The microfluidic devices are widely used in many applications such as ultra-precision machine tool spindles, inertial navigation system (INS), medical devices, and hard disk drives (HDDs) in micro-electro-mechanical systems (MEMS). Since the development of microsystems engineering technology, gas journal microbearing has been generally preferred over the electromagnetic bearing and rolling bearing owing to its advantages of simple structure, high rotary accuracy, high running speed, low friction power loss, and wide working temperature range [

1,

2,

3]. The real surface of a mechanical part produced by various machining and finish operations is composed of a large number of distributed peaks and valleys. The lubricating film thickness between the surfaces of shaft and bearing has continually decreased, which results in the increase of the roughness heights that are of the same order of magnitude as the minimum clearance gap. Thus, the assumptions that the gas flow is typically treated as a continuum flow with no slip boundary condition and the bearing surface roughness is considered negligible in classical fluid mechanics have to be revised at microscales [

4,

5,

6,

7]. The interaction between rarefaction and surface roughness in microbearing can influence the reliability and operational efficiency of micro rotating machinery obviously, so the gas journal microbearing performance should be comprehensively analyzed.

In order to accurately predict the effects of surface roughness and rarefaction on the bearing characteristics, many researchers devoted numerous research efforts to study the complicated flow behaviors at very small clearances over the last few decades. In consideration of rarefaction effects in ultra-thin gas film lubrication, the Knudsen number

K_{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.

In this paper, the Weierstrass-Mandelbrot (W-M) fractal function is used to characterize the homogeneous surface roughness, and the Boltzmann slip correction model is applied to represent the rheological behavior of compressible rarefied gas film. The generalized Reynolds-type equation considering gas rarefaction, as well as roughness effect, is mathematically derived and solved by the partial derivative method and relaxation iteration algorithm. Bearing performances (including the load-carrying capacity, friction coefficient and corresponding attitude angle, dynamic stiffness, and damping coefficients) are presented and discussed in comparison with smooth surface bearings. The work is expected to elucidate the performance characteristics of gas microbearings with Poiseuille flow and random asperities, which is conducive to understand the fluid mechanisms of very low clearance gas films for microfluidics devices.