M. Bulicek, Charles University in Prague, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovska 83, 186 75 Prague 8, Czech Republic, e-mail: mbul8060@karlin.mff.cuni.cz; J. Malek, Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovska 83, 186 75 Prague 8, Czech Republic, e-mail: malek@karlin.mff.cuni.cz; K. R. Rajagopal, Department of Mechanical Engineering, Texas A&M University College Station, TX 77843, USA, e-mail krajagopal@mengr.tamu.edu
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.
Keywords: existence, weak solution, incompressible fluid, pressure-dependent viscosity, shear-dependent viscosity, spatially periodic problem
Classification (MSC 2000): 35Q30, 76A05, 76D03, 76D05
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