A new extended Reynolds equation for gas bearing lubrication based on the method of moments

Microfluidics and Nanofluidics, Jan 2016

An extended Reynolds equation based on the regularised 13 moment equations and lubrication theory is derived for gas slider bearings operating in the early to upper transition regime. The new formulation performs well beyond the capability of the conventional Reynolds equation modified with simple velocity-slip models. Both load capacity and pressure distribution can be reliably predicted by the extended Reynolds equation and are in good agreement with available direct simulation Monte Carlo data. In addition, the equations are able to provide both velocity and stress information, which is not conveniently recovered from available kinetic models. Tests indicate that the equations can provide accurate data for Knudsen numbers up to unity but, as expected, begin to deteriorate afterwards.

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A new extended Reynolds equation for gas bearing lubrication based on the method of moments

Microfluid Nanofluid A new extended Reynolds equation for gas bearing lubrication based on the method of moments Xiao‑Jun Gu 0 1 Haijun Zhang 0 1 David R. Emerson 0 1 0 College of Mechanical-Electrical Engineering, Jiaxing University , Jiaxing 314001, Zhejiang , China 1 Scientific Computing Department, STFC Daresbury Laboratory , Warrington WA4 4AD , UK An extended Reynolds equation based on the regularised 13 moment equations and lubrication theory is derived for gas slider bearings operating in the early to upper transition regime. The new formulation performs well beyond the capability of the conventional Reynolds equation modified with simple velocity-slip models. Both load capacity and pressure distribution can be reliably predicted by the extended Reynolds equation and are in good agreement with available direct simulation Monte Carlo data. In addition, the equations are able to provide both velocity and stress information, which is not conveniently recovered from available kinetic models. Tests indicate that the equations can provide accurate data for Knudsen numbers up to unity but, as expected, begin to deteriorate afterwards. Reynolds equation; Lubrication theory; Slider bearing; Moment method; Knudsen number; Non-equilibrium 1 Introduction In modern computer hard disk drives, the gas between the read-write head and the surface of a spinning disk forms an air slider bearing that supports the head floating above the disk. Traditionally, the classical Reynolds lubrication equation has been used to model slider bearings. In the theory of lubrication, the continuum description is assumed and inertial terms are negligible (Reynolds 1886) . A pressure equation can then be obtained with one dimension less than the Navier–Stokes (NS) equations. However, as modern read-write heads are typically floating 50 nm or less above the moving disk, the characteristic length scale involved is comparable to the molecular mean free path, , and the continuum assumption is no longer considered to be valid. Any gas in such a narrow spacing will suffer from significant non-equilibrium effects, which can be measured by the Knudsen number, Kn, and relates the ratio of the mean free path, , to the characteristic length scale of the device. For convenience, it is possible to classify four distinct flow regimes based on the Knudsen number (Barber and Emerson 2006) : (1) Kn < 10−3, represents the continuum flow regime where no-slip boundary conditions can generally be applied; (2) 10−3 < Kn < 10−1, indicates that the flow is in the slip regime and boundary conditions need to account for velocity-slip and temperature-jump conditions; (3) 10−1 < Kn < 10, represents the transition flow regime and results obtained using continuum-based equations are no longer considered reliable; and (4) Kn > 10, is the free molecular or collisionless flow regime and the continuum hypothesis is not valid. In the case of slider bearings operating in the slip-flow regime, it is usually adequate to modify the Reynolds equation with a velocity-slip boundary condition (Burgdorfer 1959; Hsia and Domoto 1983; Mitsuya 1993; Bahukudumbi and Beskok 2003; Chen and Bogy 2010) . However, for many problems, especially those related to disk-drive heads, the operating conditions lie well within the transition regime. This is a region where non-equilibrium effects come not only from the wall but also from the Knudsen layer (Cercignani 2000; Lilley and Sader 2008; Gu et al. 2010) . Modification of the slip boundary condition alone is not sufficient to compensate for the substantial deviation from the continuum assumption. The Reynolds lubrication equation with a first-order velocity-slip boundary condition will over-predict the pressure rise (Burgdorfer 1959) , while a second-order boundary condition consistently under-predicts the pressure rise (Hsia and Domoto 1983) . To account for the non-equilibrium effects from the wall and the Knudsen layer, Fukui and Kaneko (1988) derived a generalised lubrication equation (FK model) for arbitrary Knudsen number based on the linearised Boltzmann equation with the Bhatnagar–Gross–Krook (BGK) collision model (Bhatnagar et al. 1954) . Their derivation focused on the Poiseuille flow rate and no analytical expressions for the velocity profile and shear stress were given, although they can be recovered from the solution of the pressure equation. Due to the complexity of the original expression of the FK model, a piecewise curve fit formula was provided for its practical use (Fukui and Kaneko 1990) . However, Cercignani et al. (2004) pointed out that the database given by Fukui and Kaneko (1990) is inaccurate. Furthermore, as noted by Chen and Bogy (2010) , there is a contact pressure singularity in the FK model. Cercignani et al. (2007) derived a Reynolds equation on the basis of the BGK and ellipsoidal statistical BGK Boltzmann equation. Its solution of pressure requires an accurate evaluation of t (...truncated)


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Xiao-Jun Gu, Haijun Zhang, David R. Emerson. A new extended Reynolds equation for gas bearing lubrication based on the method of moments, Microfluidics and Nanofluidics, 2016, pp. 23, Volume 20, Issue 1, DOI: 10.1007/s10404-015-1697-7