Coupled problems on stationary non-isothermal flow of electrorheological fluids

We set up and investigate a coupled problem on stationary non-isothermal flow of electrorheological fluids. The problems 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 consider original and regularized coupled problems. In the regularized problem the dissipation of energy is defined by the regularized velocity field which leads to a nonlocal model. We introduce the notions of generalized solutions for the original and regularized problems. The existence of the generalized solution of the regularized problem is proved by using the methods of monotonicity, compactness, and topological degree. We prove that there exists a solution of the original problem where the domain of flow is two-or three-dimensional (in the case of a three-dimensional domain an extra condition is being assumed). It is shown that the solution of the original problem is a limiting point of the set of solutions of the regularized problems in which the parameter of regularization tends to zero.