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Derivation of ferrofluid lubrication equation for slider bearings with variable magnetic field and rotations of the carrier liquid as well as magnetic particles

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Abstract

Based on the Shliomis model for ferrofluid flow and continuity equation, modified Reynolds equation for the study of lubrication of different slider bearings, is derived by considering the effects of oblique radially variable magnetic field and squeeze velocity. The Shliomis model is important because it includes the effects of rotations of the carrier liquid as well as magnetic particles. The variable magnetic field is important because of its advantage of generating maximum field at the required active contact zone. Using Reynolds equation, expressions for dimensionless load-carrying capacity, frictional force, coefficient of friction and center of pressure are obtained. Using these expressions, results for different slider bearings are computed for different parameters and compared.

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Abbreviations

a :

\( \frac{{h_{2} }}{{h_{1} }} \)

A :

Bearing length (m)

B :

Bearing breadth (m)

f :

Coefficient of friction

\( \bar{f} \) :

Dimensionless coefficient of friction defined in Eq. (29)

F :

Frictional force on the slider (N)

\( \bar{F} \) :

Dimensionless frictional force defined in Eq. (28)

h :

Film thickness (m)

\( \bar{h} \) :

Dimensionless quantity defined in Eq. (21)

\( h_{1} ,h_{2} \) :

Minimum and maximum values of h (m)

\( \dot{h}_{1} \) :

Squeeze velocity, \( \frac{{dh_{1} }}{dt}\;({\text{m}}\;{\text{s}}^{ - 1} ) \)

H :

Magnetic field strength (A m−1)

H :

Magnetic field vector

I :

Sum of moments of inertia of the particles per unit volume (N s2 m−2)

\( I^{*} ,I^{**} \) :

Integrals defined in Eqs. (31, 32)

K :

Quantity defined in Eq. (12) (A m−3)

\( k_{B} \) :

Boltzmann constant (J(°K)−1)

m :

Magnetic moment of a particle (A m2)

M :

Magnetization vector

\( M_{0} \) :

Saturation magnetization (A m−1)

n :

Number of particles per unit volume (m−3)

p :

Film pressure (N m−2)

\( \bar{p} \) :

Dimensionless film pressure defined in Eq. (21)

q = (u, 0, w):

Fluid velocity vector

t :

Time (s)

T :

Temperature (°K)

U :

Slider velocity (ms−1)

W :

Load-carrying capacity (N)

\( \bar{W} \) :

Dimensionless load-carrying capacity defined Eq. (27)

x,y,z :

Coordinates

\( \bar{x} \) :

x-Coordinates of the center of pressure (m)

\( \bar{\bar{x}} \) :

Dimensionless center of pressure defined in Eq. (30)

X :

\( \frac{x}{A} \)

\( \upalpha \) :

Inclination of the magnetic field with the x-axis

\( \upbeta \) :

Squeeze velocity parameter defined in Eq. (21)

\( \updelta \) :

Central thickness of the convex pad (m)

\( \bar{\updelta } \) :

\( \frac{\updelta }{{h_{1} }} \)

\( \upeta \) :

Viscosity of the suspension (N s m−2)

\( \upeta_{0} \) :

Viscosity of the liquid carrier (N s m−2)

\( \uplambda \) :

Magnetic field strength parameter defined in Eq. (23)

\( \upmu_{0} \) :

Permeability of free space

\( \upxi \) :

Dimensionless field strength (Langevin’s parameter)

\( \uptau \) :

Rotational viscosity parameter

\( \bar{\uptau } \) :

Defined in Eq. (10)

\( \uptau_{B} \) :

Brownian relaxation time (s)

\( \uptau_{s} \) :

Magnetic moment relaxation time (s)

\( \bar{\Omega } \) :

\( \tfrac{1}{2}\nabla \times \;{\mathbf{q}} \)

ϕ:

Volume concentration of the particles

References

  1. Shliomis MI (1972) Effective viscosity of magnetic suspensions. Sov Phys JETP 34(6):1291–1294

    ADS  Google Scholar 

  2. Jenkins JT (1971) Some simple flows of a para-magnetic fluid. Journal de Physique 32:931–938

    Article  Google Scholar 

  3. Jenkins JT (1972) A theory of magnetic fluids. Arch Ration Mech Anal 46(1):42–60

    Article  MathSciNet  MATH  Google Scholar 

  4. Huang W, Wang X (2016) Ferrofluids lubrication: a status report. Lubr Sci 28:3–26

    Article  ADS  Google Scholar 

  5. Neuringer JL, Rosensweig RE (1964) Ferrohydrodynamics. Phys Fluids 7(12):1927–1937

    Article  ADS  MathSciNet  Google Scholar 

  6. Tipei N (1982) Theory of lubrication with ferrofluids: application to short bearings. Trans ASME 104:510–515

    Google Scholar 

  7. Agrawal VK (1986) Magnetic fluid based porous inclined slider bearing. Wear 107:133–139

    Article  Google Scholar 

  8. Chi CQ, Wang ZS, Zhao PZ (1990) Research on a new type of ferrofluid-lubricated journal bearing. J Magn Magn Mater 85:257–260

    Article  ADS  Google Scholar 

  9. Prajapati BL (1995) Magnetic-fluid-based porous squeeze films. J Magn Magn Mater 149:97–100

    Article  ADS  Google Scholar 

  10. Shah RC, Bhat MV (2004) Ferrofluid squeeze film in a long journal bearing. Tribol Int 37:441–446

    Article  Google Scholar 

  11. Ahmad N, Singh JP (2007) Magnetic fluid lubrication of porous-pivoted slider bearings with slip velocity. J Eng Tribol 221:609–613

    Google Scholar 

  12. Shah RC, Patel NI (2015) Impact of various and arbitrary porous structure in the study of squeeze step bearing lubricated with magnetic fluid considering variable magnetic field. J Eng Tribol 229(5):646–659

    Google Scholar 

  13. Shah RC, Kataria RC (2016) On the squeeze film characteristics between a sphere and a flat porous plate using ferrofluid. Appl Math Model 40:2473–2484

    Article  MathSciNet  Google Scholar 

  14. Shah RC, Patel DA (2016) On the ferrofluid lubricated squeeze film characteristics between a rotating sphere and a radially rough plate. Meccanica 51:1973–1984

    Article  MathSciNet  MATH  Google Scholar 

  15. Ram P, Verma PDS (1999) Ferrofluid lubrication in porous inclined slider bearing. Indian J Pure Appl Math 30(12):1273–1281

    MathSciNet  MATH  Google Scholar 

  16. Maugin GA (1980) The principle of virtual power: application to coupled fields. Acta Mech 35:1–70

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Shah RC, Bhat MV (2002) Exponential Slider bearing lubricated with ferrofluid. In: Mediterraneantrib Conference, Kayseri, Turkey, pp 99–103

  18. Shah RC, Bhat MV (2003) Lubrication of a slider bearing with magnetic fluid. J Balk Tribol Assoc 9(2):280–284

    Google Scholar 

  19. Shah RC, Patel DB (2014) Magnetic fluid lubrication of porous pivoted slider bearing with slip and squeeze velocity. Int J Ind Math 6(3):199–206

    Google Scholar 

  20. Shukla JB, Kumar D (1987) A theory for ferromagnetic lubrication. J Magn Magn Mater 65:375–378

    Article  ADS  Google Scholar 

  21. Shah RC, Bhat MV (2005) Ferrofluid squeeze film between curved annular plates including rotation of magnetic particles. J Eng Math 51:317–324

    Article  MathSciNet  MATH  Google Scholar 

  22. Shah RC, Parikh KS (2014) Comparative study of ferrofluid lubricated various designed slider bearings considering rotation of magnetic particles and squeeze velocity. Int J Theor Math Phys 4(2):63–72

    Google Scholar 

  23. Singh UP, Gupta RS (2012) Dynamic performance characteristics of a curved slider bearing operating with ferrofluids. Adv Tribol 6, Article ID 278723

  24. Lin JR (2013) Dynamic characteristics of magnetic fluid based sliding bearings. Mechanika 19(5):554–558

    Article  Google Scholar 

  25. Khonsari MM, Booser ER (2001) Applied Tribology: bearing design and lubrication. Wiley, New York

    Google Scholar 

  26. Szeri AZ (1998) Fluid film lubrication. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

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Acknowledgement

The authors are thankful to the Reviewer, Associate Editor and Editor in Chief for their valuable comments.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Technology and Engineering, The Maharaja Sayajirao University of Baroda, Vadodara, Gujarat, 390 001, India

    Rajesh C. Shah

  2. Department of Applied Mathematics, Polytechnic, The Maharaja Sayajirao University of Baroda, Vadodara, Gujarat, 390 002, India

    Rajiv B. Shah

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  1. Rajesh C. Shah
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Correspondence to Rajesh C. Shah.

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Shah, R.C., Shah, R.B. Derivation of ferrofluid lubrication equation for slider bearings with variable magnetic field and rotations of the carrier liquid as well as magnetic particles. Meccanica 53, 857–869 (2018). https://doi.org/10.1007/s11012-017-0788-9

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