Research ArticleThe Performance of Squeeze Film between Parallel TriangularPlates with a Ferro-Fluid Couple Stress Lubricant

Akbar Toloian1 Maghsood Daliri 1 and Nader Javani2

1Department of Mechanical Engineering Islamic Azad University Bonab Branch Bonab Iran2Faculty of Mechanical Engineering Yildiz Technical University Besiktas Istanbul Turkey

Correspondence should be addressed to Maghsood Daliri dmaghsoodyahoocom

Received 19 August 2019 Revised 1 December 2019 Accepted 27 December 2019 Published 30 January 2020

Academic Editor GunHee Jang

Copyright copy 2020 Akbar Toloian et al )is 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 isproperly cited

)e present study aims at investigating a couple stress ferrofluid lubricant effects on the performance of the squeezed film when auniform external magnetic field is applied For this purpose Shliomis ferrohydrodynamic and couple stress fluid models areemployed )e considered geometry is parallel triangular plates )e effects of couple stress volume concentration and Langevinparameters on squeeze film characteristics including time vs height relationship and load-carrying capacity are investigatedAccording to the results employing couple stress ferrofluid lubricant in the presence of the magnetic field leads to an increasedperformance of the squeeze film

1 Introduction

)e squeeze film technology has numerous applications indifferent fields of industry ranging from disc clutches torotating devices including almost all types of power plantwhere turbomachines are used In order to overwhelm therelated issues and making improvements magnetic fieldbase ferrofluid couple stress fluids are employed)e presentstudy aims at investigating the couple stress ferrofluid lu-bricant effects on the performance of the squeezed film oncea uniform external magnetic field is available For thispurpose Shliomis ferrohydrodynamic and couple stressfluid models are employed )e considered geometry isparallel triangular plates )e effects of couple stress volumeconcentration and Langevin parameters on squeeze filmcharacteristics including time-height relationship and load-carrying capacity are investigated

Daniel et al [1] applied a numerical method to analyzethe performance of an elastohydrodynamic journal bearingwhich is lubricated with ferrofluids using couple stresses Byincreasing the couple stresses and magnetism temperaturein the bearing increases Also the effect of Prandtl andEckert numbers on temperature is investigated in their

research Daniel et al [2] also analyzed magneto-electro-hydrodynamics by deriving the Reynolds equation throughemploying continuum and momentum equations Pressuredistribution in the journal bearing was obtained which in-dicated an increase in the pressure by enhancement ofmagnetism and couple stresses Daliri and Javani [3] in-vestigated a squeezing motion in conical plates where thelubricant was ferrofluid couple stress along with the effect ofconvective fluid inertia )ey concluded that an enhance-ment in convective fluid inertia increases the performance ofsqueezing film On the contrary Maxwell equations can beapplied in this case Shah and Bhat [4] and Lin [5] alsodeveloped a procedure to derive the governing equationwhich has been followed in the current study Daliri et al [6]investigated the characteristics of a squeezed film in con-junctions of parallel rectangular plates A couple stress in-compressible fluid was considered once a magnetic field isapplied Stokes microcontinuum in a couple-stress fluidalong with the modified Reynolds equation were combinedto obtain an analytical solution )ey also showed thatmagnetohydrodynamic couple stress fluids perform better ina steady load rather than a transient load conditions Forlubricated parallel annular discs combined with squeeze

HindawiAdvances in TribologyVolume 2020 Article ID 8151069 8 pageshttpsdoiorg10115520208151069

film and shear motions Daliri et al [7] showed that contactload-carrying capacity increases by the presence of mag-netohydrodynamic couple stress fluids For the same ge-ometry of parallel annular plates however increasedrotational inertia can diminish the squeeze film charac-teristics [8] Parallel disc lubrication film geometry withrough surfaces where the lubricant is piezoviscous couplestress is also studied by Daliri and Jalali-Vahid [9] )eyshowed that surface roughness pattern and the effect ofpressure on viscosity improve the squeeze film charac-teristics Another geometry of interest is parallel-steppedplates with porous surfaces where couple stress fluids areemployed for the lubrication purpose Biradar [10] derivedthe modified Reynolds equation using Stokes modelingcouple stress fluid with additives in the squeeze film Re-sults of his study show an increase in the squeeze film load-carrying capacity and its performance once the respondtime decreases compared with a classical Newtonian lu-bricant For a narrow journal bearing with transversalrough porous surface the effect of a slip on the hydro-magnetic squeeze film is investigated elsewhere [11]insisting the importance of the slip minimization in orderto improve the performance of these bearings RecentlyHanumagowda et al [12] studied the effects of viscosity-pressure dependency on the non-Newtonian squeeze filmperformance between parallel-stepped circular platesconsidering effects of surface roughness )ey concludedthat the using of couple stress fluid and considering theviscosity-pressure dependency increased the squeeze filmcharacteristics It was also found that considering thesurface roughness effects has significant effect on theperformance of the squeeze film

Parallel circular disc lubrication film geometry withrough surfaces considering effects of rotational inertiawhere the lubricant is ferrofluid couple stress is alsostudied by Daliri [13] He showed that using of both fer-rofluid and couple stress lubricants improve the squeezefilm characteristics It was also found that with increasingrotational inertia the squeeze film performance is de-creased In a recent study of triangular plates the effects ofviscosity-pressure dependency together with couple stressfluid on the squeeze film characteristics were investigatedby AminKhani and Daliri [14] According to their researchit was concluded that the dependency of lubricant viscosityon pressure increases the squeeze film characteristics Onthe contrary it was found that the couple stress fluidimproves the squeeze film performance compared withNewtonian fluid

According to the earlier studies analysis of squeeze filmcharacteristics between triangular plates with ferrofluidcouple stress lubricant has not been investigated previ-ously )e motivation of the present study is due to theapplication of triangular platesrsquo geometry in industryFriction discs used in wet clutches normally have somekind of groove pattern (triangular shape) on the frictionsurface to allow the lubricant to flow over the surface andcool down the clutch pack )erefore one of the practical

situations where the geometry of triangular plates could beused is wet clutch

In this research squeeze film performance is investigatedbetween parallel triangular plates which are lubricated by acouple stress ferrofluid lubricant in the presence of a uni-form external magnetic field (Ho) to obtain the character-istics of the squeeze film )e load-carrying capacity and therelation between time and height are studied for differentcouple stress values along with the Langevin and volumeconcentration parameters as outcomes of squeeze filmcharacteristics

2 Governing Equations

)e geometry of triangular plates which are lubricatedunder a couple stress ferro-fluid is shown in Figure 1Transverse magnetic field is dominant in the system Twoplates are approaching to each other at a relative velocity ofminus dhdt in which the upper plates is moving down Con-sidering the available geometry and dimensions it isreasonable to apply the prevalence of thin-film lubricationtheory In order to formulate the x-direction component ofthe velocity (u) and y-direction velocity component (v)Shliomis [15 16] model of ferro-hydrodynamic and Stokesmicro-continuum theory [17] are employed )ereforecontinuity and momentum equations are expressed asfollows

0 minuszp

zx+ η

z2u

zz2 + ητz2u

zz2 minus ηc

z4u

zz4 (1)

0 minuszp

zy+ η

z2v

zz2 + ητz2v

zz2 minus ηc

z4v

zz4(2)

zp

zz 0 (3)

)e equation of continuity is

zu

zx+

zv

zy+

zw

zz 0 (4)

where

τ 32Φξ minus tanh ξξ + tanh ξ

(5)

Here τ represents the rotational viscosity η is thesuspension viscosity and ηc is a constant related to thecouple stress fluid property Also Φ and ξ represent thevolumetric concentration of the magnetic particles andLangevin parameter w is the z-direction component of thevelocity the suspension viscosity is approximated as follows[16]

η η0(1 + 25Φ) (6)

2 Advances in Tribology

)e main liquid velocity is η0 and the boundary con-ditions of velocity components can be formulated as follows

z 0

u 0

z2u

zz2 0

v 0

z2v

zz2 0

w 0

(7)

z h

u 0

z2u

zz2 0

v 0

z2v

zz2 0

w dh

dt

(8)

Considering no-slip condition at z 0 and z h the velocity(u) is equal to zero At the same time no-couple stress con-ditions at the surface implies that z2uzz2 0 and z2vzz2 0for z 0 and z h [3] Since the lower plate is not moving thenfor z 0 both v andw velocities are zero Upper plate squeezingmotion will lead to v 0 at z h and w dhdt

Equations (1) and (2) will lead to the velocity compo-nents of u and v under the boundary conditions described in(7) and (8)

u 1

2η0(1 + τ)(1 + 25Φ)

zp

zxz2

minus hz1113872 11138731113896

+2l2c

(1 + τ)(1 + 25Φ)

middot 1 minuscosh ((2z minus h)

(1 + τ)(1 + 25Φ)

1113968)2lc1113872 1113873

cosh (h(1 + τ)(1 + 25Φ)

1113968)2lc1113872 1113873

⎡⎢⎣ ⎤⎥⎦⎫⎬

⎭

v 1

2η0(1 + τ)(1 + 25Φ)

zp

zyz2

minus hz1113872 11138731113896

+2l2c

(1 + τ)(1 + 25Φ)

middot 1 minuscosh ((2z minus h)

(1 + τ)(1 + 25Φ)

1113968)2lc1113872 1113873

cosh (h(1 + τ)(1 + 25Φ)

1113968)2lc1113872 1113873

⎡⎢⎣ ⎤⎥⎦⎫⎬

⎭

(9)

where lc ηc η0 shows characteristics length of the addi-tives responsible for couple stress fluid

Continuity equation integration over the thickness of thelubricant film will give the following expression

z

zx1113946

h

0udz1113888 1113889 +

z

zy1113946

h

0vdz1113888 1113889

zh

zt (10)

Once u and v are substituted in equation (10) Reynolds-type coupled stress ferro-fluid squeeze-film can be obtainedas follows

z2p

zx2 +z2p

zy2 minus 2η0A2(dhdt)

f lc τΦ h( 1113857 (11)

Z

X

Y Ho

parttparth

Figure 1 Squeeze film geometry between parallel triangular plates

Advances in Tribology 3

where A (1 + τ)(1 + 25Φ)

1113968f(lc τΦ h) minus (h36) +

(2l2c A2)[h minus (2lcA)tanh(Ah2lc)]

)eboundary conditions for pressure field are as follows

p x1 y1( 1113857 0 (12)

where

x1 minus a( 1113857 x1 minus3

radicy1 + 2a( 1113857 x1 +

3

radicy1 + 2a( 1113857 0 (13)

)e equilateral triangle side a is expressed as

(x minus a)(x minus3

radicy + 2a)(x +

3

radicy + 2a) 0 (14)

)e intersection point of the triangle medians is selectedto be the origin and thereafter equations (11) and (12) can besolved to obtain the pressure equation as follows

p +2η0A2(dhdt)

f lc τΦ h( 1113857

a2

31 minus

34a2x

2minus

34a2y

2minus

14a3x

3+

34a3 xy

21113874 1113875

(15)

Dimensionless form of the pressure distribution willhave the following format

plowast

minus ph3

oη0(dhdt)3

3

radica2

minus 2A2

93

radicflowast llowast τΦ hlowast( )

middot 1 minus xlowast

( 1113857 1 +

3

radic

2ylowast

+xlowast

21113888 1113889 1 minus

3

radic

2ylowast

+xlowast

21113888 11138891113896

(16)

where

llowast

lc

ho

xlowast

x

a

ylowast

y

a

hlowast

h

ho

flowast

llowast τΦ h

lowast( 1113857

f lc τΦ h( 1113857

h3o

flowast

llowast τΦ h

lowast( 1113857 minus

hlowast3

6+2llowast2

A2 hlowast

minus2llowast

Atanh

Ahlowast

2llowast1113888 11138891113890 1113891

(17)

where ho is the initial film thickness)e expression for the load-carrying capacity is obtained

by

w 1113946a

minus 2a1113946

2a+x1( )radic3

minus 2a+x1( )radic3p dy1dx1

w 9

3

radicη0A2(dhdt)a4

10f lc τΦ h( 1113857

(18)

)e expression for the load-carrying capacity indimensionless form is

wlowast

minus wh3

o27η0(dhdt)a4

minus3

radicA2

30flowast llowast τΦ hlowast( ) (19)

Introducing the dimensionless response time

wlowast

wh2

oη0a4 t (20)

)e time-height relationship can be derived in the di-mensionless form as follows

dhlowast

dtlowast minus

1Wlowast

(21)

Equation (21) is an ordinary but nonlinear differentialequation To obtain an accurate result 4th order Run-gendashKutta method is applied through employing the fol-lowing initial condition

hlowast

1 at tlowast

0 (22)

3 Results and Discussion

From the above analysis the couple stress ferrofluid-lu-bricated squeeze film performances are characterized bythree parameters the non-Newtonian couple stress pa-rameter llowast the volume concentration of particles ϕ and theLangevin parameter ξ )e special case can be obtained fromspecific values of these parameters

(i) ϕ 0 and ξ 0 the couple stress nonferrofluid casewithout magnetic fields )e nondimensional loadcapacity in equation (19) reduces to

(ii) limϕ⟶0 ξ⟶0wlowast

3

radic(5 hlowast3 minus 60 llowast2hlowast + 120llowast3

tanh(hlowast2llowast))

(iii) )is is the derivation of Fathima et al [18] whenneglecting the MHD effects(the Hartmann numbertends to zero)

)e geometry of parallel triangular plates with a ferro-fluid lubricant of couple stress is investigated )e results ofthe study three parameters including the non-Newtoniancouple stress parameter llowast the volume concentration ofparticles Φ and Langevin parameter ξ are employed to

Table 1 Rheological parameters of lubricantInitial film thickness ho 12times10minus 5mTriangle geometry a 02mViscosity of main liquid η0 27times10minus 5 pamiddotsCouple stress material constants ηc 3888times10minus 15NmiddotsSqueezing velocity V 32msCharacteristic length of the additives lc 12times10minus 6mCouple stress parameter llowast 01)e free space permeability μp 4π times10minus 7 kgmiddotmmiddotsminus 2middotAmpminus 2

Magnetic moment of a particle m 004Magnetic field Ho 16363times10minus 5 Ampmiddotmminus 1

)e Boltzmann constant KB 138times10minus 23 kgmiddotm2middotsminus 2middotkminus 1

)e absolute temperature T 298k)e Langevin parameter ξ 10

4 Advances in Tribology

ϕ = 0 ξ = 0 llowast = 0 hlowast = 04ϕ = 002 ξ = 5 llowast = 0 hlowast = 04ϕ = 004 ξ = 5 llowast = 0 hlowast = 04

ϕ = 004 ξ = 10 llowast = 0 hlowast = 04ϕ = 004 ξ = 10 llowast = 006 hlowast = 04

0

5

10

15

20

plowast

ndash15 ndash1 ndash05 0 05 1ndash2xlowast

Figure 2 Pressure distribution for different values of Φ ξ and llowast at hlowast 04

ϕ = 0 ξ = 0 llowast = 0 hlowast = 03ϕ = 002 ξ = 5 llowast = 0 hlowast = 03ϕ = 004 ξ = 5 llowast = 0 hlowast = 03

ϕ = 004 ξ = 10 llowast = 0 hlowast = 03ϕ = 004 ξ = 10 llowast = 006 hlowast = 03

ndash15 ndash1 ndash05 0 05 1ndash2xlowast

0

10

20

30

40

50

plowast

Figure 3 Pressure distribution for different values of Φ ξ and llowast at hlowast 03

ξ = 0 llowast = 006 hlowast = 04ξ = 2 llowast = 006 hlowast = 04

ξ = 5 llowast = 006 hlowast = 04ξ = 10 llowast = 006 hlowast = 04

001 002 003 004 005 006 007 0080ϕ

65

7

75

8

85

wlowast

Figure 4 Variations of dimensionless load capacity versus volume concentration Φ for different values of ξ at llowast 006 and hlowast 04

Advances in Tribology 5

ϕ = 002 llowast = 006 hlowast = 04ϕ = 004 llowast = 006 hlowast = 04

ϕ = 006 llowast = 006 hlowast = 04ϕ = 008 llowast = 006 hlowast = 04

2 4 6 8 10 12 14 16 18 200ξ

7

75

8

85

wlowast

Figure 5 Variations of dimensionless load capacity versus Langevin parameter ξ for different values of Φ at llowast 006 and hlowast 04

ϕ = 004 ξ = 10 hlowast = 05ϕ = 004 ξ = 10 hlowast = 04ϕ = 004 ξ = 10 hlowast = 03

0

20

40

60

80

100

120

140

wlowast

005 01 015 02 025 030llowast

Figure 6 Dimensionless load capacity versus couple stress parameter llowast for different values of hlowast at ξ 10 and Φ 004

ϕ = 004 ξ = 10 llowast = 0ϕ = 004 ξ = 10 llowast = 01

ϕ = 004 ξ = 10 llowast = 02ϕ = 004 ξ = 10 llowast = 03

03 04 05 06 07 08 09 102hlowast

0

10

20

30

40

50

60

tlowast

Figure 7 Dimensionless lubricant film thickness variations as a function of dimensionless time at s of 004 and ξ 10 for different values ofllowast

6 Advances in Tribology

characterize the squeeze film performance Table 1 shows therheological parameters of lubricant At hlowast 04 and hlowast 03lubricant film pressure distribution for altered values of Φ ξare shown in Figures 2 and 3 respectively By increasing thevalues of Φ ξ and llowast there will occur an enhancement in themaximum pressure of lubricant

Dimensionless load-carrying capacity Wlowast variation withrespect to the volumetric concentration (Φ) Langevin (ξ)and couple stress llowast are shown in Figures 4ndash6 Knowing thatthese parameters lead to a higher film pressure it can be seenthat the integrated load-carrying capacity will increasesubsequently )e load-carrying capacity can be wiselyconsidered as a function of Φ ξ and llowast which is shown inFigures 4ndash6 where increasing Φ ξ and llowast will lead to anincrease in load-carrying capacity which refers to the leastload-carrying capacity in a Newtonian fluid

Dimensionless film thickness (hlowast) changes versus di-mensionless response time (tlowast) at altered coupled stresses(llowast) is illustrated in Figure 7 It is also concluded that in-creasing llowast will lead to an increase in the response time

4 Conclusion

In this research the effects of couple stress ferrofluid lu-bricant in the presence of magnetic field on the squeeze filmperformance are investigated Shliomis ferrofluid and Stokescouple stress fluid models are used and the consideredgeometry is triangular plates By solving Reynolds equationthe pressure distribution is obtained and also with inte-grating on pressure in the film region load-carrying capacityis derived A 4th order RungendashKutta method is used to solvethe non linear differential equation between lubricant filmthickness and time Furthermore the variations of lubricantfilm thickness with the time are presented According to theresults it is found that using both ferrofluid and couplestress fluid as a lubricant will increase characteristics of thesqueeze film such as load-carrying capacity pressure dis-tribution and triangular plates moving time significantly)e obtained results are beneficial in designing of thebearings in the engineering applications

Nomenclature

a )e equilateral triangle sideu v w Velocity component in the Cartesian coordinate

systemΦ )e volume concentration of particlesWlowast Dimensionless load-carrying capacity

Wh3o27η0a4V

t tlowast Response time (Wh2oη0a4)t

η0 Viscosity of main liquidlc Characteristic length of the additives (ηcη0)x y z Cartesian coordinatesllowast )e couple stress parameter lchoHo Applied magnetic fieldm Magnetic moment of a particleT )e absolute temperatureξ )e Langevin parameter ξ μpmHoKBT

ho Initial lubricant film thickness

xlowast ylowast Dimensionless coordinates xlowast xa ylowast yaV Squeezing velocity minus dhdt

p plowast Squeeze film pressure ph3oη0V3

3

radica2

τ Rotational viscosity parameterηc Couple stress fluid indexh hlowast Film thickness hho(dimensionless)η Viscosity of the suspensionμp )e free space permeabilityKB )e Boltzmann constant

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare no conflicts of interest

References

[1] K Daniel K Mathew and K Mark ldquoInvestigation of Tem-perature effects for a finite elasto-hydrodynamic journalbearing lubricated by ferro fluids with couple stressesrdquoJournal of Computations amp Modelling vol 5 no 3 pp 79ndash902015

[2] K Daniel K Mathew and K Mark ldquoInvestigation of pressuredistributions for a finite elastohydrodynamic journal bearinglubricated by Ferro fluids with couple stressesrdquo AmericanJournal of Applied Mathematics vol 2 no 4 pp 135ndash1402014

[3] M Daliri and N Javani ldquoSqueeze film characteristics of ferro-coupled stress inertial fluids in conical platesrdquo IndustrialLubrication and Tribology vol 70 no 5 pp 872ndash877 2018

[4] R C Shah and M V Bhat ldquoFerrofluid squeeze film betweencurved annular plates including rotation of magnetic parti-clesrdquo Journal of Engineering Mathematics vol 51 no 4pp 317ndash324 2005

[5] J-R Lin ldquoDerivation of ferrofluid lubrication equation ofcylindrical squeeze films with convective fluid inertia forcesand application to circular disksrdquo Tribology Internationalvol 49 pp 110ndash115 2012

[6] M Daliri D Jalali-Vahid and H Rahnejat ldquoSqueeze filmlubrication of coupled stress electrically conducting inertialfluids in wide parallel rectangular conjunctions subjected to amagnetic fieldrdquo Proceedings of the Institution of MechanicalEngineers Part J Journal of Engineering Tribology vol 228no 3 pp 288ndash302 2014

[7] M Daliri D Jalali-Vahid and H Rahnejat ldquoMagneto-hy-drodynamics of couple stress lubricants in combined squeezeand shear in parallel annular disc viscous coupling systemsrdquoProceedings of the Institution of Mechanical Engineers Part JJournal of Engineering Tribology vol 229 no 5 pp 578ndash5962015

[8] M Daliri and D Jalali-Vahid ldquoInvestigation of combinedeffects of rotational inertia and viscosityndashpressure dependencyon the squeeze film characteristics of parallel annular plateslubricated by couple stress fluidrdquo Journal of Tribology vol 137no 3 Article ID 031702 2015

[9] M Daliri and D Jalali-Vahid ldquoCombined effects of pie-zomdashviscous coupled stress lubricant and rotational inertiaupon squeeze film performance of rough circular discsrdquo In-dustrial Lubrication and Tribology vol 67 no 6 pp 564ndash5712015

Advances in Tribology 7

[10] T V Biradar ldquoSqueeze film lubrication between porousparallel stepped plates with couple stress fluidsrdquo TribologyOnline vol 8 no 5 pp 278ndash284 2013

[11] S Shukla and G Deheri ldquoHydromagnetic squeeze film inrough porous narrow journal bearing a study of slip effectrdquoTribology Online vol 12 no 4 pp 177ndash186 2017

[12] B Hanumagowda B Raju J Santhosh Kumar andK Vasanth ldquoCombined effect of surface roughness andpressure-dependent viscosity over couple-stress squeeze filmlubrication between circular stepped platesrdquo Proceedings ofthe Institution of Mechanical Engineers Part J Journal ofEngineering Tribology vol 232 no 5 pp 525ndash534 2018

[13] M Daliri ldquoInvestigation of squeeze film performance in roughparallel circular discs by ferro-fluid couple stress lubricantconsidering effects of rotational inertiardquo Industrial Lubrica-tion and Tribology vol 70 no 7 pp 1201ndash1208 2018

[14] H Aminkhani and M Daliri ldquoEffects of piezo-viscous-cou-pled stress lubricant on the squeeze film performance ofparallel triangular platesrdquo Proceedings of the Institution ofMechanical Engineers Part J Journal of Engineering Tribology2019

[15] M I Shliomis ldquoEffective viscosity of magnetic suspensionsrdquoSoviet Physics Journal of Experimental and8eoretical Physicsvol 34 no 6 pp 1291ndash1294 1972

[16] M I Shliomis ldquoMagnetic fluidsrdquo Soviet Physics Uspekhivol 17 no 2 pp 153ndash169 1974

[17] V K Stokes ldquoCouple stresses in fluidsrdquo Physics of Fluidsvol 9 no 9 pp 1709ndash1715 1966

[18] S T Fathima N B Naduvinamani B N Hanumagowda andJ Santhosh Kumar ldquoModified Reynolds equation for differenttypes of finite plates with the combined effect of MHD andcouple stressesrdquo Tribology Transactions vol 58 no 4pp 660ndash667 2015

8 Advances in Tribology

)e main liquid velocity is η0 and the boundary con-ditions of velocity components can be formulated as follows

z 0

u 0

z2u

zz2 0

v 0

z2v

zz2 0

w 0

(7)

z h

u 0

z2u

zz2 0

v 0

z2v

zz2 0

w dh

dt

(8)

Considering no-slip condition at z 0 and z h the velocity(u) is equal to zero At the same time no-couple stress con-ditions at the surface implies that z2uzz2 0 and z2vzz2 0for z 0 and z h [3] Since the lower plate is not moving thenfor z 0 both v andw velocities are zero Upper plate squeezingmotion will lead to v 0 at z h and w dhdt

Equations (1) and (2) will lead to the velocity compo-nents of u and v under the boundary conditions described in(7) and (8)

u 1

2η0(1 + τ)(1 + 25Φ)

zp

zxz2

minus hz1113872 11138731113896

+2l2c

(1 + τ)(1 + 25Φ)

middot 1 minuscosh ((2z minus h)

(1 + τ)(1 + 25Φ)

1113968)2lc1113872 1113873

cosh (h(1 + τ)(1 + 25Φ)

1113968)2lc1113872 1113873

⎡⎢⎣ ⎤⎥⎦⎫⎬

⎭

v 1

2η0(1 + τ)(1 + 25Φ)

zp

zyz2

minus hz1113872 11138731113896

+2l2c

(1 + τ)(1 + 25Φ)

middot 1 minuscosh ((2z minus h)

(1 + τ)(1 + 25Φ)

1113968)2lc1113872 1113873

cosh (h(1 + τ)(1 + 25Φ)

1113968)2lc1113872 1113873

⎡⎢⎣ ⎤⎥⎦⎫⎬

⎭

(9)

where lc ηc η0 shows characteristics length of the addi-tives responsible for couple stress fluid

Continuity equation integration over the thickness of thelubricant film will give the following expression

z

zx1113946

h

0udz1113888 1113889 +

z

zy1113946

h

0vdz1113888 1113889

zh

zt (10)

Once u and v are substituted in equation (10) Reynolds-type coupled stress ferro-fluid squeeze-film can be obtainedas follows

z2p

zx2 +z2p

zy2 minus 2η0A2(dhdt)

f lc τΦ h( 1113857 (11)

Z

X

Y Ho

parttparth

Figure 1 Squeeze film geometry between parallel triangular plates

Advances in Tribology 3

where A (1 + τ)(1 + 25Φ)

1113968f(lc τΦ h) minus (h36) +

(2l2c A2)[h minus (2lcA)tanh(Ah2lc)]

)eboundary conditions for pressure field are as follows

p x1 y1( 1113857 0 (12)

where

x1 minus a( 1113857 x1 minus3

radicy1 + 2a( 1113857 x1 +

3

radicy1 + 2a( 1113857 0 (13)

)e equilateral triangle side a is expressed as

(x minus a)(x minus3

radicy + 2a)(x +

3

radicy + 2a) 0 (14)

)e intersection point of the triangle medians is selectedto be the origin and thereafter equations (11) and (12) can besolved to obtain the pressure equation as follows

p +2η0A2(dhdt)

f lc τΦ h( 1113857

a2

31 minus

34a2x

2minus

34a2y

2minus

14a3x

3+

34a3 xy

21113874 1113875

(15)

Dimensionless form of the pressure distribution willhave the following format

plowast

minus ph3

oη0(dhdt)3

3

radica2

minus 2A2

93

radicflowast llowast τΦ hlowast( )

middot 1 minus xlowast

( 1113857 1 +

3

radic

2ylowast

+xlowast

21113888 1113889 1 minus

3

radic

2ylowast

+xlowast

21113888 11138891113896

(16)

where

llowast

lc

ho

xlowast

x

a

ylowast

y

a

hlowast

h

ho

flowast

llowast τΦ h

lowast( 1113857

f lc τΦ h( 1113857

h3o

flowast

llowast τΦ h

lowast( 1113857 minus

hlowast3

6+2llowast2

A2 hlowast

minus2llowast

Atanh

Ahlowast

2llowast1113888 11138891113890 1113891

(17)

where ho is the initial film thickness)e expression for the load-carrying capacity is obtained

by

w 1113946a

minus 2a1113946

2a+x1( )radic3

minus 2a+x1( )radic3p dy1dx1

w 9

3

radicη0A2(dhdt)a4

10f lc τΦ h( 1113857

(18)

)e expression for the load-carrying capacity indimensionless form is

wlowast

minus wh3

o27η0(dhdt)a4

minus3

radicA2

30flowast llowast τΦ hlowast( ) (19)

Introducing the dimensionless response time

wlowast

wh2

oη0a4 t (20)

)e time-height relationship can be derived in the di-mensionless form as follows

dhlowast

dtlowast minus

1Wlowast

(21)

Equation (21) is an ordinary but nonlinear differentialequation To obtain an accurate result 4th order Run-gendashKutta method is applied through employing the fol-lowing initial condition

hlowast

1 at tlowast

0 (22)

3 Results and Discussion

From the above analysis the couple stress ferrofluid-lu-bricated squeeze film performances are characterized bythree parameters the non-Newtonian couple stress pa-rameter llowast the volume concentration of particles ϕ and theLangevin parameter ξ )e special case can be obtained fromspecific values of these parameters

(i) ϕ 0 and ξ 0 the couple stress nonferrofluid casewithout magnetic fields )e nondimensional loadcapacity in equation (19) reduces to

(ii) limϕ⟶0 ξ⟶0wlowast

3

radic(5 hlowast3 minus 60 llowast2hlowast + 120llowast3

tanh(hlowast2llowast))

(iii) )is is the derivation of Fathima et al [18] whenneglecting the MHD effects(the Hartmann numbertends to zero)

)e geometry of parallel triangular plates with a ferro-fluid lubricant of couple stress is investigated )e results ofthe study three parameters including the non-Newtoniancouple stress parameter llowast the volume concentration ofparticles Φ and Langevin parameter ξ are employed to

Table 1 Rheological parameters of lubricantInitial film thickness ho 12times10minus 5mTriangle geometry a 02mViscosity of main liquid η0 27times10minus 5 pamiddotsCouple stress material constants ηc 3888times10minus 15NmiddotsSqueezing velocity V 32msCharacteristic length of the additives lc 12times10minus 6mCouple stress parameter llowast 01)e free space permeability μp 4π times10minus 7 kgmiddotmmiddotsminus 2middotAmpminus 2

Magnetic moment of a particle m 004Magnetic field Ho 16363times10minus 5 Ampmiddotmminus 1

)e Boltzmann constant KB 138times10minus 23 kgmiddotm2middotsminus 2middotkminus 1

)e absolute temperature T 298k)e Langevin parameter ξ 10

4 Advances in Tribology

ϕ = 0 ξ = 0 llowast = 0 hlowast = 04ϕ = 002 ξ = 5 llowast = 0 hlowast = 04ϕ = 004 ξ = 5 llowast = 0 hlowast = 04

ϕ = 004 ξ = 10 llowast = 0 hlowast = 04ϕ = 004 ξ = 10 llowast = 006 hlowast = 04

0

5

10

15

20

plowast

ndash15 ndash1 ndash05 0 05 1ndash2xlowast

Figure 2 Pressure distribution for different values of Φ ξ and llowast at hlowast 04

ϕ = 0 ξ = 0 llowast = 0 hlowast = 03ϕ = 002 ξ = 5 llowast = 0 hlowast = 03ϕ = 004 ξ = 5 llowast = 0 hlowast = 03

ϕ = 004 ξ = 10 llowast = 0 hlowast = 03ϕ = 004 ξ = 10 llowast = 006 hlowast = 03

ndash15 ndash1 ndash05 0 05 1ndash2xlowast

0

10

20

30

40

50

plowast

Figure 3 Pressure distribution for different values of Φ ξ and llowast at hlowast 03

ξ = 0 llowast = 006 hlowast = 04ξ = 2 llowast = 006 hlowast = 04

ξ = 5 llowast = 006 hlowast = 04ξ = 10 llowast = 006 hlowast = 04

001 002 003 004 005 006 007 0080ϕ

65

7

75

8

85

wlowast

Figure 4 Variations of dimensionless load capacity versus volume concentration Φ for different values of ξ at llowast 006 and hlowast 04

Advances in Tribology 5

ϕ = 002 llowast = 006 hlowast = 04ϕ = 004 llowast = 006 hlowast = 04

ϕ = 006 llowast = 006 hlowast = 04ϕ = 008 llowast = 006 hlowast = 04

2 4 6 8 10 12 14 16 18 200ξ

7

75

8

85

wlowast

Figure 5 Variations of dimensionless load capacity versus Langevin parameter ξ for different values of Φ at llowast 006 and hlowast 04

ϕ = 004 ξ = 10 hlowast = 05ϕ = 004 ξ = 10 hlowast = 04ϕ = 004 ξ = 10 hlowast = 03

0

20

40

60

80

100

120

140

wlowast

005 01 015 02 025 030llowast

Figure 6 Dimensionless load capacity versus couple stress parameter llowast for different values of hlowast at ξ 10 and Φ 004

ϕ = 004 ξ = 10 llowast = 0ϕ = 004 ξ = 10 llowast = 01

ϕ = 004 ξ = 10 llowast = 02ϕ = 004 ξ = 10 llowast = 03

03 04 05 06 07 08 09 102hlowast

0

10

20

30

40

50

60

tlowast

Figure 7 Dimensionless lubricant film thickness variations as a function of dimensionless time at s of 004 and ξ 10 for different values ofllowast

6 Advances in Tribology

characterize the squeeze film performance Table 1 shows therheological parameters of lubricant At hlowast 04 and hlowast 03lubricant film pressure distribution for altered values of Φ ξare shown in Figures 2 and 3 respectively By increasing thevalues of Φ ξ and llowast there will occur an enhancement in themaximum pressure of lubricant

Dimensionless load-carrying capacity Wlowast variation withrespect to the volumetric concentration (Φ) Langevin (ξ)and couple stress llowast are shown in Figures 4ndash6 Knowing thatthese parameters lead to a higher film pressure it can be seenthat the integrated load-carrying capacity will increasesubsequently )e load-carrying capacity can be wiselyconsidered as a function of Φ ξ and llowast which is shown inFigures 4ndash6 where increasing Φ ξ and llowast will lead to anincrease in load-carrying capacity which refers to the leastload-carrying capacity in a Newtonian fluid

Dimensionless film thickness (hlowast) changes versus di-mensionless response time (tlowast) at altered coupled stresses(llowast) is illustrated in Figure 7 It is also concluded that in-creasing llowast will lead to an increase in the response time

4 Conclusion

In this research the effects of couple stress ferrofluid lu-bricant in the presence of magnetic field on the squeeze filmperformance are investigated Shliomis ferrofluid and Stokescouple stress fluid models are used and the consideredgeometry is triangular plates By solving Reynolds equationthe pressure distribution is obtained and also with inte-grating on pressure in the film region load-carrying capacityis derived A 4th order RungendashKutta method is used to solvethe non linear differential equation between lubricant filmthickness and time Furthermore the variations of lubricantfilm thickness with the time are presented According to theresults it is found that using both ferrofluid and couplestress fluid as a lubricant will increase characteristics of thesqueeze film such as load-carrying capacity pressure dis-tribution and triangular plates moving time significantly)e obtained results are beneficial in designing of thebearings in the engineering applications

Nomenclature

a )e equilateral triangle sideu v w Velocity component in the Cartesian coordinate

systemΦ )e volume concentration of particlesWlowast Dimensionless load-carrying capacity

Wh3o27η0a4V

t tlowast Response time (Wh2oη0a4)t

η0 Viscosity of main liquidlc Characteristic length of the additives (ηcη0)x y z Cartesian coordinatesllowast )e couple stress parameter lchoHo Applied magnetic fieldm Magnetic moment of a particleT )e absolute temperatureξ )e Langevin parameter ξ μpmHoKBT

ho Initial lubricant film thickness

xlowast ylowast Dimensionless coordinates xlowast xa ylowast yaV Squeezing velocity minus dhdt

p plowast Squeeze film pressure ph3oη0V3

3

radica2

τ Rotational viscosity parameterηc Couple stress fluid indexh hlowast Film thickness hho(dimensionless)η Viscosity of the suspensionμp )e free space permeabilityKB )e Boltzmann constant

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare no conflicts of interest

References

[1] K Daniel K Mathew and K Mark ldquoInvestigation of Tem-perature effects for a finite elasto-hydrodynamic journalbearing lubricated by ferro fluids with couple stressesrdquoJournal of Computations amp Modelling vol 5 no 3 pp 79ndash902015

[2] K Daniel K Mathew and K Mark ldquoInvestigation of pressuredistributions for a finite elastohydrodynamic journal bearinglubricated by Ferro fluids with couple stressesrdquo AmericanJournal of Applied Mathematics vol 2 no 4 pp 135ndash1402014

[3] M Daliri and N Javani ldquoSqueeze film characteristics of ferro-coupled stress inertial fluids in conical platesrdquo IndustrialLubrication and Tribology vol 70 no 5 pp 872ndash877 2018

[4] R C Shah and M V Bhat ldquoFerrofluid squeeze film betweencurved annular plates including rotation of magnetic parti-clesrdquo Journal of Engineering Mathematics vol 51 no 4pp 317ndash324 2005

[5] J-R Lin ldquoDerivation of ferrofluid lubrication equation ofcylindrical squeeze films with convective fluid inertia forcesand application to circular disksrdquo Tribology Internationalvol 49 pp 110ndash115 2012

[6] M Daliri D Jalali-Vahid and H Rahnejat ldquoSqueeze filmlubrication of coupled stress electrically conducting inertialfluids in wide parallel rectangular conjunctions subjected to amagnetic fieldrdquo Proceedings of the Institution of MechanicalEngineers Part J Journal of Engineering Tribology vol 228no 3 pp 288ndash302 2014

[7] M Daliri D Jalali-Vahid and H Rahnejat ldquoMagneto-hy-drodynamics of couple stress lubricants in combined squeezeand shear in parallel annular disc viscous coupling systemsrdquoProceedings of the Institution of Mechanical Engineers Part JJournal of Engineering Tribology vol 229 no 5 pp 578ndash5962015

[8] M Daliri and D Jalali-Vahid ldquoInvestigation of combinedeffects of rotational inertia and viscosityndashpressure dependencyon the squeeze film characteristics of parallel annular plateslubricated by couple stress fluidrdquo Journal of Tribology vol 137no 3 Article ID 031702 2015

[9] M Daliri and D Jalali-Vahid ldquoCombined effects of pie-zomdashviscous coupled stress lubricant and rotational inertiaupon squeeze film performance of rough circular discsrdquo In-dustrial Lubrication and Tribology vol 67 no 6 pp 564ndash5712015

Advances in Tribology 7

[10] T V Biradar ldquoSqueeze film lubrication between porousparallel stepped plates with couple stress fluidsrdquo TribologyOnline vol 8 no 5 pp 278ndash284 2013

[11] S Shukla and G Deheri ldquoHydromagnetic squeeze film inrough porous narrow journal bearing a study of slip effectrdquoTribology Online vol 12 no 4 pp 177ndash186 2017

[12] B Hanumagowda B Raju J Santhosh Kumar andK Vasanth ldquoCombined effect of surface roughness andpressure-dependent viscosity over couple-stress squeeze filmlubrication between circular stepped platesrdquo Proceedings ofthe Institution of Mechanical Engineers Part J Journal ofEngineering Tribology vol 232 no 5 pp 525ndash534 2018

[13] M Daliri ldquoInvestigation of squeeze film performance in roughparallel circular discs by ferro-fluid couple stress lubricantconsidering effects of rotational inertiardquo Industrial Lubrica-tion and Tribology vol 70 no 7 pp 1201ndash1208 2018

[14] H Aminkhani and M Daliri ldquoEffects of piezo-viscous-cou-pled stress lubricant on the squeeze film performance ofparallel triangular platesrdquo Proceedings of the Institution ofMechanical Engineers Part J Journal of Engineering Tribology2019

[15] M I Shliomis ldquoEffective viscosity of magnetic suspensionsrdquoSoviet Physics Journal of Experimental and8eoretical Physicsvol 34 no 6 pp 1291ndash1294 1972

[16] M I Shliomis ldquoMagnetic fluidsrdquo Soviet Physics Uspekhivol 17 no 2 pp 153ndash169 1974

[17] V K Stokes ldquoCouple stresses in fluidsrdquo Physics of Fluidsvol 9 no 9 pp 1709ndash1715 1966

[18] S T Fathima N B Naduvinamani B N Hanumagowda andJ Santhosh Kumar ldquoModified Reynolds equation for differenttypes of finite plates with the combined effect of MHD andcouple stressesrdquo Tribology Transactions vol 58 no 4pp 660ndash667 2015

8 Advances in Tribology

where A (1 + τ)(1 + 25Φ)

1113968f(lc τΦ h) minus (h36) +

(2l2c A2)[h minus (2lcA)tanh(Ah2lc)]

)eboundary conditions for pressure field are as follows

p x1 y1( 1113857 0 (12)

where

x1 minus a( 1113857 x1 minus3

radicy1 + 2a( 1113857 x1 +

3

radicy1 + 2a( 1113857 0 (13)

)e equilateral triangle side a is expressed as

(x minus a)(x minus3

radicy + 2a)(x +

3

radicy + 2a) 0 (14)

)e intersection point of the triangle medians is selectedto be the origin and thereafter equations (11) and (12) can besolved to obtain the pressure equation as follows

p +2η0A2(dhdt)

f lc τΦ h( 1113857

a2

31 minus

34a2x

2minus

34a2y

2minus

14a3x

3+

34a3 xy

21113874 1113875

(15)

Dimensionless form of the pressure distribution willhave the following format

plowast

minus ph3

oη0(dhdt)3

3

radica2

minus 2A2

93

radicflowast llowast τΦ hlowast( )

middot 1 minus xlowast

( 1113857 1 +

3

radic

2ylowast

+xlowast

21113888 1113889 1 minus

3

radic

2ylowast

+xlowast

21113888 11138891113896

(16)

where

llowast

lc

ho

xlowast

x

a

ylowast

y

a

hlowast

h

ho

flowast

llowast τΦ h

lowast( 1113857

f lc τΦ h( 1113857

h3o

flowast

llowast τΦ h

lowast( 1113857 minus

hlowast3

6+2llowast2

A2 hlowast

minus2llowast

Atanh

Ahlowast

2llowast1113888 11138891113890 1113891

(17)

where ho is the initial film thickness)e expression for the load-carrying capacity is obtained

by

w 1113946a

minus 2a1113946

2a+x1( )radic3

minus 2a+x1( )radic3p dy1dx1

w 9

3

radicη0A2(dhdt)a4

10f lc τΦ h( 1113857

(18)

)e expression for the load-carrying capacity indimensionless form is

wlowast

minus wh3

o27η0(dhdt)a4

minus3

radicA2

30flowast llowast τΦ hlowast( ) (19)

Introducing the dimensionless response time

wlowast

wh2

oη0a4 t (20)

)e time-height relationship can be derived in the di-mensionless form as follows

dhlowast

dtlowast minus

1Wlowast

(21)

Equation (21) is an ordinary but nonlinear differentialequation To obtain an accurate result 4th order Run-gendashKutta method is applied through employing the fol-lowing initial condition

hlowast

1 at tlowast

0 (22)

3 Results and Discussion

From the above analysis the couple stress ferrofluid-lu-bricated squeeze film performances are characterized bythree parameters the non-Newtonian couple stress pa-rameter llowast the volume concentration of particles ϕ and theLangevin parameter ξ )e special case can be obtained fromspecific values of these parameters

(i) ϕ 0 and ξ 0 the couple stress nonferrofluid casewithout magnetic fields )e nondimensional loadcapacity in equation (19) reduces to

(ii) limϕ⟶0 ξ⟶0wlowast

3

radic(5 hlowast3 minus 60 llowast2hlowast + 120llowast3

tanh(hlowast2llowast))

(iii) )is is the derivation of Fathima et al [18] whenneglecting the MHD effects(the Hartmann numbertends to zero)

)e geometry of parallel triangular plates with a ferro-fluid lubricant of couple stress is investigated )e results ofthe study three parameters including the non-Newtoniancouple stress parameter llowast the volume concentration ofparticles Φ and Langevin parameter ξ are employed to

Table 1 Rheological parameters of lubricantInitial film thickness ho 12times10minus 5mTriangle geometry a 02mViscosity of main liquid η0 27times10minus 5 pamiddotsCouple stress material constants ηc 3888times10minus 15NmiddotsSqueezing velocity V 32msCharacteristic length of the additives lc 12times10minus 6mCouple stress parameter llowast 01)e free space permeability μp 4π times10minus 7 kgmiddotmmiddotsminus 2middotAmpminus 2

Magnetic moment of a particle m 004Magnetic field Ho 16363times10minus 5 Ampmiddotmminus 1

)e Boltzmann constant KB 138times10minus 23 kgmiddotm2middotsminus 2middotkminus 1

)e absolute temperature T 298k)e Langevin parameter ξ 10

4 Advances in Tribology

ϕ = 0 ξ = 0 llowast = 0 hlowast = 04ϕ = 002 ξ = 5 llowast = 0 hlowast = 04ϕ = 004 ξ = 5 llowast = 0 hlowast = 04

ϕ = 004 ξ = 10 llowast = 0 hlowast = 04ϕ = 004 ξ = 10 llowast = 006 hlowast = 04

0

5

10

15

20

plowast

ndash15 ndash1 ndash05 0 05 1ndash2xlowast

Figure 2 Pressure distribution for different values of Φ ξ and llowast at hlowast 04

ϕ = 0 ξ = 0 llowast = 0 hlowast = 03ϕ = 002 ξ = 5 llowast = 0 hlowast = 03ϕ = 004 ξ = 5 llowast = 0 hlowast = 03

ϕ = 004 ξ = 10 llowast = 0 hlowast = 03ϕ = 004 ξ = 10 llowast = 006 hlowast = 03

ndash15 ndash1 ndash05 0 05 1ndash2xlowast

0

10

20

30

40

50

plowast

Figure 3 Pressure distribution for different values of Φ ξ and llowast at hlowast 03

ξ = 0 llowast = 006 hlowast = 04ξ = 2 llowast = 006 hlowast = 04

ξ = 5 llowast = 006 hlowast = 04ξ = 10 llowast = 006 hlowast = 04

001 002 003 004 005 006 007 0080ϕ

65

7

75

8

85

wlowast

Figure 4 Variations of dimensionless load capacity versus volume concentration Φ for different values of ξ at llowast 006 and hlowast 04

Advances in Tribology 5

ϕ = 002 llowast = 006 hlowast = 04ϕ = 004 llowast = 006 hlowast = 04

ϕ = 006 llowast = 006 hlowast = 04ϕ = 008 llowast = 006 hlowast = 04

2 4 6 8 10 12 14 16 18 200ξ

7

75

8

85

wlowast

Figure 5 Variations of dimensionless load capacity versus Langevin parameter ξ for different values of Φ at llowast 006 and hlowast 04

ϕ = 004 ξ = 10 hlowast = 05ϕ = 004 ξ = 10 hlowast = 04ϕ = 004 ξ = 10 hlowast = 03

0

20

40

60

80

100

120

140

wlowast

005 01 015 02 025 030llowast

Figure 6 Dimensionless load capacity versus couple stress parameter llowast for different values of hlowast at ξ 10 and Φ 004

ϕ = 004 ξ = 10 llowast = 0ϕ = 004 ξ = 10 llowast = 01

ϕ = 004 ξ = 10 llowast = 02ϕ = 004 ξ = 10 llowast = 03

03 04 05 06 07 08 09 102hlowast

0

10

20

30

40

50

60

tlowast

Figure 7 Dimensionless lubricant film thickness variations as a function of dimensionless time at s of 004 and ξ 10 for different values ofllowast

6 Advances in Tribology

characterize the squeeze film performance Table 1 shows therheological parameters of lubricant At hlowast 04 and hlowast 03lubricant film pressure distribution for altered values of Φ ξare shown in Figures 2 and 3 respectively By increasing thevalues of Φ ξ and llowast there will occur an enhancement in themaximum pressure of lubricant

Dimensionless load-carrying capacity Wlowast variation withrespect to the volumetric concentration (Φ) Langevin (ξ)and couple stress llowast are shown in Figures 4ndash6 Knowing thatthese parameters lead to a higher film pressure it can be seenthat the integrated load-carrying capacity will increasesubsequently )e load-carrying capacity can be wiselyconsidered as a function of Φ ξ and llowast which is shown inFigures 4ndash6 where increasing Φ ξ and llowast will lead to anincrease in load-carrying capacity which refers to the leastload-carrying capacity in a Newtonian fluid

Dimensionless film thickness (hlowast) changes versus di-mensionless response time (tlowast) at altered coupled stresses(llowast) is illustrated in Figure 7 It is also concluded that in-creasing llowast will lead to an increase in the response time

4 Conclusion

In this research the effects of couple stress ferrofluid lu-bricant in the presence of magnetic field on the squeeze filmperformance are investigated Shliomis ferrofluid and Stokescouple stress fluid models are used and the consideredgeometry is triangular plates By solving Reynolds equationthe pressure distribution is obtained and also with inte-grating on pressure in the film region load-carrying capacityis derived A 4th order RungendashKutta method is used to solvethe non linear differential equation between lubricant filmthickness and time Furthermore the variations of lubricantfilm thickness with the time are presented According to theresults it is found that using both ferrofluid and couplestress fluid as a lubricant will increase characteristics of thesqueeze film such as load-carrying capacity pressure dis-tribution and triangular plates moving time significantly)e obtained results are beneficial in designing of thebearings in the engineering applications

Nomenclature

a )e equilateral triangle sideu v w Velocity component in the Cartesian coordinate

systemΦ )e volume concentration of particlesWlowast Dimensionless load-carrying capacity

Wh3o27η0a4V

t tlowast Response time (Wh2oη0a4)t

η0 Viscosity of main liquidlc Characteristic length of the additives (ηcη0)x y z Cartesian coordinatesllowast )e couple stress parameter lchoHo Applied magnetic fieldm Magnetic moment of a particleT )e absolute temperatureξ )e Langevin parameter ξ μpmHoKBT

ho Initial lubricant film thickness

xlowast ylowast Dimensionless coordinates xlowast xa ylowast yaV Squeezing velocity minus dhdt

p plowast Squeeze film pressure ph3oη0V3

3

radica2

τ Rotational viscosity parameterηc Couple stress fluid indexh hlowast Film thickness hho(dimensionless)η Viscosity of the suspensionμp )e free space permeabilityKB )e Boltzmann constant

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare no conflicts of interest

References

[1] K Daniel K Mathew and K Mark ldquoInvestigation of Tem-perature effects for a finite elasto-hydrodynamic journalbearing lubricated by ferro fluids with couple stressesrdquoJournal of Computations amp Modelling vol 5 no 3 pp 79ndash902015

[2] K Daniel K Mathew and K Mark ldquoInvestigation of pressuredistributions for a finite elastohydrodynamic journal bearinglubricated by Ferro fluids with couple stressesrdquo AmericanJournal of Applied Mathematics vol 2 no 4 pp 135ndash1402014

[3] M Daliri and N Javani ldquoSqueeze film characteristics of ferro-coupled stress inertial fluids in conical platesrdquo IndustrialLubrication and Tribology vol 70 no 5 pp 872ndash877 2018

[4] R C Shah and M V Bhat ldquoFerrofluid squeeze film betweencurved annular plates including rotation of magnetic parti-clesrdquo Journal of Engineering Mathematics vol 51 no 4pp 317ndash324 2005

[5] J-R Lin ldquoDerivation of ferrofluid lubrication equation ofcylindrical squeeze films with convective fluid inertia forcesand application to circular disksrdquo Tribology Internationalvol 49 pp 110ndash115 2012

[6] M Daliri D Jalali-Vahid and H Rahnejat ldquoSqueeze filmlubrication of coupled stress electrically conducting inertialfluids in wide parallel rectangular conjunctions subjected to amagnetic fieldrdquo Proceedings of the Institution of MechanicalEngineers Part J Journal of Engineering Tribology vol 228no 3 pp 288ndash302 2014

[7] M Daliri D Jalali-Vahid and H Rahnejat ldquoMagneto-hy-drodynamics of couple stress lubricants in combined squeezeand shear in parallel annular disc viscous coupling systemsrdquoProceedings of the Institution of Mechanical Engineers Part JJournal of Engineering Tribology vol 229 no 5 pp 578ndash5962015

[8] M Daliri and D Jalali-Vahid ldquoInvestigation of combinedeffects of rotational inertia and viscosityndashpressure dependencyon the squeeze film characteristics of parallel annular plateslubricated by couple stress fluidrdquo Journal of Tribology vol 137no 3 Article ID 031702 2015

[9] M Daliri and D Jalali-Vahid ldquoCombined effects of pie-zomdashviscous coupled stress lubricant and rotational inertiaupon squeeze film performance of rough circular discsrdquo In-dustrial Lubrication and Tribology vol 67 no 6 pp 564ndash5712015

Advances in Tribology 7

[10] T V Biradar ldquoSqueeze film lubrication between porousparallel stepped plates with couple stress fluidsrdquo TribologyOnline vol 8 no 5 pp 278ndash284 2013

[11] S Shukla and G Deheri ldquoHydromagnetic squeeze film inrough porous narrow journal bearing a study of slip effectrdquoTribology Online vol 12 no 4 pp 177ndash186 2017

[12] B Hanumagowda B Raju J Santhosh Kumar andK Vasanth ldquoCombined effect of surface roughness andpressure-dependent viscosity over couple-stress squeeze filmlubrication between circular stepped platesrdquo Proceedings ofthe Institution of Mechanical Engineers Part J Journal ofEngineering Tribology vol 232 no 5 pp 525ndash534 2018

[13] M Daliri ldquoInvestigation of squeeze film performance in roughparallel circular discs by ferro-fluid couple stress lubricantconsidering effects of rotational inertiardquo Industrial Lubrica-tion and Tribology vol 70 no 7 pp 1201ndash1208 2018

[14] H Aminkhani and M Daliri ldquoEffects of piezo-viscous-cou-pled stress lubricant on the squeeze film performance ofparallel triangular platesrdquo Proceedings of the Institution ofMechanical Engineers Part J Journal of Engineering Tribology2019

[15] M I Shliomis ldquoEffective viscosity of magnetic suspensionsrdquoSoviet Physics Journal of Experimental and8eoretical Physicsvol 34 no 6 pp 1291ndash1294 1972

[16] M I Shliomis ldquoMagnetic fluidsrdquo Soviet Physics Uspekhivol 17 no 2 pp 153ndash169 1974

[17] V K Stokes ldquoCouple stresses in fluidsrdquo Physics of Fluidsvol 9 no 9 pp 1709ndash1715 1966

[18] S T Fathima N B Naduvinamani B N Hanumagowda andJ Santhosh Kumar ldquoModified Reynolds equation for differenttypes of finite plates with the combined effect of MHD andcouple stressesrdquo Tribology Transactions vol 58 no 4pp 660ndash667 2015

8 Advances in Tribology

ϕ = 0 ξ = 0 llowast = 0 hlowast = 04ϕ = 002 ξ = 5 llowast = 0 hlowast = 04ϕ = 004 ξ = 5 llowast = 0 hlowast = 04

ϕ = 004 ξ = 10 llowast = 0 hlowast = 04ϕ = 004 ξ = 10 llowast = 006 hlowast = 04

0

5

10

15

20

plowast

ndash15 ndash1 ndash05 0 05 1ndash2xlowast

Figure 2 Pressure distribution for different values of Φ ξ and llowast at hlowast 04

ϕ = 0 ξ = 0 llowast = 0 hlowast = 03ϕ = 002 ξ = 5 llowast = 0 hlowast = 03ϕ = 004 ξ = 5 llowast = 0 hlowast = 03

ϕ = 004 ξ = 10 llowast = 0 hlowast = 03ϕ = 004 ξ = 10 llowast = 006 hlowast = 03

ndash15 ndash1 ndash05 0 05 1ndash2xlowast

0

10

20

30

40

50

plowast

Figure 3 Pressure distribution for different values of Φ ξ and llowast at hlowast 03

ξ = 0 llowast = 006 hlowast = 04ξ = 2 llowast = 006 hlowast = 04

ξ = 5 llowast = 006 hlowast = 04ξ = 10 llowast = 006 hlowast = 04

001 002 003 004 005 006 007 0080ϕ

65

7

75

8

85

wlowast

Figure 4 Variations of dimensionless load capacity versus volume concentration Φ for different values of ξ at llowast 006 and hlowast 04

Advances in Tribology 5

ϕ = 002 llowast = 006 hlowast = 04ϕ = 004 llowast = 006 hlowast = 04

ϕ = 006 llowast = 006 hlowast = 04ϕ = 008 llowast = 006 hlowast = 04

2 4 6 8 10 12 14 16 18 200ξ

7

75

8

85

wlowast

Figure 5 Variations of dimensionless load capacity versus Langevin parameter ξ for different values of Φ at llowast 006 and hlowast 04

ϕ = 004 ξ = 10 hlowast = 05ϕ = 004 ξ = 10 hlowast = 04ϕ = 004 ξ = 10 hlowast = 03

0

20

40

60

80

100

120

140

wlowast

005 01 015 02 025 030llowast

Figure 6 Dimensionless load capacity versus couple stress parameter llowast for different values of hlowast at ξ 10 and Φ 004

ϕ = 004 ξ = 10 llowast = 0ϕ = 004 ξ = 10 llowast = 01

ϕ = 004 ξ = 10 llowast = 02ϕ = 004 ξ = 10 llowast = 03

03 04 05 06 07 08 09 102hlowast

0

10

20

30

40

50

60

tlowast

Figure 7 Dimensionless lubricant film thickness variations as a function of dimensionless time at s of 004 and ξ 10 for different values ofllowast

6 Advances in Tribology

characterize the squeeze film performance Table 1 shows therheological parameters of lubricant At hlowast 04 and hlowast 03lubricant film pressure distribution for altered values of Φ ξare shown in Figures 2 and 3 respectively By increasing thevalues of Φ ξ and llowast there will occur an enhancement in themaximum pressure of lubricant

Dimensionless load-carrying capacity Wlowast variation withrespect to the volumetric concentration (Φ) Langevin (ξ)and couple stress llowast are shown in Figures 4ndash6 Knowing thatthese parameters lead to a higher film pressure it can be seenthat the integrated load-carrying capacity will increasesubsequently )e load-carrying capacity can be wiselyconsidered as a function of Φ ξ and llowast which is shown inFigures 4ndash6 where increasing Φ ξ and llowast will lead to anincrease in load-carrying capacity which refers to the leastload-carrying capacity in a Newtonian fluid

Dimensionless film thickness (hlowast) changes versus di-mensionless response time (tlowast) at altered coupled stresses(llowast) is illustrated in Figure 7 It is also concluded that in-creasing llowast will lead to an increase in the response time

4 Conclusion

In this research the effects of couple stress ferrofluid lu-bricant in the presence of magnetic field on the squeeze filmperformance are investigated Shliomis ferrofluid and Stokescouple stress fluid models are used and the consideredgeometry is triangular plates By solving Reynolds equationthe pressure distribution is obtained and also with inte-grating on pressure in the film region load-carrying capacityis derived A 4th order RungendashKutta method is used to solvethe non linear differential equation between lubricant filmthickness and time Furthermore the variations of lubricantfilm thickness with the time are presented According to theresults it is found that using both ferrofluid and couplestress fluid as a lubricant will increase characteristics of thesqueeze film such as load-carrying capacity pressure dis-tribution and triangular plates moving time significantly)e obtained results are beneficial in designing of thebearings in the engineering applications

Nomenclature

a )e equilateral triangle sideu v w Velocity component in the Cartesian coordinate

systemΦ )e volume concentration of particlesWlowast Dimensionless load-carrying capacity

Wh3o27η0a4V

t tlowast Response time (Wh2oη0a4)t

η0 Viscosity of main liquidlc Characteristic length of the additives (ηcη0)x y z Cartesian coordinatesllowast )e couple stress parameter lchoHo Applied magnetic fieldm Magnetic moment of a particleT )e absolute temperatureξ )e Langevin parameter ξ μpmHoKBT

ho Initial lubricant film thickness

xlowast ylowast Dimensionless coordinates xlowast xa ylowast yaV Squeezing velocity minus dhdt

p plowast Squeeze film pressure ph3oη0V3

3

radica2

τ Rotational viscosity parameterηc Couple stress fluid indexh hlowast Film thickness hho(dimensionless)η Viscosity of the suspensionμp )e free space permeabilityKB )e Boltzmann constant

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare no conflicts of interest

References

[1] K Daniel K Mathew and K Mark ldquoInvestigation of Tem-perature effects for a finite elasto-hydrodynamic journalbearing lubricated by ferro fluids with couple stressesrdquoJournal of Computations amp Modelling vol 5 no 3 pp 79ndash902015

[2] K Daniel K Mathew and K Mark ldquoInvestigation of pressuredistributions for a finite elastohydrodynamic journal bearinglubricated by Ferro fluids with couple stressesrdquo AmericanJournal of Applied Mathematics vol 2 no 4 pp 135ndash1402014

[3] M Daliri and N Javani ldquoSqueeze film characteristics of ferro-coupled stress inertial fluids in conical platesrdquo IndustrialLubrication and Tribology vol 70 no 5 pp 872ndash877 2018

[4] R C Shah and M V Bhat ldquoFerrofluid squeeze film betweencurved annular plates including rotation of magnetic parti-clesrdquo Journal of Engineering Mathematics vol 51 no 4pp 317ndash324 2005

[5] J-R Lin ldquoDerivation of ferrofluid lubrication equation ofcylindrical squeeze films with convective fluid inertia forcesand application to circular disksrdquo Tribology Internationalvol 49 pp 110ndash115 2012

[6] M Daliri D Jalali-Vahid and H Rahnejat ldquoSqueeze filmlubrication of coupled stress electrically conducting inertialfluids in wide parallel rectangular conjunctions subjected to amagnetic fieldrdquo Proceedings of the Institution of MechanicalEngineers Part J Journal of Engineering Tribology vol 228no 3 pp 288ndash302 2014

[7] M Daliri D Jalali-Vahid and H Rahnejat ldquoMagneto-hy-drodynamics of couple stress lubricants in combined squeezeand shear in parallel annular disc viscous coupling systemsrdquoProceedings of the Institution of Mechanical Engineers Part JJournal of Engineering Tribology vol 229 no 5 pp 578ndash5962015

[8] M Daliri and D Jalali-Vahid ldquoInvestigation of combinedeffects of rotational inertia and viscosityndashpressure dependencyon the squeeze film characteristics of parallel annular plateslubricated by couple stress fluidrdquo Journal of Tribology vol 137no 3 Article ID 031702 2015

[9] M Daliri and D Jalali-Vahid ldquoCombined effects of pie-zomdashviscous coupled stress lubricant and rotational inertiaupon squeeze film performance of rough circular discsrdquo In-dustrial Lubrication and Tribology vol 67 no 6 pp 564ndash5712015

Advances in Tribology 7

[10] T V Biradar ldquoSqueeze film lubrication between porousparallel stepped plates with couple stress fluidsrdquo TribologyOnline vol 8 no 5 pp 278ndash284 2013

[11] S Shukla and G Deheri ldquoHydromagnetic squeeze film inrough porous narrow journal bearing a study of slip effectrdquoTribology Online vol 12 no 4 pp 177ndash186 2017

[12] B Hanumagowda B Raju J Santhosh Kumar andK Vasanth ldquoCombined effect of surface roughness andpressure-dependent viscosity over couple-stress squeeze filmlubrication between circular stepped platesrdquo Proceedings ofthe Institution of Mechanical Engineers Part J Journal ofEngineering Tribology vol 232 no 5 pp 525ndash534 2018

[13] M Daliri ldquoInvestigation of squeeze film performance in roughparallel circular discs by ferro-fluid couple stress lubricantconsidering effects of rotational inertiardquo Industrial Lubrica-tion and Tribology vol 70 no 7 pp 1201ndash1208 2018

[14] H Aminkhani and M Daliri ldquoEffects of piezo-viscous-cou-pled stress lubricant on the squeeze film performance ofparallel triangular platesrdquo Proceedings of the Institution ofMechanical Engineers Part J Journal of Engineering Tribology2019

[15] M I Shliomis ldquoEffective viscosity of magnetic suspensionsrdquoSoviet Physics Journal of Experimental and8eoretical Physicsvol 34 no 6 pp 1291ndash1294 1972

[16] M I Shliomis ldquoMagnetic fluidsrdquo Soviet Physics Uspekhivol 17 no 2 pp 153ndash169 1974

[17] V K Stokes ldquoCouple stresses in fluidsrdquo Physics of Fluidsvol 9 no 9 pp 1709ndash1715 1966

[18] S T Fathima N B Naduvinamani B N Hanumagowda andJ Santhosh Kumar ldquoModified Reynolds equation for differenttypes of finite plates with the combined effect of MHD andcouple stressesrdquo Tribology Transactions vol 58 no 4pp 660ndash667 2015

8 Advances in Tribology

ϕ = 002 llowast = 006 hlowast = 04ϕ = 004 llowast = 006 hlowast = 04

ϕ = 006 llowast = 006 hlowast = 04ϕ = 008 llowast = 006 hlowast = 04

2 4 6 8 10 12 14 16 18 200ξ

7

75

8

85

wlowast

Figure 5 Variations of dimensionless load capacity versus Langevin parameter ξ for different values of Φ at llowast 006 and hlowast 04

ϕ = 004 ξ = 10 hlowast = 05ϕ = 004 ξ = 10 hlowast = 04ϕ = 004 ξ = 10 hlowast = 03

0

20

40

60

80

100

120

140

wlowast

005 01 015 02 025 030llowast

Figure 6 Dimensionless load capacity versus couple stress parameter llowast for different values of hlowast at ξ 10 and Φ 004

ϕ = 004 ξ = 10 llowast = 0ϕ = 004 ξ = 10 llowast = 01

ϕ = 004 ξ = 10 llowast = 02ϕ = 004 ξ = 10 llowast = 03

03 04 05 06 07 08 09 102hlowast

0

10

20

30

40

50

60

tlowast

Figure 7 Dimensionless lubricant film thickness variations as a function of dimensionless time at s of 004 and ξ 10 for different values ofllowast

6 Advances in Tribology

characterize the squeeze film performance Table 1 shows therheological parameters of lubricant At hlowast 04 and hlowast 03lubricant film pressure distribution for altered values of Φ ξare shown in Figures 2 and 3 respectively By increasing thevalues of Φ ξ and llowast there will occur an enhancement in themaximum pressure of lubricant

Dimensionless load-carrying capacity Wlowast variation withrespect to the volumetric concentration (Φ) Langevin (ξ)and couple stress llowast are shown in Figures 4ndash6 Knowing thatthese parameters lead to a higher film pressure it can be seenthat the integrated load-carrying capacity will increasesubsequently )e load-carrying capacity can be wiselyconsidered as a function of Φ ξ and llowast which is shown inFigures 4ndash6 where increasing Φ ξ and llowast will lead to anincrease in load-carrying capacity which refers to the leastload-carrying capacity in a Newtonian fluid

Dimensionless film thickness (hlowast) changes versus di-mensionless response time (tlowast) at altered coupled stresses(llowast) is illustrated in Figure 7 It is also concluded that in-creasing llowast will lead to an increase in the response time

4 Conclusion

In this research the effects of couple stress ferrofluid lu-bricant in the presence of magnetic field on the squeeze filmperformance are investigated Shliomis ferrofluid and Stokescouple stress fluid models are used and the consideredgeometry is triangular plates By solving Reynolds equationthe pressure distribution is obtained and also with inte-grating on pressure in the film region load-carrying capacityis derived A 4th order RungendashKutta method is used to solvethe non linear differential equation between lubricant filmthickness and time Furthermore the variations of lubricantfilm thickness with the time are presented According to theresults it is found that using both ferrofluid and couplestress fluid as a lubricant will increase characteristics of thesqueeze film such as load-carrying capacity pressure dis-tribution and triangular plates moving time significantly)e obtained results are beneficial in designing of thebearings in the engineering applications

Nomenclature

a )e equilateral triangle sideu v w Velocity component in the Cartesian coordinate

systemΦ )e volume concentration of particlesWlowast Dimensionless load-carrying capacity

Wh3o27η0a4V

t tlowast Response time (Wh2oη0a4)t

η0 Viscosity of main liquidlc Characteristic length of the additives (ηcη0)x y z Cartesian coordinatesllowast )e couple stress parameter lchoHo Applied magnetic fieldm Magnetic moment of a particleT )e absolute temperatureξ )e Langevin parameter ξ μpmHoKBT

ho Initial lubricant film thickness

xlowast ylowast Dimensionless coordinates xlowast xa ylowast yaV Squeezing velocity minus dhdt

p plowast Squeeze film pressure ph3oη0V3

3

radica2

τ Rotational viscosity parameterηc Couple stress fluid indexh hlowast Film thickness hho(dimensionless)η Viscosity of the suspensionμp )e free space permeabilityKB )e Boltzmann constant

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare no conflicts of interest

References

[1] K Daniel K Mathew and K Mark ldquoInvestigation of Tem-perature effects for a finite elasto-hydrodynamic journalbearing lubricated by ferro fluids with couple stressesrdquoJournal of Computations amp Modelling vol 5 no 3 pp 79ndash902015

[2] K Daniel K Mathew and K Mark ldquoInvestigation of pressuredistributions for a finite elastohydrodynamic journal bearinglubricated by Ferro fluids with couple stressesrdquo AmericanJournal of Applied Mathematics vol 2 no 4 pp 135ndash1402014

[3] M Daliri and N Javani ldquoSqueeze film characteristics of ferro-coupled stress inertial fluids in conical platesrdquo IndustrialLubrication and Tribology vol 70 no 5 pp 872ndash877 2018

[4] R C Shah and M V Bhat ldquoFerrofluid squeeze film betweencurved annular plates including rotation of magnetic parti-clesrdquo Journal of Engineering Mathematics vol 51 no 4pp 317ndash324 2005

[5] J-R Lin ldquoDerivation of ferrofluid lubrication equation ofcylindrical squeeze films with convective fluid inertia forcesand application to circular disksrdquo Tribology Internationalvol 49 pp 110ndash115 2012

[6] M Daliri D Jalali-Vahid and H Rahnejat ldquoSqueeze filmlubrication of coupled stress electrically conducting inertialfluids in wide parallel rectangular conjunctions subjected to amagnetic fieldrdquo Proceedings of the Institution of MechanicalEngineers Part J Journal of Engineering Tribology vol 228no 3 pp 288ndash302 2014

[7] M Daliri D Jalali-Vahid and H Rahnejat ldquoMagneto-hy-drodynamics of couple stress lubricants in combined squeezeand shear in parallel annular disc viscous coupling systemsrdquoProceedings of the Institution of Mechanical Engineers Part JJournal of Engineering Tribology vol 229 no 5 pp 578ndash5962015

[8] M Daliri and D Jalali-Vahid ldquoInvestigation of combinedeffects of rotational inertia and viscosityndashpressure dependencyon the squeeze film characteristics of parallel annular plateslubricated by couple stress fluidrdquo Journal of Tribology vol 137no 3 Article ID 031702 2015

[9] M Daliri and D Jalali-Vahid ldquoCombined effects of pie-zomdashviscous coupled stress lubricant and rotational inertiaupon squeeze film performance of rough circular discsrdquo In-dustrial Lubrication and Tribology vol 67 no 6 pp 564ndash5712015

Advances in Tribology 7

[10] T V Biradar ldquoSqueeze film lubrication between porousparallel stepped plates with couple stress fluidsrdquo TribologyOnline vol 8 no 5 pp 278ndash284 2013

[11] S Shukla and G Deheri ldquoHydromagnetic squeeze film inrough porous narrow journal bearing a study of slip effectrdquoTribology Online vol 12 no 4 pp 177ndash186 2017

[12] B Hanumagowda B Raju J Santhosh Kumar andK Vasanth ldquoCombined effect of surface roughness andpressure-dependent viscosity over couple-stress squeeze filmlubrication between circular stepped platesrdquo Proceedings ofthe Institution of Mechanical Engineers Part J Journal ofEngineering Tribology vol 232 no 5 pp 525ndash534 2018

[13] M Daliri ldquoInvestigation of squeeze film performance in roughparallel circular discs by ferro-fluid couple stress lubricantconsidering effects of rotational inertiardquo Industrial Lubrica-tion and Tribology vol 70 no 7 pp 1201ndash1208 2018

[14] H Aminkhani and M Daliri ldquoEffects of piezo-viscous-cou-pled stress lubricant on the squeeze film performance ofparallel triangular platesrdquo Proceedings of the Institution ofMechanical Engineers Part J Journal of Engineering Tribology2019

[15] M I Shliomis ldquoEffective viscosity of magnetic suspensionsrdquoSoviet Physics Journal of Experimental and8eoretical Physicsvol 34 no 6 pp 1291ndash1294 1972

[16] M I Shliomis ldquoMagnetic fluidsrdquo Soviet Physics Uspekhivol 17 no 2 pp 153ndash169 1974

[17] V K Stokes ldquoCouple stresses in fluidsrdquo Physics of Fluidsvol 9 no 9 pp 1709ndash1715 1966

[18] S T Fathima N B Naduvinamani B N Hanumagowda andJ Santhosh Kumar ldquoModified Reynolds equation for differenttypes of finite plates with the combined effect of MHD andcouple stressesrdquo Tribology Transactions vol 58 no 4pp 660ndash667 2015

8 Advances in Tribology

characterize the squeeze film performance Table 1 shows therheological parameters of lubricant At hlowast 04 and hlowast 03lubricant film pressure distribution for altered values of Φ ξare shown in Figures 2 and 3 respectively By increasing thevalues of Φ ξ and llowast there will occur an enhancement in themaximum pressure of lubricant

Dimensionless load-carrying capacity Wlowast variation withrespect to the volumetric concentration (Φ) Langevin (ξ)and couple stress llowast are shown in Figures 4ndash6 Knowing thatthese parameters lead to a higher film pressure it can be seenthat the integrated load-carrying capacity will increasesubsequently )e load-carrying capacity can be wiselyconsidered as a function of Φ ξ and llowast which is shown inFigures 4ndash6 where increasing Φ ξ and llowast will lead to anincrease in load-carrying capacity which refers to the leastload-carrying capacity in a Newtonian fluid

Dimensionless film thickness (hlowast) changes versus di-mensionless response time (tlowast) at altered coupled stresses(llowast) is illustrated in Figure 7 It is also concluded that in-creasing llowast will lead to an increase in the response time

4 Conclusion

In this research the effects of couple stress ferrofluid lu-bricant in the presence of magnetic field on the squeeze filmperformance are investigated Shliomis ferrofluid and Stokescouple stress fluid models are used and the consideredgeometry is triangular plates By solving Reynolds equationthe pressure distribution is obtained and also with inte-grating on pressure in the film region load-carrying capacityis derived A 4th order RungendashKutta method is used to solvethe non linear differential equation between lubricant filmthickness and time Furthermore the variations of lubricantfilm thickness with the time are presented According to theresults it is found that using both ferrofluid and couplestress fluid as a lubricant will increase characteristics of thesqueeze film such as load-carrying capacity pressure dis-tribution and triangular plates moving time significantly)e obtained results are beneficial in designing of thebearings in the engineering applications

Nomenclature

a )e equilateral triangle sideu v w Velocity component in the Cartesian coordinate

systemΦ )e volume concentration of particlesWlowast Dimensionless load-carrying capacity

Wh3o27η0a4V

t tlowast Response time (Wh2oη0a4)t

η0 Viscosity of main liquidlc Characteristic length of the additives (ηcη0)x y z Cartesian coordinatesllowast )e couple stress parameter lchoHo Applied magnetic fieldm Magnetic moment of a particleT )e absolute temperatureξ )e Langevin parameter ξ μpmHoKBT

ho Initial lubricant film thickness

xlowast ylowast Dimensionless coordinates xlowast xa ylowast yaV Squeezing velocity minus dhdt

p plowast Squeeze film pressure ph3oη0V3

3

radica2

τ Rotational viscosity parameterηc Couple stress fluid indexh hlowast Film thickness hho(dimensionless)η Viscosity of the suspensionμp )e free space permeabilityKB )e Boltzmann constant

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare no conflicts of interest

References

[1] K Daniel K Mathew and K Mark ldquoInvestigation of Tem-perature effects for a finite elasto-hydrodynamic journalbearing lubricated by ferro fluids with couple stressesrdquoJournal of Computations amp Modelling vol 5 no 3 pp 79ndash902015

[2] K Daniel K Mathew and K Mark ldquoInvestigation of pressuredistributions for a finite elastohydrodynamic journal bearinglubricated by Ferro fluids with couple stressesrdquo AmericanJournal of Applied Mathematics vol 2 no 4 pp 135ndash1402014

[3] M Daliri and N Javani ldquoSqueeze film characteristics of ferro-coupled stress inertial fluids in conical platesrdquo IndustrialLubrication and Tribology vol 70 no 5 pp 872ndash877 2018

[4] R C Shah and M V Bhat ldquoFerrofluid squeeze film betweencurved annular plates including rotation of magnetic parti-clesrdquo Journal of Engineering Mathematics vol 51 no 4pp 317ndash324 2005

[5] J-R Lin ldquoDerivation of ferrofluid lubrication equation ofcylindrical squeeze films with convective fluid inertia forcesand application to circular disksrdquo Tribology Internationalvol 49 pp 110ndash115 2012

[6] M Daliri D Jalali-Vahid and H Rahnejat ldquoSqueeze filmlubrication of coupled stress electrically conducting inertialfluids in wide parallel rectangular conjunctions subjected to amagnetic fieldrdquo Proceedings of the Institution of MechanicalEngineers Part J Journal of Engineering Tribology vol 228no 3 pp 288ndash302 2014

[7] M Daliri D Jalali-Vahid and H Rahnejat ldquoMagneto-hy-drodynamics of couple stress lubricants in combined squeezeand shear in parallel annular disc viscous coupling systemsrdquoProceedings of the Institution of Mechanical Engineers Part JJournal of Engineering Tribology vol 229 no 5 pp 578ndash5962015

[8] M Daliri and D Jalali-Vahid ldquoInvestigation of combinedeffects of rotational inertia and viscosityndashpressure dependencyon the squeeze film characteristics of parallel annular plateslubricated by couple stress fluidrdquo Journal of Tribology vol 137no 3 Article ID 031702 2015

[9] M Daliri and D Jalali-Vahid ldquoCombined effects of pie-zomdashviscous coupled stress lubricant and rotational inertiaupon squeeze film performance of rough circular discsrdquo In-dustrial Lubrication and Tribology vol 67 no 6 pp 564ndash5712015

Advances in Tribology 7

[10] T V Biradar ldquoSqueeze film lubrication between porousparallel stepped plates with couple stress fluidsrdquo TribologyOnline vol 8 no 5 pp 278ndash284 2013

[11] S Shukla and G Deheri ldquoHydromagnetic squeeze film inrough porous narrow journal bearing a study of slip effectrdquoTribology Online vol 12 no 4 pp 177ndash186 2017

[12] B Hanumagowda B Raju J Santhosh Kumar andK Vasanth ldquoCombined effect of surface roughness andpressure-dependent viscosity over couple-stress squeeze filmlubrication between circular stepped platesrdquo Proceedings ofthe Institution of Mechanical Engineers Part J Journal ofEngineering Tribology vol 232 no 5 pp 525ndash534 2018

[13] M Daliri ldquoInvestigation of squeeze film performance in roughparallel circular discs by ferro-fluid couple stress lubricantconsidering effects of rotational inertiardquo Industrial Lubrica-tion and Tribology vol 70 no 7 pp 1201ndash1208 2018

[14] H Aminkhani and M Daliri ldquoEffects of piezo-viscous-cou-pled stress lubricant on the squeeze film performance ofparallel triangular platesrdquo Proceedings of the Institution ofMechanical Engineers Part J Journal of Engineering Tribology2019

[15] M I Shliomis ldquoEffective viscosity of magnetic suspensionsrdquoSoviet Physics Journal of Experimental and8eoretical Physicsvol 34 no 6 pp 1291ndash1294 1972

[16] M I Shliomis ldquoMagnetic fluidsrdquo Soviet Physics Uspekhivol 17 no 2 pp 153ndash169 1974

[17] V K Stokes ldquoCouple stresses in fluidsrdquo Physics of Fluidsvol 9 no 9 pp 1709ndash1715 1966

[18] S T Fathima N B Naduvinamani B N Hanumagowda andJ Santhosh Kumar ldquoModified Reynolds equation for differenttypes of finite plates with the combined effect of MHD andcouple stressesrdquo Tribology Transactions vol 58 no 4pp 660ndash667 2015

8 Advances in Tribology

[10] T V Biradar ldquoSqueeze film lubrication between porousparallel stepped plates with couple stress fluidsrdquo TribologyOnline vol 8 no 5 pp 278ndash284 2013

[11] S Shukla and G Deheri ldquoHydromagnetic squeeze film inrough porous narrow journal bearing a study of slip effectrdquoTribology Online vol 12 no 4 pp 177ndash186 2017

[12] B Hanumagowda B Raju J Santhosh Kumar andK Vasanth ldquoCombined effect of surface roughness andpressure-dependent viscosity over couple-stress squeeze filmlubrication between circular stepped platesrdquo Proceedings ofthe Institution of Mechanical Engineers Part J Journal ofEngineering Tribology vol 232 no 5 pp 525ndash534 2018

[13] M Daliri ldquoInvestigation of squeeze film performance in roughparallel circular discs by ferro-fluid couple stress lubricantconsidering effects of rotational inertiardquo Industrial Lubrica-tion and Tribology vol 70 no 7 pp 1201ndash1208 2018

[14] H Aminkhani and M Daliri ldquoEffects of piezo-viscous-cou-pled stress lubricant on the squeeze film performance ofparallel triangular platesrdquo Proceedings of the Institution ofMechanical Engineers Part J Journal of Engineering Tribology2019

[15] M I Shliomis ldquoEffective viscosity of magnetic suspensionsrdquoSoviet Physics Journal of Experimental and8eoretical Physicsvol 34 no 6 pp 1291ndash1294 1972

[16] M I Shliomis ldquoMagnetic fluidsrdquo Soviet Physics Uspekhivol 17 no 2 pp 153ndash169 1974

[17] V K Stokes ldquoCouple stresses in fluidsrdquo Physics of Fluidsvol 9 no 9 pp 1709ndash1715 1966

[18] S T Fathima N B Naduvinamani B N Hanumagowda andJ Santhosh Kumar ldquoModified Reynolds equation for differenttypes of finite plates with the combined effect of MHD andcouple stressesrdquo Tribology Transactions vol 58 no 4pp 660ndash667 2015

8 Advances in Tribology