DynamicPerformanceCharacteristicsofaCurvedSlider...

Hindawi Publishing CorporationAdvances in TribologyVolume 2012, Article ID 278723, 6 pagesdoi:10.1155/2012/278723

Research Article

Dynamic Performance Characteristics of a Curved SliderBearing Operating with Ferrofluids

Udaya P. Singh1 and R. S. Gupta2

1 Department of Applied Science & Humanities, Ambalika Institute of Management & Technology, Lucknow 227305, India2 Department of Mathematics, Kamala Nehru Institute of Technology, Sultanpur 228118, India

Correspondence should be addressed to Udaya P. Singh, [email protected]

Received 18 May 2012; Revised 15 July 2012; Accepted 22 July 2012

Academic Editor: Michel Fillon

Copyright © 2012 U. P. Singh and R. S. Gupta. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

In the present theoretical investigation, the effect of ferrofluid on the dynamic characteristics of curved slider bearings ispresented using Shliomis model which accounts for the rotation of magnetic particles, their magnetic moments, and the volumeconcentration in the fluid. The modified Reynolds equation for the dynamic state of the bearing is obtained. The results of dynamicstiffness and damping characteristics are presented. It is observed that the effect of rotation of magnetic particles improves thestiffness and damping capacities of the bearings.

1. Introduction

In the field of engineering and technology, slider bearingsare often designed to bear the transverse loads. The study ofperformance characteristics of slider bearings with differentshape and different lubricants has been done from time totime by the researchers. Gupta and Kavita [1] analysed theeffect of frame rotation for porous slider bearing, Singh andGupta [2] investigated the performance of a pivoted curvedslider bearings for pseudoplastic and dilatant lubricants,Pascovici et al. [3] presented an experimental evidence ofcavitation effects in a Rayleigh step slider, Venkateswarlu andRodkiewicz [4] discussed the thrust bearing characteristicsconsidering the terminal speed of the slider, Williams andSymmons [5] analysed the performance of hydrodynamicslider bearings for non-Newtonian lubricants, and Sharmaand Pandey [6] presented an experimental comparison of theslider bearing performance for different shapes.

In last few decades, the researches shown that the per-formances of the bearings can be improved, and enhancedpressure and load carrying capacity can be obtained byuse of the magnetic lubricants and magnetic fields [7–10].The applications of magnetic lubricant are widely found indampers, seals, sensors, loudspeakers, gauges, steppers, and

coating systems [11]. Investigators have used the Jenkinsmodel [12] for the lubricant flow. On the other hand,Shliomis [13, 14] proposed a ferrofluid flow model, in whichthe effects of rotation of magnetic particles, their magneticmoments, and the volume concentration are included. Ramand Verma [15] used the Shliomis model to investigate theperformance of a porous inclined slider bearing and reportedan increased pressure and load capacity. Shah and Bhat[16] used this model to study the ferrofluid-based squeezefilm characteristics of curved annular plates and obtainedsimilar results. Yamaguchi [17] presented a detailed analysisand simplification of the Shliomis model for differentlubricating conditions. Recently, the Shliomis model wasused by Lin [18] to study the squeeze film characteristicsof circular plates under magnetic field. It was observed thatthe volume concentration and the intensity of magnetic fieldprovide an increase in the load capacity and the time ofapproach.

All these researchers have analyzed the steady-state char-acteristics of the bearings lubricated with magnetic fluids.However, the study of the dynamic (damping and stiffness)characteristics of the bearings lubricated with magneticfluids is yet to be investigated for its importance in thebearing design.

2 Advances in Tribology

C.P.

Pivot

PAD

B

Hch1Z

Y

XO

h

Ho

h2

U

Figure 1: Schematic diagram of a curved slider bearing underapplied magnetic field with a pivot at the centre of pressure (C.P.)and Hc at B/2.

In the present theoretical analysis, an attempt has beenmade to investigate the effects of the particle rotation andvolume concentration of magnetic particles on the dynamiccharacteristics of curved slider bearings lubricated with mag-netic fluids in the presence of transversely uniform magneticfields.

2. Constitutive Equations andBoundary Conditions

The physical configuration of a curved slider bearing lubri-cated with a magnetic fluid in the presence of a transversemagnetic field is described in Figure 1. It is assumed thatthe thin film lubrication theory is applicable for the presentanalysis.

After the ferrohydrodynamic flow model by Shliomis [13,14], the electromagnetic field equations are

∇× �H = 0,

∇ ·(�H + �M

)= 0,

(1)

where �H is the applied magnetic field and �M is the magneti-zation vector.

Constitutive equations for incompressible magnetic fluidwith internal rotation of magnetic particles [13, 14, 17] are

∇ ·�v = 0,

ρD�vDt

= −∇p + η∇2�v + μo(�M · ∇

)�H +

12τs∇×

(�S− I�Ω

),

(2)

where �v is the fluid velocity vector, ρ is the fluid density, t isthe time, p is the pressure, η is the viscosity of the suspension,μo is the free space permeability, τs is the spin relaxation time,and I is the sum of moments of inertia of the particles per

unit volume. The local angular velocity of the �Ω, equilibrium

magnetization �M, and the internal angular moment �S aregiven as

�Ω = 12∇×�v,

�M =Mo

�HHo

+τBI

(�S× �M

),

�S = I�Ω + μoτs(�M × �H

).

(3)

where τB is the Brownian relaxation time of magneticparticles.

Assuming the slider to be infinitely wide, (2) for one di-mensional flow under isothermal condition [15, 16] are

∂u

∂x+∂w

∂z= 0, (4)

∂2u

∂z2= 1

η(1 + μoMoHoGτ/4η

) ∂p∂x

, (5)

where

τ = τB1 + μoτBτsMoHo/I

. (6)

In the Langevin relationship, the magnetization dependsupon the intensity of magnetic fields for a system of sphericalparticles [13, 14] and it is given as

Mo = nm(

Coth(α)− 1α

),

α = μomHo

kBT,

τs = 6ηφI

,

τB = 3ηφnkBT

,

(7)

where n is the number of magnetic particles per unitvolume, m is the magnetic moment of a particle, α is theLangevin parameter, kB is the Boltzmann constant, T is thetemperature, and φ is the volume fraction of the dispersedsolid phase.

Substituting (6) and (7) in (5) and introducing the mag-netoviscosity, the additional viscosity (ηm) due to rotation ofmagnetic particles [13, 14]:

ηm = 32ηφ

α− tanh(α)α + tanh(α)

, (8)

(5) can be rewritten as

∂2u

∂z2= 1

η + ηm

∂p

∂x. (9)

Advances in Tribology 3

Integrating (9) under the no-slip boundary conditions:

u = U at z = 0,

u = 0 at z = h,(10)

the expression of u is obtained as

u = 1η + ηm

∂p

∂x

(z2 − zh

)+ U

(1− z

h

). (11)

Substituting (11) in (4) and integrating under the boundaryconditions:

w = 0 at z = 0,

w = V = −∂h

∂tat z = h,

(12)

the modified Reynolds equation is obtained as

∂

∂x

(h3 ∂p

∂x

)= 6

(η + ηm

)U∂h

∂x+ 12

(η + ηm

)∂h∂t

. (13)

Here, the effective viscosity η [14] is given by

η = ηo

(1 +

4α + tanh(α)α + tanh(α)

φ)

, (14)

where ηo denotes the viscosity of the carrier fluid.For the one-dimensional curved slider bearings (Fig-

ure 1), the film thickness is taken as suggested by Abramovitz[19] in the form:

h = Hc

[4(x

B− 1

2

)2

− 1

]+[

(h1 − h2)(

1− x

B

)]+ h2,

(15)

which can be rewritten as

h(x, t) = hs(x) + hm(t), (16)

where hs determines the steady state film profile, given by

hs = (h1 − h2 + 4Hc)2

16Hc− (h1 − h2 + 4Hc)

Bx +

4Hc

B2x2, (17)

and hm(t) is the time dependent minimum film thicknesswhich, in steady state, can be given as

hm(0) =[

1− (1− r)2

− (1− r)2

16δ− δ

]h2, (18)

where, δ = Hc/h2 and r = h1/h2.Introducing the dimensionless quantities:

x = x

B, h = h

h2, hm = hm

h2,

hs = hsh2

, p = ph22

ηoUB,

t = Ut

B, V = BV

hU= dhm

dt,

η = η

ηo, ηm =

ηmηo

,

(19)

the modified Reynolds equation can be expressed in dimen-sionless form:

∂

∂x

(h

3 ∂p

∂x

)= −6

(η + ηm

)[{(1− 4δ − r) + 8δx} + 2V

],

(20)

h = hS + hm = (r − 1 + 4δ)16δ

− (r − 1 + 4δ)x + 4δx2 + hm(t).

(21)

Integrating (20) under the pressure boundary conditions p =0 at x = 0, 1, the film pressure is obtained as

p = − 6(η + ηm

)[(1− 4δ − r + 2V

)f1(x)

+ 4δ f2(x)]

+ C f3(x),(22)

where

f1(x) =∫ x

0

x

h3 dx,

f2(x) =∫ x

0

x2

h3 dx,

f3(x) =∫ x

0

1

h3 dx,

C = 6(η + ηm

)(

1− 4δ − r + 2V)f1(1) + 4δ f2(1)

f3(1).

(23)

The film force is obtained by integrating the film pressureover the slider length as follows:

F = L∫ B

x=0p dx. (24)

It takes the dimensionless form:

F =∫ 1

0p dx

= − 6(η + ηm

)[(1− 4δ − r + 2V

)∫ 1

0f1(x) dx

+ 4δ∫ 1

0f2(x) dx

]+ C

∫ 1

0f3(x)dx,

(25)

where F = (h22/ηoULB2)F.


The steady state load capacity (Fs) and centre of pressure(x) can be obtained from the film force and film pressure,respectively, under the steady state (i.e., hm = hm(0) and V =0) as follows:

FS = [F](t=0,V=0),

x =(∫ B

0 xp dx∫ B0 p dx

)

(t=0,V=0)

.(26)

The dynamic damping coefficient CD and the dynamicstiffness coefficient SD can be obtained as

CD =(∂F

∂V

)

(t=0,V=0),

SD =(∂F

∂hm

)

(t=0,V=0),

(27)

which take the dimensionless form:

CD = h32

ηoLB3CD

= − 12(η + ηm

)[∫ 1

0f1(x)dx

−(f1(1)f3(1)

)∫ 1

0f3(x)dx

]

t=0

,

SD = h32

ηoULB3SD

= − 6(η + ηm

)[(1− 4δ − r)

∫ 1

0

{∂

∂hmf1(x)

}dx

+ 4δ∫ 1

0

{∂

∂hmf2(x)

}dx

]

t=0

+

{C∫ 1

0

∂

∂hmf3(x)dx

}

t=0

+

{∂C

∂hm

∫ 1

0f3(x)dx

}

t=0

.

(28)

3. Results and Discussion

To study the ferrohydrodynamic effects on the dynamic char-acteristics of curved slider bearings, the numerical resultsfor the coefficients of dynamic damping (CD) and stiffness(SD) have been calculated. The effect of magnetic field isanalyzed by use of parameters φ and α which stand for theamount of volume concentration of particles and Langevinparameter, respectively. In the analysis, α = 0 and φ = 0 referto the cases of “no magnetic field” and common lubricants(nonmagnetic lubricant), respectively. In order to validatethe present results and to enhance its practical applicability,the steady state load carrying capacity of simple inclinedslider (δ = 0) is compared with the results of Taylor andDowson [20] in Figure 2.

The values of the parameters taken in the present analysisare as follows:

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 1.5 2 2.5 3

Taylor and Dowson

r

Fs

α = 5; φ = 0.2α = 2.5; φ = 0.2α = 5; φ = 0.1

α = 2.5; φ = 0.1α = 0; φ = 0

Figure 2: Variation of steady-state load capacity of plane inclinedslider with respect to film thickness ratio r for different values of αand φ.

(i) α: 0–10, (ii) φ: 0–1.0, (iii) δ: 0–0.5, and (iv) r: 1.0–2.0,Figure 2 shows the variation of dimensionless steady state

load capacity with respect to the inlet outlet film thicknessratio r for plane inclined slider (δ = 0) compared withthe results of Taylor and Dowson [20]. It is clear from thefigure that the load carrying capacity obtained in the presentanalysis for ferrohydrodynamic parameters α = 0, φ =0 is in the good agreement with the results obtained byTaylor and Dowson [20]. Further, it is observed that the loadcapacity increases with the increase of Langevin parameterα as well as the volume concentration parameter φ.

Figure 3 presents the variation of dimensionless dynamicdamping coefficient CD with respect to the volume con-centration parameter φ for Langevin parameter α =0, 2.5, 5, 7.5, 10 and curvature parameter δ = 0, 0.5. It isobserved that the coefficient of dynamic damping (CD)increases with the increase of φ in both the cases: (i) when themagnetic field is absent (α = 0) and (ii) when the uniformtransverse magnetic field is applied (α > 0). However, thepresence of magnetic field increases the damping coefficientwhich further increases with the increase of intensity ofmagnetic field (α), that is, the value of damping coefficientis higher for higher values of Langevin parameter α. Further,the nature of variation of damping coefficient for plain andcurved sliders (δ = 0 and δ = 0.5) is found similar.

Figure 4 presents the variation of dimensionless dynamicstiffness coefficient SD with respect to the Langevin parame-ter α for φ = 0, 0.1, 0.2, 0.3, 0.4 and curvature parameter δ =0, 0.5. It is observed that the coefficient of stiffness obtained

Advances in Tribology 5

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1φ

CD

α = 10α = 7.5α = 5α = 2.5

α = 0δ = 5δ = 0

Figure 3: Variation of dynamic damping coefficient with respect tovolume concentration parameter φ for film thickness ratio r = 2and different values of α.

with common lubricants (φ = 0) is less than that of aferrofluid based lubricant regardless of the intensity of themagnetic field (α). Further, with the ferrofluid analysis, thestiffness increases with the increase in Langevin parameter(α). It is also observed that the stiffness increases with α at ahigher rate upto α ≈ 2 and for α > 4, the effect of increasingthe field intensity (i.e., increasing α) decays in a sequentialmanner. The results are still sustained for both the plain andcurved sliders (δ = 0 and δ = 0.5).

4. Conclusions

In the present theoretical investigation, the dynamic charac-teristics of curved slider bearings are studied in the presenceof transverse magnetic field. The ferrofluid model given byShliomis is adopted to obtain modified Reynolds equation.Based on the results, so obtained, the following conclusionshave been drawn.

(1) Magnetic fluid under the transverse magnetic fieldprovides an improvement in the dynamic stiffnessand damping characteristics of slider bearings.

(2) On comparing with the conventional lubricants, boththe dynamic stiffness and the dynamic dampingcoefficients are higher for ferrofluids even if themagnetic field is absent.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10α

S D

φ = 0.4φ = 0.3φ = 0.2φ = 0.1

φ = 0

δ = 5δ = 0

Figure 4: Variation of dynamic stiffness coefficient with respect toLangevin parameter α for film thickness ratio r = 2 and differentvalues of φ.

(3) Under the applied magnetic field, both the stiffnessand the damping increase with the increase of volumeconcentration of magnetic particles.

Nomenclature

B: Length of the bearingCD,CD: Dynamic damping coefficients,

CD = (h32/ηoLB

3)CD

F,F: Film force, F = (h22/ηoUB2L)F

Fs: Dimensionless steady state load capacityh,h: Film thickness, h = h/h2

hm,hm: Time dependent minimum film thickness,hm = hm/h2

hs,hs: Steady state film thickness, hs = hs/h2

�H : Applied magnetic fieldI : Sum of moments of inertia of the particles

per unit volumekB: Boltzmann constantL: Width of the bearingm: Magnetic moment of a particle�M: Magnetization vectorn: Number of magnetic particles per unit

volumep, p: Film pressure, p = ph2

2/(ηoUB)r: h1/h2,�S: Internal angular moment


SD, SD: Dynamic stiffness coefficients,SD = (h3

2/ηoULB3)SDt, t: Time, t = Ut/BT : Temperature�v: Velocity vectorV : dhm/dtx, x: Coordinate distance, x = x/Bx: Centre of pressureα: Langevin parameterδ: Hc/h2

η,η: Effective viscosity (Shliomis model),η = η/ηo

ηm,ηm: Additional viscosity due magnetization,ηm = ηm/ηo

ηo: Viscosity of the carrier fluidμo: Free space permeability�Ω: Local angular velocityφ: Volume fraction of the dispersed solid phaseρ: Fluid densityτs: Spin relaxation timeτB: Brownian relaxation time of magnetic

particles.

Acknowledgment

The authors, hereby, thank Dr. V. K. Kapur (Former Profes-sor and Chairman, K. N. Institute of Technology, Sultanpur,India) for providing useful materials and the valuableguideline to enhance the content of this paper.

References

[1] R. S. Gupta and P. Kavita, “Analysis of rotation in the lubrica-tion of a porous slider bearing: small rotation,” Wear, vol. 111,no. 3, pp. 245–258, 1986.

[2] U. P. Singh and R. S. Gupta, “On the performance ofpivoted curved slider bearings: rabinowitsch fluid model,” inProceedings of the National Tribology Conference (NTC ’11), p.24, IIT Roorkee, 2011.

[3] M. D. Pascovici, A. Predescu, T. Cicone, and C. S. Popescu,“Experimental evidence of cavitational effects in a Rayleighstep slider,” Proceedings of the Institution of Mechanical Engi-neers, Part J, vol. 225, no. 6, pp. 527–537, 2011.

[4] K. Venkateswarlu and C. M. Rodkiewicz, “Thrust bearingcharacteristics when the slider is approaching terminal speed,”Wear, vol. 67, no. 3, pp. 341–350, 1981.

[5] P. D. Williams and G. R. Symmons, “Analysis of hydrodynamicslider thrust bearings lubricated with non-newtonian fluids,”Wear, vol. 117, no. 1, pp. 91–102, 1987.

[6] R. K. Sharma and R. K. Pandey, “Experimental studies ofpressure distributions in finite slider bearing with single con-tinuous surface profiles on the pads,” Tribology International,vol. 42, no. 7, pp. 1040–1045, 2009.

[7] V. K. Kapur, “Magneto-hydrodynamic pivoted slider bearingwith a convex pad surface,” Japanese Journal of Applied Physics,vol. 8, no. 7, pp. 827–835, 1969.

[8] M. E. Shimpi and G. M. Deheri, “A study on the performanceof a magnetic fluid based squeeze film in curved porous rotat-ing rough annular plates and deformation effect,” TribologyInternational, vol. 47, pp. 90–99, 2012.

[9] N. C. Das, “A study of optimum load-bearing capacity forslider bearings lubricated with couple stress fluids in magneticfield,” Tribology International, vol. 31, no. 7, pp. 393–400,1998.

[10] R. B. Kudenatti, D. P. Basti, and N. M. Bujurke, “Numericalsolution of the MHD reynolds equation for squeeze film lubri-cation between two parallel surfaces,” Applied Mathematicsand Computation, vol. 218, pp. 9372–9382, 2012.

[11] K. Raj, B. Moskowitz, and R. Casciari, “Advances in ferrofluidtechnology,” Journal of Magnetism and Magnetic Materials, vol.149, no. 1-2, pp. 174–180, 1995.

[12] J. T. Jenkins, “A theory of magnetic fluids,” Archive for RationalMechanics and Analysis, vol. 46, no. 1, pp. 42–60, 1972.

[13] M. I. Shliomis, “Effective viscosity of magnetic suspensions,”Soviet Physics, vol. 34, pp. 1291–1294, 1972.

[14] M. I. Shliomis, “Magnetic fluids,” Soviet Physics, vol. 17, pp.153–169, 1974.

[15] P. Ram and P. D. S. Verma, “Ferrofluid lubrication in porousinclined slider bearing,” Indian Journal of Pure and AppliedMathematics, vol. 30, no. 12, pp. 1273–1281, 1999.

[16] R. C. Shah and M. V. Bhat, “Ferrofluid squeeze film betweencurved annular plates including rotation of magnetic parti-cles,” Journal of Engineering Mathematics, vol. 51, no. 4, pp.317–324, 2005.

[17] H. Yamaguchi, Engineering Fluid Mechanics, Springer, Amster-dam, The Netherlands, 2008.

[18] J. R. Lin, “Derivation of ferrofluid lubrication equation of cyl-indrical squeeze films with convective fluid inertia forces andapplication to circular disks,” Tribology International, vol. 49,pp. 110–115, 2012.

[19] S. Abramovitz, “Theory for a slider bearing with a convex padsurface; side flow neglected,” Journal of the Franklin Institute,vol. 259, no. 3, pp. 221–233, 1955.

[20] C. M. Taylor and D. Dowson, “Turbulent lubrication theory—application to design,” Journal of Lubrication Technology, vol.96, no. 1, pp. 36–47, 1974.

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