On the Navier–Stokes flows for heat-conducting fluids with mixed boundary conditions

We consider a coupled model for steady flows of viscous incompressible heat-conducting fluids with temperature dependent material coefficients in a fixed three-dimensional open cylindrical channel. We introduce the Banach spaces X and Y to be the space of possible solutions of this problem and the space of its data, respectively. We show that the corresponding operator of the problem acting between X and Y is Fréchet differentiable. Applying the local diffeomorphism theorem we get the local solvability results for a variational formulation.