Hydrodynamic Lubrication of Micro-Grooved Gas Parallel Slider Bearings with Parabolic Grooves

Yanjun LU***, Fuxi LIU***, Yongfang ZHANG**** *Xi’an University of Technology, Key Laboratory of Manufacturing Equipment of Shananxi Province, Xi’an 710048, China, E-mail: yanjunlu@xaut.edu.cn **Huazhong University of Science and Technology, State Key Laboratory of Digital Manufacturing Equipment and Technology, Wuhan 430074, China, E-mail: yanjunlu@xaut.edu.cn ***Xi’an University of Technology, School of Mechanical and Precision Instrument Engineering, Xi’an 710048, China, E-mail: liufx28@163.com (corresponding author) ****Xi’an University of Technology, School of Printing, Packaging Engineering and Digital Media Technology, Xi’an 710048, China, E-mail: zhangyf@xaut.edu.cn (corresponding author)


Introduction
Opening micro-grooves in the surface of a mechanical component is thought to be an effective method for enhancing the performance of a mechanical component, such as a cylinder liner [1], piston ring [2], thrust bearing [3], journal bearing [4], or slider bearing [5].In recent years, several theoretical and experimental studies concerning micro-grooved mechanical components have been conducted.Ali et al. [6] theoretically investigated the effect of micro-grooves on friction behaviour under elastohydrodynamic lubrication point contacts and established a good agreement with the experimental results.Yuan et al. [7] investigated the effect of groove orientation on sliding friction.The experimental results showed that grooves parallel or perpendicular to the sliding direction have an important influence on sliding friction.Experimentally, Zum Gahr et al. [8] examined the film thickness and friction coefficient of a micro-grooved oil-lubricated ceramic/steel friction pair.It was observed that grooves perpendicular to the sliding direction have the ability to generate a greater film thickness and lower friction coefficient than those parallel to the sliding direction.Luo et al. [9] used a unidirectional ball-on-disk sliding wear tester to study the wear resistance of micro-grooved surfaces, which demonstrated that the wear mass loss of micro-grooved surfaces generally increases as the groove spacing varies from 100 μm to 1000 μm.To improve the cutting performance, Obikawa et al. [10] applied a photolithography process to fabricate micro-grooves in the cutting tool surface.It was revealed that the presence of micro-grooves could reduce the friction force and friction coefficient.When the real topography of the cylinder liner was considered, Mezghani et al. [11] presented a friction model to investigate the friction performance of a micro-grooved piston ring-cylinder liner combination.They showed that the friction coefficient of the piston ring-cylinder liner combination could decrease by optimizing the geometric parameters of the microgrooves.Kango et al. [12] investigated the steady-state performance of micro-grooved oil-lubricated hydrodynamic journal bearings.Du et al. [13] investigated the load performance of micro-grooved aerostatic journal bearings.Opening micro-grooves in the bearing surface could increase the load-carrying capacity in both of the studies.To analyze the hydrodynamic lubrication property of micro-grooved oil-lubricated parallel slider bearings, Shi and Ni [14] established a two-dimensional computational fluid dynamics simulation model.It was found that both loadcarrying capacity and friction force increase as the sliding speed increased, and load-carrying capacity increases faster than friction force.
The effect of micro-grooves on hydrodynamic pressure has been systematically analyzed for microgrooved oil-lubricated parallel slider bearings [14,15].However, no investigations detailing the effect of microgrooves on hydrodynamic pressure seem to exist in the available literature for micro-grooved gas parallel slider bearings.Micro-grooved gas parallel slider bearings and micro-grooved oil-lubricated parallel slider bearings have different Reynolds equations.Therefore, investigating the hydrodynamic pressure of micro-grooved gas parallel slider bearings is necessary.In a previous investigation, Fu et al. [15] have employed parabolic grooves to evaluate the effects of minimum film thickness, groove width, depth, spacing, and orientation angle on the hydrodynamic pressure of micro-grooved oil-lubricated parallel slider bearings.In the present investigation, parabolic grooves are also employed to evaluate the hydrodynamic pressure of micro-grooved gas parallel slider bearings, and special attention is paid to the effects of sliding speed, groove width, depth, spacing, and orientation angle on average pressure.

Theory
Fig. 1 shows a schematic diagram of microgrooved gas parallel slider bearings; the lower slider is grooved and fixed, the upper slider is smooth and moves along the x direction, the speed of the upper slider is U, the minimum film thickness between the two sliders is c and each groove has a depth of g h .Fig. 2 shows the geometric model of a microgrooved slider, where x and y are the coordinates in a global Cartesian coordinate system, l is the slider length,  is the groove orientation angle, g w is the groove width, g s is the groove spacing, and g l is the distance between two adjacent grooves.
The upper slider and lower slider are separated by the gas and the bearing works under the steady-state condition.The lubricant is a Newtonian fluid and the gas flow is isothermal and laminar.Hence, the compressible Reynolds equation for micro-grooved gas parallel slider bearings is expressed by: 33 where h is the gas-film thickness, p is the gas-film pressure, and  is the fluid viscosity.The dimensionless variables are defined as follows: where 0 w is the referenced groove width and a p is the ambient pressure.
By submitting Eq. ( 2) into Eq.( 1), the dimensionless Reynolds equation is obtained by: where is the bearing number.The dimensionless gas-film thickness H is determined by the groove orientation angle  .In the case of , the dimensionless gas-film thickness H is given by [15]: , elsewhere while in the case of  90   , the dimensionless gas-film thickness H is given by [15]: where , fix is a function returning a value towards the nearest integer, is the dimensionless groove depth, The boundary conditions of Eq. ( 3) are expressed by: where 0 / w l L  is the dimensionless slider length.The multi-grid finite element method solves Eq. ( 3), obtaining the dimensionless pressure P. 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 grids to the fine grids and the restriction from the fine grids to the coarse grids, which is the finite Newton-Raphson iteration.In the present investigation, a multi-grid W-cycle and the finest grids with 1025 1025  nodes are adopted, where the number of the layers of the W-cycle is 4. The multi-grid finite element method is implemented using the MATLAB software.More detailed discussions regarding the multi-grid finite element method could be found in Liu et al. [4].The converging condition is expressed by: (8)

Results and discussions
Once investigating the average pressure of microgrooved gas parallel slider bearings, some calculation parameters are constant:  are increased by approximately 24.6% and 41.6%, respectively.
In order to better demonstrate the effect of orientation angle  on hydrodynamic pressure, Fig. 4 shows the dimensionless gas-film thickness distribution and dimensionless gas-film pressure distribution along the X direction at 25  Y .It is observed that the maximum pressure produced by each groove could be found near the right side of the groove along the X direction and is greatly affected by the orientation angle.The reason for this phenomenon could be explained by the fact that the right side of the groove is the end of convergence clearance and the orienta-tion angle obviously controls the resistance of fluid influx.The results correlate with those obtained by Fu et al. [15].Fig. 5 shows the effect of orientation angle  on dimensionless average pressure at various values of dimensionless groove depth With the increase of dimensionless groove depth, the dimensionless average pressure initially increases, reaches a maximum value, and then gradually decreases.This maximum value is dependent of the dimensionless groove width.The results show that the depth of the grooves should be limited in order to obtain the maximum dimensionless average pressure.The pneumatic hammer vibration [13] of the gasfilm could be efficiently avoided by limiting the depth of For different values of dimensionless groove spacing g S , Fig. 7 shows the effect of dimensionless groove width g W on dimensionless average pressure.The results indicate that there is an optimum groove width to maximize the average pressure for any given groove spacing.This optimum value is dependent of the groove spacing.The results differ from those obtained by Fu et al. [15], which may be explained by the fact that the groove width has a different effect on dimensionless average pressure for different lubricants.The present study uses a compressible lubricant, but Fu et al. [15]  results and those obtained by Fu et al. [15] are not in agreement.the average pressure initially increases, reaches a maximum value, and then gradually decreases.However, the average pressure increases with increasing sliding speed when 6  g H and 8.The imply that the optimum sliding speed for maximizing the average pressure is dependent of the groove depth.Furthermore, there is an optimum value of groove depth to maximize the average pressure for any given sliding speed, and the value is dependent of the sliding speed.Therefore, an appropriate groove depth should be chosen according to the practical sliding speed in order to obtain the maximum average pressure.

Conclusions
A numerical study of micro-grooved gas parallel slider bearings with parabolic grooves is conducted.The following conclusions are made based on an analysis of the results: 1.The groove orientation angle controls the average pressure.Parabolic grooves should be perpendicular to the sliding direction in order to obtain the maximum average pressure.
2. The optimum groove depth for maximizing the average pressure is dependent of the groove width and the optimum groove width for maximizing the average pressure is dependent of the groove spacing.However, the optimum groove spacing for maximizing the average pressure is independent of the groove depth.
3. The sliding speed has a different effect on average pressure for different groove depths.Therefore, an appropriate groove depth should be chosen according to the practical sliding speed to obtain the maximum average pressure.
HYDRODYNAMIC LUBRICATION OF MICRO-GROOVED GAS PARALLEL SLIDER BEARINGS WITH PARABOLIC GROOVES S u m m a r y The hydrodynamic lubrication performance of micro-grooved gas parallel slider bearings with parabolic grooves is investigated in this paper.By using the multigrid finite element method, the pressure distribution between a micro-grooved slider and a smooth slider is obtained.The geometric parameters of the parabolic grooves are optimized to maximize the average pressure under a given sliding speed.The numerical results show that geometric parameters such as groove depth, width, spacing, and orientation angle have an important influence on hydrodynamic pressure.Furthermore, the effect of sliding speed on hydrodynamic pressure is investigated under a given set of groove depths.It is observed that there is an optimum value of sliding speed to maximize the average pressure for any given groove depth and that the optimum value is dependent of the groove depth.The results of this study indicate that the average pressure could be improved by employing the optimized groove depth according to the practical sliding speed.Keywords: gas parallel slider bearings, micro-grooves, hydrodynamic lubrication, multi-grid finite element method.

Fig. 1 Fig. 2
Fig.1Schematic diagram of micro-grooved gas parallel slider bearings[15] numbers of nodes in the x and y direction, respectively,

..
the dimensionless gas-film pressure distributions over the slider surface in the case of It is noted that the hydrodynamic pressure distribution is significantly affected by the geometric parameters of the grooves, closely correlating with the orientation angle.For both cases, the hydrodynamic pressure behaviours are similar and the optimum hydrodynamic pressure values are reached at the end of convergence clearance.However, the maximum pressure for  Because there is no hydrodynamic lubrication effect for smooth gas parallel slider bearings, the values of the dimensionless pressure and dimensionless average pressure of smooth gas parallel slider bearings are both equal to 1.When compared with smooth gas parallel slider bearings, the maximum pressures in the cases of

gH.Fig. 6
Fig. 6 presents the effect of dimensionless groove depth g H on dimensionless average pressure for different values of dimensionless groove width g W .With the increase of dimensionless groove depth, the dimensionless average pressure initially increases, reaches a maximum value, and then gradually decreases.This maximum value is dependent of the dimensionless groove width.The results show that the depth of the grooves should be limited in order to obtain the maximum dimensionless average pressure.The pneumatic hammer vibration[13] of the gasfilm could be efficiently avoided by limiting the depth of The results coincide with those obtained by Fu et al.[15].

Fig. 8
Fig. 4 (a) Dimensionless gas-film thickness distribution along the X direction at 25  Y ; (b) dimensionless gas-film pressure distribution along the X direction at 25  Y (