We propose a relaxation framework for general fluid models which can be understood as a natural ex- tension of the Suliciu approach in the Euler setting. In particular, the relaxation system may be totally degenerate. Several stability properties are proved. The relaxation procedure is shown to be efficient in the numerical approximation of the entropy weak solutions of the original PDEs. The numerical method is particularly simple in the case of a fully degenerate relaxation system for which the solution of the Riemann problem is explicit. Indeed, the Godunov solver for the homogeneous relaxation system results in an HLLC-type solver for the equilibirum model. Discrete entropy inequalities are established under a natural Gibbs principle.