*2.2. Load Sharing Concept*

According to Johnson [39], in the case of a mixed lubrication regime (ML), the total normal load (*FT*) is equal to the load carried by the boundary lubrication BL force component (*FC*) plus the hydrodynamic lubrication HL force component (*FH*), therefore:

$$F\_T = F\_C + F\_H \tag{2}$$

Based on (Equation (2)), coefficients (γ1) and (γ2) are introduced:

$$
\gamma\_1 = \frac{F\_T}{F\_H}, \text{ yz} = \frac{F\_T}{F\_C} \tag{3}
$$

The two coefficients (γ<sup>1</sup> and γ2) are dependent of each other through the equation:

$$1 = \frac{1}{\gamma^1} + \frac{1}{\gamma^2} \tag{4}$$

In the deterministic asperity contact model, for the contact between a rigid flat surface against a rough surface, the Stribeck curve can be calculated by employing the two (γ1) and (γ2) parameters, where these two coefficients are defined in Equation (3). In the work of Gelinck [41], these two parameters are presented in terms of pressure (see Equation (5)). By combining the well-known Greenwood and Williamson [40] contact model under a classical hypothesis of the Reynolds isothermal equation, the entire Stribeck curve can be calculated.

$$
\gamma\_1 = \frac{p\_T}{p\_H}, \gamma\_2 = \frac{P\_T}{p\_C} \tag{5}
$$

where *pT*, is the total pressure carried by the contact, *pC* is the pressure on asperities, and *pH* is the pressure carried by the hydrodynamic component of mixed lubrication [48].

In mixed lubricated contacts, in order to calculate the coefficient of friction, the asperity and hydrodynamic load components (*FC* and *FH*), as well as the related film thickness, must be determined. By solving the following three equations, it is possible to determine the above-mentioned three parameters:

I. The first equation is Equation (2) (*FT* = *FC* + *FH*), in order to consider the load components in the BL component and the HL component, and their relation with the total load.

II. The second equation is the film thickness relation. In this study, the film thickness calculation is based on solving the Reynolds equation for textured surfaces [57].

This Reynolds equation is derived from the Navier-Stokes equation by taking the narrow gap assumption into account. In the Cartesian coordinate system, the Reynolds equation can be written as:

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{\rho h^3}{\eta} \frac{\partial p}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial y} \left( \frac{\rho h^3}{\eta} \frac{\partial p}{\partial y} \right) = 6(u\_0) \frac{\partial (\rho h)}{\partial \mathbf{x}} + 6\rho h \frac{\partial (u\_0)}{\partial \mathbf{x}} + 12 \frac{\partial (\rho h)}{\partial t} \tag{6}$$

where *p* is the lubricant pressure, η is the viscosity, *h* is the film thickness, and *u*<sup>0</sup> is the sum velocity. In absence of textures over the surface in parallel sliding contacts, the right side of the Reynolds equation equals zero; therefore there will be no pressure build up in contact, and no film can get formed.

At the outlet of the cavity, the lubricant is dragged through a converging region, and as a result, pressure is generated. The flow divergence at the entry of the cavity results in a negative pressure. This negative pressure is suppressed by cavitation, and as a cavitation product, vapour bubbles are appearing in the lubricant film. The Jakobsson-Floberg-Olsson model is dividing the lubrication film into two zones. The first zone is the lubricant film without the cavitation effect; therefore no vapour bubble exists in this zone, and the Reynolds equation is valid. In the second zone, where cavitation

does take place, the lubricant occupies just a fraction of the film gap, and the vapour bubbles exist in the void fraction. In this zone the pressure is taken as a constant [58].

By suggesting the use of a switch function, Elrod introduced a universal solution for cavitated and full film zones (see Equation (7)). In Equation (7), ϕ represents a dimensionless dependent variable, and *F* is the aforementioned switch function, and these parameters are defined as in a liquid zone, where *F* = 1, ϕ ≥ 0 and *p* = ϕ, and in the cavitated region, *F* = 0, ϕ < 0 and *p* = 0.

The steady-state mass-conservation Reynolds equation, taking the Elrod cavitation algorithm into account, can be written in a Cartesian coordinate system as (Equation (7)) [59]:

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{\hbar^3}{\eta} \frac{\partial (F\wp)}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{x}} \left( \frac{\hbar^3}{\eta} \frac{\partial (F\wp)}{\partial \mathbf{y}} \right) = \frac{6u\_0}{p\_\mathbf{a} - p\_\mathbf{c}} \frac{\partial ((1 + (1 - F)\wp)h)}{\partial \mathbf{x}} \tag{7}$$

**Figure 2.** Geometrical scheme of patterns: (**a**) Chevron, (**b**) Groove (Reproduced with the permission of Mingfeng Qiu, Bret R. Minson, Bart Raeymaekers, *Tribology International*, published by Elsevier, 2013) [60].

In this study, chevron and groove patterns have been investigated. Figure 2, shows the different cavity shapes and the parameters characterizing their geometry. The chevron pattern is defined by two similar equilateral triangles of different sizes. When the inner edge length of the chevron approaches zero, the chevron pattern transforms to a triangular pocket. For these two cases, the centre of the unit cell coincides with the midpoint of the altitude line of the triangle or chevron shape; (see also [60]). All patterns have a rectangular cross-sectional profile; (see Figure 3). The general film thickness formula can be written as (Equation (8)):

$$h = h\_0 + h\_{\text{macro}} + h\_{\text{texture}} \tag{8}$$

**Figure 3.** Schematic illustration of the cavity profile.

When both interacting surfaces of contact are flat, then *hmacro* is negligible. In case of a flat on flat sliding contact, the film thickness (Equation (8)) reduces to (Equation (9)) [60,61]:

$$\frac{h(\mathbf{x},\mathbf{y})}{h\_0(\mathbf{x},\mathbf{y})} = 1 + H(\mathbf{x},\mathbf{y}) \tag{9}$$

The film thickness formula for the chevron can be written as (Equation (10)):

$$H(\mathbf{x}, y) = \begin{cases} 0, & \text{(X, Y \notin \Omega)} \\ \frac{T\_d}{h\_0}, & \text{(X, Y \in \Omega)} \end{cases} \\ \Omega: -\frac{3}{4} \le X \le \frac{3}{4} \\ \text{and} \\ \begin{cases} \frac{1}{\sqrt{3}}X + \frac{\sqrt{3}}{2} \left| K - \frac{1}{2} \right| \le \frac{1}{\sqrt{3}} \\\ Y \le \frac{1}{\sqrt{3}}X + \frac{\sqrt{3}}{4} \\\ -\frac{1}{\sqrt{3}}X - \frac{\sqrt{3}}{4} \le \\\ Y \le -\frac{1}{\sqrt{3}}X + \frac{\sqrt{3}}{2} \left| \frac{1}{2} - K \right| \end{cases} \tag{10}$$

The film thickness formula for the grooves is given in (Equation (11)):

$$\Omega \,\Omega : -\frac{1}{2} \le X \le \frac{1}{2} \text{ and } \frac{1}{2} \le Y \le \frac{1}{2} \tag{11}$$

In this simulation *rp* is the characteristic radius for the chevron patterns and the half width of the grooves.

To solve the Elrod cavitation algorithm for Reynolds equation (Equation (7)), the tri-diagonal matrix algorithm (TMDA) is used, and in order to reduce the storage needed for calculation, the line-by-line TDMA solver (Patankar [62]) is employed. The TDMA is a direct method for a one-dimensional situation, but by solving it iteratively line-by-line, it is possible to apply it for two- and three-dimensional problems, as well. [63]. The algorithm for this numerical solution is presented in Appendix A.

III. The third equation is derived from the equilibrium of the modified relation for the central pressure and average contact pressure carried by asperities, represented as [48].

$$F\_{\mathbb{C}} = \sum\_{i=1}^{N} \frac{2}{3} E' R\_i (z\_i - d) \tag{12}$$

In Equation (12), *E* is the combined elasticity modulus and *Ri* is the reduced radius of the cylinder. The reduced elastic modulus is given by:

$$\frac{2}{E'} = \frac{1 - v\_1^2}{E\_1} + \frac{1 - v\_2^2}{E\_2} \tag{13}$$

where *E*<sup>1</sup> = *E*<sup>2</sup> = *E* and ν<sup>1</sup> = ν<sup>2</sup> = ν.

The total friction force (*Ff*) in the ML regime can be calculated as the sum of the shear force of the lubricant (*Ff H*) and friction force of the contacting asperities and (Equation (14)):

$$F\_f = \sum\_{i=1}^{N} \iint\_{ci} \pi\_{ci} dA\_{ci} + F\_{fH} \tag{14}$$

In Equation (14), τ*ci* is the shear stress at the asperity contact, *N* is the number of contacting asperities and *Aci* the area of contact of a single asperity contact. For the hydrodynamic component of shear force (*Ff H*), the friction force can be written as:

$$F\_{f\!\!\!H} = \pi\_{\!\!\!H} A\_{\!\!\!H} \tag{15}$$

where τ*<sup>H</sup>* is the shear stress of the lubricant and *AH* is the contact area of the hydrodynamic component. Friction is assumed to be of the Coulomb type for the contacting asperities:

$$f\_{\rm Ci} = \frac{\tau\_{\rm Ci}}{p\_{\rm Ci}} \tag{16}$$

with *pCi* the average contact pressure on the *i* th asperity, and *fCi* the coefficient of friction which is assumed constant for all asperities; then the double integral in Equation (14) can be written as:

$$\sum\_{i=1}^{N} \iint f\_{\mathbb{C}} p\_{ci} dA\_{ci} = f\_{\mathbb{C}} F\_{\mathbb{C}} \tag{17}$$

The value of *fC* is measureable from experiments, and in these calculations this value is set to 0.1 based on the measurements presented in Appendix B. The coefficient of friction is given by:

$$f = \frac{F\_f}{F\_N} = \frac{f\_\mathbb{C} F\_\mathbb{C} + F\_{fH}}{F\_N} \tag{18}$$

It is worth mentioning that, in the absence of texturing features in parallel sliding, no lubricant film in contact can be formed, and the two sliding surfaces will stick to each other. Therefore, the coefficient of friction will be constant and equal to the coefficient of friction in the boundary lubrication regime (*fC*).

In order to study the effect of starvation for textured surfaces with different texturing patterns, and to investigate the frictional behaviour based on the different texturing parameters, several simulations are carried out.
