  In this paper, we present a new model for heat transfer in compressible fluid
flows. The model is derived from Hamilton's principle of stationary action in
Eulerian coordinates, in a setting where the entropy conservation is recovered
as an Euler--Lagrange equation. The governing system is shown to be hyperbolic.
It is asymptotically consistent with the Euler equations for compressible heat
conducting fluids, provided the addition of suitable relaxation terms. A study
of the Rankine--Hugoniot conditions and the Clausius--Duhem inequality reveals
that contact discontinuities cannot exist while expansion waves and compression
fans are possible solutions to the governing equations. Evidence of these
properties is provided on a set of numerical test cases.
