Хмельник Соломон Ицкович : другие произведения.

Navier-Stokes equations. On the existence and the search method for global solutions

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    In this book we formulate and prove the variational extremum principle for viscous incompressible fluid, from which principle follows that the Naviet-Stokes equations represent the extremum conditions of a certain functional. We describe the method of seeking solution for these equations, which consists in moving along the gradient to this functional extremum. We formulate the conditions of reaching this extremum, which are at the same time necessary and sufficient conditions of this functional global extremum existence. Then we consider the so-called closed systems. We prove that for them the necessary and sufficient conditions of global extremum for the named functional always exist. Accordingly, the search for global extremum is always successful, and so the unique solution of Naviet-Stokes is found. We contend that the systems described by Naviet-Stokes equations with determined boundary solutions (pressure or speed) on all the boundaries, are closed systems. We show that such type of systems include systems bounded by impenetworkrable walls, by free space under a known pressure, by movable walls under known pressure, by the so-called generating surfaces, through which the fluid flow passes with a known speed. The book is supplemented by open code programs in the MATLAB system - functions realizing the calculation method and test programs. Links on test programs are given in the text of the book when the examples are described. The programs may be obtained from the author by request at solik@netvision.net.il
  
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  Contents
  
  Foreword of the Reviewer \ 8
  Introduction \ 10
  Сhapter 1. Principle extremum of full action \ 13
  1.1. The Principle Formulation \ 13
  1.2. Electrical Engineering \ 15
  1.3. Mechanics \ 16
  1.4. Electrodynamics \ 17
  1.4.1. The power balance of electromagnetic field \ 17
  1.4.2. Building the functional for Maxwell equations \ 19
  1.4.3. Splitting the functional for Maxwell equations \ 21
  Chapter 2. Principle extremum of full action for hydrodynamics \ 23
  2.1. Hydrodynamic equations for viscous incompressible fluid \ 23
  2.2. The power balance \ 23
  2.3. Energian and quasiextremal \ 26
  2.4. The split energian \ 27
  2.5. About sufficient conditions of extremum \ 29
  2.6. Boundary conditions \ 31
  2.6.1. Absolutely hard and impenetrable walls \ 31
  2.6.2. Systems with a determined external pressure \ 32
  2.6.3. Systems with generating surfaces \ 33
  2.6.4. A closed systems \ 34
  2.7. Modified Navier-Stokes equations \ 35
  2.8. Conclusions \ 37
  Chapter 3. Computational Algorithm \ 38
  Chapter 5. Stationary Problems \ 39
  Chapter 6. Dynamic Problems \ 40
  6.1. Absolutely closed systems \ 42
  6.2. Closed systems with variable mass forces and external pressures \ 43
  Chapter 7. An Example: Computations for a Mixer \ 42
  7.1. The problem formulation \ 42
  7.2. Polar coordinates \ 44
  7.3. Cartesian coordinates \ 44
  7.4. Mixer with walls \ 46
  7.5. Ring Mixer \ 47
  7.6. Mixer with bottom and lid \ 50
  7.7. Acceleration of the mixer \ 52
  Chapter 8. An Example: Flow in a Pipe \ 54
  8.1. Ring pipe \ 54
  8.2. Long pipe \ 57
  8.3. Variable mass forces in a pipe \ 61
  8.4. Long pipe with shutter \ 62
  8.5. Variable mass forces in a pipe with shutter \ 65
  8.6. Pressure in a long pipe with shutter \ 67
  Chapter 9. Compressible Fluid \ 71
  Discussion \ 72
  Supplements \ 74
  Supplement 1. Certain formulas \ 74
  Supplement 2. Excerpts from the book of Nicholas Umov \ 80
  Supplement 3. Proof that Integral (2.84) is of Constant Sign \ 88
  Supplement 4. Solving Variational Problem with Gradient Descent Method \ 89
  Supplement 5. The Surfaces of Constant Lagrangian \ 92
  Supplement 6. Discrete Modified Navier-Stokes Equations \ 94
  1. Discrete modified Navier-Stokes equations for stationary flows \ 94
  2. Discrete modified Navier-Stokes equations for dynamic flows \ 97
  Supplement 7. An electrical model for solving the modified Navier-Stokes equations \ 98
  References \ 101
  
  Introduction
  In his previous works [6-25] the author presented the full action extremum principle, allowing to construct the functional for various physical systems, and, which is most important, for dissipative systems. This functional has a global saddle point, and so the gradient descent to saddle point method may be used for the solution of physical systems with such functional. As the global extremum exists, the solution also always exists.
  The first step in the construction of such functional consists in writing the equation of energy conservation or the equation of powers balance for a certain physical system. Here we must take into account the energy losses (such as friction or heat losses), and also the energy flow into the system or from it.
  This principle has been used by the author in electrical engineering, electrodynamics, mechanics. In this book we make an attempt to extend the said principle to hydrodynamics.
  
  In Chapter 1 the full action extremum principle is stated and its applicability in electrical engineering theory, electrodynamics, mechanics is shown.
  In Chapter 2 the full action extremum principle is applied to hydrodynamics for viscous incompressible fluid. It is shown that the Naviet-Stokes equations are the conditions of a certain functional's extremum. A method of searching for the solution of these equations, which consists in moving along the gradient towards this functional's extremum. The conditions for reaching this extremum are formulated, and they are proved to be the necessary and sufficient conditions of the existence of this functional's global extremum.
  Then the closed systems are considered For them it is proved that the necessary and sufficient conditions of global extremum for the named functional are always valid. Accordingly, the search for global extremum is always successful, and thus the unique solution of Naviet-Stokes is found.
  It is stated that the systems described by Naviet-Stokes and having determined boundary conditions (pressures or speeds) on all the bounds, belong to the type of closed systems. It is shown that such type includes the systems that are bounded by:
  o Impenetworkrable walls,
  o Free surfaces, находящимися под известным давлением,
  o Movable walls being under a known pressure,
  o So-called generating surfaces through which the flow passes with a known speed.
  Thus, for closed systems it is proved that there always exists an unique solution of Naviet-Stokes equations.
  In Chapter 3 the numerical algorithm is briefly described.
  In Chapter 5 the numerical algorithm for stationary problems is described in detail.
  In Chapter 6 the algorithm for dynamic problems solution is suggested, as a sequence of stationary problems solution, including problems with jump-like and impulse changes in external effects.
  Chapter 7 shows various examples of solving the problems in calculations of a mixer by the suggested method.
  In chapter 8 we consider the fluid movement in pipe with arbitrary form of section.. It is shown that regardless of the pipe section form the speed in the pipe length is constant along the pipe and is changing parabolically along its section, if there is a constant pressure difference between the pipe's ends. Thus, the conclusion reached by the proposed method for arbitrary profile pipe is similar to the solution of a known Poiseille problem for round pipe.
  In Chapter 9 it is shown tat the suggested method may be extended for viscous compressible fluids.
  Into Supplement 1 some formulas processing was placed in order not to overload the main text.
  For the analysis of energy processes in the fluid the author had used the book of Nikolay Umov, some fragments of which are sited in Supplement 2 for the reader's convenience.
  In Supplement 3 there is a deduction of a certain formula used for proving the necessary and sufficient condition for the existence of the main functional's global extremum.
   In Supplement 4 the method of solution for a certain variational problem by gradient descend is described.
  In Supplement 5 we are giving the derivation of some formulas for the surfaces whose Lagrangian has a constant value and does not depend on the coordinates.
  In Supplement 6 dealt with a discrete version of modified Navier-Stokes equations and the corresponding functional.
  In Supplement 7 we discuss an electrical model for solving modified Navier-Stokes equations and the solution method for these equations which follows this model.
  
  
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