# Article

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Keywords:
{existence, weak solution, incompressible fluid, pressure-dependent viscosity, shear-dependent viscosity, spatially periodic problem}
Summary:
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.
References:
[1] Andrade, C.: Viscosity of liquids. {Nature} 125 309-310 (1930) \JFM 56.1264.10. DOI 10.1038/125309b0
[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 10.1080/05698190500414391
[3] Bair, S., Kottke, P.: Pressure-viscosity relationships for elastohydrodynamics. {Tribology transactions} 46 289-295 (2003). DOI 10.1080/10402000308982628
[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 10.1512/iumj.2007.56.2997 | MR 2305930 | Zbl 1129.35055
[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 2515781
[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 10.1098/rspa.2004.1360 | MR 2121929
[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 10.1016/S0378-4754(02)00085-X | MR 1984133
[10] Leray, J.: Sur le mouvement d'un liquide visquex emplissant l'espace. {Acta Math.} 63 193-248(1934)\JFM 60.0726.05. DOI 10.1007/BF02547354 | MR 1555394
[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 10.1007/s00205-002-0219-4 | MR 1941479
[12] Málek, J., Nečas, J., Rokyta, M., Růžička, M.: {Weak and Measure-valued Solutions to Evolutionary PDEs}. Chapman & Hall, London (1996). MR 1409366
[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).
[14] Rajagopal, K. R.: On implicit constitutive theories. {Appl. Math.} 48(4) 279-319 (2003). DOI 10.1023/A:1026062615145 | MR 1994378 | Zbl 1099.74009
[15] Rajagopal, K. R.: On implicit constitutive theories for fluids. {J. Fluid Mech.} 550 243-249 (2006). DOI 10.1017/S0022112005008025 | MR 2263984 | Zbl 1097.76009
[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 10.1098/rspa.2004.1385 | MR 2165511 | Zbl 1186.74008
[17] Schaeffer, D. G.: Instability in the evolution equations describing incompressible granular flow. {J. Differential Equations} 66(1) 19-50 (1987). DOI 10.1016/0022-0396(87)90038-6 | MR 0871569 | Zbl 0647.35037
[18] Schaeffer, D. G.: Instability in the evolution equations describing incompressible granular flow. {J. Differential Equations} 66(1) 19-50 (1987). DOI 10.1016/0022-0396(87)90038-6 | MR 0871569 | Zbl 0647.35037

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