## 1 Introduction  
Gas journal microbearing is of great importance in various microelectromechanical systems (MEMS) devices such as ultrasonic motors, micro-gas turbines, microaccelerometers, inertial navigation systems and magnetic hard disk drives (HDD) [1; 2; 3]. Due to the continuous development of rotating machinery for high precision, miniaturization, low energy consumption and long life span. Aerodynamic microbearings are widely used owing to their advantages of simple structure, low friction resistance, no contamination and minimal temperature limitation in comparison with the oil-lubricated bearings and electromagnetic bearings [4; 5; 6; 7]. In these microfluidic applications, the lubricating films between the journals and bearing pads usually become very small and are found to be even smaller than the molecular mean free path [8; 9], and the continuum flow of Reynolds equation is no longer valid for describing the rarefied gas flow characteristics. Thus the effect of gas rarefaction must be taken into consideration. The rarefied gas flow can be classified into three categories [10; 11] on the basis of the Knudsen number \(K_{n}\), which is defined as the ratio of the gas molecular mean free path \(\lambda_{0}\) to the characteristic length scale \(L\), i.e., the slip region (\(0.001<K_{n}<0.1\)), the transition region (\(0.1<K_{n}<10\)), and the free molecular region (\(K_{n}>10\)). In the slip region, the velocity slip and temperature jump occur at the solid wall [12; 13; 14]. In the transition region, both the mutual collisions between the molecules and between gas molecule and boundary wall surface should be considered. The variation of microscopic particle velocity which results from collision can be ignored in the free molecular region and the Boltzmann equation without collision term and the Maxwellian distribution can be employed to study the gas flow in this region. In most of the theoretical investigation of ultra-thin gas film lubrication, the Poiseuille flow rate coefficient, which is an important flow correction term, is incorporated into the compressible Reynolds equation to account for gaseous rarefaction effects. [PERSON] [15] firstly introduced the first-order slip velocity boundary condition into the Reynolds equation for small Knudsen numbers in 1959. Other noted researchers, including [PERSON] [16], [PERSON] [17], [PERSON], [PERSON] [18], and [PERSON] [19], derived the modified Reynolds equations by incorporating their different Poiseuille flow factors for arbitrary Knudsen numbers. Subsequently, a number of approaches, based on the kinetic theory, to the practical analysis of rarefaction effects on micro fluidic systems have been reported that have created discussion. [PERSON] et al. [20] proposed a new modified Reynolds equation which includes three adjustable coefficients in the velocity boundary conditions, and the results showed that the model produces good approximation to the Boltzmann model for a wide range of Knudsen numbers. [PERSON] et al. [21] extended the databases of Poiseuille flow rate and Couette flow rate by introducing new terms into system equations at track recording head/disk interface in hard disk drives by using the Bhatnagar-Gross-Krook kinetic equation. In 2007, [PERSON] et al. [22] proposed a new first-order slip model which includes an additional term by utilizing the Chapman-Enskog solution of Boltzmann equation. In recent years, the problems connected with rarefaction effects for gas journal microbearings had also been investigated taking into account surface roughness, bearing pad deformation, turbulent flow, bearing thermal expansions and shaft misalignment effects. [PERSON] [23] developed an analytical method that employs the perturbation method and multiple-scale analysis to discuss the influences of surface roughness and gas rarefaction on low clearance gas bearing with high wave number and high bearing number. [PERSON] et al.