, Volume 53, Issue 1–2, pp 209–228 | Cite as

Numerical simulations of an incompressible piezoviscous fluid flowing in a plane slider bearing

  • Martin Lanzendörfer
  • Josef Málek
  • Kumbakonam R. Rajagopal


We provide numerical simulations of an incompressible pressure-thickening and shear-thinning lubricant flowing in a plane slider bearing. We study the influence of several parameters, namely the ratio of the characteristic lengths \(\varepsilon >0\) (with \(\varepsilon \searrow 0\) representing the Reynolds lubrication approximation); the coefficient of the exponential pressure–viscosity relation \(\alpha ^*\ge 0\); the parameter \(G^*\ge 0\) related to the Carreau–Yasuda shear-thinning model and the modified Reynolds number \({\mathrm {Re}}_\varepsilon \ge 0\). The finite element approximations to the steady isothermal flows are computed without resorting to the lubrication approximation. We obtain the numerical solutions as long as the variation of the viscous stress \(\varvec{S}=2\eta (p,{{\mathrm{tr}}}\,\varvec{D}^2)\varvec{D}\) with the pressure is limited, say \(|\partial \varvec{S}/\partial p|\le 1\). We show conclusively that the existing practice of avoiding the numerical difficulties by cutting the viscosity off for large pressures leads to results that depend sorely on the artificial cut-off parameter. We observe that the piezoviscous rheology generates pressure differences across the fluid film.


Finite element approximation Incompressible fluid Pressure-thickening Shear-thinning Thin-film flow Channel flow Fluid mechanics 



J. Málek acknowledges the support of the ERC-CZ Project LL1202 financed by MŠMT (Ministry of Education, Youth and Sports of the Czech Republic). K. R. Rajagopal thanks the National Science Foundation, United States for its support.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Almqvist T, Larsson R (2002) The Navier–Stokes approach for thermal EHL line contact solutions. Tribol Int 35(3):163–170. doi: 10.1016/S0301-679X(01)00112-8 CrossRefGoogle Scholar
  2. 2.
    Almqvist T, Larsson R (2008) Thermal transient rough EHL line contact simulations by aid of computational fluid dynamics. Tribol Int 41(8):683–693. doi: 10.1016/j.triboint.2007.11.004 CrossRefGoogle Scholar
  3. 3.
    Almqvist T, Almqvist A, Larsson R (2004) A comparison between computational fluid dynamic and Reynolds approaches for simulating transient EHL line contacts. Tribol Int 37(1):61–69. doi: 10.1016/S0301-679X(03)00131-2 CrossRefGoogle Scholar
  4. 4.
    Bair S (2006) Reference liquids for quantitative elastohydrodynamics: selection and rheological characterization. Tribol Lett 22(2):197–206. doi: 10.1007/s11249-006-9083-y CrossRefGoogle Scholar
  5. 5.
    Bair S (2007) High pressure rheology for quantitative elastohydrodynamics. Tribol. interface eng. Elsevier Science, AmsterdamGoogle Scholar
  6. 6.
    Bair S, Khonsari M, Winer WO (1998) High-pressure rheology of lubricants and limitations of the Reynolds equation. Tribol Int 31(10):573–586CrossRefGoogle Scholar
  7. 7.
    Bayada G, Cid B, García G, Vázquez C (2013) A new more consistent Reynolds model for piezoviscous hydrodynamic lubrication problems in line contact devices. Appl Math Modell 37(18–19):8505–8517. doi: 10.1016/j.apm.2013.03.072 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bruneau CH, Fabrie P (1996) New efficient boundary conditions for incompressible Navier–Stokes equations: a well-posedness result. RAIRO Math Modell Numer Anal 30(7):815–840MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bruyere V, Fillot N, Morales-Espejel GE, Vergne P (2012) Computational fluid dynamics and full elasticity model for sliding line thermal elastohydrodynamic contacts. Tribol Int 46(1):3–13. doi: 10.1016/j.triboint.2011.04.013 CrossRefGoogle Scholar
  10. 10.
    Buckholz RA (1987) The effect of lubricant inertia near the leading edge of a plane slider bearing. J Tribol 109(1):60–64. doi: 10.1115/1.3261328 CrossRefGoogle Scholar
  11. 11.
    Bulíček M, Málek J, Rajagopal KR (2009) Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling \(\nu (p,\cdot )\rightarrow +\infty\) as \(p\rightarrow +\infty\). Czechoslov Math J 59(2):503–528MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bulíček M, Málek J, Rajagopal KR (2009) Mathematical analysis of unsteady flows of fluids with pressure, shear-rate and temperature dependent material moduli that slip at solid boundaries. SIAM J Math Anal 41(2):665–707MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bulíček M, Majdoub M, Málek J (2010) Unsteady flows of fluids with pressure dependent viscosity in unbounded domains. Nonlinear Anal: Real World Appl 11(5):3968–3983. doi: 10.1016/j.nonrwa.2010.03.004 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Davies TA (2004) UMFPACK version 4.3 user guide. Tech rep REP-2004-349, University of Florida. http://www.ciseufledu/research/sparse/umfpack
  15. 15.
    Davies AR, Li XK (1994) Numerical modelling of pressure and temperature effects in viscoelastic flow between eccentrically rotating cylinders. J Non-Newton Fluid Mech 54:331–350CrossRefGoogle Scholar
  16. 16.
    Franta M, Málek J, Rajagopal KR (2005) On steady flows of fluids with pressure- and shear-dependent viscosities. Proc R Soc Lond A 461(2055):651–670. doi: 10.1098/rspa.2004.1360 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gresho PM, Sani RL (2000) Incompressible flow and the finite element method, vol 2: isothermal laminar flow. Wiley, LondonzbMATHGoogle Scholar
  18. 18.
    Gustafsson T, Rajagopal KR, Stenberg R, Videman J (2015) Nonlinear Reynolds equation for hydrodynamic lubrication. Appl Math Modell 39(17):5299–5309. doi: 10.1016/j.apm.2015.03.028 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gwynllyw DR, Davies AR, Phillips TN (1996) On the effects of piezoviscous lubricant on the dynamics of a journal bearing. J Rheol 40:1239–1266ADSCrossRefGoogle Scholar
  20. 20.
    Hartinger M, Dumont ML, Ioannides S, Gosman D, Spikes H (2008) CFD modeling of a thermal and shear-thinning elastohydrodynamic line contact. J Tribol 130(4):041,503. doi: 10.1115/1.2958077 CrossRefGoogle Scholar
  21. 21.
    Heywood JG, Rannacher R, Turek S (1996) Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations. Int J Numer Methods Fluids 22(5):325–352MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hirn A, Lanzendörfer M, Stebel J (2012) Finite element approximation of flow of fluids with shear-rate- and pressure-dependent viscosity. IMA J Numer Anal 32(4):1604–1634. doi: 10.1093/imanum/drr033 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hron J, Málek J, Rajagopal KR (2001) Simple flows of fluids with pressure-dependent viscosities. Proc R Soc Lond A 457(2011):1603–1622ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Hron J, Málek J, Průša V, Rajagopal KR (2011) Further remarks on simple flows of fluids with pressure-dependent viscosities. Nonlinear Anal: Real World Appl 12(1):394–402MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Janečka A, Průša V (2014) The motion of a piezoviscous fluid under a surface load. Int J Nonlinear Mech 60:23–32. doi: 10.1016/j.ijnonlinmec.2013.12.006 CrossRefGoogle Scholar
  26. 26.
    Knauf S, Frei S, Richter T, Rannacher R (2013) Towards a complete numerical description of lubricant film dynamics in ball bearings. Comput Mech 53(2):239–255. doi: 10.1007/s00466-013-0904-1 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kračmar S, Neustupa J (2001) A weak solvability of a steady variational inequality of the Navier–Stokes type with mixed boundary conditions. Nonlinear Anal: Theory Methods Appl 47(6, Part 6 Sp. Iss. SI):4169–4180. doi: 10.1016/S0362-546X(01)00534-X MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lanzendörfer M (2009) On steady inner flows of an incompressible fluid with the viscosity depending on the pressure and the shear rate. Nonlinear Anal: Real World Appl 10(4):1943–1954. doi: 10.1016/j.nonrwa.2008.02.034 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lanzendörfer M, Stebel J (2011) On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities. Appl Math 56(3):265–285. doi: 10.1007/s10492-011-0016-1 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Li XK, Davies AR, Phillips TN (2000) A transient thermal analysis for dynamically loaded bearings. Comput Fluids 29(7):749–790. doi: 10.1016/S0045-7930(99)00035-3 CrossRefzbMATHGoogle Scholar
  31. 31.
    Lugt PM, Morales-Espejel GE (2011) A review of elasto-hydrodynamic lubrication theory. Tribol Trans 54(3):470–496. doi: 10.1080/10402004.2010.551804 CrossRefGoogle Scholar
  32. 32.
    Málek J, Rajagopal KR (2007) Mathematical properties of the solutions to the equations governing the flow of fluids with pressure and shear rate dependent viscosities. In: Friedlander S, Serre D (eds) Handbook of mathematical fluid dynamics, vol IV, 1st edn. North Holland, Amsterdam, pp 407–444 (Chap. 7)CrossRefGoogle Scholar
  33. 33.
    Neustupa T (2016) A steady flow through a plane cascade of profiles with an arbitrarily large inflow—the mathematical model, existence of a weak solution. Appl Math Comput 272 Part 3:687–691. doi: 10.1016/j.amc.2015.05.066 MathSciNetCrossRefGoogle Scholar
  34. 34.
    Průša V, Rajagopal KR (2013) A note on the modeling of incompressible fluids with material moduli dependent on the mean normal stress. Int J Nonlinear Mech 52:41–45. doi: 10.1016/j.ijnonlinmec.2013.01.003 CrossRefGoogle Scholar
  35. 35.
    Rajagopal KR (2015) Remarks on the notion of “pressure”. Int J Nonlinear Mech 71:165–172. doi: 10.1016/j.ijnonlinmec.2014.11.031 CrossRefGoogle Scholar
  36. 36.
    Rajagopal KR, Szeri AZ (2003) On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication. Proc R Soc Lond A 459:2771–2787ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Řehoř M, Průša V (2016) Squeeze flow of a piezoviscous fluid. Appl Math Comput 274:414–429. doi: 10.1016/j.amc.2015.11.008 MathSciNetGoogle Scholar
  38. 38.
    Reynolds O (1886) On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments, including an experimental determination of the viscosity of olive oil. Phil Trans R Soc Lond 177:157–234CrossRefzbMATHGoogle Scholar
  39. 39.
    Sani RL, Gresho PM, Lee RL, Griffiths DF (1981) The cause and cure of the spurious pressures generated by certain FEM solutions of the incompressible Navier–Stokes equations. Int J Numer Methods Fluids 1:17–43 (Part I)–171–204 (Part II)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Szeri AZ (2011) Fluid film lubrication: theory and design, 2nd edn. Cambridge University Press, CambridgezbMATHGoogle Scholar
  41. 41.
    Szeri AZ, Snyder V (2006) Convective inertia effects in wall-bounded thin film flows. Meccanica 41(5):473–482. doi: 10.1007/s11012-006-0006-7 CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Mathematical Institute, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic
  2. 2.Department of Mechanical EngineeringTexas A&M UniversityCollege StationUSA

Personalised recommendations