Lubrication-Contact Boundary Conditions for Lubrication Problems

Under sever conditions, asperity contacts take place, surrounded by fluid lubrication. Mathematical descriptions are needed to describe local connection between fluid lubrication and solid contacts. In order to differentiate them from conventional boundary conditions, these conditions are called lubrication-contact boundary conditions (LCBCs), which have not been sufficiently addressed by previous studies. In this work, a set of LCBCs formulations are constructed based on local flow continuity from the continuum mechanics point of view, together with pressure inequalities. Numerical implementations are developed and tested with problems involving simple geometries, and they are expected to be integrated with mixed/boundary EHL solvers, as well as be applied to deterministic sub-models of stochastic models in order to obtain flow factors that consider solid contacts.

are ignored in these work [10,12,[14][15][16][17][18]. Deolalikar et al. [21] treated lubrication and solid-contact regions separately, using film thickness comparison against a fixed small value to differentiate these two kinds of regions. They explicitly applied a no-flow boundary condition to boundaries where the fluid regions are upstream of the solid-contact regions. When a control volume has this boundary as its east wall, flow through the control volume is modified. LCBCs are dealt with to some extent in these works [11,21]. It should be noted that references [10][11][12][13][14][15][16][17][18][19][20][21] have successfully used surface gap values to determine whether a node is under fluid-film lubrication or is a solid contact, and thus temporary boundaries between film lubrication and solid contacts can be obtained before each iteration. Once the iteration process is completed, this partition becomes final.
Wang and Zhu [22] summarized their contributions to the mixed lubrication simulations using the finite difference method in a monograph. Venner and Lubrecht [23] and Habchi [24] in their monographs describe in detail how to apply the multigrid method and the finite element method, respectively, to solve EHL problems. However, so far, existing simulation work has either failed to address or has only partially dealt with the LCBCs as mentioned above.
Furthermore, Zhou et al. [25] applied EHL simulation to study lubrication between case-hardened gears, and AL-Mayali et al. [26] studied micropitting initiation with experiments and numerical simulations. Hansen et al. [27] proposed a new updated film parameter.
Mohammadpour et al. [28] investigated exterior boundary conditions of point EHL contacts with a combined numerical-experimental analysis, however, the author is not aware of any experimental studies about lubrication-contact boundaries of EHL problems. Such a situation motivates a series of explorations about the LCBCs through theoretical analyses. Liu et al. [26] studied the flow continuity of line-contact EHL results and revealed a flow-continuity view of the discrete Reynolds equation. Liu [27] derived analytical solutions to several lubrication problems with flow blockages. Based on these insights and engineering principles, a set of LCBCs are Tribology Letter proposed in this work and numerical implementations are developed and verified with problems involving simple geometries, without the complication of elastic deformation/cavitation/wall-slip.
These LCBCs are expected to be integrated with mixed/boundary EHL solvers, as well as be applied to deterministic sub-models of stochastic models in order to obtain flow factors considering solid contacts. Fig. 1 Side view illustration of a 1D problem with a lubrication-contact boundary

Boundary conditions
From the continuum point of view, the boundary conditions between fluid and solid contacts are discussed in this section, and a set of boundary conditions is explicitly defined to describe the lubrication-contact interaction in a typical mixed lubrication problem. Numerical implementations of these boundary conditions will be presented next in a discrete format. This set of LCBCs includes two parts: pressure and flow. Figure 1 illustrates such boundaries where both lubrication and solid contact co-exist in a one-dimensional (1D) problem. The coordinate of the boundary is xΓ, and on the two sides of it there are fluid pressure, , and contact pressure, , where − is the location to the left but very close to the boundary and + is the location to the right but very close to the boundary. Engineering principles suggest that at the boundary, In fact, due to the flow continuity, zero net flow is true everywhere, and If the gap is continuous around the boundary of ℎ( ) = 0, ℎ( − ) can be infinitely small. Note that the pressure gradient in Eq. (3a) can be infinitely large. In reality, many more features of the lubricant itself (non-Newtonian behavior, for example) and/or interactions between the lubricant and the bonding surface (slip boundary, for example) may be considered, but those are beyond the scope of this paper.
The lubricant velocity direction in 2D problems can be arbitrary, but in this paper, the entraining-velocity direction is simply along the x axis. One can use the same treatment to handle the velocity component along the y axis. The 2D LCBCs related to flows can be expressed in the following four scenarios, should be added to the last two equations. These are all the necessary LCBCs from the continuum point of view to describe lubrication problems involving lubrication-contact boundaries.

Numerical Implementation
In this section, the newly proposed LCBCs for lubrication problems involving contacts are implemented in a discrete format so that they can be incorporated into a numerical algorithm.

1D problems
The dimensionless Reynolds equation can be expressed as and the dimensionless mass flow can be written as Which has the Couette flow Moreover, the derivative of the Couette flow can also be expressed in a separate form [22,31,32], which can be discretized with the following options ("S" in the notation stands for "separate"): i. S1B scheme: If the 1B scheme is used for the derivatives and * and take values at i, which has one more term than what is in Eq. (8). This scheme is used in Eq. (11) of [32].
ii. If the 1B scheme is used for the derivatives but * and take averages values between i and i-1, one obtains the same equation as Eq. (8).
iii. S2C scheme: If the 2C scheme is used for the derivatives and * and take values at i, which has two more terms than what is in Eq. (10).
iv. If the 2C scheme is used for the derivatives with * and taking the averages values between i+1 and i-1, one obtains the same equation as Eq. (10).
v. Using the 2B scheme for the derivatives with * and taking averages values between i+1 and i-1, one obtains, Tribology Letter vi. Using the 2B scheme for the derivatives with * and take values at i, one has, Liu et al. [29] presented a flow-continuity view of the discrete Reynolds equation, which is extended in this work to handle the LCBC. Regardless of which differential scheme is used for the derivative of the Couette flow, the discrete Reynolds equation can be expressed in four flow terms as follows, where the two terms of flow leaving the node of i are, and the two terms of flow entering the node of i, Q −0.5 and Q −0.5 , are obtained by replacing i with i-1, except no such a replacement for the subscripts of * − −1 * in the S1B, e.g., Furthermore, for the S1B scheme, one could use * + * ( − − ) in Eq. (17) and subscripts on the right-hand sides, except those of the bold font are smaller than the subscripts in the left-hand side. For example, the 1B scheme of Eq. (17) has a subscript of i on the right-hand side, which is smaller than the subscript of + 0.5 on the left-hand side. Therefore, if subscript values increase from the left to the right in the discretization (Fig. 3), the differential schemes of 1B, 2B, or S1B use Couette flow values slightly to the left to approximate the Couette flow terms needed at the two ends of the element (In Fig. 3, the thick line represents one element and these Couette flow terms are illustrated with hollow arrows).
The discrete 1D lubrication problem involving contacts is shown in Fig. 3. Each node represents a length that is one half intervals in front of and behind the node, represented by the thick line on the horizontal axis in Fig. 3. Note that if the boundary location, XΓ, is known, one could arrange two nodes with XΓ right at the center. However, XΓ could be unknown at the beginning and change during iterations. In a discrete mesh, if XΓ is located between two neighboring nodes, the center of them is used to approximate XΓ. Of course, a smaller interval between nodes (finer mesh in other words) could reduce error of such approximation. A dashed line is used in Fig. 3 to represent the local geometry. iΓ is used to represent the fluid lubrication node closest to the discrete boundary. empty circles--solid contact, with pf and pd for the fluid and contact pressures, respectively.
In the following, the no-flow constraint is integrated into the discrete Reynolds equation in order to determine the pressure at iΓ. If the film thickness at +1 is the first zero value tracing from inlet, the discrete boundary is between node and node + 1. The no flow boundary condition at the i+0.5 node is Q +0.5 + Q +0.5 = 0, reducing Eq. (15) to, Therefore, based on the differential schemes for the Couette flow, the following can be obtained Since the net flow is zero everywhere, these recursion formulae are also valid for other nodes in the lubrication region. Once the lubrication and contact have been solved, at that i Γ node, the following inequality needs to be true, Otherwise, the location of the boundary has to be adjusted.

2D problems
For 2D problems, assuming that the motion is along the horizontal axis for convenience, Fig. 4 shows a discrete mixed lubrication example, where the grey box is a representative boundary in the continuous space. In the discrete space, once a node is in contact, the material inside a rectangle area around it is in contact and the same is true for the nodes in fluid lubrication. The With Q c by 1B/S1B

North
With Q c by 2B Descriptions of the differential and finite difference schemes in 1D problems can be expanded. If node (i, j) and its surrounding four nodes are all under fluid lubrication (see the illustration in the top-right of Fig. 4), the discrete Reynolds equation for such a node has a standard form. Equation where , , is the fluid pressure at the node of (i, j). Totally five nodes are involved in these discrete Tribology Letter For the S1B scheme, one can also use , * , + , * ( , − − , ) in Eq. (31) and −1, * −1, + −1, * ( , − − , ) in Eq. (32). Note that subscripts in the right hand side of Eqs. (31)(32) are also slightly shifted from the subscript of Q c (see Fig. 4b). In other words, the right-hand side of the expressions in Eq. (31) are: (i) independent of the node (i+1, j) when the 1B, 2B, or S1B scheme is selected. When node (i+1,

Tribology Letter
In total, four flows are involved in these discrete equations. When one or more of these four surrounding nodes is under solid contact, these equations should be adjusted according to the LCBCs mentioned in section 2. In the following, different cases are discussed in detail.

Control volumes with one side on the boundary
where the definitions of the Couette flow terms, Q , +0.5, and Q , −0.5, , are given in Eqs. (31)(32).
, , is used for fluid and , , for solid contact pressures. nodes is on the east side, Q , +0.5, is set to zero. After these necessary treatments, Eq. (36) can then be used to obtain the updated pressure at the current node of the control volume.
If a line-by-line pressure evaluation is executed, one can use this format [22,31] , −1, + , , + , +1, − = 0 where αi, βi, γi, and are coefficients. These coefficients need to be modified accordingly when one or more surrounding nodes are in a solid contact region to make sure both the Poiseuille and

Pressure constraints
During iteration, a node can change between film lubrication and solid contact. If a node in solid contact has one, two, three, or even four surrounding nodes in lubrication (in Fig. 4 and 5, there are various cases illustrated), its solid contact pressure value should be larger or equal to these fluid-lubrication pressure values. Otherwise, this node may very likely become a node in lubrication in the subsequent iteration. In a final result, these pressure constraints have to be satisfied.
In summary, various modifications, detailed in sections 3.2.2 to 3.2.5, have been derived to satisfy the lubrication-contact boundary conditions and the discrete Reynolds equation. These modifications are applied in the following section to obtain numerical solutions.

Numerical results and discussion
Steady-state lubrication problems are used to explain the numerical implementation and validate the proposed lubrication-contact boundary conditions in term of flow constraints without evaluating solid-contact pressures. The pressure constraints are not implemented in this section, but can be activated for EHL problems once solid-contact pressures are available. Due to their simplicity, 1D problems are not included here.
A fixed incline pad bearing is shown in Figure 6 with a rigid block. It is assumed that (

Blockage with a rigid cylinder pillar
In this example, the square column in Fig. 6 is replaced with a cylinder pillar, which has a radius of 4mm, and its center is at (35, 0). The modified fluid pressure result is depicted in Fig. 8 with the same color map as in Fig. 7. values between the blockage and the exit. The square column builds a higher highest pressure right in front of the blockage than the cylindrical pillar, as expected. When the exit side is completely blocked, an analytical solution has been derived by Liu [30] and the corresponding values at selected locations labeled by "B-Exit ana." in the legend are shown in Fig. 9 with filled square marks. Numerical results obtained by the iteration process and labeled by "B-Exit num." Tribology Letter are depicted too and they match the analytical solution very well. One can see clearly in the region close to the inlet that the ranking of pressure values from high to low are square-column blockage, cylinder-pillar blockage, exit blockage, and no blockage. When a node is not close to the boundary where the gap is zero, similar numerical results are obtained independent of mesh size. However, for the nodes that are close to the boundary, finer mesh produces much higher pressure values. This could be physically true to some extent, however, when the gap is around several nanometers, lubricant is not a continuum and the Reynolds equation is not valid. For these regions with ultra-small gaps right around the lubrication-contact boundaries, a multi-scale approach together with molecular dynamics at the nanoscale [33], should be developed to understand the true story there.

Conclusions
Lubrication-contact boundary conditions (LCBCs) between fluid and solid contacts in lubrication problems are investigated because of their importance to numerical simulations.
Expressions for Couette flows entering or leaving a control volume have been clearly defined for several differential schemes. This work proposes a set of lubrication-contact boundary conditions,

Acknowledgements
The author would express his sincere thanks to Dr. Q. Jane Wang at Northwestern University for her valuable discussion and proofreading during manuscript preparation, and to Dr. Ashlie Martini at University of California, Merced for her proofreading.