||In this work, we propose a tensorial description for a thermodynamics of dissipative continua compatible with the Galilean physics. With this aim in view, we emphasize the role of Bargmann’s group, a central extension of Galilei’s one. We introduce a new divergence-free 2-rank mixed momentum tensor gathering the energy, the linear momentum and mass density. We recover the balances of energy, linear momentum and mass. From an additive decomposition of this momentum tensor, we deduce the invariance of the production of entropy.