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Nonlinear Engineering 2019; 8: 397–406
Vishwanath B. Awati* and Ashwini Kengangutti
Surface roughness eect on thermohydrodynamic
analysis of journal bearings lubricated with
couple stress fluids
https://doi.org/10.1515/nleng-2018-0017
Received October 7, 2017; revised May 18, 2018; accepted July 6,
2018.
Abstract: The paper presents, surface roughness eect
for thermo-hydrodynamic analysis of journal bearings ex-
tended to couple stress lubricants with high polymer addi-
tives. A modied energy equation is simultaneously solved
with heat transfer equation as well as modied Reynolds
equation by using Multigrid method. The eects of cou-
ple stress and surface roughness on the performances of
a nite journal bearing are presented in detail. Further, it
is shown that lubricants with couple stress and surface
roughness, not only increases the load capacity and de-
creases the friction coecient, but also generates a lower
bearing temperature eld. Thus, the lubricant with couple
stress improves the performance of journal bearings. The
characteristics of bearing are compared with numerical re-
sults.
1Introduction
The performance of journal bearings with thermal eects
in lubrication process is analysed by thermal hydrody-
namic lubrication (THD). Since it produces peak bearing
temperature, bearing failure can be estimated at the de-
sign when maximum temperature exceeds a certain limit.
Ferron et al. [1] solved Dowson’s [2] generalized Reynolds
equation simultaneously with energy equation and heat
transfer equation and produced excellent results. THD
solutions under various boundary conditions, consider-
ing the mixing of recirculation of uid and supply oil
is obtained by Khonsari and Beaman [3]. Ott and Par-
adissiadis [4] applied Jakobsson–Floberg–Olsson (JFO)
*Corresponding Author: Vishwanath B. Awati, Rani Channamma
University, NH-4 Bhootaramanahatti Belagavi, 591156, India, E-mail:
awati_vb@yahoo.com
Ashwini Kengangutti, Belagavi, 591156, India, E-mail:
ashwini7.k9@gmail.com
boundary conditions, which conserve mass within cavi-
tated region as well as boundary, to determine the cavita-
tion region and used same energy equation for both full
lm and cavitation regions. In recent years, more com-
prehensive THD analysis are developed by Paranjpe and
Han [5], Khonsari et al. [6], Ramesh et al. [7], and Shi and
Wang [8]. However all these studies do not include more
complex uids. The use of complex uids as lubricants is
getting more and more important to full the desires of
modern machines. For example, high polymer additives
are added in lubricating oils as a kind of viscosity index
improver so that viscosity variation with temperature de-
creases. On the other hand, addition of polymers exhibit
non-newtonian behaviour, leading to classical continuum
theory which becomes inapplicable.
The classical continuum uid mechanics neglects size
of uid particles to describe the ow of uids. The micro-
continuum uid mechanics accounts for intrinsic motion
of material constituents in order to extend the range of ap-
plication, for example, to describe the ow of complex u-
ids, polymer molecules are used in polymeric suspensions.
The classical continuum uid mechanics is generalized by
non-polar theory of uids, which is characterized by clas-
sical Cauchy stress. The micro-continuum uid mechanics
is generalized by polar theory of complex uids, which is
characterized by classical Cauchy stress as well as by cou-
ple stress [9]. Recently several studies are carried out to de-
scribe the ow of complex uids. The couple stress theory
of uids was presented by Stokes [10] to examine simplest
generalization of classical theory that would allow for po-
lar eects. This model denes the rotation eld in terms of
velocity eld; the rotation vector is equated to one-half of
curl of velocity vector. The couple stress uid model has
been widely used in lubrication for its relative mathemati-
cal simplicity. A study of a journal bearing lubricated with
a couple stress uids including elasticity of the linear is
presented by Mokhiamer et al. [11]. This showed that the
inuence of couple stress uid on the bearing characteris-
tics is signicantly apparent and that lubricants with cou-
ple stresses produce an increase in the load carrying ca-
pacity, decrease in attitude angle, the frictional force and
398 |Vishwanath B. Awati and Ashwini Kengangutti, Thermohydrodynamic analysis of journal bearings
side leakage. Lin [12] analysed the study of squeeze lm
characteristics of nite journal bearings assuming the lu-
bricant to be an incompressible uid with couple stress.
The studies showed that couple stress eects provide a
longer time to prevent journal bearing contact and result-
ing in longer bearing life. Das [13] studied the slider bear-
ings lubricated with couple stress uids in a magnetic eld
and observed that maximum load capacity increases with
an increase in both magnetic and couple stress parame-
ters. Also, the ratio of inlet to outlet lm heights for max-
imum load capacity increases with magnetic parameter
and decreases with couple stress parameter.
In most of the theoretical studies of lm lubrication it
was more or less assumed that the bearing surfaces could
be presented by smooth mathematical planes. But this ap-
pears to be an unrealistic assumption particularly in bear-
ing, working with small lm thickness. However, bearing
surfaces particularly after they have received some run –
in and wear, seldom exhibit a type of roughness approxi-
mated by this model. Christensen and Tonder investigated
the hydrodynamic lubrication of rough surfaces having -
nite width. In 1975 Christensen et al. [17] obtained a gen-
eralised Reynolds equation applicable to rough surface
by assuming that the lm thickness function is govern-
ing a stochastic process. Bujurke et al. [14] studied the ef-
fect of surface roughness on squeeze lm poroelastic bear-
ings with special reference to synovial joints. A simplied
mathematical model is developed for understanding com-
bined eect of surface roughness and couple stress. Sur-
face roughness eects in a short porous journal bearing
with a couple stress uid is studied by Naduvinamani et
al. [15]. Thermohydrodynamic analysis of journal bearings
with couple stress uids is analysed by Xiao et al. [16].
All the applications of couple stress models are limited to
isothermal case as noticed up to now.
Multigrid method is used to nd the solution more
accurately than the nite dierence method which used
in earlier study. Hydrodynamic bearings are usually as-
sumed to have a smooth surface. For most practical bear-
ings this is a good approximation since the inuence of
the roughness on the bearing performance is negligible.
Under normal operating conditions the surface roughness
heights are small compared to the thickness of the lubri-
cation lm. However, when surface roughness amplitudes
are of the same range as that of lubrication lm thickness,
the surface roughness eects can no longer be neglected.
Since most of the available literature on micro asperity lu-
brication is based on the smooth surface assumption, the
present work extends the existing literature to cover the ef-
fect of random surface roughness on micro asperity hydro-
dynamic lubrication. The aim of this paper is to present the
THD performance of journal bearings lubricated with cou-
ple stress uids with surface roughness eect. The energy
equation together with heat transfer equation and modi-
ed Reynolds equation were numerically solved by using
multigrid method. The various eects which include tem-
perature eld, load carrying capacity, friction force, fric-
tion coecient and end leakage are analysed in detail for
dierent couple stress parameters.
2Theoretical analysis
On the basis of Micro-continuum theory of Stokes [10],
body forces and body couples are neglected; the basic
equations of motion of uids with couple stresses can be
expressed as
˙
ρ+ρ vi,i= 0 (1)
ρ˙
vi=σji,j(2)
Mji,j+eijk σj k = 0 (3)
The constitutive equations for stress tensor and couple
stress tensor are given as [10]
σij = (−p+λ vk,k)δij +µ(vi,j+vj,i)−1
2eijk Mr k,r(4)
Mrk =1
3Mnn δrk + 4ηωk,r+ 4η′ωr,k(5)
where λ=−2µ/3. A nite journal bearing lubricated with
an incompressible couple stress uid is considered and is
shown in the Fig. 1. The mathematical formulation for lm
thickness with surface roughness is
H=h+hs=C(1 + εcos θ) + hs(θ,z,ξ)(6)
By the assumptions of hydrodynamic lubrication to
thin lms, the equations of motion are reduced as
∂p
∂x =µ∂2u
∂y2−η∂4u
∂y4(7a)
∂p
∂y = 0 (7b)
∂p
∂z =µ∂2w
∂y2−η∂4w
∂y4(7c)
The velocity boundary conditions at the shaft surface are
u|y=0 =U,v|y=0 = 0,w|y=0 = 0 ,
∂2u
∂y2y=0
= 0,∂2w
∂y2y=0
= 0 (7d)
Vishwanath B. Awati and Ashwini Kengangutti, Thermohydrodynamic analysis of journal bearings |399
Fig. 1: The schematic diagram of journal bearing
The velocity boundary conditions at the bushing surface
are
u|y=H= 0,v|y=H= 0,w|y=H= 0 ,
∂2u
∂y2y=H
= 0,∂2w
∂y2y=H
= 0 (7e)
Integrating Eqs. (7a) and (7c) with above boundary condi-
tions, the velocity components can be obtained as
u=U1−y
H+1
2µ
∂p
∂x y(y−H)
+2l21−cosh 2y−H
2lcosh H
2l−1 (8)
w=1
2µ
∂p
∂z y(y−H)
+2l21−cosh 2y−H
2lcosh H
2l−1 (9)
where l=η/µ
Substituting Eqs. (7) and (8) into Eq. (1), we get the fol-
lowing modied Reynolds equation in the form
∂
∂x f(l,H)
µ
∂p
∂x +∂
∂z f(l,H)
µ
∂p
∂z = 6U∂H
∂x (10)
where
f(l,H) = H3−12 l2H+ 24 l3tanh H
2l.(11)
Let f(hs)be the probability density function of
stochastic lm thickness hs. Taking stochastic average of
Eq. (10) with respect to f(hs), we get
∂
∂x Ef(l,H)
µ∂E(p)
∂x +∂
∂z Ef(l,H)
µ∂E(p)
∂z
= 6 UE ∂H
∂x (12)
where expectancy operator E(•)is dened by
E(•) =
∞
−∞
(•)f(hs)dhs.
The probability density function [17] is given as
f(hs) = 35
32c7(c2−hs2)3,−c<hs<c
0,elsewhere
where c= 3σ1.
This non dimensional modied Reynolds equation
based on a couple stress model [20] can be obtained by in-
troducing following dimensionless variables
θ=x
R,y*=y
H,z*=z
R,E(H*) = E(H)
C= 1 + εcos θ,
µ=µ
µ0
,ω=U
R,E(p*) = E(p)C2
µoωR2,l*=l
C.
The dimensionless form of modied Reynolds equa-
tion can be written as
∂
∂θ E(f*(l*,H*)
µ*av
)∂E(p*)
∂θ +∂
∂z*E(f*(l*,H*)
µ*av
)∂E(p*)
∂z*
=−6E(εsin θ)(13)
where
E(f*(l*,H*)) = E(H*3)−12l2E(H*)
+ 24l*3tanh E(H*)
2l*.(14)
The lubricant viscosity µ*
av in Eq. (13) is a cross-lm av-
erage value. Lubrication in a steady-state and an incom-
pressible couple stress uid is considered as shown in
Fig. 1. According to Tucker and Keogh [18], the viscosity
cross lm average can oer a close approximation for vis-
cosity in lubrication analysis.
The pressure boundary conditions are:
∂E(p*)
∂z*z*=0
=E(p*)z*=±B/2R= 0,E(p*)θ=−δ=E(p0*),
∂E(p*)
∂θ θ=θcav
=E(p*)θ=θcav
= 0 (15)
where θcav represents location of starting point for the cav-
itation zone.
The rst law of thermodynamics gives energy equation
in uids and is written as
ρDE
Dt =∇·(k∇T) + Φ−p∇·
⇀
V(16)
where Eis the specic internal energy of uid and Φis the
heat dissipation, it can be written as
Φ=p∇·
V+
σ:∇
V.(17)
400 |Vishwanath B. Awati and Ashwini Kengangutti, Thermohydrodynamic analysis of journal bearings
Substituting Eqs. (4) and (5) in Eq. (17), we get
Φ=λ(∇iVi)2+1
2µ(∇iVj+∇jVi)2
−η(∇r∇rekmn∇mVn)(ekij ∇iVj)(18)
Considering an incompressible couple stress uid and
lubrication in steady state, Eqs. (16) and (18) can be written
as
ρ cfu∂T
∂x +v∂T
∂y +w∂T
∂z =Kf
∂2T
∂y2+Φ(19)
where
Φ=µ∂u
∂y 2
+∂w
∂y 2−l2∂u
∂y
∂3u
∂y3+∂w
∂y
∂3w
∂w2
(20)
when ltends to zero, Eq. (20) becomes
Φ=µ∂u
∂y 2
+∂w
∂y 2(21)
The non dimensional modied energy equation can
be obtained by introducing following dimensionless vari-
ables as
u*=u
U,w*=w
U,T*=T
T0
,ρ*=ρ
ρ0
.
and the derivatives of u*,w*are given in Appendix A.
The dimensionless form of energy equation becomes
Peρ*u*∂E(T*)
∂θ +Γ*∂E(T*)
∂y*+ρ*w*∂E(T*)
∂z*
=1
E(H*2)
∂2E(T*)
∂y*2+Φ*(22)
where Pe=ρ0cfUC2
KfR
E(Γ*) =
−1
E(H*)⎡
⎢
⎣
∂
∂x*(E(H*)
y*
0
ρ*u*dy*) + ∂
∂z*(E(H*)
y*
0
ρ*w*dy*)⎤
⎥
⎦
β=µ0U2
KfT0
(23)
3Discretization of modied
Reynolds and Energy equations
The modied Reynolds equation and energy equation are
solved numerically by using one of the most ecient -
nite dierence method i.e. multigrid method. By using sec-
ond order nite dierence scheme, the derivative terms in
Eqs. (13) and (19) can be approximated as
Pi,j=A1Pi+1,j+A2Pi−1,j+A3Pi,j+1 +A4Pi,j−1+A5(24)
where
A0=r2(fi+0.5,j+fi=0.5,j+fi,j+0.5+fi,j−0.5)
A1=fi+0.5,j/µi+0.5,jA0,A2=fi−0.5,j/µi−0.5,jA0,
A3=fi,j+0.5/µi,j−0.5A0,A4=fi,j=0.5/µi,j=0.5A0
A5=−6∆z*2εicos θi,r=∆z*
∆θ .
The discretized boundary conditions becomes
pi,j−pi−1,j= 0,pi,j= 0,pi,j=p0i,j
The energy equation can be written in discretized form
as
Peρ*µ*T*
i,j−T*
i−1,j
θi−θi−1
+Γ*T*
i,j−T*
i−1,j
yi−yi−1
+ρ*w*T*
i,j−T*
i−1,j
zi−zi−1
=1
H*2
i,j−H*2
i−1,j
T*2
i,j−T*2
i−1,j
y2
i−y2
i−1
+Φ*
i(25)
The discretized temperature boundary conditions are
Ti,j=Tmixi,j,Ti,j−Ti,j−1= 0
3.1 Heat transfer equation
The heat transfer equation in bush for a steadily loaded
journal bearing is dened as
∂2E(T*)
∂r*2+1
r*
∂E(T*)
∂r*+1
r*2
∂E(T*)
∂θ2+∂2E(T*)
∂z*2= 0 (26)
The temperature boundary conditions at inlet groove
and central plane are respectively given as
E(T)*‖θ=−δ=E(T*
mix),∂E(T*)
∂z*‖z*=0 = 0 (27)
where E(T*
mix)is dened as homogeneous mixing temper-
ature of re-circulating oil [3].
At the oil-shaft interface and oil-bush interface, the
boundary conditions for THD becomes
q*=
2π
0
1
H*
∂T*
∂y*‖y*=ydθ = 0 (28)
∂T*
∂r*‖r*=1 =K(θ)R
KbH*
∂T*
∂y*‖y*=1 (29)
Vishwanath B. Awati and Ashwini Kengangutti, Thermohydrodynamic analysis of journal bearings |401
where K(θ)is the thermal conductivity of uid which is
equal to Kfin active zone and is variable in inactive zone
of the lm. In this area the thermal conductivity is given by
K(θ) = Ka−B(θ)
B(Ka−Kf)(30)
In Eq. (30), represents the eective bearing length cov-
ered with lubricant. It can be obtained by using continuity
equation.
B(θ) = H*(θcav)
H*(θ)B(31)
For the outer surface and lateral surface of bush, the free
convection and radiation hypothesis gives
∂T*
∂r*‖r*=R*
b=−Nu(T*‖r*=R*
b−T*
a)(32)
∂T*
∂z*‖z*=±B/2R=−Nu(T*‖z*=±B/2R−T*
a)(33)
where Nu =hbR/Kb.
3.2 Viscosity- temperature relation
The variation of lubricant viscosity with temperature can
be represented by Walther equation and written as
log log µ
ρ+ 0.6=k1−k2log(T)(34)
where k1and k2are constants.
3.3 Load carrying capacity
The uid lm force and its attitude angle are given as:
W=W2
1+W2
2,ϕ= arcsin W2
W(35)
where W1=−2R4µ0ω
C22R
0B/2R
0E(p) cos θdzdθ,W2=
−2R4µ0ω
C22R
0B/2R
0E(p) sin θdzdθ.
3.4 Friction force and friction coecient
The shear stress along journal surface is expressed as [11]
τ=µ∂u
∂y ‖y=0 −η∂3u
∂y3‖y=0 (36)
The friction force can be obtained by integrating shear
stress around journal surface and it is written as
Fr=2ωµ0R3
C
2π
0
B/2R
0µ1
H+h
2R
∂P
∂θ dzdθ.(37)
The coecient of friction can be obtained by dividing fric-
tion force by lm force
Cf=FfR
WC (38)
3.5 Side leakage flow
The side leakage ow for a journal bearing is given as
Q=−CR2ω
6
2π
0
1
µ
∂E(p)
∂z z=B/2RH*3−12H*l2+ 24l3tanh H*
2ldθ.
(39)
4Multigrid method
The uid velocity components and their derivatives are
evaluated for the assumed initial pressure distribution.
Also, energy equation and heat transfer equation are
solved simultaneously, thus producing temperature eld
in oil and bushing. Further, the cross lm average temper-
ature can be obtained as
T*av (θ,z*) =
1
0
T*av (θ,y*,z*)dy*(40)
The modied Reynolds equation is solved by Multigrid
method then new pressure distribution is yielded. The dis-
cretized equation can be written in more general form as
Ahuh=fh
where Ais a linear operator, uis the solution, fis the right
hand side and his the mesh size.
The algorithm for Vcycle multigrid method as dis-
cussed in [25–29] and is given as
•Relaxing Ahuh=fh, 2 times with initial guess uh
0
•Compute the residual rh=fh−Ahuh
•Restrict f2h=I2hrh,to zeros to get u2h
•Relaxing A2hu2h=f2h, 2 times with guess u2h
•Compute the residual, r2h=f2h−A2hu2h
•Restrict, f4h=I4h
2hr2h, to zeros to get u4h
.
.
.
•Solve the error equation, eH= (AH)−1fH
402 |Vishwanath B. Awati and Ashwini Kengangutti, Thermohydrodynamic analysis of journal bearings
.
.
.
•Interpolate e2h←I2h
4hu4h.
•Correcting the ne grid approximation u2h←u2h+
e2h
•Interpolate eh←Ih
2hu2h.
•Correcting uh←uh+eh.
The initial approximations are taken on nest grid and
two iterations of Gauss-Seidel method is applied on nest
grid for smoothing errors. To transfer the calculated resid-
ual to next coarse grid level, the half weighting restric-
tion is used. Repeating the procedure till coarsest grid level
reaches to single point. The solution is obtained at coars-
est level. The bilinear interpolation is applied to prolon-
gation of solution obtained from coarsest level to the next
ne grid level and then applying two time Gauss- Seidel it-
erations. Repeating the procedure until original nest grid
level is reached. This is referred to as one V-cycle. The iter-
ations are repeated until it reaches to convergence.
Flowchart
Table 1: Bearing variables used in the analysis
Bearing length, B (m) 0.028
Radial clearance, C (m) 5.2×10−5
Bush specic heat, cb(J kg−1K−1) 465
Lubricant specic heat, cf(J Kg−1K−1) 2022
Shaft radius, R (m) 0.04
Bush outside diameter, Rb(m) 0.043
Heat transfer coecient, hb(W m−2K−1) 230
Air thermal conductivity, Ka(W m−1K−1) 0.025
Bush thermal conductivity, Kb(W m−1K−1) 49
Lubricant thermal conductivity, Kf(W m−1K−1) 0.141
Rotational speed, n (rpm) 3300
Lubricant inlet pressure, P0(Pa) 2×104
Lubricant inlet temperature, T0(K) 333
Bush density, ρb(kg m−3) 7800
Inlet attitude angle, (∘) 24
k17.8212
k23.0145
5Results and discussion
The eect of surface roughness for thermo-hydrodynamic
analysis of journal bearings extended to couple stress lu-
bricants with high polymer additives is presented. The
modied Reynolds and energy equations are solved by -
nite dierence method i.e. multigrid method.
Couple stress mainly depends on the polar additives in
the lubricant. The presence of micro structure in the uid
lm inuences the couple stress of the uid which in turn
increases the uid pressure. The increase in lm pressure
results in an increase in the load carrying capacity. It is ob-
served that lm pressure is high for a low eccentricity ratio.
This shows that load carrying capacity is high for a low ec-
centricity ratio. Minimum uid lm thickness increases for
the bearing operate under couple stress lubricants. Sti-
ness and damping coecients are larger than the couple
stress uids lubricated bearings and threshold speed im-
proves when the bearings are lubricated by couple stress
lubricants.
The temperature contour in the centre plane at the ec-
centricity ratio 0.3 and roughness parameter 0.1 is shown
in Fig. 2. The journal bearing variables used in computer
simulations to the ow with eccentricity ratio for dier-
ent couple stress parameters are listed in Table 1. The
shaft temperature and the maximum oil temperature are
aected due to couple stress in uids for dierent eccen-
tricity ratios are shown in Figs. 3 and 4. It is observed
that maximum oil temperature and shaft temperature de-
creases on increasing l*, and it increases with the addi-
tion of roughness parameter 0.1, the inuence of couple
stress is more signicant at a higher eccentricity ratio is
Vishwanath B. Awati and Ashwini Kengangutti, Thermohydrodynamic analysis of journal bearings |403
Fig. 2: Temperature contour in the centre plane at eccentricity ratio
0.3 and roughness parameter 0.1
Fig. 3: Shaft temperature versus eccentricity ratio for dierent val-
ues of the couple stress parameter with roughness parameter 0.1
predicted. Fig. 5 shows that for any eccentricity ratio, load
capacity increases with increasing l*and roughness pa-
rameter 0.1 load capacity increases for all values of l*.
The friction force remains in increasing trend with in-
creasing l*and it increases with roughness parameter 0.1
is depicted in Fig. 6, while Fig. 7 shows that friction coe-
cient decreases with increase in l*but it is seen that friction
coecient increases for the roughness parameter 0.1. The
variation in the side leakage ow with eccentricity ratio for
dierent couple is showed in Fig. 8. It shows that the side
leakage ow of couple stress with roughness parameter 0.1
is almost same as that for dierent couple stress parameter
l*.
To verify the accuracy of the present model, the cir-
cumferential pressure distribution of the lubricant lm
with smooth case is compared with that of surface rough-
Fig. 4: Maximum temperature versus eccentricity ratio for dierent
values of the couple stress parameter with roughness parameter 0.1
Fig. 5: Load capacity versus eccentricity ratio for dierent values of
the couple stress parameter with roughness parameter 0.1
Fig. 6: Friction force versus eccentricity ratio for dierent values of
the couple stress parameter with roughness parameter 0.1
404 |Vishwanath B. Awati and Ashwini Kengangutti, Thermohydrodynamic analysis of journal bearings
Fig. 7: Friction coecient versus eccentricity ratio for dierent val-
ues of the couple stress parameter with roughness parameter 0.1
Fig. 8: Side leakage versus eccentricity ratio for dierent values of
the couple stress parameter with roughness parameter 0.1
ness. Multigrid method is used to minimize the errors and
nd the solution in fast and ecient way. It is observed
that, the shape of temperature contour for is same as for
l*= 0.3but in case of temperature eld for l*= 0.3it is
lower than that for l*= 0.0. This shows that couple stress
uids yields less heat dissipation than that of Newtonian
uids.
6Conclusion
The following conclusions are made from this study.
1. The bearing temperature eld is aected due to the
application of polymeric additives and the presence
of surface roughness. The maximum oil temperature
and the shaft temperature become lower at higher
eccentricity ratios.
2. The couple stress eects produce a higher load ca-
pacity, higher friction force and a lower friction co-
ecient but the presence of surface roughness en-
hances the coecient of friction. The side leakage
ow almost remains constant for dierent values of
couple stress parameter and surface roughness.
3. The rate of convergence of the method used is more
compared to the other nite dierence methods.
4. The surface roughness eect case is comparablewith
the earlier smooth case.
AAppendix
Velocity components and derivatives
Introducing the dimensionless variables y*=y
H,H*=H
C=
1ϵcos θand l*=l
Cin Eqs. (8) and (9), it becomes
u*= (1 −y*) + 1
2µ*
∂p*
∂θ H*2y*(y*−1)
+2l*1−cosh 2H*y*−H*
2l*cosh H*
2l*−1 (i)
∂u*
∂y*=−1 + 1
2µ*
∂p*
∂θ H*2(2y*−1)
−2l*H*sinh 2H*y*−H*
2l*cosh H*
2l*−1,(ii)
∂3u*
∂y*3=−H*3
µ*l*
∂p*
∂θ sinh 2H*y*−H*
2l*cosh H*
2l*−1
(iii)
w*=1
2µ*
∂p*
∂z*H*2y*(y*−1)
+2l*1−cosh 2H*y*−H*
2l*cosh H*
2l* (iv)
∂w*
∂y*=1
2µ*
∂p*
∂z*H*2(2y*−1)
−2l*H*sinh 2H*y*−h*
2l*cosh H*
2l*−1(v)
∂3w*
∂y*3=−H*3
µ*l*
∂p*
∂z*sinh 2H*y*−H*
2l*cosh H*
2l*−1
(vi)
Vishwanath B. Awati and Ashwini Kengangutti, Thermohydrodynamic analysis of journal bearings |405
Nomenclature
BBearing width (m)
cbBush specic heat (J kg−1K−1)
cfLubricant specic heat (J kg −1K−1)
C Radius clearance (m)
eijk Permutation tensor
FfFriction force (N)
hDimensional lm thickness (m)
h*Dimensionless lm thickness, h*=1+ϵcosθ
hbHeat transfer coecient (Wm−2K−1)
KaAir thermal conductivity (Wm−1K−1)
KbBush thermal conductivity (Wm−1K−1)
KfLubricant thermal conductivity (Wm−1K−1)
lCharacteristics length of the additives, (m)
l*Couple stress parameter, l*=l/C
Mij Couple stress tensor in uid
nRotational speed (rpm)
pHydrodynamic pressure (Pa)
p*Dimensionless hydrodynamic pressure, p*=pC2/R2
p0Lubricant inlet pressure (Pa)
QSide leakage ow (m3 s−1)
RJournal radius (m)
R*bDimensionless bush outside radius, R*b=Rb/R
TTemperature (K)
T*Dimensionless temperature, T*=T/T0
T0Lubricant inlet temperature (K)
T*aDimensionless ambient temperature, T*a=Ta/T0
T*max Dimensionless maximum bearing temperature,
T*max=Tmax/T0
T*mix Dimensionless mixing temperature of recirculating
uid and supply oil, T*mix=Tmix/T0
UShaft linear speed (m s−1)
u*,v*,w*Dimensionless components of the lubricant ve-
locity in the x,yand zdirections, respectively,
u*=u/U,v*=v/U,w*=w/U
W1, W2 Fluid lm forces in the xand zdirections (N)
x,y,zCoordinates (m)
ϵEccentricity ratio
Inlet attitude angle
θ,y*,z*Dimensionless coordinates, q=x/R,y*=y/h,z*=z/R
θ,z*,rDimensionless coordinates, q=x/R,z*=z/R,r*=r/R
θcav Location of the starting of cavitation zone
η,η′Material constant responsible for the couple stress
property (N s)
µLubricant viscosity (Pa s)
µ0Inlet lubricant viscosity (Pa s)
µ*Dimensionless lubricant viscosity (m*=m/m0)
ρLubricant density (kg m3)
ρbBush density (kg m3)
ρ0Inlet lubricant density (kg m3)
ρ*Dimensionless lubricant density, r*=r/r0
τViscous shear stress (Pa)
σij Stress tensor in uid
ωRotation vector (rad s−1)
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