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)