A sociedade do conhecimento e suas tecnologias: estudos em Ciências Exatas e Engenharias: - Volume 6

De Adailton Azevêdo Araújo Filho

Este último volume da coletânea "A sociedade do conhecimento e suas tecnologias: estudos em Ciências Exatas e Engenharias" visa compor uma coleção de diversos textos acadêmicos com uma abordagem genuinamente multidisciplinar. Teremos estudos que vão desde a teoria da turbulência até a investigação do comportamento de filtros eletrônicos. Ademais, sob a luz dos autores contemporâneos, esta obra também está recheada de conceitos e metodologias atuais com o fito de melhorar a qualidade de vida das pessoas. Em particular, nos enfocamos na aplicação de nanotecnologia a doenças neurodegenerativas, i.e., doença de Parkinson, tão bem quanto na utilização de tecnologias para a geração de energia e de recarga de equipamentos implantados.

• Idioma: Português
• Editora: Editora Dialética
• Data de lançamento: 30 de ago. de 2022
• ISBN: 9786525258454

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AN ANALYSIS OF NAVIER-STOKES EQUATIONS AND THEORY OF TURBULENCE

Edinei Cesario Zanoni

Graduado

[e-mail]

DOI 10.48021/978-65-252-5846-1-c1

ABSTRACT: The Navier-Stokes Equations are used a mathematical model for the analysis of flows since the 19th century, however, the difficulty of using and understanding the existing physical phenomena increases considerably as the Knudsen Number increases, which limits our understanding of these equations and their physical meanings. The aim of this article is to introduce new physical-mathematical concepts in the formulation of these equations to broaden the understanding of the phenomena described by them, with the intention of demonstrating that the Navier-Stokes Equations are a particular case of a set of more general equations, both for incompressible and compressible flows, so that we are able to devise a better explanation for phenomena little understood today, such as transient flow, for example. For this, I take into account the theory of Stochastic Equations in the formulation of the hypotheses of the movement of fluid particles in the flow, incorporating the concepts of mean and standard deviation in its position, as well as the introduction of a purely random term in the equations. This last term suggests that it is particularly important in flows with higher Knudsen Numbers, which introduces the explanation of the movement of particles in the transient and turbulent flow.

Keywords: Navier-Stokes Equations; Transient Flow; Turbulent Flow; Brownian Motion; Compressibility.

INTRODUCTION

Consider a small lake in any region of space, in which the water that comprises it does not flow in any direction. It is known from experience that when a pollen grain is released on the surface of this lake, it will travel unpredictable trajectories due to the existing shocks with the particles that make up the fluid. This phenomenon was explained yet through the physical-mathematical formulation of the phenomenon known as the Brownian Movement (EINSTEIN 1926). If, for whatever reason, an external force initiates the movement of water from this lake at low speeds (that is, a laminar flow begins), the model currently used to describe the flow is the one that uses these particles to start, from then on, an extremely well-ordered movement, with a succession of infinite well-ordered queues composed of infinite particles, of length and width analogous to the lengths and width of the flow, of infinitesimal height and each composed of infinite particles well aligned and ordered. So if, at this moment, we choose to evaluate the movement of these particles in this flow through an Eulerian description (considering an area small enough to pass only one particle at a time in a random region of the space through which the flow passes, then we can verify the movement of the particles that pass through this area) and think about considering the calculation of the quantity of particles that pass through this area per unit of time (that is, the frequency with which the particles pass through this area), we will notice that there will always be at least a particle passing through this area, regardless of the instant of time that we analyze or what region of the flow we imagine this area to be. To try to make this a little clearer, consider the following example: if we consider this same area close to the size of a particle in a random region of the space through which the flow passes, mark an initial time value and evaluate time intervals of one nanosecond, one microsecond, one millisecond, one second or any other variations of time, both larger and smaller, we will notice that there will be a particle crossing this area in each evaluation. This is because the particles used in the current models have no dimensions and, therefore, we will always have at least one particle passing through the section per unit of time. Physicaly, this approximation occours when the Knudsen Number of the flow is very small, lesser than 10-3(FILHO 2007). The Knuden Number is defined as the Equation (1) below.

in wich is the free middle path and is the physically representative length scale. Thus, in spite of being widely used today, the Navier-Stokes Equations have limitations that begin to become evident as the turbulence begins to have greater influence, that is, as the Knudsen Number increases. As examples of these limitations, we can mention that the Navier-Stokes Equations lose precision when evaluating bubbles in a flow, mists or any other examples that have a higher Knudsen Number. To circumvent this, we will consider that the effects of imprecision in measuring the displacement of a particle arising from the Brownian Movement is something intrinsic to the behavior of the particles, that is, we will consider that each particle in the laminar flow, for example, starts the movement maintaining among its characteristics a certain randomness and, thus, are not arranged in well-ordered sheets as in the current models: in this case, we are considering that the laminar flow is, in reality, a flow formed by particles that have influence of innumerable combinations of external actions to each of them, and that this influence is severely reduced by external forces acting on this flow and directing it, leading us to the understanding that in this flow there are queues composed of rows of particles, but they have small movements in the empty spaces between them. This means that, however much the external forces reduce the effect of