Problems on electrorheological fluid flows

In this paper, we develop and analyze a new model describing electrorheological fluid flow. In contrast to existing models, which assume the electric field to be perpendicular to the velocity field and are thus restricted to simple shear flow and flows close to it, we consider the fluid as anisotropic and introduce a general constitutive relation based on a viscosity function that depends on the shear rate, the electric field strength, and on the angle between the electric and the velocity vectors. We study general flow problems under nonhomogeneous mixed boundary conditions with given values of velocities and surface forces on different parts of the boundary. We investigate both the case where the viscosity function is continuous and the case where it is singular for vanishing shear rate. In the latter case, the problem reduces to a variational inequality. Using methods of nonlinear analysis such as fixed point theory, monotonicity, and compactness, we establish existence results for the problems under consideration. Some efficient methods for the numerical solution of the problems are presented, and numerical results for the simulation of the fluid flow in electrorheological shock absorbers are given.