**5. Results and Discussion**

Figures 2 and 3 elucidate the effect of the dimensionless van der Waals numbers on the normalized Nusselt number Nu/Nu0, as predicted by Equation (50). The numerical values of Nu/Nu0 coincide with the data obtained in our work [28] using an approximate analytical solution.

**Figure 2.** Effect of the van der Waals number Wa*a* on the normalized Nusselt number under condition Wa*<sup>b</sup>* = const: (1) Wa*<sup>b</sup>* = 0; (2) Wa*<sup>b</sup>* = 0.1; (3) Wa*<sup>b</sup>* = 0.2; (4) Wa*<sup>b</sup>* = 0.3.

**Figure 3.** Effect of the van der Waals number Wa*<sup>b</sup>* on the normalized Nusselt number under condition Wa*a* = const: (1) Wa*a* = 0; (2) Wa*a* = 0.1; (3) Wa*a* = 0.2; (4) Wa*a* = 0.3.

The parameter *a* in the van der Waals equation characterizes the additional pressure in the near-wall layer, which increases the probability of collision of real gas molecules with the wall in comparison with the ideal one. Obviously, this leads to an increase in lifting force. As a result, the velocity in the boundary layer increases, which causes an increase in heat transfer with an increase in the Wa*a* number.

As it is known, the nature of the interaction of gas molecules with the wall has a significant effect on heat transfer, which occurs due to the exchange of energy between molecules and the surface. The parameter *a* in the van der Waals equation characterizes the additional pressure in the near-wall layer, which increases the probability of a collision between real gas molecules and the wall in comparison with the ideal one. Obviously, this leads to an increase in the lifting force. As a result, the velocity in the boundary layer increases, which causes an increase in heat transfer, with an increase in the Wa*a* number.

In turn, the parameter *b* describes the additional volume not filled with molecules. With its increase, the heat transfer between the molecules and the wall decreases, which causes deterioration in the conditions of interaction between them. Consequently, as can be seen from Figures 2 and 3, with an increase in the Wa*<sup>b</sup>* number, the normalized Nusselt number decreases compared to ideal gas case.

Calculations have shown that with an increase in the Wa*<sup>a</sup>* number, the effect of the Wa*<sup>b</sup>* number noticeably weakens, and at values of Wa*<sup>a</sup>* ≥ 0.6, cannot be observed at all (Figure 2). The more pronounced effect of the Wa*a* number on the flow characteristics can be explained by the fact that the constant *a* describes the quadratic effect on the density variation in the van der Waals equation, whereas the parameter *b* describes the linear effect on the density variation.

We must, however, emphasize that Equation (16) is to be used for small values of the parameters Waa and Wab for the approximate series expansion of the full van der Waals equation to remain in force. Thus, for the larger values of Waa and Wab, Figures 2 and 3 demonstrate only qualitative trends.
