Analytical Solutions to Blocked Lubrication

When mixed/boundary lubrication problems are studied, proper boundary conditions are needed to reflect the local physical reality that occurs between fluid lubrication and solid contacts. In order to improve understanding of such reality, often in a microscopic scale hidden from direct observation, simple lubrication problems with zero net exiting flow or blocked lubrication are identified, and several analytical solutions are derived for the first time based on one- and two-dimensional Reynolds equations.

Their treatments related to LCBCs can be summarized as following: when the film thickness at a node (i, j) is less than or equal to zero, (a) the film thickness is set to zero; (b) two coefficients, "+',$ and "&',$ , need to be updated with zero film thickness; and (c) the fluid pressure is set to the solid-contact pressure.
Zhao et al. [5] simulated circular contacts start up, where the constraints (a) for the fluidfilm lubrication regions are positive film thickness, zero solid-contact pressure, and non negative fluid-film pressure, (b) for the solid-contact regions are zero film thickness, positive solidcontact pressure, and zero fluid-film pressure. Deolalikar et al. [6] explicitly applied a no-flow boundary condition to boundaries where the fluid regions are upstream of the solid-contact regions.
(2) On the other hand, pressure variables, i.e., old and new pressure arrays during iterations, cover both lubrication and solid-contact regions, and are updated within the same sweep over nodes of an iteration process. Zhu and Hu [7]  Dimensionless film thickness is compared with a fixed small value to decide the truncation in (b). This approach is practical and has been successfully applied in their subsequent work [8].
Holmes et al. [9] used regular Reynolds equation and the elastic deflection equation in a differential form to update unknown nodal values of pressure and film thickness. Once a nodal value of film thickness is negative, it is set to zero and the nodal value of pressure is updated with the elastic deflection equation only. Li and Kahraman [10], Azam et al. [11], and Wang et al. [12] utilized the unified equation system of Zhu and Hu [7].
The LCBCs in mixed/boundary EHL problems are microscopic, complex by nature, and often overlooked in experimental and fundamental studies, so it is important to understand/formulate them through focused investigations in order to enable physically meaningful numerical results. Given the situation, it is necessary to first address simpler lubrication problems under simpler configurations. In this paper, several analytical solutions are derived for the first time based on one-or two-dimensional Reynolds equations without net exiting lubricant-blocked lubrication problems. These solutions themselves may be applicable to special engineering problems or to be used to validate numerical algorithms in the mixed/boundary lubrication study. Also, they can be exercises for students in the tribology field, and can help readers to better understand lubrication where flow is blocked locally. In the future work, reasonable boundary conditions can be introduced to deal with boundaries between fluid lubrication and solid contacts, and they ultimately allow details closer to reality around solid contacts to be obtained in lubrication simulations.

Analytical Solutions
In the following steady-state lubrication problems, flow through the lubrication area is intentionally blocked in one direction, in order to gain insights of film pressure involving solid contacts in mixed lubrication. They are mostly two dimensional problems, but one three dimensional pad bearing problem is also analytically solved in section 2.5.
For simplicity, density ρ are constant. Also in sections 2.1 to 2.3, and 2.5, lubricant viscosity η is constant and bodies are rigid. In section 2.4, the Barus viscosity-pressure relationship is used with an elastic cylinder. Furthermore, referring to Figs. 1-5, it is assumed that (a) The bottom plate is always flat and horizontal moving along the x axis with a constant velocity of u, and the top plate is stationary with various shapes; (b) no air has been trapped inside for all problems; (c) a perfect sealing exists between the rigid block plate and the moving plate, and (d) there is no slip between lubricant and the plate surfaces. One could use this website https://www.wolframalpha.com/ or softwares such as mathematica® or Maple ® to help derivation.

Straight plate
Problem 1 has a tilted straight plate and a block plate, both of which are rigid surfaces (see Fig. 1). The gap in the inlet is hi and the gap on the blocked side is h0.
After substituting the film shape and another integration, Integration constant D is determined by inlet boundary condition: p = 0 at x = 0, so D = −1. After simplification, the solution of p is obtained as, When ℎ I is close to zero, the pressure there becomes very large. However, in reality the situation with near-zero ℎ I at the blocked exit is very complicated and some of assumptions mentioned above may not be valid anymore. Thus, it is beyond the scope of this work. Note that if the top plate is parallel to the bottom one, there is still pressure build up and the pressure distribution is a linear function of x.
2.2 Plate with a step Figure 2 shows a tilted plate with an interior step, where the gap value changes from h0 to h1. Equation (4) is applicable for both regions separated by the step. For the left region, the result in section 2.1 is still valid. For the right region, the film shape is

Fig. 2 Tilted plate with a step, blocked exit
After substituting the film shape into Eq. (4) and an integration, Integration constants ? should be determined by pressure continuity at x = l. One can find So, If h1 equals to h0, step disappears, this plate has a point, and this solution is still valid.

Cylinder
This problem involves a rigid cylinder with a radius of R. The minimum gap is h0, and the block plate is at x = x0, see Fig. 3. The film shape is After substituting the film shape into Eq. (4) and an integration,

Fig. 3 Cylinder, blocked exit
Where ω =`2 ℎ I . Integration constant D is determined by inlet boundary condition: After simplification, the solution of p is obtained as, One can see this solution does not contain x0. If x0 is negative, it has a convergent geometry, so this solution is valid for the entire length of l. But if x0 is positive, the portion beyond x = 0 is divergent. Even in this divergent gap, this solution is valid. When x0 = 0, pressure reaches this  [13] (also in Morales-Espejel et al., [14]) discussed the EHL inlet analyses with the Barus viscosity-pressure relationship, The reduced pressure x a gap is applied. Figure 4a illustrates an elastic cylinder (radius of R) and a rigid plate under a load. The Hertzian contact width is a. dimensionless coordinate X is defined as x/a. Without considering the deformation from the fluid pressure, the gap between these two bodies when ≤ −1 can be expressed as, In order to facilitate integration, Eq. (20) can be approximated as follows, see Eq. (10.38) of [13], It is found that this approximation is reasonable for −1.
After integration with the film shape of Eq. (21), one can obtain .5p The Beyond this location, the pressure increases dramatically. Note that when the speed is slower, X needs to be closer to −1 to reach the same pressure value. If the deformation from the fluid pressure has to be considered in the calculation, one has to use numerical simulations.
2.5 Fixed incline pad bearing blocked at the exit Figure 5 shows a fixed incline pad bearing with a blocked exit. This is a threedimensional lubrication problem and requires a two-dimensional Reynolds' equation. Following Muskat et al. [16] and Hay [17] successes, the middle line of Eq. (34) with Bessel functions will be selected for the problem in Fig. 5, and is defined with x1 the inlet coordinate to satisfy zero pressure boundary condition at x1 since Š ( ' ) = 0. Š is used to replace 'Š for simplicity. The origin of the coordinate system in  Fig. 6 with a label of "line". For the problem in section 2.2, the step is 20mm away from the inlet and the heights are 0.7 and 0.5mm, The modified pressure and gap distributions are shown in Fig. 6 with a label of "step". The last case shown in Fig. 6 with a label of "bent" is for a bent plate with the minimum gap of 0.35mm.
One can see there are pressure further built-up inside the divergent gap. The example for section 2.3 has a radius R of 20mm, l = 2mm, and h0 = 0.05mm. So ω = 2 ℎ I =√2. Figure 7 shows the gap and the modified pressure.  Table 1. The coefficients of Š = Š / I can be readily obtained. The same website can calculate the corresponding values of the Struve functions, which are also listed in Table 1. The values of Δ, Bessel functions, and the coefficients, Š are listed in Table 2. with the width of 20mm are obtained for half of the width due to symmetry, and is shown in Fig.   8. In Fig. 9, a comparison is shown, where "1d" is for the infinite width, "2d 40" is for the centerline pressure distribution with the width of 40mm, and "2d" is for the centerline pressure distribution with the width of 20mm. One can see the narrower the width, the less pressure build-up. The label "edge" is for the pressure on the edge of the plate with the width of 20mm, all of which should be zero. However, one can see numerical errors exist, particularly when x is around 20. Double precision and more summation terms can reduce such errors.

Conclusions
Several analytical solutions to lubrication problems with blocked exits have been derived.
They are critical to understand the boundary conditions between solid contact and fluid lubrication occurring in the mixed/boundary lubrication regimes.

Acknowledgements
The author would acknowledge his family's financial support during COVID-19, which make this work possible.

Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.