IMA Journal of Applied Mathematics (2015) 80, 1409–1430
doi:10.1093/imamat/hxu053
Advance Access publication on 4 December 2014
Dynamics of a parallel, high-speed, lubricated thrust bearing with
Navier slip boundary conditions
N. Y. Bailey
University Technology Centre in Gas Turbine Transmission Systems, Faculty of Engineering,
University of Nottingham, Nottingham NG7 2RD, UK
K. A. Cliffe and S. Hibberd∗
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
∗
Corresponding author: [email protected]
and
H. Power
Fuels and Power Technology Research Division, Faculty of Engineering, University of Nottingham,
Nottingham NG7 2RD, UK
[Received on 18 November 2013; revised on 26 September 2014; accepted on 29
October 2014]
An incompressible fluid flow model for a thin-film thrust bearing with slip flow is derived, leading to
a modified Reynolds equation for a highly rotating rotor that incorporates a slip length shear condition
on the bearing faces, extending previous bearing studies for new bearing applications associated with
decreasing film thickness. Mathematical and numerical modelling is applied to the coupled process of
the pressurized fluid flow through the bearing, with a Navier slip condition replacing a no-slip condition,
and the axial motion of the rotor and stator. The derived modified Reynolds equation is coupled with
the dynamic motion of the stator through the pressure exerted by the fluid film, with explicit analytical
expressions for the pressure and force determined and the equation for the bearing gap reduced to a
non-linear second-order non-autonomous ordinary differential equation. A mapping solver is used to
investigate the time-dependent bearing gap for prescribed periodic motion of the rotor. A parametric
study focuses on bearing operation under close contact motion to examine the minimum film thickness
and possibility of bearing face contact.
Keywords: incompressible; Reynolds equation; slip length; film clearance; bearing dynamics.
1. Introduction
Fluid-lubricated, thrust-bearing technology comprises two structural components such as a rotor and
a stator separated by a thin fluid film experiencing relative rotational motion. The thin fluid film is
employed to maintain a clearance between the rotating and stationary elements when subjected to external axial loads requiring a hydrodynamic force to be generated by the dynamic motion of the bearing
faces enhancing the fluid film pressure.
Common bearing geometries are slider bearings that employ a thin lubricating air film to separate
two non-parallel moving plates, as studied by Witelski (1998), journal bearings with a rotating cylindrical shaft within a supporting cylindrical shell separated by a thin air film, as considered by Belforte
c The authors 2014. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/
by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
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N. Y. BAILEY ET AL.
et al. (1999), and thrust bearings, whose dynamics have significant importance in turbomachinery applications requiring very high operating rotational speeds. The effects of inertia in a slider bearing were
examined by Wilson & Duffy (1996) in considering a moderate Reynolds number. These authors replace
a classical lubrication approach with a boundary layer approach in the channel and match the internal
flow with a uniform potential flow. Results showed the effect of the boundary layer on the pressure
at the inlet was dependent on the inlet velocity relative to a critical value of the core velocity. Garratt
et al. (2012) were the first to study the additional centrifugal inertia effects in a high-speed thrust bearing with compressible flow, where the fluid flow was coupled with the structural model. The bearing
dynamics were examined when the lower plate has prescribed periodic axial oscillations, with amplitude less than the equilibrium film thickness, and the upper plate is free to move axially in response
to the film dynamics. A similar geometry was considered by Bailey et al. (2013) with incompressible
flow, providing more extensive analytical investigations. The bearing dynamics were extended to the
axial oscillations with amplitude larger than the equilibrium film thickness. Results indicated the film
thickness can become very small, comparable with the mean free path of the fluid, possibly invalidating
the classical no-slip velocity condition.
Experimental studies by Sayma et al. (2002) examining both lift and flow leakage in mechanical
face seals, with typical gaps of under 10 µm, showed that it was not possible to attain no-slip steadystate gas flow motions under the given conditions. Typical test conditions corresponded to flow with
the film thickness comparable with the mean free path of the fluid molecules conjecturing a lack of
understanding of the dynamic behaviour and requiring greater analysis on this smaller scale.
Fluid flows at micro- and nanofluidics devices are characterized by confinement of the fluid environment to micro- and nano scales. The reduction of scale implies that surface effects start to dominate
over volume-related phenomena, requiring accurate details of the flow surface interaction. The classification of the mathematical models describing thin gas flow regimes is usually determined by the
Knudsen number, Kn = l/ĥ0 with l as the mean free path (collision distance between molecules) and
ĥ0 is the characteristic fluid thickness (for air at atmospheric conditions l = 68 nm). For small Knudsen
number (Kn 10−3 ), the fluid is considered as a continuum with no-slip boundary conditions. For a
larger Knudsen number between 10−3 and 10−1 , a continuum model with slip boundary condition is
usually employed; this is the flow regime of interest of the present work. For Knudsen number between
10−1 and 10, the flow is in a transition region and a modified continuum model needs to be considered.
Finally for larger values (Kn 10), molecular dynamics can be employed to describe the free molecular
flow, for more details see Karniadakis et al. (2005).
The existence of velocity slip was first predicted by Navier (1829). Navier proposed a constant slip
model with a linear relationship between the tangential shear rate and the fluid-wall velocity differences, i.e. proportional to the derivatives of the surface fluid velocity (first-order model), with a slip
length as the proportionality constant. This type of linear slip model has successfully been used in
reproducing the characteristics of many types of slip flows, see the work of Gad-el-Hak (2006), Wei
& Yogendra (2007) and Nieto et al. (2011). The classic theory for determining the slip coefficient
is due to Maxwell (1879). Briefly, the theory assumes that as gas flows some fraction f (momentum accommodation coefficient) of the molecules colliding with the wall are diffusely reflected with
the slip coefficient proportional to (2 − f )/f , which usually is of the same order of magnitude as the
mean free path. In order to extend the range of applicability of the slip condition beyond Kn = 10−1 ,
where first order models of the Navier type are not valid, several higher order models, including high
order derivatives of the surface fluid velocity, have been reported in the literature; for more details see
Karniadakis et al. (2002).
DYNAMICS OF THRUST-BEARING WITH NAVIER SLIP BOUNDARY CONDITIONS
1411
For liquids this classification for very thin flow regimes is not so clear. The Knudsen number is
mostly defined for gases, but an equivalent parameter can be applied to liquids. In liquids the intermolecular distance replaces the mean free path, resulting in an extremely small fluid thickness according to the
above classification. For liquid fluid motion over hydrophilic surfaces, the classical no-slip boundary
condition appears to be consistent even at the nano scale. However, when the surface is hydrophobic,
an apparent slip velocity has been observed just above the solid surface, with a slip length typically of
the order of 1 µm and extended to an order of 50 µm in the cases of superhydrophobic surface, see Choi
& Kim (2006). The numerical simulation of liquid fluid motion over hydrophilic surface with fluid film
thickness of the same order of magnitude as the irregular surface roughness becomes extremely difficult
due to the requirement of imposing the no-slip velocity condition at the corresponding irregular boundary. In these cases, it is possible to replace the no-slip boundary condition over the irregular surface by
an effective slip boundary condition at an equivalent smooth surface, with the slip length of the order of
the size of the surface roughness (see Miksis & Davis, 1994; Sarkar & Prosperetti, 1996). Homogenization theory has also been employed by several authors to find an asymptotically equivalent slip length
for small-scale variations of the boundary (see Dalibard & Gerard-Varet, 2011).
Slip flow through a bearing has widely been studied in a variety of different geometries. A gaslubricated inclined plane slider bearing was examined by Burgdofer (1959) considering the slip flow
regime using a first-order model with a boundary slip velocity given at a mean free path distance
from the wall. A corresponding Reynolds equation for compressible flow was developed and analytical
results were obtained. Hsia & Domoto (1983) modified the analysis by incorporating a second-order
slip model. A more generalized approach by Fukui & Kaneko (1988) derived a lubrication equation
valid at all Knudsen numbers from a linearized Boltzmann equation. Numerical results at large Knudsen numbers revealed that the first-order slip model overestimated the load carrying capacity, whereas
the second-order slip model underestimated the load carrying capacity. These predictions were confirmed by experimental studies carried out by Kato et al. (1990) on a rigid disk drive with film thickness
decreasing from 0.15 µm to less than 0.04 µm. Results motivated (Mitsuya, 1993) to propose a modified second-order slip model for flow slippage taking into account additional physical considerations,
referred to as a 1.5 slip model, with excellent agreement with experimental results for an inclined slider
bearing. Additional modifications of the slip model and implementation for slider bearings have also
been reported, for example Sun et al. (2002) presented analytical investigations of slip flow in a hard
disk drive between the flying head and disk.
Investigation of the slip effects on the fluid flow in gas journal bearing, with the slip boundary
conditions used in Burgdofer (1959), was examined by Malik (1984) for compressible flow. Predictions
gave increasing slip impairing the bearing performance at low journal speeds; however, increasing the
journal speed reduces the slip effect with the author indicating slip could have a beneficial effect at high
journal speed. On the other hand, Maureau et al. (1997) examined a journal bearing with incompressible
flow finding the force and torque on the load bearing inner cylinder decreasing with increased slip.
Experimenting with regions of slip and no-slip on journal bearing faces, Aurelian et al. (2011) showed
well chosen no-slip and slip regions on the surface can considerably improve the bearing behaviour, but
an inadequate no-slip and slip pattern can lead to deteriorating bearing behaviour.
Park et al. (2008) considered a non-axisymmetric thrust bearing with foil pads on the rotor face,
developing a classical Reynolds equation with slip for the gas flow coupled with the bearing structure.
The bearing dynamics were examined for rotor displacement of small amplitude in comparison with
the film thickness, using perturbation analysis. Results were presented for a no-slip and slip condition,
where a smaller load carrying capacity was associated with a slip condition due to a decrease in the
1412
N. Y. BAILEY ET AL.
linear velocity, causing the hydrodynamic pressure to decrease. Similarly the stiffness and damping
coefficients for axial perturbations are reduced with a slip condition compared with a no-slip condition.
The slip boundary effect on a gas journal bearing dynamics where the flow behaviour was coupled to
the structural model of the bearing was examined by Huang (2007). The subsequent effect of increasing
slip gave an increased gas flow rate but decreased gas film pressure and load carrying capacity. The
stability of the bearing was found to be lowered through reduced dynamic (stiffness and damping)
coefficients leading to possible contact with the housing when the rotor has a small disturbance. A
corresponding study by Zhang et al. (2008), with a new slip model derived taking a more physical
approach, reported similar outcomes.
The work by Park et al. (2008), Huang (2007) and Zhang et al. (2008) investigates the dynamics of
the bearing fluid structure interaction. The increasing advantages for bearings to operate with reduced
gap between the rotor and stator motivated the derivation of a thin film model valid for slip flow to
investigate if the fluid film is maintained under extreme operating conditions. This work develops a
slip length model which couples the flow dynamics to a thrust-bearing structure induced by the fluid
force exerted, when the rotor has prescribed periodic axial oscillations. In an extension to previous slip
models, terms are included relevant to high rotational speed bearing operation through incorporating the
leading-order centrifugal fluid inertia. In Section 2 the formulation of the coupled governing equations is
presented; namely the Reynolds equation (thin film approximation) in terms of a slip boundary condition
characterized by a slip length parameter, and the stator displacement equation (spring-mass-damper
system) given that the rotor has prescribed periodic motion. A representative single non-linear secondorder non-autonomous ordinary differential equation for the bearing gap is derived in Section 3. Further,
a stroboscopic map solver is described and numerical procedure identified to solve for the periodic
bearing face clearance iteratively, for varying slip length parameter. Detailed evaluation of steady flow
motions, i.e. without axial periodic rotor displacement, is given in Section 4. In Sections 5 and 6 the
effect of the slip velocity on the bearing dynamic and minimum gap is examined through selected
parameter studies. The analysis of the influence of the slip velocity on the minimum bearing gap is one
of the main objectives of the present work, given that in our previous work it is shown that contact cannot
occur for a no-slip condition on the bearing faces, although the predicted minimum gap can become very
small (see Bailey et al., 2013). Asymptotic analysis is carried out to verify the numerical solution in the
limit of small face clearance. The present work investigates the effect of large axial rotor displacement
extending previous studies, e.g. Park et al. (2008) and the concurrent effect on the minimum bearing
gap and possibility of contact. To verify the complex and extended algebra for correctness, two different
symbolic computations were used.
2. Geometric configuration
A simplified mathematical model of a fluid lubricated bearing containing parallel faces, as developed
in Bailey et al. (2013) is extended by incorporating a slip condition on the bearing faces. The coaxial
annular rotor and stator can move axially, having displacement heights ĥr and ĥs , respectively; the rotor
also has rotational speed Ω̂. A differential pressure is imposed at the inner and outer radii of the bearing,
allowing a pressure gradient to drive a radial flow.
The axisymmetric rotor-stator clearance is given by
ĝ(t̂) = ĥs (t̂) − ĥr (t̂) = ĥs (t̂) − ĥ0 sin(ω̂t̂),
(2.1)
DYNAMICS OF THRUST-BEARING WITH NAVIER SLIP BOUNDARY CONDITIONS
1413
with the imposed axial motion of the rotor modelled by ĥr (t̂) = ĥ0 sin(ω̂t̂), with amplitude ĥ0 where
ĥ0 is the equilibrium film thickness.
Deriving the velocity boundary conditions for a slip bearing in coordinate system (ẑ, r̂, θ̂ ) with velocities û = (ŵ, û, v̂) requires the normal and tangential velocities on the rotor and stator to be specified,
due to the additional slip component on the bearing face. The rotor normal vector n̂r and stator normal
vector n̂s are in the ẑ positive and negative directions, respectively, whilst the two orthogonal tangential
components on the face of the rotor, t̂1 and t̂2 are in the radial and azimuthal direction, respectively.
Similarly the two orthogonal tangential components on the face of the stator t̂3 and t̂4 are in the radial
and azimuthal direction, respectively.
Using the Navier slip condition, the fluid velocity on the bearing face comprises a jump in the
tangential velocities across the fluid–solid interface which is equal to the slip velocity induced due to
wall shear. Continuity of the normal velocity across the fluid–solid interface is imposed along with a no
flux condition. Thus the velocity boundary conditions on the rotor, denoted by superscript r, and stator,
denoted by superscript s, are given by
û · t̂1 − ûr · t̂1 = 2lˆs êij n̂j t̂1,i ,
û · t̂2 − ûr · t̂2 = 2lˆs êij n̂j t̂2,i ,
û · n̂r − ûr · n̂r = 0,
at z = hˆr ,
û · t̂3 − ûr · t̂3 = 2lˆs êij n̂j t̂3,i ,
û · t̂4 − ûr · t̂4 = 2lˆs êij n̂j t̂4,i ,
û · n̂s − ûr · n̂s = 0,
at z = hˆs .
(2.2)
For a rotor velocity ûr = (∂ hˆr /∂ t̂, 0, Ω̂ r̂) and stator velocity ûs = (∂ hˆs /∂ t̂, 0, 0), the velocity boundary conditions (2.2) can be written as:
∂ û ∂ ŵ
∂ v̂
∂ hˆr
+
, v̂ − Ω̂ r̂ = 2lˆs êθz n̂rz t̂θ1 = lˆs , ŵ =
cos β̂ at z = hˆr ,
∂ ẑ
∂ r̂
∂ ẑ
∂t
∂ û ∂ ŵ
∂hs
∂ v̂
s 3
ˆ
ˆ
+
û = 2ls êrz n̂z t̂r = −ls
at z = hˆs ,
, v̂ = 2lˆs êθz n̂sz t̂θ4 = −2lˆs , ŵ =
∂ ẑ
∂ r̂
∂ ẑ
∂ t̂
û = 2lˆs êrz n̂rz t̂r1 = lˆs
(2.3)
where the rate of strain tensor components as given in Batchelor (1967) were used, and where azimuthal
partial derivatives do not appear as the bearing configuration is axisymmetric.
A model for incompressible fluid flows through the bearing gap is derived from the Navier–Stokes
momentum and continuity equations in axisymmetrical coordinates. To reformulate in dimensionless
variables, a typical bearing radius r̂0 , rotor velocity Ω̂ r̂ and time scale T̂ = 1/ω̂ is taken, with dimensionless time variable t = ω̂t̂ where ω̂ is an axial disturbance frequency. The dimensionless velocities
are taken to be û/Û, v̂/Ω̂ r̂0 and ŵ/ĥ0 T̂ −1 with dimensionless radius and height given by r = r̂/r̂0 and
z = ẑ/ĥ0 , respectively. The dimensionless slip length is given by ls = lˆs /ĥ0 .
The geometry in Fig. 1 is in the dimensionless coordinate system (r, θ , z), with rotor and stator
heights given by hr (t) and hs (t), respectively, giving the axisymmetric rotor-stator clearance as g(t) =
hs (t) − hr (t). The axial rotor motion is prescribed by periodic oscillations hr (t) = sin t, with prescribed
amplitude . The internal and external pressure are given by pI and pO , respectively, and the inner and
outer radius are rI and rO , respectively.
The associated radial and azimuthal Reynolds numbers and their ratio are given, respectively, as
ReU =
ρ̂r̂0 Û
,
μ̂
ReΩ =
ρ̂r̂02 Ω̂
μ̂
and
Re∗ =
ReΩ
Ω̂ r̂0
.
=
ReU
Û
(2.4)
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N. Y. BAILEY ET AL.
Fig. 1. Geometry and notation of bearing.
The aspect ratio δ0 , squeeze number σ̃ and Froude number Fr are defined as
δ0 =
ĥ0
,
r̂0
σ̃ =
r̂0
Û T̂
and
Û
Fr = ,
ĝĥ0
(2.5)
respectively. Here ĝ is the acceleration due to gravity, ρ̂ is the density and μ̂ the dynamic viscosity.
For thin film bearings δ0 1. To ensure that the effects of viscosity are retained at leading order
the pressure is scaled as P̂ = μ̂r̂0 Û/ĥ20 . Classical lubrication theory neglects inertia due to the reduced
Reynolds number ReU δ02 1, however, as in Garratt et al. (2012), the centrifugal inertia is retained to
include cases of high rotational speed bearing operations for which terms of the order ReU δ02 (Re∗ )2 are
considered to be of O(1), with (Re∗ )2 1. The squeeze number σ̃ characterizes any time dependent
effects whilst the Froude number Fr parametrizes the importance of the gravitational effects relative
to the radial flow although gravity can be neglected with ReU δ02 Fr−2 1; this is consistent with lubrication theory provided the Froude number is O(1). In Appendix A, it is shown that under the present
approximation the flow field of very thin film air bearing (nano scale gap) must satisfy the usual incompressible solenoidal condition.
To leading order, where terms O(δ0 ) are neglected, the governing lubrication equations become
−
∂ 2v
v2
∂p ∂ 2 u
+ 2 = −η ,
= 0,
∂r
∂z
r
∂z2
1 ∂
∂w
(ru) + σ̃
= 0,
r ∂r
∂z
∂p
= 0,
∂z
(2.6)
(2.7)
where the speed parameter η = ReU δ02 (Re∗ )2 = ρ̂r̂0 ĥ20 Ω̂ 2 /μ̂Û characterizes the importance of centrifugal inertia. For a full derivation, see Bailey et al. (2013).
DYNAMICS OF THRUST-BEARING WITH NAVIER SLIP BOUNDARY CONDITIONS
1415
Re-scaling the velocity boundary conditions (2.3) in terms of dimensionless variables gives
∂u
∂v
2 ∂w
u = ls
+ δ0 σ̃
, v = r + ls ,
∂z
∂r
∂z
∂v
∂w
∂u
u = −ls
+ δ02 σ̃
, v = −ls ,
∂z
∂r
∂z
w=
∂hr
,
∂t
at z = hr ,
w=
∂hs
,
∂t
at z = hs ,
(2.8)
and to a leading order approximation, for δ0 1, becomes
∂u
∂v
, v = r + ls ,
∂z
∂z
∂u
∂v
u = −ls , v = −ls ,
∂z
∂z
u = ls
∂hr
,
∂t
∂hs
w=
,
∂t
w=
at z = hr ,
(2.9)
at z = hs .
Analytical responses for the velocity components u, v and w can be found readily in terms of the
unknown stator position hs (t) from (2.6) and (2.7), with the full expressions given in Appendix B,
equations (B.1–B.3).
A governing equation for bearing flow is readily obtained by integrating the leading order continuity
equation (2.7) between the rotor and stator, and applying the axial velocity boundary condition in (2.9)
to give a modified Reynolds equation, including slip effects, as
r2
λ ∂
70 3 2
∂p 3
2
5
4
2 3
g + 10g ls + g ls + 20g ls
= 0,
r (g + 6ls g ) +
∂r
r ∂r (g + 2ls )2
3
(2.10)
with λ = 3/10η and σ = 12σ̃ of O(1). This modified Reynolds equation expresses the relationship
between the internal bearing flow pressure p(r, ls , t) and bearing gap g(t).
Pressure boundary conditions at the inner and outer radii of the bearing are defined in dimensionless
variables as
dg 1 ∂
−
σ
dt
r ∂r
p = pI
at r = a
and
p = pO
at r = 1.
(2.11)
Axial displacement of the stator hs (t) is modelled using a standard spring-mass-damper model incorporating the bearing pressure variation in dimensionless variables as
d2 hs
dhs
+ Kz (hs − 1) = αF(ls , λ, t) = α2π
+ Da
2
dt
dt
1
(p − pa )r dr,
(2.12)
a
where pa is the ambient pressure above the stator and the force coupling dimensionless parameter given
by α = μ̂Û/m̂ω̂2 δ03 , considered to be of O(1). Bearing quantities Da = D̂a /m̂ω̂ and Kz = K̂z /m̂ω̂2 are
dimensionless linear damping and effective restoring force coefficients, respectively, with D̂a and K̂z as
their corresponding dimensional values and m̂ the mass of the stator.
The corresponding radial flux through the bearing is obtained from the integration of the radial
flow velocity over an azimuthal bearing cross section and is given in Appendix B as equation (B.4).
An expression for the steady-state stream function along a radial cross section is obtained from the
integration of the velocity components (u, w) and is given by (B.5).
1416
N. Y. BAILEY ET AL.
3. Bearing gap equation
Displacement of the stator is coupled by the periodic forcing of the rotor through the film flow, requiring
the Reynolds equation (2.10) and stator equation (2.12) to be solved simultaneously. Integrating the
modified Reynolds equation (2.10) and imposing the pressure boundary conditions (2.11) gives
p(r, ls , λ, t) = pO − (pO − pI )
ln r
ln a
ln r
+ r − 1 + (1 − a )
ln a
2
2
dg
g3 + 10g2 ls + (70/3)gls 2 + 20ls 3
σ
+λ
.
4g2 (g + 6ls ) dt
2(g + 6ls )(g + 2ls )2
(3.1)
The pressure limit as the slip length goes to infinity ls → ∞ can be computed analytically representing
the pressure when the bearing has zero shear.
The dimensionless force on the stator is obtained from the integration of the pressure field (3.1) over
the surface of the stator and is given by
F(ls , λ, t) = π
dg
A(g, ls , λ) + B(g, ls ) .
dt
(3.2)
Expressions for A(g, ls , λ) and B(g, ls ) are given by
(1 − a2 )
A(g, ls , λ) = (1 − a )(pI − pa ) + (pO − pI )
+1
2 ln a
g3 + 10g2 ls + (70/3)gls 2 + 20ls 3
(1 − a2 )2
4
−λ
+
1
−
a
,
4(g + 6ls )(g + 2ls )2
ln a
σ
(1 − a2 )2
B(g, ls ) = − 2
1 − a4 +
.
8g (g + 6ls )
ln a
2
(3.3)
Expression (3.3) identifies that for a steady bearing with negligible inertial effects, λ = 0, the pressure
and force are independent of the slip length.
Substituting the force expression (3.2) and stator height specified by hs (t) = g(t) + hr (t), with axial
displacement hr (t) = sin(t), into equation (2.12) gives a non-linear second-order non-autonomous
ordinary differential equation for the periodic time-dependent bearing gap as
d2 g
dg
+ D(g, ls ) + S(g, ls , λ) = Υ sin(t + φ),
2
dt
dt
(3.4)
where
D(g, ls ) = Da − απ B(g, ls ),
S(g, ls , λ) = Kz (g − 1) − απ A(g, ls , λ),
Υ sin(t + φ) = ((1 − Kz ) sin t − Da cos t).
(3.5)
Dynamically equation (3.4) corresponds to a harmonically forced oscillator with non-linear damping
coefficient D(g, ls ), which is independent of the parameter λ, and effective restoring force S(g, ls , λ). The
1417
DYNAMICS OF THRUST-BEARING WITH NAVIER SLIP BOUNDARY CONDITIONS
total stiffness of the system is defined as
ls 2 g(7g + 30ls )
dS
dA(g, ls , λ)
= Kz − απ
= Kz + λαπ
KzT =
dg
dg
3(g + 6ls )2 (g + 2ls )3
(1 − a2 )2
1−a +
ln a
4
,
(3.6)
where Kz is the structural component of the stiffness and the other terms comprise the fluid stiffness
component, which is zero for ls = 0 or λ = 0.
Solutions to equation (3.4) are denoted by the vector g(g(t), z(t)), for given initial conditions g(t0 ) =
g0 , z(t0 ) = z0 and fixed value of slip length ls and are sought from an equivalent system of two first-order
differential equations
dg
=z
dt
and
dz
= −D(g)z − S(g) + Y sin(t + φ).
dt
(3.7)
Considering a forcing of period T, it is expected that for some values of slip length, ls , the system of
equations (3.7) has periodic solutions. Thus a stroboscopic map solver is formulated, as used by Abashar
(2011) using Newton’s method to find periodic solutions. To compute solutions for an increased value
of the slip length ls + ls an Euler–Newton scheme (parameter continuation) is developed, as used by
Cliffe (1983).
4. Effect of slip velocity on steady-state bearing motion
In this section the result of the slip effects on the fluid flow in a steady-state bearing are examined. The
rotor and stator are fixed axially at hr = 0 and hs = 1, respectively, and the rotor has constant azimuthal
velocity. Three classes of bearing pressurization with exemplar values are used: internal pressurization
with pI = 2, pO = 1, ambient pressure pI = 1, pO = 1 and external pressurisation pI = 1, pO = 2. Results
reported here correspond to a = 0.2, i.e. a wide bearing of width (1 − a) = 0.8. Corresponding results
are obtained for other values of a, with smaller variations on the pressure profile and larger variations
on the velocity profiles as the value of a increases.
The pressure field is investigated for the existence of non-monotonic behaviour by examining the
derivative of the pressure given in (3.1) with respect to r, giving a local minimum occurring at the radial
position
2
2)
1
−
p
)(g
+
6l
)(g
+
2l
)
(1
−
a
(p
I
O
s
s
+
rmin = −
.
(4.1)
ln a λ(g3 + 10g2 ls + 70
2
gls 2 + 20ls 3 )
3
The pressure field is monotonic for internal and external pressurization bearing with negligible inertia
effects λ = 0, independently of the slip length. Under ambient pressure a minimum occurs at the same
position though the bearing, independently of the values of λ and ls . In the cases of external and internal
pressurization for increasing speed parameter, λ =
| 0, a minimum develops at the inner and outer radius,
respectively, before moving into the bearing, as shown in Fig. 2 column a, for increasing speed parameter under the three classes of pressurizations. Substituting the location of the minimum rmin into the
pressure equation (3.1) gives the values of the pressure pmin as shown in column b. For a bearing with a
high-speed parameter and a no-slip condition imposed, the minimum pressure decreases monotonically
and can become negative. The pressure field is increased with increased slip length having a limiting
value of ls → ∞ denoted by case ls∞ .
1418
N. Y. BAILEY ET AL.
(a1)
(b1)
(a2)
(b2)
(a3)
(b3)
Fig. 2. The radial position rmin and pressure value pmin at the point of a minimum in the pressure field for increasing speed
parameter 0 λ 10 and slip length 0 ls ∞ in the case of (1) external, (2) ambient and (3) internal pressurisation; = 0,
σ = 1.
The radial flux in a steady bearing can be examined by integrating the steady-state Reynolds
equation from (2.10) to give
∂p
r2 g2
− r g2 (g + 6ls ) + λ
∂r
(g + 2ls )2
70 2
3
3
2
g + 10g ls + gls + 20ls = C,
3
(4.2)
where the left-hand side of (4.2) is proportional, by a factor of π/6, to the expression for the flux (B.4).
The constant C is calculated from substituting the steady pressure field (3.1) into (4.2), giving the steady
flux as
λ(1 − a2 ) (g3 + 10g2 ls + 70/3gls 2 + 20ls 3 )
π g2
2
.
(4.3)
(pO − pI )(g + 6ls ) −
Q=
6 ln a
2
(g + 2ls )2
A bearing with negligible inertia effects and ambient pressure has zero flux, corresponding to the flow
only in the azimuthal direction, see velocity field expressions (B.1) to (B.3). Imposing internal or external pressurization across the bearing gives the radial flux as unidirectional with outwards or inwards
flow, respectively. For non-zero speed parameters λ =
| 0, a critical speed parameter λc can arises where
DYNAMICS OF THRUST-BEARING WITH NAVIER SLIP BOUNDARY CONDITIONS
(a)
(b)
1419
(c)
Fig. 3. Streamlines for a bearing under external pressurization with increasing slip length (a) ls = 0, (b) ls = 0.5 and (c) ls = 1 in
the case of the critical speed parameter for a no-slip bearing λ = 2.0833; a = 0.2.
zero flux through the bearing is achieved, found from (4.3), giving
λc =
2(pO − pI )(g + 6ls )(g + 2ls )2
.
(1 − a2 )(g3 + 10g2 ls + 70/3gls 2 + 20ls 3 )
(4.4)
The critical speed parameter λc corresponds to the flow condition when the inward flow driven by the
differential pressure exactly balances the outward flow due to high-speed rotation, with expression (4.4)
giving zero flux exciting for an external pressurized bearing only. Hence for low-speed parameter values
λ < λc the differential pressure drives the flow inwards (Q < 0), whereas for higher speed parameter
λ > λc , inertial effects drive the flow outwards (Q > 0).
The streamlines in Fig. 3 give the radial path of the fluid flow through the bearing for an externally
pressurized bearing in the case of increasing slip length. Choosing the critical speed parameter for a noslip bearing, which is consistent with the estimated value, gives zero flux through the bearing. Increasing
the slip length but keeping the same speed parameter causes a negative flux in the bearing, increasing
in magnitude as the slip length increases.
Owing to the complexity of the fluid velocity equations, the fluid flow is studied in more detail by
examining the velocity field in the asymptotic limit of large finite slip length. Leading order velocity
components are given by
u=
ls h
2r ln a
pO − pI −
5λ(1 − a2 )
12
,
r
v= ,
2
w = 0,
(4.5)
where in the resulting expression for w, the steady-state asymptotic limit of the Reynolds equation is
substituted to obtain the corresponding zero value. The azimuthal velocity has rigid body motion taking
the value of the average between the azimuthal velocity of the rotor and the stationary stator. The radial
velocity is proportional to the slip length and independent of the axial coordinate. These limits are
consistent with numerical evaluation of the full expressions.
Examining the flux in the limit of large finite slip length gives
π ls g 2
Q=
ln a
5λ
2
(1 − a ) ,
pO − pI −
12
(4.6)
1420
N. Y. BAILEY ET AL.
which gives the flux proportional to the slip length. The corresponding critical speed parameter is given
by
12(pO − pI )
.
(4.7)
λc =
5(1 − a2 )
Thus only an externally pressurized bearing can be maintained with a zero flux, as previously observed
in the general case. Equation (4.7) is consistent with the limiting value of (4.4).
5. Effect of slip velocity on dynamic bearing motion
Forcing the rotor with a prescribed axial periodic oscillation allows the dynamic behaviour of the bearing to be investigated. Unlike in the steady case, the dynamic case has the pressure and force dependent
upon the slip length even with negligible inertial effect λ = 0; however, the fluid stiffness is zero for
λ = 0, see (3.6). Numerical periodic solutions for the bearing gap g(t) are found using the stroboscopic
map solver. In this section, all results are given for a larger amplitude of the rotor oscillations = 1.2.
Figure 4 shows the periodic solution for the fluid stiffness, total damping, force on the stator, stator
height and bearing gap for increasing slip length 0 ls 1 in the case of a wide bearing under internal
pressurization with inertial effects and speed parameter λ = 1. The bearing gap g(t) initially decreases
as the rotor moves sinusoidally towards the stator, then within a region of close proximity between
t ∼ 1 and t ∼ 2.4 the stator follows the rotor whilst maintaining a thin fluid film until the gap increases.
Increasing the slip length causes the minimum face clearance to decrease markedly from gmin = 0.1569
for no-slip to gmin = 0.02775 for slip length ls = 1. The force F(g) has a maximum when the rotor
first becomes close to the stator, which initially keeps the plates apart. This is due to the non-linear
term B(g) dg/dt in the force equation (3.2), where B(g) ∝ g−2 becomes asymptotically unbounded as
the bearing gap tends to zero and consequently dg/dt becomes asymptotically small to make the term
B(g) dg/dt finite. The total damping D(g) takes the underlying structural value Da = 1, unless the face
clearance is small where a maximum occurs, which initially decreases as the slip length increases before
reaching a minimum value around ls ∼ 0.1 and then increases. This behaviour is due to the term (g2 (g +
6ls ))−1 in B(g, ls ) given by (3.3), with g as a unknown function of ls , which initially decreases before
increasing with increasing slip length. A no-slip bearing has zero fluid stiffness Kzf , but increasing the
slip length causing a maximum to develop when the face clearance is small. This increases in magnitude
until ls = 0.0495 after which the maximum decreases in magnitude and develops a slight dip in the
middle until it is of small magnitude throughout the time period, apart from when the face clearance is
smallest where there is effectively no fluid stiffness. This is supported by the analytic analysis which
shows that fluid stiffness Kzf given in equation (3.6) has a maximum close to zero.
Detailed evaluation for different parameter values show similar dynamics with associated characteristic values for transitions and extrema, where some parameters have a more dominant effect than
others. Examining a bearing with increased speed parameter, λ = 10, gives similar dynamics but with
the overall magnitude of the fluid stiffness, damping and force larger even though their extrema values
are lower. At slip length ls = 1, gmin = 0.00642 causing a larger maximum in the force and the stator
closely follows the rotor for longer. Whereas if the bearing is put under external pressurization the
extrema occur at larger values and the fluid stiffness, damping and force having the same order as in
Fig. 4 but slighter smaller values. The stator follows the rotor for a shorter time and sits further away,
with the minimum bearing gap gmin = 0.0674 at ls = 1, giving the force a smaller peak when the two
plates initially become close.
A main focus of this work is to examine the magnitude of the minimum face clearance gmin and the
effect the main parameters (speed parameter, bearing width and slip length) have on gmin are given in
DYNAMICS OF THRUST-BEARING WITH NAVIER SLIP BOUNDARY CONDITIONS
1421
Fig. 4. Fluid stiffness, total damping, force, stator height and bearing gap for increasing slip length 0 ls 1 in the case of a
wide bearing under internal pressurisation; a = 0.2, = 1.2, λ = 1, σ = 1, Kz = 10, α = 1 and Da = 1.
Table 1 Minimum face clearance gmin for wide a = 0.2 and narrow a = 0.8
bearing in the case of increasing speed parameter 0 λ 1 for a no-slip condition ls = 0 and slip length ls = 1; = 1.2, σ = 1, Kz = 10, α = 1 and Da = 1.
a = 0.2
a = 0.2
a = 0.8
a = 0.8
ls = 0
ls = 1
ls = 0
ls = 1
λ=0
0.167
0.0338
0.0428
0.00102
λ=1
0.157
0.0277
0.0427
0.00101
λ = 10
0.105
0.00642
0.0416
0.000958
λ = 30
0.0550
0.00157
0.0393
0.000831
Table 1 for an internally pressurized bearing with large amplitude of rotor oscillations, = 1.2. A narrow
bearing has a smaller minimum bearing gap than a wide bearing and increasing the speed parameter
causes the minimum face clearance gmin to decrease. Introducing a slip condition on the bearing faces
also causes the minimum face clearance gmin to decrease.
Bearing characteristics under external pressurization give dynamics similar to the case of internal
pressurization with the stator sitting slightly further away from the rotor and remaining in very close
proximity for less of the time period. The extrema in the total damping, fluid stiffness and force have
decreased magnitude and occur for less time. These effects are small for a narrow bearing but larger in
a wide bearing.
1422
N. Y. BAILEY ET AL.
6. Effect of slip velocity on minimum bearing gap
An important physical aspect is the effect of the slip condition on maintaining a bearing gap when the
stator is subjected to axial disturbances. In this section the bearing gap behaviour is examined when the
amplitude of the axial rotor displacement is equal to, or larger than, the equilibrium bearing gap 1.
Investigation evaluates if contact occurs for a wide range of slip lengths. Figure 5 shows a log-log plot
of the minimum bearing gap against increasing slip length 10−3 ls 106 for axial rotor oscillations
of amplitude 1.0 1.2. The bearing has two distinctly different asymptotic behaviours for increasing slip length depending on the magnitude of axial rotor oscillations . Region I has the minimum gap
approaching a constant value for increasing slip length. For region II the minimum gap decreases monotonically with slip length, approximately linearly with gradient −1, and contact occurs in the limit of
ls → ∞. A critical amplitude c is identified which separates the regions, given by c = 1.0533 in Fig. 5.
The plot was obtained by running the stroboscopic map solver for each value of rotor oscillation with
increasing discrete values of the slip length between 10−4 ls 106 using parameter continuation in
the numerical scheme, and evaluation of the minimum bearing gap for each parameter set.
The basic dynamics as in Fig. 5 are maintained across a range of parameter values with associated
changes in critical amplitude c . For example, with large speed parameter λ = 10 the critical value is
around c = 0.802, external pressurization gives c = 1.192 and a narrow bearing has c = 1.0517.
6.1
Asymptotic analysis
Further evaluation of the behaviour of the minimum face clearance and verification of numerical solutions and asymptotic analysis for large slip lengths is presented. The existence of the two possible
asymptotic regions in Fig. 5 is verified by examining the bearing dynamics for large amplitude rotor
oscillations when the minimum face clearance is assumed very small g(t) 1, and in the limit of large
finite slip length. The expressions for the damping D and effective restoring force S as defined in (3.5)
are given to leading order in ls as
απ σ (1 − a2 )
(1 − a2 ) + 2 ln a
(1 − a2 ) −
,
48
ln a
(1 − a2 ) + 2 ln a
2
S(λ) = −Â, where  = Kz + απ (pI − pa )(1 − a ) + (pO − pI )
2 ln a
2 2
1 − a + 2 ln a
5λ(1 − a )
+
(1 − a2 ) −
.
24
ln a
D(g, ls ) = Da +
D̄
,
g 2 ls
where D̄ = −
(6.1)
Thus S(λ) is independent of the bearing gap and slip length giving it of O(1), whereas the leading order
of D(g, ls ) depends on the size of the bearing gap relative to the value of the slip length. Correspondingly,
the behaviour of the bearing is split into three different asymptotic regimes: g(t) ls −1/2 , g(t) ∼ ls −1/2
and g(t) ls −1/2 .
For the region of close contact, g(t) ls −1/2 the modified stator equation (3.4) becomes to leading
order in the limit of large finite slip length
D̄ dg
d2 g
+ 2
− Â = Υ sin(t + φ),
2
dt
g ls dt
for rotor displacement amplitude Υ of O(1).
(6.2)
DYNAMICS OF THRUST-BEARING WITH NAVIER SLIP BOUNDARY CONDITIONS
1423
Fig. 5. Minimum face clearance against increasing slip lengths 10−4 ls 106 for increasing rotor amplitudes 1.00 1.10
for a periodic wide parallel internally pressurised bearing; a = 0.2, λ = 1, α = 1, Kz = 10, σ = 1 and Da = 1.
Fig. 6. Asymptotic regions for scaling when g(t) 1 and g(t) ls −1/2 .
To solve equation (6.2) the region of small face clearance is split into three further regions as illustrated in Fig. 6 with approximate transition times t0 , t1 and t2 . Regions I and III are the transition to and
from region II, respectively, where region II has g(t) as approximately constant with minimum gap gmin .
Examining the first and second derivatives of g(t) indicates region I has dg/dt < 0 and |d2 g/dt2 | 1,
region II has dg/dt 1 and region III is initiated by dg/dt > 0.
In region I, |d2 g/dt2 | 1 so locally a short-time behaviour is comparable with an inner boundary
layer, where initial conditions become g(t0 ) = g0 and dg(t0 )/dt = −ξ , for given initial time t0 and ξ > 0,
say.
In region I the stator equation in (6.2) becomes
d2 g D̄ 1 dg
= 0.
+
dt2
ls g2 dt
(6.3)
1424
N. Y. BAILEY ET AL.
Introducing c = ξ + D̄/g0 ls , a new inner time scale τ = D̄ls c2 (t − t0 ) and rescaling y(τ ) = ls cg, equation
(6.2) becomes to leading order
d2 y
1 dy
= 0,
(6.4)
+ 2
dτ 2
y dτ
taking y(τ = 0) = ls cg0 and dy/dτ (τ = 0) = −ξ/D̄c.
Solving (6.4) subject to the initial conditions gives an intrinsic algebraic relationship for y = Y (τ ; g0 )
as
ln(1 − D̄ls cg0 ) − ln(1 − D̄y)
τ = cls g0 − y +
,
(6.5)
D̄
which is readily solved for y numerically by Newton’s method. Matching of the solution in region I to
region II is when τ → ∞ with solution y → 1/D̄.
In region II, equation (6.2) has a balance between damping, stiffness and forcing which are all of
O(1) since dg/dt 1. Taking new time scale t̄ + t0 = t and φ̄ + t0 = φ, where t̄ = O(1), in region II
equation (6.2) is given to leading order in the limit of large finite slip length with g(t) ls −1/2 and
dg/dt 1 as
D̄ 1 dg
(6.6)
− Â = Υ sin(t̄ + φ̄ + 2t0 ),
ls g2 dt̄
with solution
g(t̄) =
D̄
ls
(D̄2 c
− Ât̄ + Υ (cos(t̄ + φ̄ + 2t0 ) − cos(φ̄ + 2t0 )))
,
on matching the solution g(t̄ = 0) = 1/D̄ls c with region I.
The composite solution in regions I and II in terms of the original variables is
D̄
1 Y (D̄ls c2 (t − t0 ); g0 )
+
g(t) =
.
ls
c
−Â(t − t0 ) + Υ (cos(t + φ) − cos(φ + t0 ))
(6.7)
(6.8)
It can be shown that the composite solution (6.8) is always positive, since the inner and outer solutions are both positive as  < 0. Thus no contact occurs in the bearing when g(t) ls −1/2 and the face
clearance g(t) is inversely proportional to the slip length if c is of O(1), i.e. g0 is larger than O(ls −1 ).
This agrees with the numerical results shown in region II of Fig. 5.
A comparison of the bearing gap over a period is provided in Fig. 7 and shows a close agreement
of the full numerical solution and the asymptotic composite solution in the limit g(t) ls −1/2 for the
region, 1.05 < t < 2, i.e. t0 = 1.094, and parameters as listed. The minimum bearing gap is consistent
with region II in Fig. 5. The composite solution has increasing discrepancy with the numerical solution
as the gap grows, since region III is not included in the analysis.
For the region g(t) ls −1/2 , D(g) is to leading order in the limit of large finite slip length of O(1)
with the expression for S(λ) remaining the same as in (6.1); note the formulation does not restrict the
size of the face clearance g(t) other than non-negative.
The modified stator equation (3.4) in the limit of large finite ls with g(t) ls −1/2 is given by
d2 g
dg
+ Kz g − Â = ((1 − Kz ) sin t − Da cos t),
+ Da
dt2
dt
which is of O(1).
(6.9)
DYNAMICS OF THRUST-BEARING WITH NAVIER SLIP BOUNDARY CONDITIONS
1425
Fig. 7. Comparison of bearing gap g(t) between the full numerical and composite solutions for a bearing under internal pressurization with achieved gmin = 4.66 × 10−6 ; a = 0.2, = 1.2, λ = 1, α = 1, Kz = 10, σ = 1, Da = 1, ls = 104 , σ̄ = 22, D̄ = 0.0279,
 = 10.52, γ = 22, g0 = 0.00001 and φ = 0.
Fig. 8. Comparison of bearing gap g(t) between the full numerical and asymptotic solutions g(t) for a bearing under internal
pressurisation with achieved gmin = 3.34 × 10−3 ; a = 0.2, = 1.05, λ = 1, α = 1, Kz = 10, σ = 1, Da = 1, ls = 106 , ξ̄ = 1.05,
 = 10.53, g0 = 1.053.
Solving equation (6.9) with initial conditions g(0) = g0 , dg(0)/dt = −ξ̃ gives
Â
−Da + Da 2 − 4Kz
Da − Da 2 − 4Kz
t + C2 exp −
t +
g(t) = C1 exp
− sin t ,
2
2
Kz
(6.10)
with constants of integration in (6.10) determined as
Â
1
Da + Da 2 − 4Kz
g0 −
C1 = − ξ̂ + ,
Kz
2
Da 2 − 4Kz
(6.11)
Â
1
−Da + Da 2 − 4Kz
C2 = g0 −
+ ξ̂ − .
Kz
2
Da 2 − 4Kz
This asymptotic solution shows that in the limit g(t) ls −1/2 , with ls very large, the minimum face
clearance becomes independent of the magnitude of the slip length, corresponding to region I in Fig. 5.
As before, verification of the periodic numerical solution is given in Fig. 8 showing the full numerical
solution and the asymptotic solution over a period of forcing.
1426
N. Y. BAILEY ET AL.
Asymptotic analysis confirms the numerical studies of the bearing behaviour for increasing values
of ls which depends upon the amplitude of forcing of the bearing in determining the magnitude of
the minimum gap. In region II where g(t) ls −1/2 the minimum gap decreases inversely proportionally
with the slip length if g0 > O(ls −1 ). When g(t) ls −1/2 then the full numerical equation must be solved
to capture the full dynamics of the problem. For region I, when g(t) ls −1/2 , the minimum bearing gap
asymptotes to a value independent of the slip length.
7. Summary and conclusions
A modified Reynolds equation for incompressible flow is derived for flow within a bearing that incorporates the slip effect appropriate for very small bearing face separation. A slip length formulation for a
modified surface boundary condition is imposed on the faces of the bearing, with dual limits of a viscous
no-slip boundary for ls = 0 and total slip flow with ls → ∞. An axisymmetric lubrication approximation is used incorporating the leading order effect of centrifugal inertia relevant for high rotational speed
flows. A fluid flow and structure interaction dynamic model for the stator equation is derived from a
spring-mass-damper system.
The fluid–rotor–stator interactions are investigated by examining the fully coupled unsteady bearing
where the stator is free to move axially in response to the fluid film dynamics. Re-writing the modified
Reynolds equation and the stator equation in terms of a new variable, the time-dependent bearing gap
g(t) allows explicit analytic expressions for the pressure and force. A non-linear second-order nonautonomous ordinary differential modified stator equation is derived and a stroboscopic map solver
implemented to find periodic solutions. Post-processing allows computation of the force on the stator,
stator height, fluid stiffness and total damping.
A steady bearing with axially fixed rotor and stator was investigated by examining the existence of
non-monotonic behaviour in the pressure field. Results for increasing slip length tend towards the limit
of infinite slip length are given for representative bearing parameter choices, with internal and ambient pressure having similar characteristics for non-negligible speed parameter. Externally pressurised
bearings can exist with no overall flux through the bearing at a critical speed parameter, where pressure
effects directly balance inertial effects.
The dynamics of the bearing are investigated for large rotor amplitude = 1.2 to simulate possible
destabilising behaviour. Distinctions between no-slip and slip bearings is in the development of fluid
stiffness and the minimum bearing gap decreasing with the stator sitting closer to the rotor for longer.
Increasing the slip length causes a maximum in the fluid stiffness and minimum in the force and total
damping. The peak in the force when the rotor initially becomes close to the stator keeps the faces apart
initially and the large fluid damping ensures the plates remain apart. Increasing the speed parameter
in a narrower bearing causes the minimum bearing gap to decrease, whereas externally pressurising a
bearing causes the stator to sit further from the rotor.
A log-log plot of the minimum bearing gap against slip length showed contact does not occur for
finite slip length and there are three regions of behaviour: g(t) ls −1/2 , g(t) ∼ ls −1/2 and g(t) ls −1/2 .
Asymptotic analysis confirmed there are two asymptotic solutions, one in the region g(t) ls −1/2 and
g(t) 1 where g(t) ∝ ls −1 and the other in the region g(t) ls −1/2 is independent of the slip length.
Therefore, if the amplitude of the rotor oscillations are smaller than a critical value c the minimum
bearing gap asymptotes of to a constant value. For the rotor amplitude larger than the critical the
minimum bearing gap will decrease inversely proportional to the slip length until there is contact in the
limit of ls → ∞.
DYNAMICS OF THRUST-BEARING WITH NAVIER SLIP BOUNDARY CONDITIONS
1427
Funding
This work was supported by funding from the EPSRC studentship grant No. EP/J500483/1 and was
carried out at the University Technology Centre in Gas Turbine Transmission Systems at the University
of Nottingham with financial support from Rolls-Royce plc, Aerospace Group. The views expressed
in this paper are those of the authors and not necessarily those of Rolls-Royce plc, Aerospace Group.
Funding to pay the Open Access publication charges for this article was provided by University of
Nottingham Open Access Policy.
References
Abashar, M. E. E. (2004) Dynamic behavior of two-phase systems in physical equilibrium. Chem Eng Jurnal, 97,
183–194.
Aurelian, F., Patrick, M. & Mohamed, H. (2011) Wall slip effects in (elasto) hydrodynamic journal bearings.
Tribol. Int., 44, 868–877.
Bailey, N. Y., Cliffe, K. A., Hibberd, S. & Power, H. (2014) On the dynamics of a high speed coned fluid
lubricated bearing. IMA J. Appl. Math., 79, 535–561.
Batchelor, G. K. (1967) An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press.
Belforte, G., Raparelli, T. & Viktorov, V. (1999) Theoretical investigation of fluid inertia effects and stability
of self-acting gas journal bearings. J. Tribol., 121, 836–843.
Burgdofer, A. (1959) The influence of the molecular mean free path on the performance of hydrodynamic gas
lubricated bearings. ASME J. Basic Eng., 81, 94–99.
Choi, C. H. & Kim, C. J. (2006) Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface.
Phys. Rev. Lett., 96, 066001.
Cliffe, K. A. (1983) Numerical calculations of two-cell and single-cell Taylor flows. J. Fluid Mech., 135, 219–233.
Dalibard, A. L. & Gerard-Varet, D. (2011) Effective boundary condition at a rough surface starting from a slip
condition. J. Differential Equations, 251, 3450–3487.
Etison, I. (1982) Dynamic analysis of noncontacting face seals. Trans. ASME, 104, 460–468.
Fukui, S. & Kaneko, R. (1988) Analysis of ultra-thin gas film lubrication based on linearized Boltzmann equation:
first report-derivation of a generalized lubrication equation including thermal creep. J. Tribol., 110, 253–262.
Gad-el-Hak, M. (2006) MEMS: Introduction and Fundamentals, 2nd edn. Boca Raton: Taylor and Francis.
Garratt, J. E., Cliffe, K. A., Hibberd, S. & Power, H. (2012) Centrifugal inertia effects in high speed hydrostatic
air thrust bearings. J. Engrg. Math., 76, 59–80.
Hsia, Y. T. & Domoto, G. A. (1983) An experimental investigation of molecular rarefaction effects in gas lubricated
bearings at ultra low clearances. J. Lubrication Technol. Trans. ASME, 105, 120–130.
Huang, H. (2007) Investigation of slip effect on the performance of micro gas bearings and stability of micro
rotor-bearing systems. Sensors, 7, 1399–1414.
Jie, D., Diao, X., Cheong, K. B. & Yong, L. K. (2000) Navier-Stokes simulations of gas flow in micro devices. J.
Micromech. Microeng, 10, 372–379.
Karniadakis, G., Beskok, A. & Aluru, N. (2005) Microflows and Nanoflows: Fundamentals and Simulation.
New York: Springer.
Karniadakis, G., Beskok, A. & Gad-el-Hak, M. (2002) Microflows and nanoflows: fundamentals and simulation.
Appl. Mech. Rev., 55, 76.
Kato, T., Ohkubo, T. & Kishigami, J. (1990) High density perpendicular recording at very low flying height under
rarefied pressure condition. IEEE Trans. Magnetics, 26, 2433–2435.
Malik, M. (1984) Theoretical considerations of molecular mean free path influenced slip in self-acting gaslubricated plain journal bearings. Proc. Inst. Mech. Eng. C J. Mech. Engr. Sci., 198, 25–31.
Maureau, J., Sharatchandra, M. C., Sen, M. & Gad-el-Hak, M. (1997) Flow and load characteristics of
microbearings with slip. J. Micromech. Microeng., 7, 55–64.
Maxwell, J. C. (1897) On stresses in rarified gases arising from inequalities in temperature. Philos. Trans. R. Soc.,
170, 231–256.
1428
N. Y. BAILEY ET AL.
Miksis, M. J. & Davis, S. H. (1994) Slip over rough and coated surfaces. J. Fluid Mech., 273, 125–139.
Mitsuya, Y. (1993) Modified Reynolds-equation for ultra-thin film gas lubricated using 1.5-order slip-flow model
and considering surface accommodation coefficient. J. Tribol. Trans. ASME, 115, 289–294.
Navier, C. L. M. H. (1829) M’emoire sur les lois du mouvement des fluids. Mem. Acad. Sci. Inst. Fr., 6, 389–416.
Nieto, C., Giraldo, M. & Power, H. (2011) Boundary integral equation approach for stokes slip flow in rotating
mixers. Dynam. Syst. Series B, 15, 1019–1044.
Park, D. J., Kim, C. H., Jang, G. H. & Lee, Y. B. (2008) Theoretical considerations of static and dynamic characteristics of air foil thrust bearing with tilt and slip flow. Tribol. Int., 41, 282–295.
Sarkar, K. & Prosperetti, A. (1996) Effective boundary conditions for Stokes flow over a rough surface. J. Fluid
Mech., 316, 223–240.
Sayma, A. I., Bréard, C., Vahdati, M. & Imregun, M. (2002) Aeroelasticity analysis of air-riding seals for
aero-engine applications. J. Tribol., 124, 607–616.
Sun, Y., Chan, W. K. & Liu, N. (2002) A slip model with molecular dynamics. J. Micromech. Micoeng., 12,
316–322.
Wei, X. & Yogendra, J. (2007) Experimental and numerical study of sidewall profile effects on flow and heat
transfer inside microchannels. Int. J. Heat Mass Transfer, 50, 4640–4651.
Wilson, S. K. & Duffy, B. R. (1996) On lubrication with comparable viscous and inertia forces. Quart. J. Mech.
Appl. Math., 51, 105–124.
Witelski, T. P. (1998) Dynamics of air bearing sliders. Phys. Fluids, 10, 698–708.
Zhang, W. M., Meng, G., Huang, H., Zhou, J. B., Chen, J. Y. & Chen, D. (2008) Characteristics analysis and
dynamic responses of micro-gas-lubricated journal bearings with a new slip model. J. Phys. D Appl. Phys.,
41, 155305 (16 pp).
Appendix A. Compressibility conditions
The condition for a compressible flow to be dynamically represented as an incompressible one, i.e. with
negligible density changes, requires the conservation of mass to reduce to the statement of solenoidal
velocity field. For axisymmetric flow, this implies
1 Dρ̂ 1 ∂
∂ ŵ ρ̂ Dt̂ r̂ ∂ r̂ (r̂û) + ∂ ẑ .
(A.1)
Following Batchelor (1967, p. 167) and considering no change in density due to internal dissipation
of heat or molecular conduction of heat (isentropic conditions), the total rate of change of density can
be given in terms of the rate of change of pressure as
1 Dρ̂
1 Dp̂
= 2
,
ρ̂ Dt̂
ρ̂ ĉ Dt̂
(A.2)
with ĉ2 = (∂ p̂/∂ ρ̂)s as the square of the speed of sound with entropy per unit mass S.
Using the momentum equation to represent the gradient of the pressure field in terms of the velocity
field, the above condition for the solenoidal velocity field can be written as
1 ∂ p̂
1 Dû2
1
1
∂ ŵ 1 ˆ ˆ 1 ∂
2
ˆ
+ 2 û · ĝ +
μ̂û · ∇ û + ∇ ∇ · û
ρ̂ ĉ2 ∂ t̂ − 2ĉ2 Dt̂
r̂ ∂ r̂ (r̂û) + ∂ ẑ .
ĉ
ρ̂ ĉ2
3
(A.3)
DYNAMICS OF THRUST-BEARING WITH NAVIER SLIP BOUNDARY CONDITIONS
1429
Substituting in our dimensional variables, each of the terms in brackets on the left-hand side of the
above inequality has a maximum dimension of
μ̂ Û r̂0 Û 2
1
Û
Û
σ̃ ,
= 2
2 2
2
r̂0
ρ̂ ĉ0 ĥ0 T̂
ĉ0 ReU δ0 r̂0
(A.4)
1 Ω̂ 2 r̂02
1
Û
Û
= 2 Ω̂ 2 r̂02 σ̃ ,
r̂0
r̂0
ĉ20 T̂
ĉ0
(A.5)
1 ĝĥ0 ĝĥ0 Û
Û
= 2 σ̃ ,
r̂0
ĉ20 T̂
ĉ0 r̂0
(A.6)
μ̂ Ω̂ 2 r̂02 Ω̂ 2 r̂02 1 Û
Û
= 2
,
r̂0
ρ̂ ĉ20 ĥ20
ĉ0 ReU δ02 r̂0
(A.7)
respectively, and the right-hand side has a maximum dimension of Û/r̂0 .
Therefore, for compressible flow be dynamically represented as an incompressible one, the following four inequalities need to be satisfied simultaneously
Û 2
ReU δ02 ,
ĉ20
Ω̂ 2 r̂02
1,
ĉ20
ĝĥ0
1,
ĉ20
Ω̂ 2 r̂02
ReU δ02 .
ĉ20
(A.8)
As commented by Bachelor, the third condition in (A.8) will be satisfied for all fluid motion with thickness smaller than a few hundred of metres, in particular for thin film flows. Consequently, in our approximation Reu δ0 2 ∼ O(δ0 ), it is only necessary that the first and fourth condition in (A.8) are satisfied
simultaneously. Therefore, the conditions required for a compressible flow to behave as an incompressible flow are given by
Û 2
δ0 ,
ĉ20
Ω̂ 2 r̂02
δ0 .
ĉ20
(A.9)
A possible case of a bearing configuration is Û/ĉ0 ∼ O(δ0 ) and Ω̂ r̂0 /ĉ0 ∼ O(δ0 ) resulting in (Re)∗ ∼
O(1), giving the incompressible model satisfying compressible flow if inertial effects are negligible. A
second example of a bearing configuration is having Û/ĉ0 ∼ O(δ02 ) again but Ω̂ r̂0 /ĉ0 ∼ O(δ0 ) giving
(Re)∗ ∼ O(1/δ0 ) and therefore the incompressible model satisfies the compressible flow with inertial
effects included.
Appendix B. Velocities, flux and streamfunction
The radial, azimuthal and axial fluid velocities are given, respectively, as
u=
1 ∂p 2
(z − (hs + hr )z + hs hr − ls g)
2 ∂r
5λr
−
((z − hr )(z − hs )(z2 + (hr − 3hs )z + 3h2s − 3hs hr + h2r )
18(g + 2ls)2
+ ls ((z − hs )(−4(z − hs )2 + 6g2 ) − g3 ) + ls2 (6(z − hs )(z − hr ) − 6g2 ) − 6gls3 ),
(B.1)
1430
N. Y. BAILEY ET AL.
r
(z − hs − ls ),
(B.2)
(g + 2ls )
∂hr
1 ∂
∂p
w=
−
r (2(z − hr )3 − 3(z − hr )2 g − 6(z − hr )gls )
∂t
σ r ∂r
∂r
r2
λ ∂
+
(2(z − hr )5 − 10(z − hr )4 g + 20(z − hr )3 g2 − 15(z − hr )2 g3 )
3σ r ∂r (g + 2ls )2
v=−
+ 10ls (−(z − hr )4 + 4(z − hr )3 g − 3(z − hr )2 g2 − 3(z − hr )g3 )
+ 10ls2 (2(z − hr )3 − 3(z − hr )2 g − 6(z − hr )g2 ) + 10ls3 (−6(z − hr )g) .
The flux is by
λr2
70 3 2
r ∂p 3
6ls
5
4
2 3
g 1+
Q = 2π
g + 10g ls + g ls + 20g ls −
.
12(g + 2ls )2
3
12 ∂r
g
(B.3)
(B.4)
The streamfunction is
Ψ (r, z) =
r ∂p
(2(z − hr )3 − 3(z − hr )2 g − 6(z − hr )gls )
12 ∂r
λr2
(2(z − hr )5 − 10(z − hr )4 g + 20(z − hr )3 g2 − 15(z − hr )2 g3 )
−
36(g + 2ls )2
+ 10ls (−(z − hr )4 + 4(z − hr )3 g − 3(z − hr )2 g2 − 3(z − hr )g3 )
+ 10ls2 (2(z − hr )3 − 3(z − hr )2 g − 6(z − hr )g2 ) + 10ls3 (−6(z − hr )g)),
which is found using equations (B.1) and (B.3) at steady-state condition.
(B.5)