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Renormalization of the fragmentation equation: Exact self-similar solutions and turbulent cascades

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Using an approach developed earlier for renormalization of the Boltzmann collision integral (V.L. Saveliev and K.Nanbu, Phys. Rev. E 65, 051205, 2002), we derive an exact divergence form for the fragmentation operator. Then we reduce the fragmentation equation to the continuity equation in size-space, with the flux given explicitly. This allows us to obtain new selfsimilar solutions, and to find the new integral of motion for these solutions (we call it the bare flux). We show how these solutions can be applied as a description of cascade processes in three and two dimensional turbulence. We also suggested an empirical
cascade model of impact fragmentation of brittle materials.

Author(s):

Vladimir Leonidovich Saveliev    
Institute of Ionosphere, National Center of Space Research and Technology
Kazakhstan

Mikhail Arkadievich Gorokhovski    
Laboratoire de Mécanique des Fluids et d’Acoustique, CNRS - Ecole Centrale de Lyon – INSA Lyon – Université Claude Bernard Lyon
France