Problem on stationary flow of electrorheological fluids at the generalized conditions of slip on the boundary

General conditions of slip of a fluid on the boundary are derived and a problem on stationary flow of the electrorheological fluid in which the terms of slip are specified on one part of the boundary and surface forces are given on the other is formulated and studied. Existence of a generalized (weak) solution of this problem is proved by using the methods of penalty functions, monotonicity and compactness. It is shown that the method of penalty functions and the Galerkin approximations can be used for the approximate solution of the problem under consideration. The existence and the uniqueness of the smooth classical solution of the problem is proved in the case that the conditions of slip are prescribed on the whole of the boundary.