Recent Developments in Steady Electrorheological Fluid Flows

Electrorheological fluids(ERFs) are special materials characterized by their ability to highly change in their mechanical properties when an electric field is applied. In the model, the constitutive law for the extra stress tensor of the fluid reduces to a nonstandard growth condition of p(x; t)-type. The purpose of this book is to give an introduction to the recent developments in the existence and regularity theory for steady flows of ERFs obtained by the author. Firstly the existence and a higher integrability of weak solutions for steady flows of ERFs with non-homogeneous Dirichlet boundary condition are proved. Secondly, global regularity of weak solutions for steady flows of ERFs is proved. Most of the results not only are new even for a generalized Newtonian fluid with shear-dependent viscosity and but also can be applied to the study about a non-Newtonian fluid of which behavior is similar to one of ERFs.