We introduce a constitutive equation of electrorheological fluids such that a fluid is considered as a viscous anisotropic one with the viscosity depending on the second invariant of the rate of strain tensor, on the module of the vector of electric field strength, and on the angle between the vectors of velocity and electric field. We study the general problem on the flow of such fluids at nonhomogeneous mixed boundary conditions, wherein values of velocities and surface forces are given on different parts of the boundary. We consider the cases where the viscosity function is continuous and singular, equal to infinity, when the second invariant of the rate of strain tensor is equal to zero. In the second case the problem is reduced to a variational inequality.
The singular viscosity function is approximated by a continuous bounded one with a parameter of regularization. By using the methods of a fixed point, monotonicity, and compactness we prove the existence of a solution of the regularized problem and that the solutions of the regularized problems converge to the solution of the variational inequality as the parameter of regularization tends to zero. We establish also conditions under which the solutions of the regularized problems and the variational inequality are unique.
We also address efficient methods for numerical solution of the problems under consideration.