Let Ek denote the kinetic energy of a material volume:
Although the kinetic energy is an extensive variable it is not a quantity which is conserved. Thus the balance equation of kinetic energy does not express a conservation law. It is not an axiom of mechanics and can be derived from the continuity and the momentum equations.
In this equation the first term on the left-hand side represents the rate of change of kinetic energy within the fixed volume V. The next term represents the convective
kinetic energy flux which crosses the bounding surface of this volume. The third term represents the loss of kinetic energy. This rate of work is transformed into internal energy; this transformation of work into heat is irreversible. The balance of kinetic energy equation can be also written in differential form. The material derivative of the specific kinetic energy may be separated into the local and the convective parts
