# Interactive Effects of Rarefaction and Surface Roughness on Aerodynamic Lubrication of Microbearings

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{n}is utilized to describe the rarefied gas flow, which is defined as the ratio of molecular mean free path λ

_{0}to the characteristic length of gas film thickness. Burgdorfer [8], Hsia et al. [9], and Mitsuya [10], based on the slip velocity boundary condition, derived the classical first order, second order, and 1.5 order slip models in the slider/disk interface for HDDs to take into account the effect of gas rarefaction. Fukui and Kaneko [11,12] developed a Poiseuille flow rate database for a wide Knudsen number range to modify the compressible Reynolds type equation including thermal creep flow and accommodation coefficient from linearized Boltzmann equation. In order to account for the effect of surface roughness, Christensen and Tønder [13,14] presented a stochastic model of hydrodynamic lubrication for finite width journal bearing in which they considered the lubricant film thickness as a stochastic process. The operating characteristics of bearing was theoretically analyzed with roughness pattern, nominal geometric features, and statistical properties by surface averaging techniques. Via linear transformation of random matrices, the Gaussian or non-Gaussian distribution of surface heights were generated by Patir [15] using the prescribed autocorrelation functions and frequency density functions. Patir and Cheng [16] further derived the average Reynolds equation suitable for various roughness structure and discussed the effect of roughness on mean hydrodynamic pressure, mean viscous friction, and mean bearing inflow in finite slider bearings. The average flow model of Patir and Cheng was extended by Tripp [17], in which the statistical expectation of flow factors were calculated with a perturbation expansion of the film pressure. The results showed that the flow factors are closely correlated with roughness parameters. White et al. [18] introduced the transverse sinusoidal roughness pattern to study the influence of surface roughness on steady-state pressure profiles of wedge bearing by variable grid implicit finite difference method and found that the load capacity could be decreased to a limiting value at higher bearing numbers. For the applications of perturbation technique and mapping function, Li et al. [19] studied the effects of roughness orientations and rarefaction on static performance of magnetic recording systems. The results demonstrated that the flow factors changed with the orientation angle and Peklenik number, and the effect of moving surface on surface characteristics is more significant than that of the stationary surface. Turaga et al. [20] proposed the stochastic finite method to solve Reynolds equation and obtained the static and dynamic performance of hydrodynamic journal bearings with the longitudinal, transverse and isotropic roughness pattern. Naduvinamani et al. [21] established the surface roughness by a stochastic random variable with nonzero mean, variance and skewness, and the average Reynolds equations were adopted to analyze the performance of porous step-slider bearings with Stokes couple stress fluid. Zhang et al. [22,23] presented the modified Reynolds equation by including fractal roughness effect and velocity slip boundary condition and concluded that the flow behaviors in gas-lubricated journal microbearings was appreciably affected by Knudsen number, bearing number and fractal dimension. The coupling effects of non-Newtonian micropolar fluids and roughness on the dynamic characteristics of plane slider bearings were investigated by Lin et al. [24] on the basis of the microcontinuum theory and Christensen stochastic roughness model. They indicated that the transverse roughness serves to somewhat increase bearing dynamic property, whereas the longitudinal roughness would tend to decrease the dynamic coefficients. Jao et al. [25] examined the influences of surface roughness and anisotropic slips on hydrodynamic lubrication of journal bearings. They described the lubricant flow in rough bearing surface by the product of flow factors and flow in nominal film thickness, and also identified that boundary slip reduced the effect of surface roughness. Kalavathi et al. [26] reported a generalized Reynolds equation for finite porous slider bearing with both longitudinal and transverse roughness. The authors showed the surface roughness enhanced the pressure distribution and load carrying capacity while the permeability parameter diluted the load. Quiñonez [27] utilized the linear superposition of perturbation method and Flourier transformation to provide a solution for the flow characteristics of wide exponential land slider bearings with rough surfaces. The results were in good agreement with the cases of sinusoidal and single Gaussian dent. The linear perturbation method was used by Wang et al. [28] to solve the unsteady Reynolds equation for rough aerostatic journal bearings during the iterative process, and the dynamic performance was obtained by taking into account the interactions of journal rotation and surface waviness. However, likely due to the nonlinear and complexity of dynamic flow behavior, previous papers were mainly focusing on steady-state characteristics in rough journal bearings, and the dynamic characteristics of hydrodynamic gas-lubricated microbearings were seldom reported in the research literature. Moreover, the statistical parameters (such as root mean square of asperity heights, surface slope, curvature, skewness, and kurtosis), which are conventionally applied to characterize surface roughness, vary with the sampling length and resolution of measuring equipment. A scale-invariant surface characterization should be considered. Hence, the analytical studies of surface roughness effect on dynamic characteristics of gas slider bearings with rarefaction coefficients in microfluidic engineering devices is motivated.

## 2. Characterization of Fractal Rough Surface

_{r}(x,y) is the height of rough surface, x and y are the measure distances in the vertical and horizontal position, respectively. L is the sampling length of the profile of surface. D

_{f}is the fractal dimension, varying from 2 to 3 in three-dimensional surface topography. G is the scaling constant that relates to the roughness profile. γ is the scaling parameter (γ > 1), which determines the spectral density, γ is equal to 1.5 for a Gaussian and isotropic surface. M is the number of overlapped ridges on the surface, m and n are the frequency index, n

_{max}= int[log(L/L

_{s})/logγ], φ

_{m}

_{,n}is the random phase, L

_{s}is the cut-off length that depends on cut-off wavelength of resolution in measuring machines.

_{f}, comparisons of the distributions of asperity heights for different D

_{f}are illustrated in Figure 1. It is seen that the heights of rough surfaces increase as the fractal dimension decreases.

## 3. Numerical Model and Solution Method

_{a}, H = h/c

_{b}, φ = x/R, λ = z/R are the dimensionless gas film pressure, the dimensionless gas film thickness, and the coordinates in the circumferential and axial direction, p

_{a}is the ambient pressure, c

_{b}is the radius clearance, R is the radius of journal, p is the local gas pressure, h is the clearance spacing of ultra-thin gas film, ε is the eccentricity ratio and ε = e/c

_{b}, e is the eccentricity. Λ = 6μωR

^{2}/(p

_{a}c

_{b}

^{2}) is the gas bearing number, μ is the viscosity coefficient, ω is the rotating angular velocity of journal, T is the dimensionless time.

_{0}and random roughness h

_{r}measured from the nominal smooth height.

^{2}= S

^{2}= Π, Equation (2) can be converted to the ellipse-type partial differential equation $-\nabla \cdot (c\nabla u)+au=f$ in the following form:

_{0},θ

_{0}), and its dynamic disturbance about (ε

_{0},θ

_{0}) are denoted as E and Θ. The positions of eccentricity ratio ε and attitude angle θ of journal at a random position are represented by the static and dynamic components as follows:

_{0}and Θ

_{0}are the small perturbation amplitude of journal eccentricity ratio and attitude angle in the complex field. The dimensionless perturbation frequency Ω, which is defined as the ratio of journal disturbance frequency ν to rotating angular velocity ω of journal, $i=\sqrt{-1}$.

_{0}is the static gas-film pressure and H

_{0}is the static gas-film thickness. Q

_{gd}, H

_{gd}are the dynamic gas film pressure and gas film thickness, respectively. ${\tilde{P}}_{0}$ and ${\tilde{H}}_{0}$ are the perturbation magnitudes in terms of complex numbers for dynamic gas film pressure and gas film thickness.

_{0}and Θ

_{0}, and combining with some mathematical transformation, the resulting dynamic PDE equations are obtained for rough surface microbearings concerning the variables P

_{E}, P

_{θ}, H

_{E}, and H

_{θ}.

_{ij}and dynamic damping coefficients D

_{ij}of gas journal microbearing for fractal rough surface can be calculated by the following formula:

## 4. Results and Discussion

_{a}= 1.033 × 10

^{5}N/m

^{2}, and the aspect ratio is B/D = 0.1.

#### 4.1. Steady-State Film Pressure

^{5}rpm. It can also be seen that the pressure randomly fluctuates when the effect of surface roughness is considered.

#### 4.2. Load-Carry Capacity and Friction Coefficient

_{f}= 2.2, 2.25, 2.3, 2.35, 2.4). As the eccentricity ratio increases, which denotes the film thickness is thinner, the load capacity increases monotonically. The higher self-affine fractal dimensions yield the smaller roughness heights distribution on rough surfaces, and it is noted that the load carrying capacity is increased gradually when compared with the smooth bearing. The reason is that the increasing values of roughness heights reduces the sidewise leakage of airflow and the flow is restricted by the surface asperities. However, as the fractal dimension decreases further for D

_{f}= 2.2, the load carrying capacity tends to decline under this condition. This is because the minimum air film clearance between rotor and bearing may become too small so that the surface roughness effect which increases the load-carrying capacity is weaker than the gaseous rarefaction effects which reduces the dimensionless load capacity, thus causing a decrease in the bearing load capacity. The attitude angle θ is found to decrease with the growing eccentricity ratio. The decrease in attitude angle is more accentuated for a rougher surface as compared to a nominally smooth surface.

^{−11}is depicted in Figure 8. It can be seen that friction coefficients monotonically increase as ε increases. Although the contact area between the rarefied gas flow and the surface asperities is larger when the homogeneous surface roughness becomes more and more obvious, the static friction coefficients show a slightly more gradual increase with fractal dimension. As the fractal dimension D

_{f}is equal to 2.2, the friction coefficient is lower than that of smooth surface for the same reason that the gaseous rarefaction effect is more pronounced and thus friction coefficient drops.

^{−10}. It is found that increasing the values of the bearing number from Λ = 3 up to Λ = 100 increases the carrying capacity and reduces the corresponding attitude angle of gas journal microbearing. The surface roughness effect in aerodynamic lubrication enhances the load capacity as compared with the smooth-bearing case, especially for the bearing operating at high bearing number. Figure 11 shows the comparison of the static friction coefficients with bearing number for different fractal dimensions. The friction coefficients exhibit a near-linear increasing trend with increasing Λ, while the increase extent in static friction coefficient is even higher at smaller D

_{f}values. Consequently, the strengthened gas-lubricated hydrodynamic effect and the bearing surface with roughness undulations have a significant influence on the skin friction at the interface.

#### 4.3. Dynamic Stiffness and Damping Coefficients

_{xx}and K

_{yy}increases as the perturbation frequency increases and the K

_{yy}is much greater than K

_{xx}because of the rarefied gas lubricating film supports the weight of journal in the vertical direction. The cross-couple stiffness K

_{xy}increases at first, then decreases slightly with the growth of Ω, while K

_{yx}decreases quickly at lower perturbation frequencies. Furthermore, the enhanced dynamic stiffness coefficients are seen for the rough bearing surface as compared to that of smooth bearing case. When Ω > 2, all the dynamic stiffness coefficients of micro gas-lubricated journal bearing at the fractal dimension D

_{f}= 2.3 are obviously greater than other rough surfaces. This is mainly due to the fact that the larger asperity heights lead to an increased Poiseuille flow component along the sliding direction and the side flow suffers the constriction resistance caused by homogeneous surface roughness. The influence of perturbation frequency on dynamic damping coefficients for different fractal roughness parameters can be observed from Figure 13. The principal damping coefficient D

_{xx}first increases with increasing dimensionless perturbation frequency, reaches a maximum, then starts to decline slowly. The absolute values of the cross-coupling terms of damping coefficients D

_{xy}and D

_{yx}decrease quickly at low Ω, then approaches to zero. The result show that the damping coefficients D

_{xx}, D

_{yx}increase with the increase in the isotropic and homogeneous roughness heights of gas slider bearing surface, and the D

_{xy}and D

_{yy}are first decreases and then dramatically increases as the fractal dimension D

_{f}decreases, whereas the difference in damping coefficients of rough and smooth cases appear to converge at higher values of Ω.

^{−11}. The higher ε corresponds to the thinner gas film thickness, which results in the increase of Knudsen number K

_{n}. With the growth of the fractal surface roughness, the stiffness coefficients first increase gradually, then decrease significantly at the same ε. As illustrated in Figure 15, the damping coefficients increase marginally as the eccentricity ratio increases and the effect of fractal dimension on damping coefficient D

_{xy}is negligible at lower eccentricity ratios. The damping coefficients become more sensitive to surface roughness at higher eccentricity ratios about ε > 0.7. It can be seen that increasing the random roughness heights increase the effect of gas rarefaction in small spacing.

^{−10}. Increment of the bearing number means the larger operating conditions. The principal stiffness K

_{yy}is near proportionally dependent on Λ and dynamic stiffness coefficients K

_{xx}, K

_{xy}and K

_{yx}increase gradually with increasing bearing number. It is also observed that the increase in the dynamic stiffness coefficients is more accentuated for the fractal roughness surface as compared to the smooth surface bearing with the enhanced aerodynamic effect in gas journal bearings. The damping coefficients exhibit similar trends to the fractal dimension, namely the principal damping D

_{xx}, D

_{yy}and cross-couple damping D

_{xy}for rough bearing surface become larger than the ones in the smooth bearing, whereas the roughness effect is rather marginal in the case of the cross-couple damping D

_{yx}in Figure 17c. The damping coefficients D

_{xx}, D

_{yy}and D

_{xy}increase quickly at first and then decreased, while the damping coefficient D

_{yx}decreases with increasing bearing number. Therefore, the dynamic stiffness and damping characteristics of gas microbearings with isotropic and homogeneous roughness show relatively high values since the aerodynamic effect of the bearing is enhanced.

## 5. Conclusions

_{yx}at large bearing number values.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Simulation of a three-dimensional fractal surface topography for different fractal dimensions. (

**a**) D

_{f}= 2.2, and G = 1 × 10

^{−10}m; (

**b**) D

_{f}= 2.3, and G = 1 × 10

^{−10}m; (

**c**) D

_{f}= 2.4, and G = 1 × 10

^{−10}m.

**Figure 4.**Comparison of dimensionless gas film pressure with Zhang et al. [20].

**Figure 5.**Pressure distributions and contour plots of the gas-lubricated microbearing for different bearing numbers and eccentricity ratios: (

**a**) ε = 0.6, Λ = 20, D

_{f}= 2.3, and G = 1 × 10

^{−10}m, (

**b**) ε = 0.6, Λ = 60, D

_{f}= 2.3, and G = 1 × 10

^{−10}m, (

**c**) ε = 0.3, Λ = 20, D

_{f}= 2.3, and G = 1 × 10

^{−10}m, and (

**d**) ε = 0.6, Λ = 20, smooth.

**Figure 6.**Non-dimensional load capacity versus eccentricity ratio for different fractal dimensions at Λ = 20, G = 1 × 10

^{−11}.

**Figure 7.**Attitude angle versus eccentricity ratio for different fractal dimensions at Λ = 20, G = 1 × 10

^{−11}.

**Figure 8.**Static friction coefficient versus eccentricity ratio for different fractal dimensions at Λ = 20, G = 1 × 10

^{−11}.

**Figure 9.**Non-dimensional load capacity versus bearing number for different fractal dimensions at ε = 0.6, G = 1 × 10

^{−10}.

**Figure 10.**Attitude angles versus bearing number for different fractal dimensions at ε = 0.6, G = 1 × 10

^{−10}.

**Figure 11.**Static friction coefficient versus bearing number for different fractal dimensions at ε = 0.6, G = 1 × 10

^{−10}.

**Figure 12.**Effect of perturbation frequency on dynamic stiffness coefficients with different fractal dimensions. (

**a**) K

_{xx}vs. Ω; (

**b**) K

_{xy}vs. Ω; (

**c**) K

_{yx}vs. Ω; (

**d**) K

_{yy}vs. Ω.

**Figure 13.**Effect of perturbation frequency on dynamic damping coefficients with different fractal dimensions. (

**a**) D

_{xx}vs. Ω; (

**b**) D

_{xy}vs. Ω; (

**c**) D

_{yx}vs. Ω; (

**d**) D

_{yy}vs. Ω.

**Figure 14.**Effect of eccentricity ratio on dynamic stiffness coefficients with different fractal dimensions. (

**a**) K

_{xx}vs. ε; (

**b**) K

_{xy}vs. ε; (

**c**) K

_{yx}vs. ε; (

**d**) K

_{yy}vs. ε.

**Figure 15.**Effect of eccentricity ratio on dynamic damping coefficients with different fractal dimensions. (

**a**) D

_{xx}vs. ε; (

**b**) D

_{xy}vs. ε; (

**c**) D

_{yx}vs. ε; (

**d**) D

_{yy}vs. ε.

**Figure 16.**Effect of bearing number on dynamic stiffness coefficients with different fractal dimensions. (

**a**) K

_{xx}vs. Λ; (

**b**) K

_{xy}vs. Λ; (

**c**) K

_{yx}vs. Λ; (

**d**) K

_{yy}vs. Λ.

**Figure 17.**Effect of bearing number on dynamic damping coefficients with different fractal dimensions. (

**a**) D

_{xx}vs. Λ; (

**b**) D

_{xy}vs. Λ; (

**c**) D

_{yx}vs. Λ; (

**d**) D

_{yy}vs. Λ.

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**MDPI and ACS Style**

Wu, Y.; Yang, L.; Xu, T.; Xu, H. Interactive Effects of Rarefaction and Surface Roughness on Aerodynamic Lubrication of Microbearings. *Micromachines* **2019**, *10*, 155.
https://doi.org/10.3390/mi10020155

**AMA Style**

Wu Y, Yang L, Xu T, Xu H. Interactive Effects of Rarefaction and Surface Roughness on Aerodynamic Lubrication of Microbearings. *Micromachines*. 2019; 10(2):155.
https://doi.org/10.3390/mi10020155

**Chicago/Turabian Style**

Wu, Yao, Lihua Yang, Tengfei Xu, and Haoliang Xu. 2019. "Interactive Effects of Rarefaction and Surface Roughness on Aerodynamic Lubrication of Microbearings" *Micromachines* 10, no. 2: 155.
https://doi.org/10.3390/mi10020155