**3. Problem Definition and Its Solution**

In order to investigate the effect of lubricant supply on the coefficient of friction for different texturing parameters and patterns, several simulations were performed.

In Figure 4., when the input film thickness (*hoil*) is smaller than the calculated film thickness for non-starved lubrication (*hs*), the contact is operated under starved conditions. The limited amount of lubricant in the input region of contact (*hoil*) can affect the coefficient of friction for the textured surfaces. To gain a better understanding of the starved lubrication phenomenon, the influence of input film thickness and texturing parameters, such as texture pitch (*Px*), texture depth (*Td*) and texture size (*S*), on the coefficient of friction, is investigated. These simulations are based on the linear groove and chevron patterns (see Figure 5).

**Figure 4.** Schematic illustration of the input film thickness (*hoil*) and the calculated film thickness (*hs*) [64].

**Figure 5.** Schematic illustration of different patterns (**a**) Grooves, (**b**) Chevron pattern [34].

In order to investigate the effect of texturing, the geometrical parameters are studied. In this parametric study, the effect of the geometrical parameters determining the texture shape and the effect of the texture area fraction upon the frictional behaviour of the starved lubrication conditions are studied. By introducing texture pitch (*Px*), it is possible to define the texture area fraction.

The pitch is calculated as:

Cavity size: *S* = 2*rp* and Pitch in x direction: *Px* = *S*/*Lgx*.

Also, texture depth (*Td*) can affect the film formation in friction in textured surfaces. It is possible to define the geometry of grooves by these three parameters. Another parameter that can help to define the geometry of chevrons is the cavity width ratio, which is represented by *K* in Figure 6.

**Figure 6.** Chevron cavity width ratio.

Prior study on film thickness [57] showed that the rectangular cross section can build a thicker lubricant film in contact, and it is more efficient in comparison with the circular cross-sectional patterns; therefore in the present article, the rectangular cross-section patterns are employed (see Figure 3). Here, linear grooves and chevrons will be analysed. The operating conditions applied in this calculation are presented in Table 1. The analyses of the roughness measurement and surface topography were performed using a Keyence Color 3D LASER Scanning Microscope (Keyence, Osaka, Japan), which uses a violet LASER λ = 388 nm.


**Table 1.** Operating conditions.

Where *K* = Inner wall length/outer wall length, and in these calculations (*K*) is constant, and equal to 0.5.

To validate the model and algorithm, a comparison was performed between the experimental measurements of Kovalchenko et al. [37] and the numerical results from the algorithm developed in this work. It is worth it to mention that, although the developed algorithm can calculate the friction in all three lubrication regimes within this section just to validate the code, the comparison takes place in the hydrodynamic lubrication regime. Therefore, the numerical and measurement results are just dealing with the hydrodynamic lubrication regime. Kovalachenko et al. [37], investigated the effect of size and the density of circular pockets on the coefficient of friction (see Figure 7). In this study the Disk 3 results are compared with results from the developed numerical algorithm.

**Figure 7.** Disk 3 dimple array, (Reproduced with permission from Andriy Kovalchenko, Oyelayo Ajayi, Ali Erdemir, et al., Tribology Transactions, published by Taylor and Francis, 2004) [37].

In Disk 3 the cavity depth is 5.5 μm, the cavity size (*S* = 2*rp*) is 78 μm and the texture density is 28%. The measurement results for this disk are presented in Figure 8.

**Figure 8.** Measurement results (Reproduced with permission from Andriy Kovalchenko, Oyelayo Ajayi, Ali Erdemir, et al., Tribology Transactions, published by Taylor and Francis, 2004) [37].

Experimental data extracted from Figure 8 and the numerical results are based on the calculation of coefficient of friction for the full film condition, when the lubricant kinematic viscosity is 1247 cSt at 40 ◦C, and the normal load of 20 N is applied.

The texture array schematically illustrated in Figure 9. The comparison between the numerical results and the experimental measurements for circular pockets with *Td* = 5 μm, *rp* = 15 μm and *Px* = 0.3, are presented in Figure 10.

**Figure 9.** (**a**) Simulated texture array and (**b**) cavity profile.

**Figure 10.** Comparison between numerical and experimental results.

From Figure 10 it is possible to see that there is a good agreement between the values of the coefficient of friction calculated by the numerical model (blue line) and the experimentally measured results (red diamonds). When the velocity is 0.5 <sup>m</sup>·s<sup>−</sup>1, the calculated value is around 1% less than the measured value for Disk 3 that is in reasonable range. Moreover, results show that the calculated coefficient of friction has the same trend as the measured results for the coefficient of friction.
