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Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling ν(p, ·) → + ∞ AS p → + ∞

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Abstract

Over a large range of the pressure, one cannot ignore the fact that the viscosity grows significantly (even exponentially) with increasing pressure. This paper concerns long-time and large-data existence results for a generalization of the Navier-Stokes fluid whose viscosity depends on the shear rate and the pressure. The novelty of this result stems from the fact that we allow the viscosity to be an unbounded function of pressure as it becomes infinite. In order to include a large class of viscosities and in order to explain the main idea in as simple a manner as possible, we restrict ourselves to a discussion of the spatially periodic problem.

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References

  1. C. Andrade: Viscosity of liquids. Nature 125 (1930), 309–310.

    Article  MATH  Google Scholar 

  2. S. Bair: A more complete description of the shear rheology of high-temperature, high-shear journal bearing lubrication. Tribology transactions 49 (2006), 39–45.

    Article  Google Scholar 

  3. S. Bair and P. Kottke: Pressure-viscosity relationships for elastohydrodynamics. Tribology transactions 46 (2003), 289–295.

    Article  Google Scholar 

  4. C. Barus: Isothermals, isopiestics and isometrics relative to viscosity. American Jour. Sci. 45 (1893), 87–96,.

    Google Scholar 

  5. P. W. Bridgman: The Physics of High Pressure. MacMillan, New York, 1931.

    Google Scholar 

  6. M. Bulíček, J. Málek, and K.R. Rajagopal: Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity. Indiana Univ. Math. J. 56 (2007), 51–86.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Bulíček, J. Málek, and K. R. Rajagopal: Mathematical analysis of unsteady flows of fluids with pressure, shear-rate and temperature dependent material moduli, that slip at solid boundaries. To appear in SIAM J. Math. Anal..

  8. M. Franta, J. Málek, and K. R. Rajagopal: On steady flows of fluids with pressure-and shear-dependent viscosities. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461(2055) (2005), 651–670.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Hron, J. Málek, J. Nečas, and K. R. Rajagopal: Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure-and shear-dependent viscosities. Math. Comput. Simulation 61(3–6) (2003), 297–315.

    Article  MATH  MathSciNet  Google Scholar 

  10. J. Leray: Sur le mouvement d’un liquide visquex emplissant l’espace. Acta Math. 63 (1934), 193–248.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Málek, J. Nečas, and K. R. Rajagopal: Global analysis of the flows of fluids with pressure-dependent viscosities. Arch. Ration. Mech. Anal. 165(3) (2002), 243–269.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. Málek, J. Nečas, M. Rokyta, and M. Růžička.: Weak and Measure-valued Solutions to Evolutionary PDEs. Chapman & Hall, London, 1996.

    MATH  Google Scholar 

  13. J. Málek and K. R. Rajagopal: Mathematical Properties of the Solutions to the Equations Govering the Flow of Fluid with Pressure and Shear Rate Dependent Viscosities. In Handbook of Mathematical Fluid Dynamics, Vol. IV, Handb. Differ. Equ. Elsevier/North-Holland, Amsterdam, 2007, pp. 407–444.

    Google Scholar 

  14. K. R. Rajagopal: On implicit constitutive theories. Appl. Math. 48(4) (2003), 279–319.

    Article  MATH  MathSciNet  Google Scholar 

  15. K. R. Rajagopal: On implicit constitutive theories for fluids. J. Fluid Mech. 550 (2006), 243–249.

    Article  MATH  MathSciNet  Google Scholar 

  16. K. R. Rajagopal and A. R. Srinivasa: On the nature of constraints for continua undergoing dissipative processes. Proc. R. Soc. A 461 (2005), 2785–2795.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. G. Schaeffer: Instability in the evolution equations describing incompressible granular flow. J. Differential Equations 66(1) (1987), 19–50.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute, Charles University in Prague, Faculty of Mathematics and Physics, Sokolovská 83, 18675, Prague 8, Czech Republic

    M. Bulíček

  2. Mathematical Institute, Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 18675, Prague 8, Czech Republic

    J. Málek

  3. Department of Mechanical Engineering, Texas A&M University College Station, TX, 77843, Texas, USA

    K. R. Rajagopal

Authors
  1. M. Bulíček
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  2. J. Málek
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  3. K. R. Rajagopal
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Corresponding author

Correspondence to M. Bulíček.

Additional information

M. Bulíček’s research is supported by the Nečas Center for Mathematical Modeling, the project LC06052 financed by MSMT

The contribution of J. Málek to this work is a part of the research project MSM 0021620839 financed by MSMT. J. Málek also thanks the Czech Science Foundation, project GACR 201/06/0352, for its support.

K. R. Rajagopal thanks the National Science Foundation for its support.

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Bulíček, M., Málek, J. & Rajagopal, K.R. Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling ν(p, ·) → + ∞ AS p → + ∞. Czech Math J 59, 503–528 (2009). https://doi.org/10.1007/s10587-009-0034-2

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  • DOI: https://doi.org/10.1007/s10587-009-0034-2

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