In this paper, we present an overview of the entropy production in fluid dynamics in a systematic way. First of all, we clarify a rigorous derivation of the incompressible limit for the Navier-Stokes-Fourier system of equations based on the asymptotic analysis, which is a very well known mathematical technique used to derive macroscopic limits of kinetic equations (Chapman-Enskog expansion and Hilbert expansion are popular methodologies). This allows to overcome the theoretical limits of assuming that the material derivative of the density simply vanishes. Moreover, we show that the fundamental Gibbs relation in classical thermodynamics can be applied to non-equilibrium flows for generalizing the entropy and for expressing the second law of thermodynamics in case of both incompressible and compressible flows. This is consistent with the thermodynamics of irreversible processes and it is an essential condition for the design and optimization of fluid flow devices. Summarizing a theoretical framework valid at different regimes (both incompressible and compressible) sheds light on entropy production in fluid mechanics, with broad implications in applied mechanics.