  We consider an alternative Navier-Stokes model for compressible viscous ideal
gases, originally proposed in \cite{Svard18}. We derive a priori estimates that
are sufficiently strong to support a weak entropy solution of the system.
Guided by these estimates, we propose a finite volume scheme, derive the
analogous estimates and demonstrate grid convergence towards a weak entropy
solution. Furthermore, this existence proof is valid for "large" initial data
and no a priori assumptions on the solution are needed.
