Geodesic
Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling $\nu (p, \cdot ) \to + \infty $ as $p \to +\infty $
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 2, pp. 503-528 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 35Q30, 76A05, 76D03, 76D05
Keywords: {existence, weak solution, incompressible fluid, pressure-dependent viscosity, shear-dependent viscosity, spatially periodic problem} 
@article{CMJ_2009_59_2_a14,
     author = {Bul{\'\i}\v{c}ek, M. and M\'alek, J. and Rajagopal, K. R.},
     title = {Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling $\nu (p, \cdot ) \to + \infty $ as $p \to +\infty $},
     journal = {Czechoslovak Mathematical Journal},
     pages = {503--528},
     year = {2009},
     volume = {59},
     number = {2},
     mrnumber = {2532387},
     zbl = {1224.35311},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2009_59_2_a14/}
}
TY  - JOUR
AU  - Bulíček, M.
AU  - Málek, J.
AU  - Rajagopal, K. R.
TI  - Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling $\nu (p, \cdot ) \to + \infty $ as $p \to +\infty $
JO  - Czechoslovak Mathematical Journal
PY  - 2009
SP  - 503
EP  - 528
VL  - 59
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMJ_2009_59_2_a14/
LA  - en
ID  - CMJ_2009_59_2_a14
ER  - 
%0 Journal Article
%A Bulíček, M.
%A Málek, J.
%A Rajagopal, K. R.
%T Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling $\nu (p, \cdot ) \to + \infty $ as $p \to +\infty $
%J Czechoslovak Mathematical Journal
%D 2009
%P 503-528
%V 59
%N 2
%U http://geodesic.mathdoc.fr/item/CMJ_2009_59_2_a14/
%G en
%F CMJ_2009_59_2_a14
Bulíček, M.; Málek, J.; Rajagopal, K. R. Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling $\nu (p, \cdot ) \to + \infty $ as $p \to +\infty $. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 2, pp. 503-528. http://geodesic.mathdoc.fr/item/CMJ_2009_59_2_a14/

[1] Andrade, C.: Viscosity of liquids. {Nature} 125 309-310 (1930) \JFM 56.1264.10. | DOI

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

[3] Bair, S., Kottke, P.: Pressure-viscosity relationships for elastohydrodynamics. {Tribology transactions} 46 289-295 (2003). | DOI

[4] Barus, C.: Isothermals, isopiestics and isometrics relative to viscosity. {American Jour. Sci.} 45 87-96, (1893).

[5] Bridgman, P. W.: {The Physics of High Pressure}. MacMillan, New York (1931).

[6] Bulíček, M., Málek, J., Rajagopal, K.R.: Navier's slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity. {Indiana Univ. Math. J.} 56 51-86 (2007). | DOI | MR | Zbl

[7] Bulíček, M., Málek, J., Rajagopal, K. R.: 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. | MR

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

[9] Hron, J., Málek, J., Nečas, J., Rajagopal, K. R.: 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) 297-315 (2003). | DOI | MR

[10] Leray, J.: Sur le mouvement d'un liquide visquex emplissant l'espace. {Acta Math.} 63 193-248(1934)\JFM 60.0726.05. | DOI | MR

[11] Málek, J., Nečas, J., Rajagopal, K. R.: Global analysis of the flows of fluids with pressure-dependent viscosities. {Arch. Ration. Mech. Anal.} 165(3) 243-269 (2002). | DOI | MR

[12] Málek, J., Nečas, J., Rokyta, M., Růžička, M.: {Weak and Measure-valued Solutions to Evolutionary PDEs}. Chapman & Hall, London (1996). | MR

[13] Málek, J., Rajagopal, K. R.: 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 407-444 Elsevier/North-Holland, Amsterdam (2007). | MR

[14] Rajagopal, K. R.: On implicit constitutive theories. {Appl. Math.} 48(4) 279-319 (2003). | DOI | MR | Zbl

[15] Rajagopal, K. R.: On implicit constitutive theories for fluids. {J. Fluid Mech.} 550 243-249 (2006). | DOI | MR | Zbl

[16] Rajagopal, K. R., Srinivasa, A. R.: On the nature of constraints for continua undergoing dissipative processes. {Proc. R. Soc. A} 461 2785-2795 (2005). | DOI | MR | Zbl

[17] Schaeffer, D. G.: Instability in the evolution equations describing incompressible granular flow. {J. Differential Equations} 66(1) 19-50 (1987). | DOI | MR | Zbl

[18] Schaeffer, D. G.: Instability in the evolution equations describing incompressible granular flow. {J. Differential Equations} 66(1) 19-50 (1987). | DOI | MR | Zbl