On existence 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 $(0,T)$. Here we are motivated by the bounded domain approach with "do-nothing" boundary conditions. In terms of the velocity, pressure and enthalpy of the fluid, such flows are described by a parabolic system with strong nonlinearities and including the artificial boundary conditions for the velocity and nonlinear boundary conditions for the so called enthalpy of the fluid. The present analysis is devoted to the proof of the existence of weak solutions for the above problem. In addition, we deal with some regularity for the velocity of the fluid.