Energy Transfer in turbulent polymer solutions

The paper addresses a set of new equations concerning the scale by scale balance of turbulent fluctuations in dilute polymers solutions. The main obstacle to this achievement is represented by the energy associated with the polymers which is not a quadratic form in terms of the traditional descriptor of the micro-structure. A different choice is however possible, which, at least for mild stretching of the polymeric chains, directly leads to an L2 structure for the total free-energy density of the system thus entailing the extension of the classical machinery to polymeric fluids. The issuing budget in spectral space is discussed, providing the spectral decomposition of the energy of the system. New equations are derived also in physical space, to provide the balance equations for the fluctuations in both the kinetic field and the micro-structure, thus extending, in a sense, the celebrated Karman-Howarth and Kolmogorov equations of classical turbulence theory. The subject covered by the paper is purposely limited to the context of homogeneous turbulence. However the necessary steps required to expand the treatment to wall bounded flows of polymeric liquids are indicated in detail in the closing section devoted to comments and perspectives.