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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),

ISSN 0976 – 6359(Online), Volume 5, Issue 11, November (2014), pp. 31-43 © IAEME

31

THERMOHYDRODYNAMIC ANALYSIS OF PLAIN

JOURNAL BEARING WITH MODIFIED VISCOSITY -

TEMPERATURE EQUATION

Kanifnath Kadam, S.S. Banwait, S.C. Laroiya

National Institute of Technical Teachers Training & Research, Sector 26, Chandigarh

ABSTRACT

The purpose of this paper is to predict the temperature distribution in fluid-film, bush housing

and journal along with pressure in fluid-film using a non-dimensional viscosity-temperature

equation. There are two main governing equations as, the Reynolds equation for the pressure

distribution and the energy equation for the temperature distribution. These governing equations are

coupled with each other through the viscosity. The viscosity decreases as temperature increases. The

hydrodynamic pressure field was obtained through the solution of the Generalized Reynolds

equation. This equation was solved numerically by using finite element method. Finite difference

method has been used for three dimensional energy equations for predicting temperature distribution

in fluid film. For finding the temperature distribution in the bush, the Fourier heat conduction

equation in the non- dimensional cylindrical coordinate has been adopted. The temperature

distribution of the journal was found out using a steady-state unidirectional heat conduction

equation.

Keywords: Journal Bearings. Reynolds Equation, Thermohydrodynamic Analysis, Viscosity-

Temperature Equation.

1. INTRODUCTION

A Journal bearing is a machine element whose function is to provide smooth relative motion

between bush and journal. In order to keep a machine workable for long periods, friction and wear of

mating parts must be kept low. The plain journal bearings are used for high speed rotating

machinery. This high speed rotating machinery fails due to failure of bearings. Due to the heavy load

and high speed, the temperature increases in the bearing. For prediction of temperature and pressure

distribution in bearing, accurate data analysis is necessary. An accurate thermo hydrodynamic

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analysis is required to find the thermal response of the lubricating fluid and bush. Therefore, a need

has been felt to carry out further investigation on the thermal effects in journal bearings.

By considering thermal effects B. C. Majumdar [1] obtained a theoretical solution for

pressure and temperature of a finite full journal bearing. D. Dowson and J. N. Ashton [2] computed a

solution of Reynolds equation for plain journal bearing configuration. Operating characteristics were

evaluated from the computed solutions and results were presented graphically. The optimum design

objective was stated explicitly in terms of the operating characteristics and was minimized within

both design and operative constraints. J. Ferron et al. [3] solved three dimensional energy, three

dimensional heat conduction equation. They computed mixing temperature by performing a simple

energy balance of recirculating and supply oil at the inlet. H. Heshmat and O. Pinkus [4]

recommended that the mixing occurs in the thin lubricant layer attached on the surface of the journal.

This implies that no mixing occurs inside the grooves. An excellent brief review of thermo

hydrodynamic analysis was presented by M. M. Khonsari [5] for journal bearings. H. N. Chandrawat

and R. Sinhasan. [6] simultaneously solved the generalized Reynolds equation along with the energy

and heat conduction equations. They studied the effect of viscosity variation due to rise in

temperature of the fluid film. Also they compared Gauss- Siedel iterative scheme and the linear

complementarity approach. M. M. Khonsari and J. J. Beaman [7] presented thermohydrodynamic

effects in journal bearing operating with axial groove under steady-state loading. In this analysis, the

recirculating fluid and the supply oil was considered. S. S. Banwait and H. N. Chandrawat [8]

proposed a non-uniform inlet temperature profiles and for correct simulation. They considered the

heat transfer from the outlet edge of the bush to fluid in the supply groove. L. Costa et al. [9]

presented extensive experimental results of the thermohydrodynamic behavior of a single groove

journal bearing. And developed the influence of groove location and supply pressure on some

bearing performance characteristics. M. Tanaka [10] had shown a theoretical analysis of oil film

formation and the hydrodynamic performance of a full circular journal bearing under starved

lubrication condition. Sang Myung Chun and Dae-Hong Ha [11] examined the effect on bearing

performance by the mixing between re-circulating and inlet oil. M. Tanaka and K. Hatakenaka [12]

developed a three-dimensional turbulent thermohydrodynamic lubrication model was presented on

the basis of the isothermal turbulent lubrication model by Aoki and Harada, this model was different

from both the Taniguchi model and the Mikami model. P. B. Kosasih and A. K. Tieu [13] considered

the flow field inside the supply region of different configurations and thermal mixing around the

mixing zone above the supply region for different supply conditions. Flows in the thermal mixing

zone of a journal bearing were investigated using the computational fluid dynamics. The complexity

and inertial effect of the flows inside the supply region of different configurations were considered.

M. Fillon and J. Bouyer [14] presented the thermohydrodynamic analysis of plain journal bearing

and the influence of wear defect. They analyzed the influence of a wear defect ranging from 10% to

50% of the bearing radial clearance on the characteristics of the bearing such as the temperature, the

pressure, the eccentricity ratio, the attitude angle or the minimum thickness of the lubricating film. L.

Jeddi et al. [15] outlined a new numerical analysis which was based on the coupling of the

continuity. This model allows to determine the effects of the feeding pressure and the runner velocity

on the thermohydrodynamic behavior of the lubricant in the groove of hydrodynamic journal bearing

and to emphasize the dominant phenomena in the feeding process. S. S. Banwait [16] presented a

comparative critical analysis of static performance characteristics along with the stability parameters

and temperature profiles of a misaligned non-circular of two and three lobe journal bearings

operating under thermohydrodynamic lubrication condition. U. Singh et al. [17]

theoretically

performed a steady-state thermohydrodynamic analysis of an axial groove journal bearing in which

oil was supplied at constant pressure. L. Roy [18] theoretically obtained steady state

thermohydrodynamic analysis and its comparison at five different feeding locations of an axially

grooved oil journal bearing. Reynolds equation solved simultaneously along with the energy



33

equation and heat conduction equation in bush and shaft. B. Maneshian and S. A. Gandjalikhan

Nassab [19] presented the computational fluid dynamic techniques. They obtained the lubricant

velocity, pressure and temperature distributions in the circumferential and cross film directions

without considering any approximations. B. Maneshian and S. A. Gandjalikhan Nassab [20]

determined thermohydrodynamic characteristics of journal bearings with turbulent flow using

computational fluid dynamic techniques. The bearing had infinite length and operates under

incompressible and steady conditions. The numerical solution of two-dimensional Navier–Stokes

equation, with the equations governing the kinetic energy of turbulence and the dissipation rate,

coupled with then energy equation in the lubricant flow and the heat conduction equation in the

bearing was carried out. N. P. Mehata et al. [21] derived a generalized Reynolds equation for

carrying out the stability analysis of a two lobe hydrodynamic bearing operating with couple stress

fluids that has been solved using the finite element method. N. P. Arab Solghar et al. [22] carried out

experimental assessment of the influence of angle between the groove axis and the load line on the

thermohydrodynamic behavior of twin groove hydrodynamic journal bearings. Mukesh Sahu et al.

[23] used computational fluid dynamic technique for predicting the performance characteristics of a

plain journal bearing. Three dimensional studies have been done to predict pressure distribution

along journal surface circumferentially as well as axially. E. Sujith Prasad et al. [24] modified

average Reynolds equation that includes the Patir and Cheng’s flow factors, cross-film viscosity

integrals, average fluid-film thickness and inertia term. This was used to study the combined

influence of surface roughness, thermal and fluid-inertia on bearing performance. Abdessamed

Nessil et al. [25] presented the journal bearings lubrication aspect analysis using non-Newtonian

fluids which were described by a power law formula and thermohydrodynamic aspect. The influence

of the various values of the non- Newtonian power-law index, �, on the lubricant film and also

analyzed the journal bearing properties using the Reynolds equation in its generalized form.

The aim of this work is to predict the pressure and temperature distribution in plain journal

bearing. Thermohydrodynamic analysis of a plain journal bearing has been presented with an

improved viscosity-temperature equation. The equation has been modified by authors to predict the

proper relation between viscosity and temperature for forecasting the correct temperature in plain

journal bearing. The pressure and temperature distribution in the journal bearing which was almost

equal to the temperature obtained by experimental results of Ferron J. et al. [3]. The results have been

validated by comparison with experimental results of Ferron J. et al. [3]. and show good agreement.

2. GOVERNING EQUATIONS

In this present work three dimensional energy equation, heat conduction and Reynolds

equation were considered for analysis of thermohydrodynamic analysis of a plain journal bearing.

This bearing having a groove of 18° extent at the load line. The geometric details of the journal

bearing system are illustrated in Fig 1. Single axial groove has been used for supplying fluid to the

bearing under, negligible pressure. The model based on the simultaneous numerical solution of the

generalized Reynolds and three dimensional energy equations within the fluid-film and the heat

transfer within the bush body.

2.1 Generalized Reynolds Equation

Navier derived the equations of fluid motion for a viscous fluid. Stokes also derived the

governing equations of motion for a viscous fluid, and the basic equations are known as Navier-

Stokes equations of motion. The Reynolds equation is a simplified version of Navier-Stokes

equation. A partial differential equation governing the pressure distribution in fluid film lubrication

is known as the Reynolds equation. This equation was first derived by Osborne Reynolds. The

hydrodynamic pressure and the velocity field within fluid flow were accurately described through the



34

solution of the complete Navier-Stokes equations. This has provided a strong foundation and basis

for the design of hydrodynamic lubricated bearings.

This paper is to deal with the finite element analysis of Reynolds’ equation. It will show how

the finite element technique is used to form an approximate solution of the basic Reynolds’ equation.

The analysis has been incorporated in a computer programme and results from it were presented. A

Reynolds equation in the following dimensionless form governs the flow of incompressible

isoviscous fluid in the clearance space of a journal bearing system. This equation in the Cartesian

coordinate system is written as,

12 2

0

3 3 hp p Fh F h F h h

tFα

∂ ∂ ∂ ∂∂ ∂+ = − +∂α ∂β ∂α∂ ∂β ∂

(1)

where the non-dimensional functions of viscosity 0 1 2, andF F F are defined by,

1 1 1

10 1 2

00 0 0

; ; andFdz z z

F F dz F z dzFµ µ µ

= = = −

∫ ∫ ∫ (2)

The non-dimensional functions of viscosity 0 1 2, andF F F report for the effect of variation in fluid

viscosity across the film thickness. And non dimensional minimum film thickness is given by,

1 cos sinj jh X Zα α= − − (3)

The above equation (1) was solved to satisfy the following boundary and complementarity

conditions:

i. On the bearing side boundaries,

( ), 0pβ λ= ± = (4)

ii. On the supply groove boundaries,

sp p= (5)

iii. In the positive pressure region, Positive pressures will be generated only when the fil

thickness is thin,

0, 0Q p= > (6)

iv. In the cavitated region,

0, 0, 0p

Q pα

∂< = =

∂ (7)

Solution of Eq. (1) with above boundary and complementary conditions gives pressure at each node.

2.2 Viscosity-Temperature Equation for predicting temperature distribution in bearings

The viscosity of fluid film was extremely sensitive to the operating temperature. With

increasing temperature the viscosity of oils falls rapidly. In some cases the viscosity of oil can fall by

about 80% with a temperature increase of 25°C. From the engineering viewpoint it is important to

know the viscosity value at the operating temperature since it determines the lubricant film thickness



35

separating two surfaces. The fluid viscosity at a specific temperature can be either calculated from

the viscosity-temperature equation or obtained from the viscosity-temperature ASTM chart.

2.2.1 Viscosity-Temperature Equations There were several viscosity-temperature equations available; some of them were purely

empirical whereas others were derived from theoretical models. The Vogel equation was most

accurate. In order to keep a machine workable for long periods, friction and wear of its parts must be

kept low. For effective lubrication, fluid must be viscous enough to maintain a fluid film under

operating conditions. Viscosity is the most important property of the fluid, which utilized in

hydrodynamic lubrication. The coefficient of viscosity of fluid and density changes with

temperature. If a large amount of heat is generated in the fluid film, the thickness of fluid film

changes with respect to temperature and viscosity. The viscosity of oil decreases with increasing

temperature. Hence, the change in viscosity cannot be ignored. Due to viscous shearing of fluid

layer, heat is generated; as significance, high temperatures may be anticipated. Under this condition

the fluid can experience a variation in temperature, so that it is necessary to predict the bearing

temperature and pressure.

Therefore, a need has been felt to carry out further investigations on analysis of the thermal

effects in journal bearings, so the viscosity-temperature relation given by Ferron J. et al. [3] has been

modified. The viscosity µ is a function of temperature and it was assumed to be dependent on

temperature. The viscosity of the lubricant was assumed to be variable across the film and around the

circumference. The variation of viscosity with the temperature in the non-dimensional two degree

equation was described by Ferron J. et al. [3]; this equation was expressed as,

0 1 20

2

f fk k T k Tµ

µµ

= = − + (8)

The authors modified and developed a two degree viscosity-temperature relation in to three

degree polynomial viscosity-temperature relation. This modified equation as illustrated below,

0 1 2 30

2 3

f f fk k T k T k Tµ

µµ

= = − + − (9)

J. Ferron et al. [3] used the viscosity coefficients, k0 = 3.287, k1 = 3.064, k2 = 0.777 while the

authors considered the following modified viscosity coefficients, k0 = 3.1286, k1 = 2.4817,

k2 = 1.1605 and k3=0.3266. The polynomial equation was found out for getting improved results.

Results obtained from viscosity-temperature equation which was developed by authors’ gives good

results when compared with experimental results of J. Ferron et al. [3]. This temperature distribution

in plain journal bearing shows very slight variation between temperature obtained by authors and

temperature obtained by J. Ferron et al. [3]. At different load the computed maximum bush

temperature and pressure are nearly equal for 1500, 2000, 3000 and 4000 rpm. The authors have

found during their investigation that the developed viscosity-temperature equation gives very close

values of the maximum bush temperature when compared with the experimental results of J. Ferron

et al. [3] at all above speeds. To verify the validity of the above equations and the computer code, the

results from the above analysis was compared with experimental values of J. Ferron et al. [3]

bearing.

2.3 Three dimensional energy equation for temperature distribution in bearing

The solution of energy equation needs the pressure field established from solution of

Reynolds equation. It is very important to carry out a three-dimensional analysis to accurately predict

the temperature distribution in bearings. Accurate prediction of various bearing characteristics, like



36

temperature distribution, is very important in the design of a bearing. The heat flows inside the solid

parts, such as the bearing and the shaft, and finally dissipates in the air. The total amount of heat that

flows out by convection and conduction is equal to the total amount of heat generated. Temperature

distribution in fluid-film is given by three-dimensional energy equation. Fluid temperature has been

obtained by solving the following three-dimensional energy equation which has been modified using

thin-film approximation and changing the shape of the fluid film into a rectangular field,

22 22

2

1( )

TT T T ff f fh u vh u w z u D Pe e

z z zhz

µα β α

∂∂ ∂ ∂∂ ∂ ∂+ν + − = + +∂ ∂ ∂ ∂ ∂ ∂

∂

(10)

The non-dimensional effective inverse Peclect number (eP ) and Dissipation number (

eD ) are

as follows,

( ) ( )22,

c

f j

p rp j

kP De e

C TC c

µ ω

ρρ ω= =

(11)

Values of the non-dimensional velocity components in circumferential and axial direction are

as follow,

2

1

00 00 0

1 zz zp z F d z d zu h d z

FFα µ µ µ

∂= − + ∫∫ ∫∂

(12)

1

0

2

0 0

z zp z F d zv h d z

Fβ µ µ

∂= −∫ ∫∂

(13)

The continuity equation is partially differentiated with respect to z to determine the non-

dimensional radial component of velocity ( w ) as,

2

20

w u v u hh z

z z zz

α β α

∂ ∂ ∂ ∂ ∂ ∂ ∂+ + − =

∂ ∂ ∂∂ ∂ ∂ ∂

(14)

Integrate the above equation with finite difference method considering the following

boundary conditions,

0 at 0 and at 1h

w z w zα

∂= = = =

∂ (15)

The three dimensional energy equations have been solved with the following boundary

conditions,

(i) On the fluid–journal interface ( 1)z =

f jT T= (16)

(ii) On the fluid–bush interface ( 0)z = ,

f bT T= (17)



37

2.4 Thermal analysis of heat conduction equation for Bush-Housing

Heat conduction analysis was performed to determine the bush temperatures. The Fourier

heat conduction equation in the form of non-dimensional cylindrical coordinate form has been solved

for the temperature distribution in the bush and is given below,

2 2 2

2 2 2 2

1 10

T T T Tb b b br r

r rβ α

∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂∂

(18)

Using following boundary conditions, heat conduction equation was solved.

i. On the interface of fluid–bush 1( 0, )z r R= = ,

Continuity of heat flux gives,

1| | 0

f fbb

k TTk

r zc hr R z

∂∂ = −∂ ∂ = =

(19)

ii. On the outer part of the bush housing 2

( )r R= , The free convection and radiation

hypothesis gives,

22

||

h RT abb T Tb ar Rkr br R

∂ = − −=∂ =

(20)

iii. On the lateral faces of the bearing ( )β λ=± ,

||

h RT abb T Tb akb β λβ λ

β

=±=±

∂ = − −∂

(21)

iv. At the outlet edge of bearing pad, free convection of heat flow from bush to fluid in the

supply groove gives,

|

( )

e

h RfbTb T Tb skbα αα

=

∂ = − −∂

(22)

eα = Circumferential coordinate of the outlet edge of bearing.

v. At the inlet edge of the bearing ( )iα α= and at the fluid supply point on the outer surface,

2|r R

T Tb s=

= (23)

In addition, a free convection of heat between fluid and housing has been assuming,

( )|

h RfbTb T Tb skbiα α α

−∂ = −∂ =

(24)

Where i

α = circumferential coordinate of the inlet edge of bearing.



38

2.5 Heat conduction equation for Journal

For finding the temperature distribution in journal, the following assumptions were made,

i. Conduction of heat in the axial direction.

ii. Journal temperature does not vary in radial or circumferential direction at any section.

iii. Heat flows out of the journal from its axial ends.

Hence the following steady state unidirectional heat conduction equation was used for a

journal,

2

02

T jk y A q

j jy

∂∆ + ∆ =

∂

(25)

Where q∆ = the heat input to the element ( )q y∆ ; y∆ = the length of element. Above equation

reduces to the following non-dimensional form,

2

02

Tjqπ

β

∂+ =

∂

(26)

where q is the non-dimensional heat input to journal per unit length,

2

0

1f f

j

k Tq d

hc k z

πα

∂= − ∫ ∂

(27)

The above equations have been solved with the following boundary condition,

At the axial ends, i.e. β λ= ± ,

||

h RT ajjT Tj ak j β λ

β λβ

= ± = ±

∂= − −

∂

(28)

2.6 Thermal mixing of fluid in a groove

It was not possible for the experimenters to maintain the inlet fluid temperature at a constant

value. Because of low supply pressures and high fluid viscosities, the inlet fluid temperature would

rise. Thermal mixing analysis of hot recirculating and incoming cold fluid from supply groove was

used to calculate the fluid temperature at the inlet of the groove. Energy balance equation is used to

estimate the mean temperature of the fluid in a groove.

In this work, the overall energy balance equation is expressed in terms of mean temperature, Tm,

Q T Q T Q Tm re re s s= + (29)

Where re

T - recirculating hot fluid, For the unit length of bearing,

( )1

0

Q h u d z= ∫ (30)

Q Q Qs re= − (31)



39

( )1

0L

Q C h u d zre = ∫ (32)

( )1

0LT Q C h u T d zre re f= ∫

(33)

Mean temperature m

T related to the assumed temperature distribution, ( )fT z across the fluid

film at the inlet of the bearing pad as below,

( )1

0

fT T z d zm = ∫

(34)

3. SOLUTION PROCEDURE

The overall solution scheme for thermohydrodynamic analysis of plain journal bearing is

depicted in Fig 2. The non dimensional coefficient of viscosity has been found out. Reynolds

equation solved by finite element method for obtaining pressure distribution in the fluid-film by

iterative technique. The negative pressure nodes were set to zero and attitude angle was modified till

convergence was achieved. Pressure and temperature fields for the initial eccentricity ratio have been

recognized. The load capacity of the journal bearing was calculated by iterative method. Values of

the fluid film velocity components were calculated in circumferential, axial and radial directions.

Coefficient of contraction of fluid-film was determined. Coefficient of contraction was assumed as

unity in positive pressure zone. The mean temperature of the fluid was calculated. By using finite

difference method three dimensional energy equation was solved for temperature distribution in

fluid-film. Heat conduction equation was solved for determination of temperature distribution in

bush housing. The above procedure was repeated till convergence was achieved. One dimensional

heat conduction equation was used for temperature distribution in journal. The journal temperature

was revised after obtaining the converged temperature for fluid and bush. The energy and Fourier

conduction equations were simultaneously solved with revised journal temperature. All the above

steps were repeated until the convergence was achieved. Using modified non dimensional viscosity-

temperature relation the non dimensional viscosity was found out and modified until convergence

was achieved. After convergence achieved the temperature of fluid, bush and journal was found. For

the next value of the eccentricity ratio once the thermohydrodynamic pressure and temperature have

been established. The data used for computation of pressure and temperature in fluid, bush and

journal were depicted in Table 1.

4. RESULTS AND DISCUSSION

Numerical calculations were performed by writing a computer program in C. The non-

dimensional governing equations were discretized for numerical solution. The global iterative

scheme was used for solving these equations. A mesh discretization for fluid film and bush with 68

nodes in the circumferential direction, 16 nodes in the axial direction and 16 nodes across the film

thickness and 16 nodes across the radius of bush thickness. For thermohydrodynamic analysis of

plain journal bearing the input parameters has been taken from Table 1. The present data was

assumed for aligned plain journal bearing. It was assumed that temperature of fluid equal to

temperature of bush at the fluid–bush interface. Journal temperature is also equal to temperature fluid

at the fluid–journal interface. The condition of mixing the recirculating fluid with the supply fluid

was also considered. Fig. 3 and Fig. 4 depicts the distribution of the maximum bush temperature

obtained with different eccentricity ratio for different speeds of plain journal bearing. The



40

experimental results of J. Ferron et al. [3] were nearly equal to theoretical results of authors as per

the modified viscosity-temperature equation. Fig. 5 and Fig. 6 predict the circumferential

temperature distribution in the mid-plane of fluid-bush interface. Theoretical predictions and

experimental results of J. Ferron et al. [3] exhibit a similar pattern, the predicted maximum

temperature value and their locations are reasonably very close to the measured values of J. Ferron et

al. [3].

Pressure variation in mid plane of plain journal bearing for various speeds and loading

conditions were shown in the Fig. 7 and Fig. 8. In the authors developed model pressure distribution

was very close to the experimental values given by J. Ferron et al. [3]. The mean journal temperature

has been computed along axial direction. Fig. 9 depicts load versus mean journal temperature at 2000

and 4000 rpm for two different loads as 4000N and 6000N respectively. The radial temperature was

negligible in the present work. Journal temperature along axial direction of the journal varies by

about one degree for 2000 rpm and very close to the 4000 rpm at 4000 N and 6000 N loads

respectively. A theoretical result predicted by authors’ modified viscosity-temperature equation gives

good agreement as compared with published experimental results of J. Ferron et al. [3].

5. CONCLUSIONS

On the basis of results and discussions presented in the earlier sections, the following major

conclusions are drawn:

• The developed viscosity-temperature equation for this work is more appropriate.

• The maximum pressure is noted at minimum film thickness of fluid.

• The temperature of fluid-film increases with increase in load and speed of shaft.

• Due to thermal effects the eccentricity ratio, attitude angle and side flow also changes.

• The effect of mixing of recirculating and supply temperatures of lubricant in the groove is

quite important.

• Heat transfer from the outlet edge of the bush to fluid in the supply groove must be considered

for correctly simulating the actual conditions.

• At higher speed and heavy load, developed model of viscosity-temperature predicts accurate

values for temperature in fluid, bush and journal.

• The authors have found during their investigation that the developed equation gives very close

values of the maximum fluid and bush temperature when compared with the experimental

results of J. Ferron et al. [3] at different speeds and loads respectively.

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41

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Nomenclature

jA Cross-sectional area of the journal ( 2Rπ )

c Radial clearance, (m); /c c R=

LC Coefficient of contraction ,

LC is unity in positive pressure region

( ) ( )1 1

0 0| |L

e

C u h dz u h dzt α

= ∫ ∫

pC

Specific heat of fluid, (J/kg °C)

D Diameter of Journal, (m)

eD

Dissipation number

e Journal Eccentricity, (m); /e cε = 0 1

2

, ,F F

F

Non dimensional Integration functions of Viscosity

h Thickness of fluid-film,(m); /h h c=

abh

Convective heat transfer coefficient

bush, (W/ m2 0

C)

ajh

Convective heat transfer coefficient

of journal, (W/m2 0

C)

fbh

Convective heat transfer coefficient from bush to fluid in groove,

(W/m2 0

C)

0 1

2 3

,

,

k k

k k

Coefficient of Viscosity

,k kf b

kj

Thermal conductivity of fluid,

bush and journal, (W/m °C)

L Length of bearing, (m)

p Pressure , s

p p p= (N/ m2)

sp

Supply pressure, (N/ m2)

eP

Peclet number,



43

q Heat input per unit length

Q Fluid-flow, (m3/s)

4( c R )Q Qs jω=

r Radial coordinate; /r r R=

R Radius of journal, (m)

1 2,R R

Inner and outer radius of bush, m

1 21 2/ , /R R R R R R= =

rT Reference temperature, (°C)

aT

Ambient temperature, (°C); /a a rT T T=

bT

Bush temperature, (°C); /b b r

T T T=

fT Fluid film temperature, (°C); /T T Tf f r=

Tj

Journal temperature,(°C); /j j rT T T=

Ts

Supply temperature , (°C); /s s rT T T=

t Time ; /

jt t ω=

, ,u v w Fluid velocity components, in

circumferential, axial and radial

directions respectively (m/s) ( ) ( ) ( )

, ,/ / /

u v wu v w

R R Rj j jω ω ω= = =

, ,x y z

Cartesian Coordinate in circumferential, axial and radial direction,

/z z h= ,j jX Z Coordinates of journal centre, (m);

sin , cosj jX Zε φ ε φ= = −

α Circumferential cylindrical

coordinate; x R

β Axial cylindrical coordinate; y R

ε Eccentricity ratio;

λ Aspect ratio; L D

φ Attitude angle (degrees) µ Viscosity of fluid, (N.s/m

2);

0µ Reference viscosity of fluid,(N-s/m

2)

ρ Mass density of fluid, (kg/m3)

jω Angular speed of the journal, (rad/s)

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