On existence, regularity and uniqueness of thermally coupled incompressible flows in a system of three dimensional pipes

We study an initial-boundary-value problem for time-dependent flows of heat-conducting viscous incompressible fluids in a system of three-dimensional pipes on a time interval (0,T). We are motivated by the bounded domain approach with “do-nothing” boundary conditions. In terms of the velocity, pressure and temperature of the fluid, such flows are described by a coupled parabolic system with strong nonlinearities and including the natural boundary conditions for the velocity and temperature of the fluid on the part of the boundary where the fluid is supposed to leave the channel. The present analysis is devoted to the proof of the existence, regularity and uniqueness of the solution for the problem described above.