Observability and Mathematics

Fluid Mechanics, Solutions of Navier-Stokes Equations, and Modeling

Boris Khots

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000466287
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

The author approaches an old classic problem - the existence of solutions of Navier-Stokes equations. The main objective is to model and derive of equation of continuity, Euler equation of fluid motion, energy flux equation, Navier-Stokes equations from the observer point of view and solve classic problem for this interpretation of fluid motion laws.

If we have a piece of metal or a volume of liquid, the idea impresses itself upon us that it is divisible without limit, that any part of it, however small, would again have the same properties. But, wherever the methods of research in the physics of matter were refined sufficiently, limits to divisibility were reached that are not due to the inadequacy of our experiments but to the nature of the subject matter.

Observability in mathematics were developed by the author based on denial of infinity idea.He introduces observers into arithmetic, and arithmetic becomes dependent on observers. And after that the basic mathematical parts also become dependent on observers. This approach permits to reconsider the fluid motion laws, analyze them and get solutions of classic problems.

Table of Contents

1. Introduction.
2. Observability and Arithmetic.
3. Observability and Vector Algebra.
4. Observability and Mathematical Analysis (Calculus).
5. Classic Fluid Mechanics equations and Observability.
6. Observability and Thermodynamical equations.
7. Observability and equation of continuity.
8. Observability and Euler equation of motion of the fluid.
9. Observability and energy flux and moment flux equations.
10. Observability and incompressible fluids.
11. Observability and Navier-Stokes equations.
12. Observability and Relativistic Fluid Mechanics.
13. Appendix: Review of publications of the Mathematics with Observers.
14. Glossary.
Bibliography
Index

Biography

Boris Khots, DrSci, lives in Iowa, USA, Independent Researcher. Alma Mater - Moscow State Lomonosov University, Department of Mathematics and Mechanics (mech-math). Creator of Observer's Mathematics. Participant of more than 30 Mathematical international congresses, conferences. In particular, participated with presentation at International Congresses of Mathematicians on 1998 (Germany), 2002 (China), 2006 (Spain), 2010 (India), 2014 (South Korea). More than 150 mathematical books and papers.

1

Introduction

DOI: 10.1201/​9781003175902-1

The Mathematics with Observers was developed by authors based on denial of infinity idea. Let's note the basic mathematical parts such that arithmetic, linear algebra, mathematical analysis (calculus), geometry, differential geometry, algebra, Lie groups and Lie algebras, functional analysis are based on infinity idea. That's about such concepts as limit, derivative, integral, continuity, line, plane, space, manifold and many others. Discussion of the infinity idea is not new in mathematics and physics. We see this discussion in works of many mathematicians and physicists, for example David Hilbert, Peter Rashevsky, and the direction in Mathematical Logic with name intuitionism of Brouwer's school, where one considers as legitimate only finite sets. We know the infinite appears in the infinite number sequences that define the real numbers. In traditional science, we always assume continuity of matter, e.g., a block of wood or a pool of water can be divided into pieces infinitely, where each resulting chunk has the same divisibility properties, however, every experiment performed in physics always has a limit to divisibility due to the nature of measurement mechanisms deployed in the real world. There is well known set-theoretical and topological result – straight line and any as much as desired small open interval of this line are homeomorphic. And it is clear that this result contradicts to atomic structure of nature. Many of these analogous situations pushed authors for Mathematics with Observers development.

We introduce Observers into arithmetic, and arithmetic becomes dependent on Observers. We consider Observer dependent ascending chain of embedded sets of decimal fractions and their Cartesian products. For every set, arithmetic operations are defined (these operations locally coincide with standard operations until the results of these operations are touching of corresponding sets borders). The system of observers becomes a finite well-ordered system. The observers are ordered by their level of “depth”, i.e. an observer with greater depth sees any observer with smaller depth, and inverse an observer with a given depth is unaware of observers with larger depth values. Probability and stochastic appear automatically because of these sets, observers and arithmetic. In difference of classic arithmetic in Mathematics with Observers additive and multiplicative associativities, distributivity may fail; multiplicative inverses do not necessarily exist, or if they do, they are not necessarily unique; square roots do not necessarily exist; there are zero divisors. And the probabilities of additive and multiplicative associativities, distributivity, multiplicative inverses existence and uniqueness, square roots existence are less than 1. And after that the basic mathematical parts also become dependent on Observers.

In this book we consider Mathematics with Observers applications to Fluid Mechanics. We follow classic Physics interpretation of Fluid Mechanics laws and contemporary problems. Based on Physics data we developed here Mathematics with Observers interpretation of the main laws of Fluid Mechanics. This new equations are derived from the basic principles of continuity of mass, momentum, and energy. We get here new thermodynamical equations, continuity equation, Euler equation of motion of the fluid, energy flux and moment flux equations, incompressible fluids equations, Navier-Stokes equations. All these equations become stochastic because depend on observers point of view.

We consider thermal natural variables: temperature and entropy, and mechanical natural variables: pressure and volume. Also we consider thermodynamic potentials as functions of their thermal and mechanical variables: the internal energy with given temperature and pressure considered as the parameters, enthalpy with given temperature and volume considered as the parameters, Helmholtz free energy with given entropy and pressure considered as the parameters, and Gibbs free energy with given entropy and volume considered as the parameters. We consider classic thermodynamical equations from Mathematics with Observers point of view. And we prove that they are correct with probabilities less than 1 in Mathematics with Observers because applied here the main Observer's tools (in particular specific properties of differentiation and integration) give these results. And thermodynamical equations in Mathematics with Observers become stochastic.

We consider a finite arbitrary volume, called a control volume, over which the basic principles of continuity of mass, momentum, and energy can be applied. The control volume can remain fixed in space or can move with the fluid. We take the ideal fluid velocity and two thermodynamic quantities - pressure and density. We consider the volume and mass of fluid in this volume and calculate the total mass of fluid flowing in and out of this volume in unit time. We calculate decrease per unit time in the mass of fluid in this volume and get the equation of continuity. Using this classic way and applying here Mathematics with Observers tools we are remodeling fluid continuity and get the equation of continuity in Mathematics with Observers. And equation of continuity in Mathematics with Observers becomes stochastic. We proved that classic equation of continuity takes a place with probability less than 1.

For modeling the motion of the ideal fluid we take pressure as a function defined on cube of the ideal fluid and calculate the total vector-force acting on this volume. By Newton second law we get the equation of motion of the volume element in the fluid. Using this classic way and applying here Mathematics with Observers tools we are remodeling motion of the ideal fluid and get this equation of the motion in Mathematics with Observers. So we get Euler equation of motion of the fluid in Mathematics with Observers. If the fluid is in gravitational field, we add an additional force and get Euler equation in Mathematics with Observers with this additional force. We get additional view of the Observer's Euler equation of motion of the fluid, in which it involves only the velocity. Also we rewrite Observer's Euler equation of motion of the fluid in gravitational field. And we write the Observer's Euler equation for the fluid at rest in uniform gravitational field. And Euler equation of motion of the fluid in Mathematics with Observers becomes stochastic. We proved that classic Euler equation takes a place with probability less than 1.

We take some volume element fixed in space and define the energy of unit volume of ideal fluid as a sum of kinetic energy and the internal energy. We calculate also the change in this energy. And finally we get energy flux equation. Using this classic way and applying here Mathematics with Observers tools we are remodeling energy picture and get Mathematics with Observers energy flux equation. We calculate the momentum of unit volume of ideal fluid and the rate of this momentum change. And finally we get momentum flux equation. In process of calculations we consider the momentum flux density matrix. Using this classic way and applying here Mathematics with Observers tools we are remodeling momentum picture and get Mathematics with Observers momentum flux equation. In particular in classic Fluid Mechanics the momentum flux density matrix is called the momentum flux density tensor, but in Mathematics with Observers it is not a tensor in classic linear algebra sense, but only matrix in the chosen coordinate system. And energy flux and moment flux equations in Mathematics with Observers becomes stochastic. We proved that classic energy flux and moment flux equations take a place with probability less than 1.

If the density of flow of liquids or gases is invariable, i.e constant throughout the volume of the fluid and throughout its motion – it is named the incompressible flow. Euler equation in Mathematics with Observers does not change in this case. But equation of continuity in Mathematics with Observers becomes more simple. And we get from equation of continuity in Mathematics with Observers the equation with Laplace's operator of potential. This equation must be supplemented by boundary conditions at the surfaces where the fluid meets solid bodies. We consider also the situation of plane flow, i.e. the velocity distribution in a moving fluid depends on only two coordinates. We introduce the complex potential in Mathematics with Observers and the complex velocity and find corresponding equation. And all incompressible fluids equations in Mathematics with Observers becomes stochastic. We proved that classic incompressible fluids equations take a place with probability less than 1.

Instead of ideal fluids we consider viscous fluids. The viscosity (internal friction) causes another, irreversible, transfer of momentum from points where the velocity is large to those where it is small. The equation of motion of a viscous fluid may therefore be obtained by adding to “ideal” momentum flux some term which gives the irreversible “viscous” transfer of momentum in the fluid. The equation of motion of a viscous fluid can now be obtained by simply adding some expression to the right hand side of Euler equation in Mathematics with Observers. And we get Navier-Stokes equations in Mathematics with Observers. We get these equations in various situations with fluid and any given externally applied force. And all these equations in Mathematics with Observers becomes stochastic. We proved that classic Navier-Stokes equations take a place with probability less than 1. We make analysis of Navier-Stokes equations in Mathematics with Observers. And we make analysis of the solution existence of these equations.

Also we consider here the interpretation of Relativistic Fluid Mechanics in Mathematics with Observers. The governing principles in Fluid Mechanics are the conservation laws for mass, momentum, and energy. And in classic Physics and Mathematics the conservation laws characterizing special relativistic fluid mechanics are invariant (in fact co-variant) under Poincare group of transformations. We consider this situation from Mathematics with Observers point of view. First of all we prove that the sets of all invertible $3\times 3$, $4\times 4$, $5\times 5$ matrices are not the Lie groups in Mathematics with Observers, and group definition's conditions take a place here with some probability less than 1. Also we prove that the sets of all orthogonal matrices O(3) is not the Lie groups in Mathematics with Observers, and group definition's conditions take a place here with some probability less than 1. And we prove that the sets of all Lorentz matrices L is not the Lie groups in Mathematics with Observers, and group definition's conditions take a place here with some probability less than 1. And finally we prove that the sets of all Poincare matrices P is not the Lie groups in Mathematics with Observers, and group definition's conditions take a place here with some probability less than 1. That means in Mathematics with Observers the probabilities of the conservation laws characterizing special relativistic fluid mechanics are invariant (in fact co-variant) under Poincare group of transformations are less than 1.

2

Observability and Arithmetic

DOI: 10.1201/​9781003175902-2

We follow now the ideas of David Hilbert (see [3]), Peter Rashevsky (see [9]) and describe the Mathematics with Observers (see [5], [4]) based on ...