The internal dynamics of flows and formation of singularities

The Lagrangian description of the space derivatives of velocity and temperature—the internal dynamics of flows—is considered. The hierarchy of closed systems of limited numbers of nonlinear ordinary differential equations for the internal dynamics is obtained. Each system of this hierarchy corresponds to new local solutions of the hydrodynamics equations. Some analytical solutions, describing deformation and rotation of fluid particles and formation of singularities, are presented. Various applications of the internal dynamics, including parametrization and numerical simulation of turbulent flows, are indicated.