## 4 Results and Discussion  
Let us now discuss the main features of the solutions of Equations (14)-(23). In Figures 3-6, the velocity field stream function, the isotherms, and iso-concentration lines are reported for Pr = 6.2 and by considering the variation of one parameter (Re, Le, \(N_{B}\), or \(N_{T}\)) while the other ones, respectively, are (100, 0, 0.1, 0.1). Figure 3 refers to different values assumed by the Reynolds number and shows that this parameter does not seem to affect sensibly the solution. Moreover, the figure shows that for increasing values of Re the stream function and isotherm gradients occur only in the channel region, not involving the open cavity, while the concentration gradients appear mainly in the cavity region. Figure 4 refers to different values of the Lewis number and shows that for increasing values of Le the concentration vortex tends to occupy the whole cavity. Also, with reference to the channel region, for increasing values of the Lewis number, the concentration gradients tend to move close to the channel walls. Figure 5 refers to different values of the Brownian diffusivity coefficient, showing that the solution is not strongly affected by this parameter: in fact, slight differences appear not only in the stream function distribution but also in the temperature and in the concentration distribution. Finally, Figure 6 refers to different values of the thermophoretic diffusivity coefficient \(N_{T}\), showing that also this parameter does not strongly affect the obtained solution. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \# & \(\mathbf{N}\) & \(\overline{u}\) & \(\overline{\Theta}\) & \(\overline{\Phi}\) & \(\overline{\mathbf{Nu}}\) \\ \hline
1 & 1246 & 0.60210 & 0.43391 & 1.04165 & 0.98558 \\
2 & 30,449 & 0.60199 & 0.43461 & 1.03690 & 0.98569 \\
**3** & **59,520** & **0.60198** & **0.43463** & **1.03601** & **0.98571** \\
4 & 77,826 & 0.60198 & 0.43463 & 1.03600 & 0.98572 \\
5 & 120,360 & 0.60198 & 0.43464 & 1.03599 & 0.98573 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Grid independence test. N is the mesh elements number, \(\overline{u}\) is the mean velocity, \(\Theta\) is the mean temperature, \(\Phi\) is the mean concentration and \(\overline{\text{Nu}}\) is the mean Nusselt number. The mesh validation is done for Re = 100, Pr = 6.2, NB = NT = 0.1 and Le = 10. Figure 2: Mesh from coarse to fine: (**left**) 1246 elements, (**center**) 30,449 elements, (**right**) 59,520 elements. Figure 4: Stream function, Isotherms and Iso-concentration for various Lewis number and Pr = 6.2, Re = 100, \(N_{B}\) = \(N_{T}\) = 0.1. Figure 3: Stream function, Isotherms and Iso-concentration for various Reynolds number and Pr = 6.2, Le = 1, \(N_{B}\) = \(N_{T}\) = 0.1. Figure 5: Stream function, Isotherms and Iso-concentration for various N\({}_{\rm B}\) number and Pr = 6.2, Re = 100, Le = 1, N\({}_{\rm T}\) = 0.1. Figure 6: Stream function, Isotherms and Iso-concentration for various N\({}_{\rm T}\) number and Pr = 6.2, Re = 100, Le = 1, \(N_{\rm B}\) = 0.1. In Figure 9 the dimensionless concentration and temperature distribution are reported as a function of the Lewis number. The righthand frame of the figure shows that the dimensionless temperature does not significantly vary: in fact, very small variations occur and are visible in the evidenced region, where the dimensionless temperature is an increasing function of Le. Moreover, a local maximum in the region close to the center of the cavity arises. On the contrary, the concentration strongly depends on the value assumed by the Lewis number. As discussed concerning Figure 7, the boundary condition imposes a concentration equal to 1 for any Le at the upper corner of the cavity. In Figure 10, the dependence of the concentration and temperature fields on the Brownian diffusivity coefficient \(\mathrm{N_{B}}\) are reported, showing that smaller values of \(\mathrm{N_{B}}\) yield higher concentrations and to less uniform behavior of this function.