  We consider the non-isothermal flow of a compressible fluid through pipes.
Starting from the full set of Euler equations, we propose a variational
characterization of solutions that encodes the conservation of mass, energy,
and entropy in a very direct manner. This variational principle is suitable for
a conforming Galerkin approximation in space which automatically inherits the
basic physical conservation laws. Three different spaces are used for
approximation of density, mass flux, and temperature, and we consider a mixed
finite element method as one possible choice of suitable approximation spaces.
We also investigate the subsequent discretization in time by a problem adapted
implicit time stepping scheme for which exact conservation of mass as well as a
slight dissipation of energy and increase of entropy are proven which are due
to the numerical dissipation of the implicit time discretization. The main
arguments of our analysis are rather general and allow us to extend the
approach with minor modification to more general boundary conditions and flow
models taking into account friction, viscosity, heat conduction, and heat
exchange with the surrounding medium.
