Influence of Geometrical Shape on the Lubrication Performance of Partially Micro-Grooved Gas-Lubricated Parallel Slider Bearings

investigated in the present study. The solution procedure for the multi-grid finite element method is that the algebraic equations formed by the finite element method are in turn smoothed by the interpolation from the coarse grid to the fine grid and the restriction from the fine grid to the coarse grid. By adopting the multi-grid finite element method, the pressure distribution of partially mi-cro-grooved gas-lubricated parallel slider bearings is obtained. The grooving parameters are optimized to obtain the maximum average pressure for different micro-groove cross-section shapes. The results show that the orientation angle of the micro-grooves, depth of the micro-grooves, width of the micro-grooves, area density of the micro-grooves, micro-grooved fraction, and cross-section shape of the micro-grooves have an important influence on the hydrodynamic pressure of partially micro-grooved gas-lubricated parallel slider bearings. The results of this study demonstrate that the lubrication performance of partially micro-grooved gas-lubricated parallel slider bearings can be obviously enhanced by adopting the appropriate groov-ing parameters and micro-groove cross-shape. The results of this study are beneficial for the design of partially mi-cro-grooved gas-lubricated parallel slider bearings.


Introduction
Opening micro-grooves has been applied to enhance the mechanical component performance for many years [1][2][3]. Recently, researchers study the performance of micro-grooved mechanical components using an incompressible lubricant. Adatepe et al. [4] experimentally and numerically studied the tribological performance of microgrooved oil-lubricated journal bearings. They found that the experimental results are in accordance with the numerical results. Kango et al. [5] theoretically investigated the friction coefficient of micro-grooved oil-lubricated journal bearings. It was shown that micro-grooves could decrease the friction coefficient. Kumar et al. [6] studied the frictional power loss of micro-grooved thrust bearings using an incompressible lubricant. They showed that the frictional power loss could be reduced by opening micro-grooves. Fu et al. [7] studied the lubrication performance of fully micro-grooved oil-lubricated parallel slider bearings. It was found that the hydrodynamic pressure could be increased by using the optimized micro-groove geometrical parameters. Ji et al. [8] studied the influence of grooving parameters and micro-groove cross-section shape on the lubrication performance of partially micro-grooved oillubricated parallel slider bearings. Their results demonstrated that grooving parameters and micro-groove crosssection shape have a significant impact on the lubrication performance of partially micro-grooved fluid-lubricated parallel slider bearings. Miao et al. [9] investigated the tribological performance of micro-grooved oil-lubricated piston ring-cylinder liner components. It was shown that micro-grooves could produce better tribological performance at proper conditions.
Researchers also investigate micro-grooved mechanical components using a compressible lubricant. Yan et al. [10] investigated the performance of micro-grooved gas-lubricated foil journal bearings. It was found that opening micro-grooves on the top foil could increase the pres-sure around the load domain and decrease the end leakage. By using the chaos-enhanced accelerated particle swarm algorithm, Yu et al. [11] studied the performance of microgrooved gas-lubricated thrust bearings. It was found that the discontinuous micro-grooves could obtain the higher load-carrying capacity. Kou et al. [12] investigated the steady and dynamic performances of micro-grooved gas seals at high-speed conditions. Their results showed that the super ellipse grooves could produce the larger statefilm stiffness and dynamic stiffness coefficients. Lu et al. [13] investigated the influence of geometrical parameters of micro-grooves on the lubrication performance of fully micro-grooved gas-lubricated parallel slider bearings. They found that the average pressure of fully micro-grooved gas-lubricated parallel slider bearings is controlled by the geometrical parameters of micro-grooves.
The influence of grooving parameters and microgroove cross-section shape on the lubrication performance of partially micro-grooved oil-lubricated parallel slider bearings has been systematically investigated [8]. However, the influence of grooving parameters and micro-groove cross-section shape on the lubrication performance of partially micro-grooved gas-lubricated parallel slider bearings is not investigated. The present study is intended to study the influence of grooving parameters and micro-groove cross-section shape on the lubrication performance of partially micro-grooved gas-lubricated parallel slider bearings. Firstly, the model of partially micro-grooved gaslubricated parallel slider bearings is established. Secondly, the multi-grid finite element method is adopted to solve the dimensionless compressible Reynolds equation for the steady-state form. Finally, the influences of the orientation angle of the micro-grooves, the depth of the micro-grooves, the width of the micro-grooves, the area density of the micro-grooves, and micro-grooved fraction on average pressure are investigated for different micro-groove crosssection shapes. Fig. 1 presents the model of partially microgrooved gas-lubricated parallel slider bearings. x axel is parallel to the sliding direction of the lower slider and y axel is perpendicular to the sliding direction of the lower slider. The lower slider and upper slider are separated by the gas and the speed of the lower slider is U. The gas between the lower slider and upper slider is a Newtonian fluid and the gas flow between the lower slider and upper slider is isothermal and laminar. The upper slider is fixed and the micro-grooves are evenly distributed in the upper slider. The minimum gas film thickness of partially microgrooved gas-lubricated parallel slider bearings is c. The length of the micro-grooved zone is lm and the depth of each micro-groove is hg. Fig. 1 Model of partially micro-grooved gas-lubricated parallel slider bearings [8,14] Fig. 2 presents the model of the partially microgrooved slider. O is the origin of the coordinate system. The length of the partially micro-grooved slider is l. The micro-groove orientation angle is θ and the micro-groove spacing is sg. The width of each micro-groove is wg. In Fig. 2, b, the cross-section shape of the micro-groove along the A-A direction is rectangle. In Fig. 2, c, the cross-section shape of the micro-groove along the A-A direction is triangle. In Fig. 2, d, the cross-section shape of the microgroove along the A-A direction is parabola.

The model
The micro-groove area density sp is defined as: The micro-grooved fraction fm is defined as: . m m The compressible Reynolds equation for the steady-state form is described as [13]: where: h is the thickness of the gas film; p is the pressure; μ is the viscosity.
The dimensionless variables are defined as: where: w0 is the reference value; pa is the ambient pressure. The dimensionless compressible Reynolds equation for the steady-state form is given by: where: Fig. 2 Geometric model of the partially micro-grooved slider [8]: a) distribution of the micro-grooves; b) cross-section shape of the rectangular groove; c) cross-section shape of the triangular groove; d) cross-section shape of the parabolic groove The dimensionless thickness H is given by: case 1: 0 90      for the rectangular grooves in Fig. 2 for the triangular grooves in Fig. 2, c: Xsin for the parabolic grooves in Fig. 2 case 2: θ = 90°for the rectangular grooves in Fig. 2 for the parabolic grooves in Fig. 2, d: The multi-grid finite element method [14] is adopted to solve Eq. (5) and the dimensionless pressure P is obtained. The dimensionless average pressure Pav is calculated by:

Results and discussions
When the effect of grooving parameters and micro-groove cross-section shape on the hydrodynamic lubrication performance of partially micro-grooved gaslubricated parallel slider bearings is investigated, some calculation parameters are constant. The constant are as follows: μ = 1.8 × 10 -5 Pa·s; l = 2.5 mm; w0 = 0.05 mm; c = 3 × 10 -4 mm; pa = 0.101325 MPa; U = 4 m/s. In this study, the program is executed by using Matlab software.
The dimensionless pressure distributions for the parabolic grooves, rectangular grooves, and triangular grooves are shown in Fig. 3. It is observed that the pressure behaviors of the three micro-groove cross-section shapes are very similar and the maximum pressures can be found at the right side of the micro-grooved zone. It is also observed that the maximum pressure generated by the rectan-gular grooves is obviously greater than those generated by the parabolic grooves and triangular grooves.  Fig. 4 shows the influence of the orientation angle of the micro-grooves on average pressure for different micro-groove cross-section shapes. It is noted that the microgrooves parallel to the sliding direction generate the maximum average pressure. The result is in accordance with that obtained by Ji et al. [8]. It is also noted that the maximum average pressure generated by the rectangular grooves is greater than those generated by the parabolic grooves and the triangular grooves. The result differs from that obtained by Ji et al. [8]. This is due to the fact that Ji et al. [8] employ an incompressible lubricant and the present study employs a compressible lubricant.  5 presents the influence of the depth of the micro-grooves on average pressure for different microgroove cross-section shapes. With increasing the depth of the micro-grooves, the average pressure first increases, then reaches a maximum value, and finally decreases. It shows that there is an optimum micro-groove depth to maximize the average pressure. Furthermore, the optimum micro-groove depth is dependent on the micro-groove cross-section shape. These results are in accordance with those obtained by Ji et al. [8].  6 shows the influence of the width of the micro-grooves on average pressure for different micro-groove cross-section shapes. The results indicate that there is an optimum micro-groove width to maximize the average pressure. The results also indicate that the optimum microgroove width is dependent on the micro-groove crosssection shape. The result differs from that obtained by Ji et al. [8]. For this reason, the area density of the micro-grooves should be chosen as large as possible for maximizing the average pressure. Furthermore, the maximum average pressure generated by the rectangular grooves is greater than those generated by the parabolic grooves and the triangular grooves. These results are in accordance with those obtained by Ji et al. [8]. Fig. 7 Influence of the area density of the micro-grooves on average pressure for different micro-groove cross-section shapes (θ = 0°; Hg = 4; Wg = 4; fm = = 0.6) Fig. 8 presents the influence of micro-grooved fraction on average pressure for different micro-groove cross-section shapes. With the increase of micro-grooved fraction, the average pressure first increases, then reaches a maximum value, and finally decreases. It shows that there is an optimum micro-grooved fraction to maximize the average pressure. Furthermore, the optimum microgrooved fraction is dependent on the micro-groove crosssection shape. These results are in accordance with those obtained by Ji et al. [8].

Conclusions
A hydrodynamic lubrication model of partially micro-grooved gas-lubricated parallel slider bearings is established to investigate the influence of grooving parameters and micro-groove cross-section shapes on the lubrication performance of partially micro-grooved gas-lubricated parallel slider bearings. The grooving parameters include orientation angle of the micro-grooves, depth of the microgrooves, width of the micro-grooves, area density of the micro-grooves, and micro-grooved fraction. The microgroove cross-section shape includes parabola, rectangle, and triangle. The average pressure is chosen as the valuation criteria, and the influence of grooving parameters on average pressure is analyzed for different micro-groove cross-section shapes. The following conclusions are obtained: 1. The micro-grooves parallel to the sliding direction could generate the maximum average pressure. However, the influence of micro-groove cross-section shapes on the maximum average pressure is different. The maximum average pressure generated by the rectangular grooves is greater than those generated by the parabolic grooves and the triangular grooves.
2. The optimum micro-groove depth for maximizing the average pressure, the optimum micro-groove width for maximizing the average pressure, and the optimum micro-grooved fraction for maximizing the average pressure are dependent on the micro-groove cross-section shape.
3. The average pressure increases with the increase of micro-groove area density. Therefore, the area density of the micro-grooves should be chosen as large as possible for obtaining the maximum average pressure. However, the influence of micro-groove cross-section shapes on the maximum average pressure is different. The maximum average pressure generated by the rectangular grooves is greater than those generated by the parabolic grooves and the triangular grooves.