Franta, M.; Málek, J.; Rajagopal, K. R. On steady flows of fluids with pressure- and shear-dependent viscosities. (English) Zbl 1145.76311 Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2055, 651-670 (2005). Summary: There are many technologically important problems such as elastohydrodynamics which involve the flows of a fluid over a wide range of pressures. While the density of the fluid remains essentially constant during these flows whereby the fluid can be approximated as being incompressible, the viscosity varies significantly by several orders of magnitude. It is also possible that the viscosity of such fluids depends on the shear rate. Here we consider the flows of a class of incompressible fluids with viscosity that depends on the pressure and shear rate. We establish the existence of weak solutions for the steady flows of such fluids subjected to homogeneous Dirichlet boundary conditions and to specific body forces that are not necessarily assumed to be small. A novel aspect of the study is the manner in which we treat the pressure that allows us to establish its compactness, as well as that of the velocity gradient. The method draws upon the physics of the problem, namely that the notion of incompressibility is an idealization that is attained by letting the compressibility of the fluid to tend to zero. Cited in 1 ReviewCited in 37 Documents MSC: 76A05 Non-Newtonian fluids 35Q35 PDEs in connection with fluid mechanics 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids Keywords:Existence; weak solution; incompressible fluid; pressure-dependent viscosity; shear-dependent viscosity; Dirichlet (No-Slip) boundary condition PDFBibTeX XMLCite \textit{M. Franta} et al., Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2055, 651--670 (2005; Zbl 1145.76311) Full Text: DOI