The entropy model of hydrodynamics

A.N.Panchenkov

Nizhniy Novgorod State Technical University

Apt. 26, 148 M.Gorky st., Nizhniy Novgorod

Russia, 603000

 

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 - complex equations of hydrodynamics and turbulence in complex spaces. The advisability 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 (cessation of existence) of a laminar flow. The article contains the mathematical rationale for the scenario of development of the theory of turbulence in the complex configurational space. Hydrodynamic flows are regarded as flows on entropy manifolds that [flows] are supported by the symmetry of conservation of general entropy. The new symmetry has been introduced and studied: the forminvariance 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.

The question of building the entropy model of hydrodynamics has attracted attention of the author due to appearance of the entropy conceptual model, methodology and tools of description of the Nature and surrounding Reality. This question is intimately connected with analytical hydrodynamics and theory of turbulence.

Analytical hydrodynamics and its important section - the theory of Navier-Stokes equations - are attracting attention of many researchers nowadays. In the theory of Navier-Stokes equations, starting from the work by Leray, the majority of researches have been devoted to the problem of "weak" (turbulent) solutions. Despite the multitude of such researches, our knowledge of the structure and properties of the solutions of Navier-Stokes equations remains rather limited. Therefore, the task of brining new ideas, concepts and methodologies into analytical hydrodynamics has become actual.

The new concept, methodology and tools for studying continuous media have been developed by A.N.Panchenkov in the book series "Entropy".

The one of characteristic features of this series is the following: in the majority of the problems the basic geometrical objects (phase spaces, configurational spaces, entropy manifolds etc. ) are complex. And what is more, the subject of the book "Inertia" - the theory of inertia - was realized above the field of complex numbers. The entropy mechanics is defined by A.N.Panchenkov in the book "Entropy mechanics" as the mechanics of flows on entropy manifolds of complex configurational spaces and fields in complex configurational spaces.

The factor determining advisability and necessity of the complexification is the existence of rotor in the virtual continuous medium. Let me remind that Navier-Stokes equations were introduced into hydrodynamics for describing laminar flows of a viscous incompressible fluid; the question of validity and dependability of their application is still open. The applicability of real Navier-Stokes equations for describing turbulent flows was questioned by several investigators. Yet this question has become especially actual lately, due to activation of researches on theory of these equations, [http://claymath.org]. The particular facts and materials determining advisability of transition to the complex Navier-Stokes equations in the problem of turbulence are contained in the book series "Entropy". The author set forth the non-classical hypothesis of turbulence origin in the chapter "Turbulence" of the book "Entropy-2: Chaotic Mechanics." The origin of turbulence is believed, usually, to be related to the loss of stability of a laminar motion. I accepted the another hypothesis, that is not connected whatsoever with the loss of stability of a laminar motion. A laminar motion can not exist forever; sooner or later, it ceases to exist. A time-point of cessation of existence (destruction) of a laminar flow is, simultaneously, the point of appearance of a turbulent flow.

According to this hypothesis, the cause of turbulence is a destruction of a laminar flow. A destruction of a laminar flow (as of a certain structure) is not directly related to stability of a motion; it is not a loss of stability that is implied, but destruction of one structure (laminar flow) followed by appearance of another structure (turbulent flow). Both destruction and appearance take place in chaos, that fact determines the key role of the chaotic mechanics.

The development of the effective theory of turbulence, based on the hypothesis of destruction of a laminar flow, requires, naturally, introducing the complex basic geometrical objects and the complex equations; such introduction has been performed in the book series "Entropy". It has been established there, that, regarding the problem of turbulence, the complex phase space is the natural environment for the turbulence.