

$$m = \frac{1}{2}, \qquad \qquad \qquad n = \frac{1}{4}, \tag{43}$$

Substituting these values into Equations (41) and (42), we obtain a system of equations for the unknown constants Ψ and Φ. Eliminating one of these unknowns, we obtain a fourth-order algebraic equation with respect to the other unknown. As a result of the solution, we have four pairs of roots, of which only one has a positive real form:

$$\Psi = 2\sqrt{\frac{zgF}{5sz + 3t\text{Pr}}} \,\text{}\tag{44}$$

$$
\Phi = \sqrt{\frac{4z}{3t} \frac{\alpha}{\Psi'}}\tag{45}
$$

Having obtained relations (44) and (45), it is possible to derive a solution for the heat transfer coefficient:

$$h = \frac{k}{\delta} \left(\frac{d\theta}{d\eta}\right)\_{\eta=0} = z\frac{k}{\delta} = k\sqrt{\frac{3zt}{4}\frac{\Psi}{a\sqrt{\chi}}} = k\sqrt{\frac{3zt}{2}\frac{1}{a\sqrt{\chi}}}\sqrt{\frac{zgF}{5sz+3t\text{Pr}}}\tag{46}$$

Equation (46) can be rewritten in the form of the normalized Nusselt number:

$$\frac{\text{Nu}}{\text{Nu}\_0} = \sqrt[4]{\frac{F}{r \text{\\$} \Delta T}}.\tag{47}$$

where

$$\text{Nu} = \frac{h\text{x}}{k} \tag{48}$$

The Nusselt number is commonly interpreted as a dimensionless heat transfer coefficient, which characterizes the heat transfer rate at the boundary between the wall and the flow [29–31].

Here, the subscript "0" refers to the ideal gas (Wa*<sup>a</sup>* = Wa*<sup>b</sup>* = 0).

The coefficient *r* is determined from the equation:

$$\lim\_{\beta \Delta T \to 0} \frac{F}{\beta \Delta T} = r \frac{1 + 3 \text{Wa}\_a - 2 \text{Wa}\_b}{1 + \text{Wa}\_a - \text{Wa}\_b} \,\text{}\tag{49}$$

For example, *r* = 1/4 for Equation (30), whereas *r* = 1/3 for Equation (34). As a result, Equation (47) can be rewritten as follows:

$$\frac{\text{Nu}}{\text{Nu}\_0} = \sqrt[4]{\frac{1 + 3\text{Wa}\_d - 2\text{Wa}\_b}{1 + \text{Wa}\_d - \text{Wa}\_b}} \,\text{}\tag{50}$$

where Nu and Nu0 are the Nusselt numbers for the van der Waals gas and the ideal gas, respectively.

In the integral method considered here, the main sought quantity is the Nusselt number. If necessary, the final relation for the velocity profile can be easily found using the profile (27) for the dimensionless velocity *w*, as well as the combination of Equations (39) and (44) for the function *U*. The profile of the local temperature *T* is found from either Equation (28) or Equation (33), whereas the gas density ρ is determined by Equation (16).
