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2017 | OriginalPaper | Chapter

1. Fundamental Equations in Fluid Dynamics

Author : Department of Earth System Science and Technology, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University

Published in: Fluid Dynamics for Global Environmental Studies

Publisher: Springer Japan

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Abstract

This chapter describes the fundamentals of fluid dynamics in order to study of the movement of gases and liquids. Because the earth’s atmosphere and oceanic movements are within the scope of fluid dynamics, this book mainly aims to describe dynamics of rotating and stratified fluids, as well as environmental fluid dynamics. This chapter is allocated for the explanation of fundamental equations in fluid dynamics. First, stress and stress tensors will be discussed, and then fluids will be characterized on their basis. Next, the mass conservation law, equation of motion (momentum conservation law), and energy equations as well as the specifications of fluid motion are discussed. The concept of vorticity and its governing equations will be discussed to complete the equation of motion for a viscous fluid. We also show that this concept is necessary to relate the deformation of fluid elements due to fluid motion with stress. In this book, incompressible flow will be considered in most cases; however, this chapter has been made as general as possible by assuming the compressible flow cases.

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next chapter Dynamics of Perfect Fluids

It will be stated simply as point \(\varvec{r}\) hereinafter.

This will be explained with a little more detail in Sect. 1.9.

\(x_{1}, x_{2}, x_{3}\) indicates x, y, z, respectively, and \(\varvec{e}_{1}, \varvec{e}_{2}, \varvec{e}_{3}\) refers to the basic vector \(\varvec{i},\varvec{j},\varvec{k}\) of the xyz axes, respectively. Hereafter, both will be used as appropriate. Moreover, even if not stated specifically, it will be assumed that an appropriate xyz Cartesian coordinate system is in place.

Under normal circumstances, it should probably be expressed as a column vector because of considerations relating to the amount of paper a row vector will be used. Instead of referring to it as vector with components \(T_{i}\), it can be referred to simply as vector \(T_{i}\).

More precisely, this is a second-order tensor. Moreover, a tensor with \(\tau _{ij}\) as components may instead be referred to simply as tensor \(\tau _{ij}\).

Matrix vertically lining up vectors \(\varvec{T}(\varvec{e}_{1})\), \(\varvec{T}(\varvec{e}_{2})\), \(\varvec{T}(\varvec{e}_{3})\), which are mappings of the basic vectors \(\varvec{e}_1\), \(\varvec{e}_2\), \(\varvec{e}_3\), respectively.

Can be obtained from the invariance against the coordinate system rotation of the characteristic polynomial obtained from (1.13).

The word fluid particle is often used to indicate an infinitesimal fluid element. Here it is considered to be a “point” that makes up the fluid substance as a continuum.

Sometimes, \(\varvec{r}_{0} = (x_{0}, y_{0}, z_{0})\) can be referred to as a material coordinate.

Can also be called a substantial derivative.

Not all experts distinguish incompressible flow and incompressible fluid in this way.

Here the subscript in (1.37) is changed from i to j and the components are summed from 1 to 3. Therefore, notation change does not influence the result.

This is because the matrix is a linear operator that relates vector \(\delta \varvec{r}\) to vector \(\delta \varvec{u}\).

In terms of molecules, this occurs because the molecules on the upper side with large horizontal momentum are transferred to the lower side by molecular thermal motion, whereas those on the lower side with smaller horizontal momentum are transferred to the upper side. In this sense, viscosity can be regarded as the diffusion of momentum.

A fluid in which tangential stress is not proportional to velocity gradient is called a non-Newtonian fluid. This book chiefly focuses on Newtonian fluids.

Materials such as liquid crystals are directional and not isotropic materials.

(1.87) is also sometimes called the Navier–Stokes equation.

Imposing such constraint conditions on the tangential component is appropriate because the differential equations are second order in space in this case.

In relative motions of viscous fluids, the fluid element is generally not in thermal equilibrium. In this situation, the thermodynamic quantities are defined as follows. Density \(\rho \) is the mass of the fluid element divided by the volume, and hence can be specified regardless of the equilibrium state. Moreover, the increase in internal energy E caused by the state change is the sum of the work done and the heat added in the process. Thus, the internal energy is also definable in the non-equilibrium state. Other thermodynamic quantities such as temperature and pressure are determined from the \(\rho \) and E on the basis of the equation of state. Here the equilibrium state is the state where the fluid element settles eventually when it is suddenly isolated thermally from its surroundings.

The fluid element moves at macroscopic velocity. Here translational motion refers to its thermal motion relative to its macroscopic velocity. The pressure of the gas is proportional to the average translational energy of its thermal motion.

The subscript of \(\partial S/\partial T\) should probably be \(p_{e}\); however, the subscript e of \(p_{e}\) is omitted.

Each fluid element is regarded as a thermodynamic system. Fluid elements are considered to undergo quasi-static state changes based on the fluid motion.

Generally, an arbitrary closed curve does not always produce a tube. Thus, we confine our discussion to simple closed curves that can be reduced to a point without passing outside the fluid.

Vortex filaments can be defined in several ways. In one definition, a vortex filament is simply a vortex tube with an infinitesimally small cross section. Here we adopt the definition in [6]. As recently recognized, an important requirement of a vortex filament is that the neighboring exterior of the filament has zero vorticity. Moreover, [6] called the limit of \(\sigma \rightarrow 0\), \(\omega \rightarrow \infty \) with keeping \(\varGamma \) constant as the line vortex and distinguished it from a vortex filament. Here this limit is also termed a vortex filament. Intuitively, a vortex filament is a solitary vortex region of filament form.

For example, if \(\partial u_i/\partial x_j\) are bounded, this holds.

This situation is sometimes said that vorticity is frozen in the fluid. In a perfectly conducting fluid (see Note 2.1), the magnetic flux density \(\varvec{B}\) satisfies (1.130); hence, the lines of magnetic force also move with the fluid. This is referred for magnetic field to be frozen in the fluid.

Here it is assumed that the fluid under consideration has not the intrinsic couples, as in ordinary fluids.

Aris, R.: Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Prentice-Hall (1962)

Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)

Imai, I.: Fluid Dynamics, vol. I. Shokabo (1973)

Jeffreys, H.: Cartesian Tensors. Cambridge University Press, Cambridge (1969)

Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Butterworth-Heinemann, Oxford (1987)

Saffman, P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992)

Whitham, G.B.: The Navier-Stokes Equations of Motion. In: Rosenhead, L. (ed.) Laminar Boundary Layers, Part I. Oxford University Press, Oxford (1963)

Title: Fundamental Equations in Fluid Dynamics
Author: Department of Earth System Science and Technology, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University
Publisher: Springer Japan
Book: Fluid Dynamics for Global Environmental Studies
Print ISBN: 978-4-431-56497-3

Electronic ISBN: 978-4-431-56499-7

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-4-431-56499-7_1