Finite element analysis of elastohydrodynamic circular journal bearing with micropolar lubricants

Finite Elements in Analysis and Design 41 (2004) 75 – 89 www.elsevier.com/locate/finel

Finite element analysis of elastohydrodynamic circular journal bearing with micropolar lubricants V.P. Sukumaran Naira , K. Prabhakaran Nairb,∗ a Department of Mechanical Engineering, NSSCE, Palakkad, India b Department of Mechanical Engineering, NIT Calicut, India

Received 30 December 2003; accepted 25 April 2004 Available online 7 August 2004

Abstract In this paper the effect of deformation of the bearing liner on the static characteristics of a circular journal bearing operating with micropolar fluid is analysed. Lubricating oil containing additives and contaminants is modeled as micropolar fluid. A modified Reynold’s equation is obtained by considering the effect of mass transfer across the fluid film in the fluid flow equations. Finite element technique has been used to solve the modified Reynold’s equation governing the flow of micropolar fluid in the clearance space of the journal bearing and the three-dimensional elasticity equations governing the displacement field in the bearing shell. The static characteristics of the bearing are presented for a wide range of deformation coefficient which takes into account the flexibility of the bearing liner. The increase in volume concentration of additives in the lubricant produces significant effects in the performance characteristics of the bearing especially when the bearing operates at higher eccentricity ratio. 䉷 2004 Elsevier B.V. All rights reserved. Keywords: Elastohydrodynamic; Micropolar lubricant; Deformation coefficient

1. Introduction In conventional bearing design the bearing characteristics are determined by assuming the lubricant as Newtonian and that the bearing shell as rigid. Several investigators [1–4] have studied the performance characteristics of circular bearings under these assumptions. ∗ Corresponding author.

E-mail address: [email protected] (K. Prabhakaran Nair). 0168-874X/$ - see front matter 䉷 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2004.04.001

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Nomenclature C c cr f h e E Kd Ks p p QZ R


q th U V , Vr and Vz W

  o  

[K] [D] [F ]

radial clearance volume concentration of additives volume concentration of the additives at the journal surface body force density film thickness bearing eccentricity modulus of elasticity diffusion coefficient mass transfer rate pressure acting on the journal nondimensional pressure acting on the journal end leakage of the bearing radius of journal attitude angle velocity vector thickness of bearing linear velocity of journal at the surface displacement components of bearing liner in the circumferential, radial and axial directions load capacity angular velocity of the journal viscosity of the lubricant containing additives viscosity of the lubricant at reference temperature and pressure Poisson’s ratio deformation coefficient stiffness matrix for displacement field of bearing bush elasticity matrix column vector for nodal force components

When bearings are subjected to heavy load, the bearing deformations may quite often affect the clearance space geometry of the bearing to an extent such that the actual performance characteristics may become significantly different from those computed with rigid bearings. Several literatures are available on the elastohydrodynamic analysis of circular journal bearings [5–10] for Newtonian lubricants. To enhance certain characteristics of the lubricants various additives i.e., solids or liquids in the form of small particles are added to the lubricant and this lubricant behaves as micropolar fluid. These additives along with contaminants form a dilute suspension of solid particles in the oil. These suspended solid particles produce thickening of lubricating oil which in turn affect various performance characteristics of journal bearings.Also, there is an increase in the viscosity of the lubricant film which is in the vicinity of the journal and bearing surfaces due to adhesion and other surface phenomena. The nonuniform distribution

V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89

77

φ θ th

Rigid housing ob

Bearing liner

Rj + C . x, x

oj

ωj

x

. y, y

hmin

y

Fig. 1. Circular bearing geometry.

of the solid suspended particles build up concentration gradient which result in mass transfer of these particles across the film thickness from the journal surface leading to a reduction of this thickening effect. Albert and Thamer [11] showed that the load capacity and the frictional force of the slider bearing increase with increase in concentration of additives and contaminants. Prakash and Prawal [12] determined the static characteristics of a journal bearing with micropolar fluids. Prabhakaran Nair et al. also determined static and dynamic characteristics of circular rigid bearings with micropolar fluids [13]. Steady state characteristics of hydrodynamic bearings considering misalignment were computed by Das et al. [14] with micropolar fluid. The existing literature shows that studies on static characteristics of a flexible journal bearing operating with miocropolar lubricant is scarce. So it is felt that there is a need to recompute the performance characteristics of a circular journal bearing considering the effect of bearing liner deformation on the performance characteristics of circular bearing (Fig. 1) in which lubricant is treated as micropolar lubricant. In the present work, static characteristics in terms of load carrying capacity, attitude angle, end leakage and frictional force are computed using finite element method for rigid and deformable circular bearings in the following cases. (i) Journal bearing with Newtonian lubricant. (ii) Journal bearing with Micropolar lubricant.

2. Hydrodynamic analysis For the hydrodynamic analysis of the bearing, a modified Reynold’s equation to include the effect of micropolar lubricant is derived from Navier–Stoke’s, continuity and Fick’s Second Law equations. Navier–Stokes equation for incompressible fluid is given by [(jq/jt) + q.∇q] = f − ∇p − ∇.∇q

(1)

where  is viscosity of lubricant containing additives. Continuity equation for steady flow of incompressible fluid is given by ∇.q = 0

(2)

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V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89

and Fick’s Second Law is (jc/jt) + q.∇c = Kd ∇ 2 c.

(3)

The viscosity  of the dilute suspension of solid particles in a lubricant may be written in the form [15]  = 0 (1 + c),

(4)

where  parameter specifies the shape, size, deformation, distribution and material of additive particles and
 y c = c r 1 + Ks . (5) Kd From Eqs. (1)–(5) a modified Reynold’s equation is derived in the following form [13]:
3 
3        h h j jp j jp Ks h Ks h + 1 − cr 1 + 1 − cr 1 + jx 120 2Kd jx jz 120 2Kd jz 
 Uj h cr Ks h = 1− . jx 2 6Kd

(6)

Changing into polar co-ordinates and nondimensionalising the above equation we get the following equation:  3       3    j h K sh jp j h K sh jp 1 − cr 1 + + 1 − cr 1 + j 12 2 j jz 12 2 jz   1 jh cr K s h = 1− . (7) 2 j 6 2.1. Finite element procedure for the solution of hydrodynamic equation The flow field in the clearance space of circular bearing has been descretized into four nodded isoparametric elements. It has been divided into 14 elements in circumferential direction and 4 elements in the axial direction. The element numbering is done in such a way that the bandwidth is a minimum. The descretization of the flow field is given in Fig. 2(a). Consider a typical four nodded element as shown in Fig. 2(b) and the Langrangian interpolation functions given by N1 = 0.25(1 − )(1 − ), N2 = 0.25(1 − )(1 + ), N3 = 0.25(1 + )(1 + ), N4 = 0.25(1 + )(1 − ). The pressure at a point in the element can be expressed approximately as p=

4 j =1

Nj p j .

(8)

V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89 10

5 4

8

3

7

2

6

1

5

15

20

25

30

35

40 45 50 55 60657075

11

16

21

26

31

36 41 46 51 56 616671

79

4

3

2

1

6

(a)

η

y 2

3

2b

ξ

(xi, yi)

1

4 2a x

(b)

Fig. 2. (a) Discretization of flow field, (b) four nodded element.

Using the approximate value of p we get 

3







  4 j  Nj p j 

K sh h 1 − cr 1 + j 12 2 j j =1         3 4 j h K h j jh cr K s h 1 s   + = Re , Nj pj  − 1 − cr 1 + 1− jz 12 2 jz 2 j 6 j

(9)

j =1

where R e is known as Residue. The element equations are obtained by applying Galerkin’s technique. According to this technique minimization of the residue is obtained by orthogonalising the residue with interpolation functions (base function).  i.e.

e

Ni R e d dz = 0.

(10)

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V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89

Thus we get       4      −3 j h K sh j   Nj p j   1 − cr 1 +  12  2 j e  j j =1       4  j h−3  K s h j   + Nj pj 1 − cr 1 +  jz 12  2 jz j =1    j h cr K s h − 1− Ni d dz = 0. j 2 6

(11)

The second order term is integrated by parts to obtain C 0 continuity. The resulting equation obtained is as follows:   
  3 h K sh jNi jNj jNi jNj 1 − cr 1 + + p j d d z 2 j j jz jz e 12     h jNi cr K s h = 1− (12) d dz. 2 j 6 e The above equation can be written in matrix form for a typical element as [K]e {p}e = {Q}e , where [K]e = and {Q}e =

  e

  e

(13)

  
 3 h K sh jNi jNj jNi jNj 1 − cr 1 + + pj d dz 12 2 j j jz jz h jNi 2 j

 1−

cr K s h

6

 d dz,

[K]e is the element coefficient matrix. e refers to domain of the eth element. The element matrices are generated for all elements and assembled to get global coefficient matrix. The right-hand side of Eq. (13) are also assembled to obtain global right-hand side. The resulting global system equations can be expressed in matrix form as follows: [K]{p} = {Q}.

(14)

The following boundary conditions are used to obtain the pressure field p(, z) = 0 at  = 0, 2 , p(, z) = 0 at z ± 1, jp (, z) = 0 at  = 2 , j

where 2 is the unknown extent of positive pressure film of fluid flow field.

(15)

V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89

81

Fig. 3. Discretization of bearing liner.

3. Finite element procedure for the determination of elastic deformation of the bearing liner The bearing liner is a finite length cylinder subjected to hydrodynamic loading due to the fluid film pressure on its internal surfaces. The distribution of fluid film pressure is such that it causes the bush to deform in all directions. The bearing liner is discretized by 8 nodded hexahedral isoparametric elements. The discretization of the bearing liner is shown in Fig. 3. Using the linear elasticity theory the expression for the potential energy of an element when bearing is subjected only to surface traction forces, Ttr is written as  e =1/2 [ ]eT [J ]eT [D]e [J]e [
]e r d dr dz  − [Ttr ]e [ ]e r d dz. (16) The displacement
for the bearing liner is   V (r, , z) ( ) = Vr (r, , z) , Vz (r, , z)

(17)

where V , Vr and Vz are displacements of bearing liner in the circumferential, radial and axial directions respectively. The displacement field [
]e in eth element in terms of nodal displacements can be written as   V e [ ] = Vr = [N]e [d]e , (18) Vz where [N ]e represent the element shape function matrix and [d]e represents nodal displacement vector. V =

m i=1

Vi N i ,

Vr =

m i=1

Vri N i ,

Vz =

m i=1

Vzi Ni

(19)

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V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89

and [d]e = [V1 , Vr1 , Vz1 , . . . , Vm , Vrm , Vzm ]T , where m is the number of nodes per element. The element equations are obtained by minimizing potential energy for each element.  j e   j V i  ne   e   j = 0, i.e.  jVri  e=1   j e  

(20)

(21)

jVzi

where e = 1, 2, . . . , ne . ne = number of elements. Using the above condition, system equations are reduced to   ne
   eT e e e [J ] [D] [J ] r d dr dz − [N] [Ttr ] r d dz = 0,

(22)

e=1

where

1j      e [J ] =      

r

0 0 j jr j jr

1 r

D2  D1  D [D] =  2  0  0 0

0 0 j jz

0 

j j

1 r

0 j jz

0 

1 r j jr

j

D2 D2 D1 0 0 0

D1 D2 D2 0 0 0

0 0 0 D2 0 0

0 

j j j jr

0 0 0 0 D3 0

      ,       0 0   0  , 0  0  D3

and D1 = E(1 − )(1 + )(1 − 2), D2 = E /(1 + ) (1 − 2), D3 = E/2(1 + ), the above element equation may be written in the matrix form as [K]e [d]e = [F ]e ,

(23)

V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89

83

where 

e

[K] = [F ]e =

 s

[J ]eT [D]e [J ]e r d dr dz, [N]eT [Ttr ]e r d dz

by nondimensionalising, and using the general assembly procedure [16], the following global system equations are obtained [K][d] = [F ],

(24)

where =

0 j (Rj /C)3 th

E

Rj

,

[K] stiffness matrix, [d] displacement matrix, [F ] force matrix. 3.1. Boundary conditions for elastohydrodynamic analysis It is assumed that the bearing liner is contained in a comparatively rigid housing. Thus the outer surface of the bearing liner, which is in contact with the inner surface of the housing, does not deform, implying that the nodes in contact with the rigid surface are restrained from moving. Hence   V i (25) V ri = {0}, V zi where ‘i’ is the number of nodes on the bearing liner of rigid housing interface. 3.2. Modification of film thickness The film thickness is then modified using the relation given below: h = 1 + e cos  + V r , where V r is the nondimensional deformation of the bearing bush in the radial direction.

(26)

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V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89

4. Performance characteristics The various static performance characteristics are determined by the following formulae. 4.1. Load capacity and attitude angle The fluid film reaction components are given by  e W = pe cos  r d dz, e W

 =

(27)

pe sin  r d dz.

(28)

These reactive forces are recalculated along x and y axes for each element in each lobe. The load carrying capacity is given by the equation 2

2

W = [W x + W Y ]1/2 .

(29)

The angle between the line of the centers and the load line is known as attitude angle and is given by
= tan−1

WY . WX

(30)

4.2. Bearing oil flow The end leakage for each bearing lobe is computed from equation  R+h  r  R+h  r QL = W r d  dr + WZ=+1 r d dr. Z=−1 Z R

0

R

(31)

0

4.3. Frictional force Frictional force at the bearing surface is obtained by the following expression:    1 h jp + F=  r d dz. h 2 j

(32)

5. Solution procedure After applying the boundary conditions Eq. (15) and initially assuming lubricant to be Newtonian, Eq. (14) is solved to obtain the pressure field of fluid flow in the clearance space of the bearing. The extent (2 ) of positive pressure fluid film is obtained by an iterative procedure. For micropolar lubricants, the viscosity is modified as per Eq. (4) and the pressure field is obtained by solving Eq. (14) for different values of Ks . For the cases of both Newtonian and micropolar lubricants, the pressure field obtained by solving hydrodynamic equations is used to obtain the nodal displacements of the bearing liner using the

V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89 18

1- ε = 0.2, KS= 0.0 2- ε = 0.2, KS= 0.4 3- ε = 0.6, KS= 0.0 4- ε = 0.6, KS= 0.4

16 Loadcarrying Capacity (W)

14

85

4 3

12 10 8 6

Present results Published results

4 2

2 1

0 0

0.2 0.4 0.6 Volume concentration of Additives (λcr)

Fig. 4. Load carrying capacity (W) vs. volume concentration of additives (cr ).

elasticity Eq. (24). Using these nodal displacements the film thickness is modified and for the modified film thickness, the pressure field is again established and for the modified pressure field, nodal displacements are again computed. This iterative process is repeated until convergence is achieved within the specified tolerance.

6. Results and discussions For rigid and deformable circular hydrodynamic journal bearings, the static characteristics in terms of load carrying capacity, end leakage, attitude angle and frictional force are computed for different values of eccentricity ratio (ε) and deformation coefficient () when the bearing operate with (1) Newtonian lubricant (2) Micropolar lubricant. The variation of load carrying with respect to volume concentration of additives is shown in Fig. 4. It is observed from Fig. 4 that load carrying capacity increases with increase in volume concentration of additives, mass transfer rate and eccentricity ratio. To authenticate the solution algorithm and computer program developed, the results obtained for rigid circular bearing with micropolar lubricant is compared with the published results [13] in Fig. 4. It is seen that the present results obtained are in good agreement with the published results. Fig. 5 shows variation of load capacity with increase in deformation coefficient for Newtonian (cr = 0, Ks =0) and micropolar lubricants (cr =0.2/0.4, Ks =0.4). For micropolar lubricants the load capacity obtained at any deformation coefficient () is greater than that obtained with Newtonian lubricant when bearing operates at any eccentricity ratio. It is also observed that in the case of micropolar lubricants, for a fixed value of mass transfer rate (Ks ), when the volume concentration of additives increases, the load capacity increases for any deformation coefficient and eccentricity ratio. At low values of deformation coefficient and higher eccentricity ratio, the variation of load capacity with increase in deformation

86

V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89 25

1–ε = 0.4, λcr = 0.0, Ks = 0.0 2–ε = 0.4, λcr = 0.2, Ks = 0.4 3–ε = 0.4, λcr = 0.4, Ks = 0.4 4–ε = 0.8, λcr = 0.0, Ks = 0.0 5–ε = 0.8, λcr = 0.2, Ks = 0.4 6–ε = 0.8, λcr = 0.4, Ks = 0.4

Load Carrying Capacity (W)

20

15

1 2 3 4

10

5 6

5

0 0

0.1 0.2 0.3 0.4 Deformation Coefficient (ψ)

0.5

Fig. 5. Load carrying capacity (W) vs. deformation coefficient ().

1.4 1.2 1

End Leakage (Qz)

1

2 3

0.8

4 5

0.6

6 1–ε = 0.4, λcr = 0.0, Ks = 0.0 2–ε = 0.4, λcr = 0.2, Ks = 0.4 3–ε = 0.4, λcr = 0.4, Ks = 0.4 4–ε = 0.8, λcr = 0.0, Ks = 0.0 5–ε = 0.8, λcr = 0.2, Ks = 0.4 6–ε = 0.8, λcr = 0.4, Ks = 0.4

0.4 0.2 0 0

0.1 0.2 0.3 0.4 Deformation Coefficient (ψ)

0.5

Fig. 6. End Leakage (Qz ) vs. deformation coefficient ().

coefficient is more significant than those values obtained at lower eccentricity ratio, for any value of volume concentration of additive. The change of end leakage with the variation of deformation coefficient is shown in Fig. 6. From Fig. 6 it is observed that the end leakage decreases with increase in deformation coefficient. It is also seen

V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89

87

70 60

1 2

Attitude Angle (φ)

50

3 4

40

5 30

6 1–ε = 0.8, λcr = 0.0, Ks = 0.0 2–ε = 0.8, λcr = 0.2, Ks = 0.4 3–ε = 0.8, λcr = 0.4, Ks = 0.4 4–ε = 0.4, λcr = 0.0, Ks = 0.0 5–ε = 0.4, λcr = 0.2, Ks = 0.4 6–ε = 0.4, λcr = 0.4, Ks = 0.4

20 10 0 0

0.1 0.2 0.3 0.4 Deformation Coefficient (ψ)

0.5

Fig. 7. Attitude angle (
) vs. deformation coefficient ().

that variation of end leakage with increase in deformation coefficient is appreciable when the bearing operates at higher eccentricity ratio compared with the values obtained at lower eccentricity ratios. In Fig. 7 the values of attitude angle with the change in deformation coefficient are shown and it is seen that the change in volume concentration and mass transfer rate do not affect the values obtained for Newtonian lubricant significantly when the bearing operates at lower eccentricity ratio for any value of deformation coefficient (). From Fig. 8 it may be noted that at higher eccentricity ratio, a large reduction in frictional force is obtained with increase in deformation coefficient. For a fixed value of mass transfer rate the frictional force increase with increase in volume concentration of additives for any value of eccentricity ratio and deformation coefficient. To have a physical feel, the characteristics are calculated for the following bearing geometry and operating conditions: Journal radius 25 mm Speed 2500 rpm Lubricant considered SAE10 Material of bearing liner Texolite Length of the bearing 50 mm Liner thickness of journal radius th /Rj = 0.1 Journal radius to clearance ratio 625 The dimensional values obtained for above bearing geometry and operating condition are given in Table 1. It is seen that at any value of eccentricity ratio the value of end leakage is almost same. This can be explained as follows. When the volume concentration of additives increases, viscosity of lubricant and pressure increase. Increase in lubricant viscosity reduces end leakage, but increase in pressure will increase the end leakage. The combined effect may produce the end leakage to be almost constant at any value of volume concentration of additives when there is no mass transfer.

88

V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89 1

30

2 3

25

Frictional Force (F)

4 5

20

6 15

10

1–ε = 0.8, λcr = 0.0, Ks = 0.0 2–ε = 0.8, λcr = 0.2, Ks = 0.4 3–ε = 0.8, λcr = 0.4, Ks = 0.4 4–ε = 0.4, λcr = 0.0, Ks = 0.0 5–ε = 0.4, λcr = 0.2, Ks = 0.4 6–ε = 0.4, λcr = 0.4, Ks = 0.4

5

0 0

0.1 0.2 0.3 0.4 Deformation Coefficient (ψ)

0.5

Fig. 8. Frictional force (F) vs. deformation coefficient (). Table 1 Dimensional values of performance characteristics for flexible bearing Dimensional values W × 10−3 N Qz × 106 m3 /s
degrees FN

cr = 0, Ks = 0

cr = 0.2, Ks = 0

cr = 0.2, Ks = 0.4

cr = 0.4,

cr = 0.4,

25.21 7.48 35.29 5.308

31.51 7.49 35.29 5.348

31.62 7.88 36.08 5.354

42.03 7.495 35.29 5.386

42.24 8.285 36.168 5.427

Ks = 0.0

Ks = 0.4

ε = 0.8,  = 0.05, L/D = 1.0.

From the results obtained, it is noted that the material of bearing liner must be selected in such a way that the deformation coefficient must be as low as possible. When the deformation coefficient of the bearing liner is high, a large reduction in load carrying capacity is observed. For e.g. at ε = 0.8, cr = 0.4 and Ks = 0.4 the loads carrying capacity (W ) decreases from 25 to 5 when  increases from 0 to 0.5.

7. Conclusions 1. For any eccentricity ratio and deformation coefficient, load carrying capacity of a circular journal bearing increases with increase in volume of concentration of additives. For a rigid bearing ( = 0) when volume concentration (cr ) increases from 0.0 to 0.4, there is approximately 60% increase of load capacity at eccentricity ratio ε = 0.8.

V.P. Sukumaran Nair, K. Prabhakaran Nair / Finite Elements in Analysis and Design 41 (2004) 75 – 89

89

2. The end leakage is independent of volume concentration of additives when the mass transfer ratio is zero. 3. The attitude angle and frictional force increase with increase in mass transfer rate at any value of volume concentration of additives and eccentricity ratio. 4. The static performance characteristics: load capacity, end leakage, attitude angle and frictional force decrease when the bearing liner flexibility increases and these changes are significant especially when the flexible bearing operates at high eccentricity ratio. 5. From the designer’s point of view, the deformation coefficient of the bearing liner must be as low as possible to reduce high decrease in load carrying capacity for any eccentricity ratio.

References [1] O. Pinkus, Solution of Reynold’s equation for arbitrarily loaded journal bearings, J. Basic Eng. Trans. ASME, Series D 3 (1961) 145–152. [2] F.K. Orcutt, E.B. Arwas, The steady state and dynamic characteristics of a full circular bearing and partial arc bearing in laminar and turbulent regimes, Trans. ASME J. Lub. Tech. 89 (1967) 143–153. [3] S.C. Soni, R. Sinhasan, D.V. Singh, Analysis by the finite element method of hydrodynamic bearings operating in the laminar and super laminar regimes, J. Wear 84 (1983) 285–296. [4] S.P. Tayal, R. Sinhasan, D.V. Singh, Analysis of hydrodynamic journal bearings having non-Newtonian lubricants, Tribol. Int. 12 (1982) 17–21. [5] T.E. Carl, The experimental investigation of a cylindrical journal bearing under constant and sinusoidal loading, Proc. Inst. Mech. Eng. London 178 (1964) 100–119. [6] C. Taylor, J.F. O’Callaghan, A numerical solution of the elastohydrodynamic problem using finite elements, J. Mech. Eng. Sci. 14 (1972) 229. [7] K.P. Oh, K.H. Huebner, Solution of elastohydrodynamic finite journal bearing problems, Trans. ASME J. Lubr. Tech. 95 (1973) 342–352. [8] S.C. Jain, R. Sinhasan, D.V. Singh, A study of elastohydrodynamic lubrication of a journal bearing with piezoviscous lubricants, ASLE Trans. 5 (1984) 168–176. [9] A.C. Stafford, R.D. Henshell, B.R. Dudley, Finite element analysis of problems in elastohydrodynamic lubrication, Proceedings of the Fifth Leeds–Lyon Symposium on Tribology, Leeds, Paper ix(iii), 1978, p. 329 [10] G.A. LaBouff, J.F. Booker, Dynamically loaded journal bearing: a finite element treatment for rigid and elastic surfaces, Trans. ASME J. Trib. 107 (1985) 505–521. [11] E. Albert Yousif, M. Thamer Ibrahim, Lubrication of a slider bearing with oils containing additives and contaminants, J. Wear 81 (1982) 33–45. [12] J. Prakash, P. Prawal Sinha, Lubrication theory for micropolar fluids and its application to a journal bearing, Int. J. Eng. Sci. 13 (1975) 217–232. [13] R. Narayanan, C.C. Narayanan, K. Prabhakaran Nair, Analysis of mass transfer effects on the performance of journal bearings using micropolar lubricant, Int. J. Heat Mass Trans. 30 (1995) 429. [14] S. Das, S.K. Guha, A.K. Chattopadhyay, On the steady-state performance of misaligned hydrodynamic journal bearings lubricated with micropolar fluids, Tribology International 35 (2002) 201–210. [15] E. Albert Yousif, Somer M. Nacy, Hydrodynamic behaviour of two-phase (liquid-solid) lubricants, Wear 66 (1981) 223–240. [16] O.C. Zienkiwicz, The Finite Element Method, McGraw-Hill, New York, 1997.