The macroscopic non-equilibrium kinetic energies of a small fluid particle

An analytical formula for the macroscopic kinetic energy per unit mass of a small macroscopic fluid particle is derived. The macroscopic kinetic energy per unit mass is presented as a sum of the macroscopic translational kinetic energy per unit mass and three Galilean invariants: the classical macroscopic internal rotational kinetic energy per unit mass, a new macroscopic internal shear kinetic energy per unit mass and a new macroscopic internal kinetic energy of a shear-rotational coupling per unit mass with a small correction. The obtained formula generalizes the classical de Groot and Mazur expression by taking into account the shear component of the velocity field. The evolution equation for the average macroscopic internal kinetic energy per unit mass is derived within the frame of the model of an incompressible homogeneous viscous Newtonian fluid for the statistical ensemble of randomly and isotropically oriented and sheared small-scale turbulent eddies. The practical significance of the macroscopic internal shear kinetic energy is evaluated by the realistic prediction of the theoretical “shear-rotational” transition energy dissipation rate per unit mass for the transition of grid-generated stratified turbulence to internal gravity waves.