  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.
