We set up and study a coupled problem on stationary non-isothermal flow of
electrorheological fluids. The problem consist in finding functions of
velocity, pressure and temperature which satisfy the motion equations, the
condition of incompressibility, the equation of the balance of thermal energy
and boundary conditions.
We introduce the notions of a $P$-generalized solution and generalized
solution of the coupled problem. In case of the $P$-generalized solution the
dissipation of energy is defined by the regularized velocity field, which leads
to a nonlocal model.
Under weak conditions, we prove the existence of the $P$ -generalized
solution of the coupled problem. The existence of the generalized solution is
proved under the conditions on smoothness of the boundary and on smallness of
the data of the problem