Title: Navier's slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity

Authors: J. Malék, M. Bulícek and K. R. Rajagopal

Issue: Volume 56 (2007), Issue 1, 51-86

Abstract: There is compelling experimental evidence for the viscosity of a fluid to depend on the shear rate as well as the mean normal stress (pressure). Moreover, while the viscosity can vary by several orders of magnitude the density suffers very minor variation, when the range of the pressure is sufficiently large, thereby providing justification for considering the fluid as being incompressible while at the same time possessing a viscosity that is dependent on the pressure.\par In this article we investigate the mathematical properties of internal unsteady three-dimensional flows of such fluids subject to Navier's slip at the boundary. We establish the long-time existence of a weak solution for large data provided that the viscosity depends on the shear rate and the pressure in a suitably specified manner. This specific relationship, however, includes the classical Navier-Stokes fluids and power law fluids (with power law index $r - 2$, $r \le 2$) as special cases. Even for these special cases, the existence results that are being presented are new.