*4.4. E*ff*ect of Input Film Thickness on the Coe*ffi*cient of Friction*

In Figure 17, the calculated film thickness for grooves, when *Td* = 10 μm, *S* = 100 μm and *Px* = 0.4, is presented. In this calculation, the effect of different input film thicknesses is studied. This effect is studied for four different input film thicknesses, *hoil* = 2.5, 4, 6 and 8 μm, the operational conditions and texturing properties are presented in Table 5.

**Table 5.** Texturing properties and operating conditions.


**Figure 17.** Effect of input film thickness on (**a**) film thickness, (**b**) coefficient of friction.

The results of the calculation of the Stribeck curve and the film thickness for different limited values of input film thickness are presented in Figure 17a,b. The tendency of the starved Stribeck curve and the corresponding film thickness as a function of input film thickness can be described as follows:

1. When the input film thickness (*hoil*) is lower than 2.5 μm, the Stribeck curve transforms into a straight line, and the film thickness for different velocities stays constant in the same level as it is in the BL regime.

2. If the *hoil* varies between 2.5 and 8 μm, the friction level in the HL and ML regimes starts to decrease, and the film thickness increases. Figure 13b shows that transitions from BL to ML and from ML to HL stay approximately at the same transition velocity for different values of *hoil*.

It concludes that, although the starvation has no influence on the transition of lubrication regimes, the friction level changes due to starved lubrication.

## **5. Discussion**

In this article, the influence of starvation and surface texturing on each other by performing a set of parametric studies has been investigated. The variation range of texturing parameters in these parametric studies was chosen based on the sizes which can give more fluctuation in the frictional behaviour of contacts; these ranges were chosen from the previous study on the effect of surface texturing on film thickness [57].

Based on the results in this section, texturing for flat-flat sliding contacts with a starved lubrication regime can be helpful for reducing the coefficient of friction. However, from results (Figures 7–13), it must be considered that increasing texturing parameters to the optimum values in the case of contacts with higher sliding velocities (lubrication regime in hydrodynamic lubrication) may not be as beneficial as at lower velocities. In other words, starvation can limit the beneficial influence of surface texturing at higher velocities. Therefore, surface texturing in the case of starved lubrication may not be advantageous to reducing the friction based on the operating conditions and any limit in lubricant input film.

From the parametric study in this article, it is possible to achieve that, although the growth in the depth of cavities to the optimum values leads to the higher film thicknesses in the case of starved lubrication, this growth in lubricant film thickness is limited due to the limit in oil supply, as is shown in Figure 9. The same frictional behaviour and merging in Stribeck curves is observable in the case of the study of the effect of size and pitch parameters. Changing the size and pitch parameters has an influence on the coefficient of friction, but for the values close to the optimum value for this parameter, this effect is limited. Furthermore, it is possible to obtain that, due to the limit in lubricant supply. The same trend of behaviour is also predictable for other patterns, i.e., triangular pockets and circular pockets when the starved lubrication regime is occurring in contact. Moreover, in the case of a chevron pattern similar to the groove pattern, the lowest coefficient of friction in the case of non-starved lubrication is achievable when the depth is around 10 μm and the pitch is around 0.5.

Based on the results for the section on the effect of the input film thickness on the coefficient of friction, this parameter can have a vital role on the efficiency of texturing. In particular, when the input film thickness (*hoil*) is lower than 2.5 μm, the Stribeck curve transforms into a straight line and the film thickness does not change compared to the boundary lubrication regime film thickness.

If the *hoil* varies between 2.5 and 8 μm, the friction level in the HL and ML regimes starts to decrease, and the film thickness increases.

The influence of roughness is more considerable in the boundary lubrication regime and mixed lubrication to the point at which the transition between ML and HL is happening. Therefore, employing surfaces with a higher roughness can shift the Stribeck curve; this shift in addition to the effect of starvation can reduce the influence of texturing and application of optimized texture properties.

## **6. Conclusions**

The goal of this investigation was to study and predict the effect of surface texturing on frictional behaviour for parallel sliding contacts under starved lubrication condition. In addition, the efficiency of surface texturing as a method to reduce the friction in starved lubricated contact is also studied. This model is a numerical algorithm based on the Reynolds equation with the Elrod cavitation algorithm formulation. By applying the value of calculated film thickness in the deterministic asperity model, the coefficient of friction is calculated. In this article, Stribeck curves for different situations are plotted. The effect of several parameters on starved regime frictional behaviour, such as depth, size and texture pitch, has been studied.

In this study, the deterministic asperity contact model is employed efficiently, considering the effect of different scales of surface features (roughness and texture) on the coefficient of friction that is not dependent on the directions of the asperities.

1. This approach allows the effect of texture and roughness to influence the friction independently; this may be beneficial in optimizing the surface texture.

2. In order to reduce the friction in starved lubrication conditions, surface texturing has a beneficial effect, and this effect is also presented in numerical study of Gu et al. [33,65].

3. The positive effect is more sensible mostly when contacts are in lower sliding velocities. When the sliding velocity reaches higher values, the effect of texturing can be influenced by the input film thickness.

4. In the case of starved lubrication, when the value of this input film thickness (*hoil*) decreases, the starvation effect gains a greater influence upon the film thickness. Therefore, the effect of variation of the texture parameters (pitch, depth and size) on the coefficient of friction is also decreasing.

5. Surface texturing in starved lubricated conditions, based on operating conditions and a limit in lubricant input film, may not be advantageous as a method to reduce the fiction.

6. In order to apply the texturing in starved lubricated contacts a simulation of the coefficient of friction and film thickness based on texturing and lubricant properties can help to avoid the unnecessary surface texturing.

It is worth mentioning that based on the operational conditions, thermal and atmosphere effects could play an important role in lubrication. These effects should be included in the numerical simulation for accurate performance predictions. However, since these effects were not considered texture-specific, which is the focus of this article, thus they have not been included in this article.

**Author Contributions:** Conceptualization, D.J.S., E.L.D. and D.B.; Methodology, D.J.S. and D.B.; Software, D.B.; Validation, D.B. and D.J.S.; Formal Analysis, D.B. and D.J.S.; Investigation, D.B. and D.J.S.; Writing—Original Draft Preparation, D.B.; Writing—Review & Editing, D.B., E.L.D., M.B.d.R. and D.J.S.; Supervision, D.J.S.

**Funding:** This research was funded by Materials Innovation Institute (M2i) grant number M21.1.11448.

**Conflicts of Interest:** The authors declare no conflict of interest.
