It has been well documented in many studies that the material parameters of a fluid may essentially depend on the pressure and that they can vary by several orders of magnitude. The material parameters, for which this dependence is observed, are mainly the viscosity (due to the internal forces in the fluid) and the friction (due to fluid–(rigid) solid interaction). In addition, these large variations with respect to the pressure in the material parameters occur although the variations of the density are almost negligible (in comparison with other parameters). Therefore it is still reasonable to describe the above mentioned phenomena in many fluids by incompressible models. Likewise, the viscosity and the drag of many fluids vary with the shear rate and such shear (rate)-dependent viscosity and friction are extensively used, ranging from geophysics, chemical engineering, and bio-material science up to the food industry, enhanced oil recovery, carbon dioxide sequestration, or extraction of unconventional oil deposits, etc.
The aim of this study is to present an overview of available results for models with very complicated rheological laws used in engineering praxes. As particular examples that fit into the class of models studied here, we refer to the Darcy model, to the Brinkman models, and to the Bingham models. Nevertheless, the aim of this study is much more ambitious and we go much beyond these standard models and present a kind of unifying theory, which is based on the use of the so-called maximal monotone graphs, which seems to be very appropriate from the point of view of mathematical analysis of the problem.
