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Performance of Infinitely Wide Parabolic and Inclined Slider BearingsLubricated with Couple Stress or Magnetic FluidsMobolaji Humphrey Oladeinde and John Ajokpaoghene AkpobiCitation:AIP Conf. Proc. 1394, 58 (2011); doi: 10.1063/1.3649936View online: http://dx.doi.org/10.1063/1.3649936View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1394&Issue=1Published by theAmerican Institute of Physics.Related ArticlesQuasi-static analysis of a ferrofluid blob in a capillary tubeJ. Appl. Phys. 111, 074901 (2012)Shape oscillations of an electrically charged diamagnetically levitated droplet

Appl. Phys. Lett. 100, 114106 (2012)Magnetic fluid dynamics in a rotating magnetic fieldJ. Appl. Phys. 111, 053901 (2012)Observations of ferrofluid flow under a uniform rotating magnetic field in a spherical cavityJ. Appl. Phys. 111, 07B313 (2012)Carbon nanotube-coated silicated soft magnetic carbonyl iron microspheres and their magnetorheologyJ. Appl. Phys. 111, 07B502 (2012)Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/Journal Information: http://proceedings.aip.org/about/about_the_proceedingsTop downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCSInformation for Authors: http://proceedings.aip.org/authors/information_for_authors

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Performance of Infinitely Wide Parabolic and

Inclined Slider Bearings Lubricated with Couple

Stress or Magnetic Fluids

Mobolaji Humphrey Oladeinde and John Ajokpaoghene Akpobi

Production Engineering Department, Faculty of Engineering, University of Benin. P.M.B II54, Benin

City, Edo State, Nigeria

Abstract. The hydrodynamic and magnetohydrodynamic (MHD) lubrication problem of infinitelywide inclined and parabolic slider bearings is solved numerically using the finite element method. The

bearing configurations are discretized into three-node isoparametric quadratic elements. Stiffness

integrals obtained from the weak form of the governing equations are solved using Gauss quadrature to

obtain a finite number of stiffness matrices. The global system of equations obtained from enforcing

nodal continuity of pressure for the bearings are solved using the Gauss-Seidel iterative scheme with a

convergence criterion of 10

-10

. Numerical computations reveal that, when compared for similar profileand couple stress parameters, greater pressure builds up in a parabolic slider compared to an inclined

slider, indicating a greater wedge effect in the parabolic slider. The parabolic slider bearing is also

shown to develop a greater load capacity when lubricated with magnetic fluids. The superior

performance of parabolic slider bearing is more pronounced at greater Hartmann numbers for identical

bearing structural parameters. It is also shown that when load carrying capacity is the yardstick for

comparison, the parabolic slider bearings are superior to the inclined bearings when lubricated with

couple stress or magnetic lubricants.

Keywords: Hydrodynamic, magnetohydrodynamic, finite element, slider

PACS: 47.10.ad

INTRODUCTIONThe contacting surfaces in slider bearings are separated by fluids to help reduce

wear and friction. Slider bearings find applications in many types of machine elements

in which rectilinear sliding motion occurs. Areas of practical application of slider

bearings include mechanical seals, machine tool ways, piston rings and plain collar

thrust bearings. The need to increase the bearing loads has led to the introduction of

the use of non-Newtonian fluids in the clearance zone of sliders. The increasing use of

non-Newtonian lubricant blended with long chain polymers has been documented, as

well as the use of electrically conducting fluids in the presence of a magnetic field

A number of researchers have investigated the effect of the couple stress fluid on

the steady-state performance of different slider bearing configurations using different

numerical schemes. In recent times, most numerical work in hydrodynamic lubrication

has involved the use of the Reynolds equation and the finite difference method [1]. A

finite difference multigrid approach was used to investigate the squeeze film behavior

of poroelastic bearing with couple stress fluid as lubricant by Bujurke et al [2]. They

reported that poroelastic bearings with couple stress fluid as lubricant provide

augmented pressure distribution and ensured significant increase in the load carrying

capacity. Serangi et al [3] solved the modified Reynolds equation extended to include

couple stress effects in lubricants blended with polar additives using the finite

Current Themes in Engineering Science 2010AIP Conf. Proc. 1394, 58-68 (2011); doi: 10.1063/1.3649936

2011 American Institute of Physics 978-0-7354-0964-4/$30.00

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FIGURE 1. Bearing geometry of a parabolic shaped slider

FIGURE 2. Bearing Geometry of inclined shaped slider

The oil film profile for the parabolic slider in non-dimensional form is as shown in

Eq. 1. The corresponding film profile equation for the inclined slider bearing is shownin Eq. 2.

* * 2(1 2 )m p mh h h h x x= + = + + (1)

( )* * 1= + mh h x (2)

*

mh is the dimensionless minimum film thickness at the exit of the slider and

represents the profile parameter of the bearing, computed using the expressionm

dh

,

where dis the shoulder height of the bearing.The non-dimensional modified Reynolds equation governing the

hydrodynamic film pressure for a slider with couple stress is given by

( )* *

* *, 6* * *

d dp dhf h l

dx dx dx

=

(3)

U

h

L

hm

d

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The function ( )* *,f h l is defined by Eq. 4 for a slider with couple stress lubricantparameter.

*l represents the dimensionless couple stress parameter which accounts forthe addition of long chain polymers to the lubricant.

( )2 ** * *3 * * *

*, 12 2 tanh

2

hf h l h l h ll

=

(4)

As the value of *l approaches zero, Eq. 4 is reduced to the classical form for a

Newtonian lubricant case. For a slider bearing lubricated by a magnetic fluid, the

function ( )* *,f h l is replaced by Eq. 5 below with the boundary conditions for thebearings given in Eq. 6. M is known as Hartmann number which accounts for the

presence of an externally applied magnetic field. The Hartmann number is defined by

the expression ( )1/ 2

0 mB h

, where 0B is the applied magnetic field, mh is the minimum

film thickness, is electrical conductivity of the fluid and is lubricant viscosity.

( ) ( )( )*

* * *

2

6, coth 0.5 2=

hf h M Mh Mh

M(5)

( ) ( )* *p 0 p 1 0= = = =x x (6)

WEAK FORMULATION

The weak form of Eq. 3 is:

( ) ( )* *

1, , , , 6 0

ee enj e e

i j i ie e

j

d d dp dhf h l M p f h l M dxdx dx dx dx

=

+ = (7)

Here, for the sake of generality, the function ( )* *,f h l has been replaced by aterm ( )* *, ,f h l M . The function reduces to Eq. 5 when a bearing with a magneticlubricant is considered; and to Eq. 4 for a slider bearing with couple stress lubricant.

The details of the weak formulation can be found in Oladeinde and Akpobi [11].

( ) ( )* *1

, , , , 6 0

=

+ =

ee en

j e eij i i

e ej

d d dp dhf h l M p f h l M dx

dx dx dx dx(7)

FINITE ELEMENT MESH

The physical domains of the one dimensional bearings under consideration are

divided into uniform meshes of 1-D linear quadratic elements. A typical element in the

mesh is depicted in Figure 3. Each element is bordered by two end nodes and an

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interior node is located at the middle of the element. Neighboring elements are

interconnected at a common node. The length of each element in the domain is

obtained by dividing the dimensionless length of the bearing by the number of

elements. The solution methodology is first carried out on a mesh with a few elements.

The finite element mesh is refined progressively by halving the length of the element

used for the previous solution, until the nodal pressure solution becomes mesh

independent. At a mesh discretization of 80 linear quadratic elements, equivalent to

161 nodes, the solution obtained becomes mesh independent. This mesh discretization

was employed for parametric studies.

FIGURE 3. A typical element showing end and interior nodes

SOLUTION METHODOLOGYThe dimensionless physical domain is divided into uniform quadratic elements of

length x .This leads to a constant transformation to a local element co-ordinate

system in which the differentials dx and ddx

are respectively given by:

dx xd= (8)

d 1 1

dx x d=

(9)

Here is the natural co-ordinate system within an element. The pressure in an element

is approximated by basis or trial functions ei over each element and is given by

( ) ( )3

1

j j

j

p p =

= (10)

The transformed element integrals obtained from the weak formulation of the

governing equation shown in Eq. 7 are evaluated numerically by Gauss quadrature

making use of the transformed integral of the form

( )

( )( )( )( )

( )

( )( ) ( )

31

1

1 1,e e e

e e

d dK f x l J d

J d J d

= (11)

2

e

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where ( )eJ is the Jacobian defined by

( )( )3

1

== = ie ei

i

ddxJ x

d d(12)

The element stiffness integrals are evaluated iteratively using three Gauss points

and weights according to the expression

( ) ( )1

11

n

nl nl

l

I d w I +

=

(13)

After writing the global integral as a sum of the individual element stiffness

integrals, a system of algebraic equations is obtained. Boundary conditions (Eq. 6.) are

imposed on the global system of equations resulting in a condensed matrix which is

solved by Gauss-Seidel iterative scheme to obtain the pressure solution. Parametric

studies are carried out to determine the effect of bearing parameters on the pressure

distribution and bearing load of the bearing configurations.The load capacity is determined by streamwise integration of the oil film

pressure in the bearing, i.e.

( )1

0

W P x dx= . (14)

NUMERICAL RESULTS AND DISCUSSION

The validity of the finite element simulation result is first examined. In particular,

we investigate the convergence characteristics of the results obtained using the finite

element method. The finite element model was built using progressively finer meshes

from 10 elements to 80 elements for both slider bearing configurations with couplestress lubricants and magnetic fluids. The bearing parameters used for the simulation

are 0.5 2.5, * 1=hm ,*0 0.6 l , and 1 10 M . For both slider bearing

configurations, the nodal pressures were found to converge fully to a mesh-

independent solution when 80 elements were used for the simulation. Consequently,

this mesh density was used for the parametric studies.

Parametric Studies

To provide information for engineers involved with slider bearing design,

comparison of the performance of the two bearing profiles when the lubricant is

couple stress and magnetic fluid is presented below.

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FIGURE 4. Dimensionless pressure profiles for infinitely wide parabolic and inclined slider bearings

with couple stress.

W

W

/

FIGURE 5. Dimensionless pressure against dimensionless distance for inclined and parabolic slider

without couple stress. ( )1 =

FIGURE 6. Dimensionless pressure against dimensionless distance for infinitely wide inclined and

parabolic slider bearings, without couple stress ( 1.4 = ).

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Figure 4 shows the dimensionless pressure generated under the same conditions for

inclined and parabolic slider bearings with couple stress. The graph shows that a lower

maximum pressure is generated using the same bearing parameters for an inclined

slider bearing with couple stress compared to a parabolic bearing. The maximum

dimensionless pressure generated for parabolic slider bearing is 0.34 compared to 0.31for the inclined slider bearing. Under the magnitude of the parameters considered, the

parabolic slider bearing generates a load capacity of 0.21 compared to 0.19 for the

inclined slider bearing.

Figure 5 shows the dimensionless pressure distribution of the two bearings without

couple stress for 1 = . The graph shows that the parabolic slider bearing retains itshigher maximum pressure as in the non-Newtonian lubricant case. However, the effect

of simulating the two bearings without couple stress is to decrease the pressure

generated from 0.34 to 0.28 and from 0.31 to 0.25 for the parabolic and inclined slider

bearings respectively. Increasing the profile parameter from 1 = (Figure 4) to

1.4 = (Figure 6) results in a decrease in the maximum pressure generated in the

parabolic slider, in contrast to the inclined case where it has a positive effect.

Increasing the profile parameter is equivalent to increasing the wedge effect,ultimately results in greater pressure in the lubricating film of the inclined slider

bearing. The net result of simulating the bearings without couple stress is to decrease

the load carrying capacity, as shown in Figure 7.

FIGURE 7. Dimensionless load capacity against dimensionless couple stress for parabolic and inclined

slider bearings.

Figure 7 shows that the load carrying capacity for a parabolic slider bearing is

higher than that for the inclined case for the same couple stress parameter. Parabolic

slider bearings are observed to be superior at higher couple stress parameters.

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FIGURE 8. Dimensionless pressure in inclined slider bearings for 0.1 = , *mh 1= , using different

values of Hartmann numbers.

Figure 8 shows that in general, the pressure generated in the bearing increases with

the Hartmann number. This finding is consistent with that by Lin and Lu [12] who

theoretically came to a similar conclusion. Figure 9 shows the pressure generated ininclined and parabolic slider bearing using 8=M , 0.1 = and *mh 1= . The pressure

generated is higher in the parabolic slider for * 0.58x . At dimensionless distance

along the slider greater than 0.58, the pressure in the inclined slider is greater. This

position at which the transition occurs is independent of M for a particular magnitude

of and PPPPKKKK . The load capacity generated in the inclined and parabolic slider bearings

for1 10 M , and 1=PPPPKKKK using two instances of profile parameter namely, 1 = and

2.8 = is shown in Figure 10. The plot shows that the load capacity increases with the

Hartmann numberM , which agrees with the finding of Ramanaiah [13]. In addition it

can be deduced from the graph that at low profile parameters, there is no significant

difference in the bearing load developed in both slider configurations. At 2.8 = ,

however, the load capacity of a parabolic slider bearing is observed to be greater thanthat developed in inclined slider bearing for a particular value of Hartmann number.

The increase in the load capacity for a parabolic slider bearing is greater at higher

values of M .

FIGURE 9. Pressure generated in inclined and parabolic slider bearings lubricated with magnetic fluid.

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FIGURE 10. Load capacity as a function of the Hartmann number for different profile parameters.

CONCLUSION

A comparative study of the pressure distribution and load capacity of infinitely

wide parabolic and inclined slider bearings has been presented. The finite element

method was used to discretize the governing equation with the associated boundary

conditions. The infinitely wide parabolic slider bearing has been shown to be superior

in terms of bearing load as a result of the greater pressure generation when lubricated

with couple stress and magnetic lubricants.

REFERENCES

1. B. P. Mitidierri, Advanced Modeling of Elastohydrodynamic Lubrication, Doctoral Thesis,

Tribology Section and thermo fluids section, Department of Mechanical Engineering, Imperial

College, London, pp 19 (2005).

2. N. M. Bujurke and B. R., Kudenati, Multigrid solution of modified Reynolds Equation

incorporating poroelasticity and Couple Stress,Journal of Porous Media, 10(2), pp 125 136

(2007).

3. M. Senangi, B. C. Majumda and A. S. Sekhar, Elastohydrodynamically lubricated ball

bearings with couple stress fluids, Part 1: Steady state analysis, Tribology Transactions,

48(3), pp. 404 414 (2005).

4. R. Linj, Squeeze film characteristics of finite journal bearings; couple stress fluid model,

Tribology International, 31(4), pp 201 207 (1998).5. A. A. Elsharkawy, Effects of misalignment on the performance of finite journal bearings

lubricated with couple stress fluids, International Journal of Computer Applications in

Technology (IJCAT), 21(3), 2004.

6. J. R. Lin, Derivation of dynamic couple-stress Reynolds equation of sliding-squeezing

surfaces and numerical solution of plane inclined slider bearings, Tribology International,

36(9), pp 679 685 (2003).

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7. G. S. Nada and T. A. Osman, Static Performance of Finite Hydrodynamic Journal Bearings

Lubricated by Magnetic Fluids with Couple Stresses, Tribology letters, 27(3), pp 261 268

(2007).

8. H. M. Bujurke and R. B. Kudenati, MHD lubrication flow between rough rectangular plates,

Fluid dynamics Research, 39(4), pp. 334 345, 2007.

9. M. I. Anwar and C. M. Rodkiewicz, Slider bearing with Segmented Electric Field, Wear,29(2), pp. 171 179, 1974.

10. F. Paulo, J. C. Pimenta, and A. Jorge, Journal Bearings Subjected to Dynamic loads: The

Analytical Mobility Method, Mecanica Experimental, 13, pp.115 127 (2006).

11. M. H. Oladeinde and J. A. Akpobi, A Comparative study of load capacity and Pressure

distribution of infinitely wide Parabolic and Inclined Slider bearing, Proceedings of the World

Congress on Engineering, London, UK, 30th

June 2nd

July 2010, pp. 1370 -1377.

12. J. R. Lin and R. F. Lu, Dynamic Characteristics for Magnetohydrodynamic Wide Slider

Bearings with exponential film profile, Journal of Marine Science and Technology, 18( 2),

268-276, 2010.

13. G. Ramanaiah, Magnetohydrodynamic step slider bearing with variable viscosity, Applied

Science Research, 12(5 6), pp. 424 434, 1965.

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akpobi bola inclinded slider bearing springer

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