## 4 Results and Discussion  
The results presented are comparisons in terms of the contact pressure, fluid pressure, and friction coefficient between the two lubrication regimes considered. The first model comprises hydrodynamic lubrication, in which the film is thicker and has no proper roughness contact. The second presents the mixed regime, where the interaction may occur between some asperities, and most regions have a fluid film and no solid-solid contact. It is possible to notice uniformly distributed contact pressures on the surface at some simulation moments. However, the contact pressure only prevails at a few peaks most of the time. It proves that due to the random dispersion of asperities on an actual rough surface, the distribution of their heights and valleys occurs in a non-homogeneous way. The present model contains the first step toward multi-scale modeling. With the estimated average properties, one can increase its resolution. In Figure 12, it is possible to observe the dissipation of stresses in the solid to demonstrate that the proposed depth is adequate for the analysis, since the stresses at the bottom of the solid (\(\sigma=16\) MPa to \(\sigma=116\) MPa) are small compared to the maximum stress (\(\sigma=418\) MPa) close to the surface. Regarding the pressures in the fluid film, a greater variation occurs in the mixed regime, where there is greater proximity between the asperities and the rigid plane. In the hydrodynamic regime, the measurements vary between \(p_{f}=102\) MPa to \(p_{f}=169\) MPa, and in the case of mixed lubrication from \(p_{f}=67.5\) MPa to \(p_{f}=172\) MPa (see Figure 13). Figure 12: A cut view presenting the dissipation of stresses through depth. Figure 13: Pressure’s analysis in the fluid film. (**a**) Hydrodynamic lubrication. (**b**) Mixed lubrication. When considering this study, and the surface roughness in contact with the fluid film, it becomes appropriate to visualize the behavior of the film thickness and the pressure plots in the fluid along a central line to demonstrate more clearly the arrangement of the results found. Thus, in Figures 14 and 15, it is possible to observe said behaviors for both the mixed lubrication and hydrodynamic lubrication regimes. Figure 14 compares the film thickness of the two lubrication regimes. It can be seen that the film thickness is almost the same, although the hydrodynamic regime started with greater fluid thickness during the working condition. At the end of the simulation, both reached the same thickness, which was expected since the rigid plane motion realized the pressure. The difference is that the film carries higher pressure under the hydrodynamic regime condition, which can be verified in Figure 15.  
In contrast, in Figure 16, the surface under the hydrodynamic regime is more protected than the surface under the mixed lubrication. Both figures being at the same scale, it can be seen that the green colors appear in a greater area on the surface, subject to a mixed film. Figure 14: Behavior of film thickness along the central line. Figure 15: Fluid pressure profile along the central line. Finally, a procedure to homogenize the results made it possible to obtain the friction coefficient for each time instant, according to the equation (11). It considered the ratio between the overall forces resisting movement, where \(F_{i}^{\tau}\) is a tangential force at each surface element, \(F_{i}^{N}\) is a normal force at each surface element and \(nel\) is the total number of surface elements. Thus, in Figure 17, there is an evolution of the coefficient of friction between the surfaces considering the hydrodynamic- and mixed-lubrication regimes. \[f=\frac{\sum\limits_{i=1}^{nel}F_{i}^{\tau}}{\sum\limits_{i=1}^{nel}F_{i}^{N}} \tag{11}\]  
It is possible to observe that in the hydrodynamic regime, the friction coefficient starts from zero at the beginning of sliding and then remains close to 0.0045. In the mixed regime, the sliding movement begins with a higher friction coefficient value and remains so, reaching close to 0.014.