Complex Spaces In Hydrodynamics: Complex Navier-Stokes Equations

by A.N. Panchenkov arXiv.org E-print Archive, 19 Sep 2006 The study is devoted to the development of new effective tools and methods of analytical hydrodynamics, including problems of existence, smoothness and structure of laminar and turbulent flows. The main problem is complex Navier-Stokes equations and turbulence in complex spaces. The necessity of introducing complex spaces in hydrodynamics is determined by the mechanism of transition of a laminar flow into a turbulent flow. The author proposes a non-traditional scenario of the transition: the cause of turbulence is in destruction of a laminar flow. The article contains the mathematical rationale for the necessity of development of the theory of turbulence in the complex configurational space: the complex configurational space is the natural area of existence of turbulence. Hydrodynamic flows are regarded as flows on entropy manifolds that [flows] are supported by the two symmetries: the symmetry of conservation of general entropy and the symmetry of duality of impulse representation. The new symmetry has been introduced and studied: the form invariance of Helmholtz matrix of impulse density. The strict foundation has been provided for the known fact of chaotic mechanics: appearance of the new structure (a turbulent flow) is a result of interaction of two entities: dissipation and vorticity. On the deep level the phenomenology of turbulence in complex spaces is based on the transition from the mechanics of a material point to the mechanics of an oriented material point, that [transition] takes place in a current period of time. Read more