Principles

of

Fluid Dynamics

of

Fluid Dynamics

Principles

of

Fluid Dynamics

of

Fluid Dynamics

by

Franco Mattioli

Franco Mattioli

by

Franco Mattioli

Franco Mattioli

Department of Physics

University of Bologna

University of Bologna

A textbook of fluid dynamics for undergraduate and graduate students
in Physics, Mathematics and Engineering

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Principles

of

Fluid Dynamics Volume I Volume II Volume III Appendices

Elements of Fluid Dynamics Elements

Principles

of

Fluid Dynamics Volume I Volume II Volume III Appendices

Elements of Fluid Dynamics Elements

Fluid Dynamics is the science that studies the
motion of the fluids, among which water and air play a fundamental role.
Fluids can flow in man-made structures or freely in nature. The interaction
of the water and air with natural obstacles or within artificial structures
is the topic of *Hydrodynamics* and *Aerodynamics*, respectively.
*Ocean and Atmosphere Dynamics*, also called *Dynamic Oceanography*
and *Dynamic Meteorology*, study the dynamics of the Ocean and the
Atmosphere.

This classification is not always respected, and many other disciplines are quoted, such as:* *

- Physics of Fluids, Fluid Mechanics, Hydrodynamics, Hydraulics, Water Wave Mechanics, Aerodynamics,

- Physical Oceanography, Marine Hydrodynamics, Coastal Oceanography, Theoretical Oceanography,

- Physics of the Atmosphere, Physical Meteorology, Meteorological Fluid Dynamics, Theoretical Meteorology,

- Atmospheric Thermodynamics, Atmospheric Diffusion, Boundary Layer Meteorology, Coastal Meteorology,

- Atmospheric-Ocean Interaction, Atmosphere-Ocean Dynamics, Geophysical Fluid Dynamics, Astrophysics,

with meanings which are slightly different from time to time, from
author to author. This is the consequence of the fact that the various parts
of Fluid Dynamics are so strictly related to each other that a clear
subdivision of the topic is not possible yet.

This situation suggested to the author of this web site the derivation of the Principles of Fluid Dynamics at the basis of all these disciplines in a systematic way, with a particular attention to the scientific rigor and clarity of the exposition. A simplified version of the first notions, entitled Elements of Fluid Dynamics is published as well, to provide the largest number of basic results with the minimal effort, without sacrificing rigor and clarity.

Any comment about my work will be highly appreciated. I am at the disposal of every reader for clarifications and discussions about the published material.

Franco Mattioli, e-mail: franco.mattioli@unibo.it

Bologna, 1 jan 2010.

An exhortation. The chapters of my book may be downloaded free and printed, but cannot be modified or edited. In particular, do not mix material taken from my book with other material taken elewhere. The use of complete chapters is not only permitted, but also recommended. Thank you.

Publication Plan

The book is organized in a series of Chapters that should be read sequentially and a series of Appendices that should be consulted when necessary.

The appendices are transversal to the main text, so that you can find in them references to material dealt with in the course of the whole book. Furthermore, appendices do not aim at being complete and detailed. Many notions are only quoted to inform the reader about the scientific background he needs to be able to understand the main text. However, the appendices do not limit themselves to quote classical results, but also include considerations and problems that are particularly oriented to the understanding of the fluid dynamics, but cannot be easily found elsewhere.

With the exception of the first introductory part, the chapters aim at being as complete as possible, proving all the derived laws with a few exceptions relative to particularly long demonstrations of little physical relevance. At the end of some chapters and appendices we provide the reader with information about the time and the authors of the main discoveries dealt with in the text, along with recent references to which the reader is addressed to deepen the various topics. They make no pretence to be accurate or complete. They have been introduced especially to stress that this text follows a logical, and not a historical order.

The date in front to the chapters refers to the first publication of the material in the context of the relative book. Later this material can have been corrected, modified, expanded, split in two or moved in another place. In the future other chapters can be inserted between two already published chapters, even if the publication is normally sequential. For this reason sometimes the sequential number of pages, chapters and parts has been interrupted, in order tho make easier future arrangements of the material. The same can be repeated for the appendices.

If you want to quote my work, you should refer to the single chapters, providing the interval of time in which the chapter has been elaborated. For example: Mattioli, F. Principles of fluid dynamics (www.fluiddynamics.it) Vol. I The states of the matter (01 jan 2010 - 01 jan 2023). Here, the former date is the date on the left of the title of the quoted chapter and the latter date is the date in which you consulted the chapter.

For the same reason we publish a News page, in order to help the reader in the identification of the new material from the momentt of his last access. The Index will be updated as frequently as possible, but some incoherence might exist for a short lapse of time, especially regarding the page numbering.

The role of Fluid Dynamics in Mathematics and Physics

Fluid Dynamics is one of the older and difficult branches of Physics. At present, it seems a still underestimated sector of physics for several reasons as stated, for example, at the beginning of the book* Fluid
Dynamics for Physicists. * Cambridge University Press, 1995, by Faber,
T.E. : "Nowadays [fluid dynamics] is partially obscured
from view by branches of more recent origin, such as relativity, atomic
physics and quantum mechanics, and students of physics pay rather little
attention to it. This is a pity, for several reasons. Firstly,
because of the engineering applications of the subject, which are many and
various: the design of airplanes and boats and automobiles, and indeed of any
structure intended to move through fluid or propel fluid or simply to
withstand the forces exerted by fluid, depends in a critical way upon the
principles of the subject. Secondly, because fluid dynamics has important
applications in other branches of physics and indeed in other realms of
science, including astronomy, meteorology, oceanography, zoology and
physiology: dripping tabs, solitary waves on canals, vortices in liquid
helium, seismic oscillations of the Sun, the great Red Spot on Jupiter, small
organisms that swim, the circulation of the blood - these are just a few of
the very varied topics involving fluid dynamics which have been occupying
research scientists and mathematicians of international reputation over the
past few decades. Thirdly, because most other subjects in the physics
curriculum are almost exclusively concerned with *
linear * processes, whereas fluid dynamics leads one into the
* non-linear * domain. And lastly, because
there are so many curious and beautiful natural phenomena, visible every day
in the world about us, which a physicist with no knowledge of fluid mechanics
in unable to appreciate to the full."

The necessity to understand the behavior of a fluid forced several authors of the past to discover new mathematical tools suitable for the scope. Thus Fluid Mechanics led to the introduction of the Partial Differential Equations, to the Singular Perturbation Theory, to the Method of Multiple Scale Analysis, to the Inverse Scattering Transform, and of other advanced mathematical techniques. Later, these techniques spread over other fields of Mathematics and Physics.

For a long time the solutions proposed for the flows of a fluid seemed very far from reality. Indeed, the interpretation of many of the most common phenomena of our everyday life involving fluids is still very poor. While Physics evolved in fields very far from the possibility of a direct observation, but rich of important applications. The field of Fluid Dynamics was essentially abandoned by physicists because of its difficulties. The development of the discipline relied on the important civil and military applications of engineering. The advent of computers revived interest in this topic, not only in its numerical aspects, but also in its theoretical and experimental components.

Today, Fluid Dynamics appears as one of the most advanced fields of Physics, characterized by a boundless number of problems linked to each other in various ways. The role of a problem inside this complex network is difficult to establish, but essential for its full understanding. Unfortunately, the complication of this matter leads very often to the presentation of the discipline as a set of uncorrelated phenomena, very far from each other, thus generating a misunderstanding not only of the discipline, but also of its true importance.

In particular, the meteo-oceanographic system appears to be as the most complicated system present in physics as stated, for example, at the beginning of their book* Introductory Physics of the
Atmosphere and Ocean. * D. Reidel Publishing Company, 1986, by Hasse, L.
& Dobson, F. : "Atmosphere and ocean are probably the
most complex system which is treated in physics, since the scales of
motions involved span such a wide range: 10,000 km for long planetary waves,
through a few millimeters for the scale of decaying turbulent eddies.".
Within this system many phenomena superimpose and interact in a complicated,
bidirectional way, so that their satisfactory understanding is still very
far. At present we know only some of the main processes, which can partially
justify a certain number of important phenomena, but we are still unable to
interpret a large number of many other phenomena, and especially the
unpredictability of the so-called extreme events
or the possible transitions to new climatic equilibria,
threatening the safety of the life of humankind on our planet.

This classification is not always respected, and many other disciplines are quoted, such as:

- Physics of Fluids, Fluid Mechanics, Hydrodynamics, Hydraulics, Water Wave Mechanics, Aerodynamics,

- Physical Oceanography, Marine Hydrodynamics, Coastal Oceanography, Theoretical Oceanography,

- Physics of the Atmosphere, Physical Meteorology, Meteorological Fluid Dynamics, Theoretical Meteorology,

- Atmospheric Thermodynamics, Atmospheric Diffusion, Boundary Layer Meteorology, Coastal Meteorology,

- Atmospheric-Ocean Interaction, Atmosphere-Ocean Dynamics, Geophysical Fluid Dynamics, Astrophysics,

This situation suggested to the author of this web site the derivation of the Principles of Fluid Dynamics at the basis of all these disciplines in a systematic way, with a particular attention to the scientific rigor and clarity of the exposition. A simplified version of the first notions, entitled Elements of Fluid Dynamics is published as well, to provide the largest number of basic results with the minimal effort, without sacrificing rigor and clarity.

Any comment about my work will be highly appreciated. I am at the disposal of every reader for clarifications and discussions about the published material.

Franco Mattioli, e-mail: franco.mattioli@unibo.it

Bologna, 1 jan 2010.

An exhortation. The chapters of my book may be downloaded free and printed, but cannot be modified or edited. In particular, do not mix material taken from my book with other material taken elewhere. The use of complete chapters is not only permitted, but also recommended. Thank you.

Publication Plan

The book is organized in a series of Chapters that should be read sequentially and a series of Appendices that should be consulted when necessary.

The appendices are transversal to the main text, so that you can find in them references to material dealt with in the course of the whole book. Furthermore, appendices do not aim at being complete and detailed. Many notions are only quoted to inform the reader about the scientific background he needs to be able to understand the main text. However, the appendices do not limit themselves to quote classical results, but also include considerations and problems that are particularly oriented to the understanding of the fluid dynamics, but cannot be easily found elsewhere.

With the exception of the first introductory part, the chapters aim at being as complete as possible, proving all the derived laws with a few exceptions relative to particularly long demonstrations of little physical relevance. At the end of some chapters and appendices we provide the reader with information about the time and the authors of the main discoveries dealt with in the text, along with recent references to which the reader is addressed to deepen the various topics. They make no pretence to be accurate or complete. They have been introduced especially to stress that this text follows a logical, and not a historical order.

The date in front to the chapters refers to the first publication of the material in the context of the relative book. Later this material can have been corrected, modified, expanded, split in two or moved in another place. In the future other chapters can be inserted between two already published chapters, even if the publication is normally sequential. For this reason sometimes the sequential number of pages, chapters and parts has been interrupted, in order tho make easier future arrangements of the material. The same can be repeated for the appendices.

If you want to quote my work, you should refer to the single chapters, providing the interval of time in which the chapter has been elaborated. For example: Mattioli, F. Principles of fluid dynamics (www.fluiddynamics.it) Vol. I The states of the matter (01 jan 2010 - 01 jan 2023). Here, the former date is the date on the left of the title of the quoted chapter and the latter date is the date in which you consulted the chapter.

For the same reason we publish a News page, in order to help the reader in the identification of the new material from the momentt of his last access. The Index will be updated as frequently as possible, but some incoherence might exist for a short lapse of time, especially regarding the page numbering.

The role of Fluid Dynamics in Mathematics and Physics

Fluid Dynamics is one of the older and difficult branches of Physics. At present, it seems a still underestimated sector of physics for several reasons as stated, for example, at the beginning of the book

The necessity to understand the behavior of a fluid forced several authors of the past to discover new mathematical tools suitable for the scope. Thus Fluid Mechanics led to the introduction of the Partial Differential Equations, to the Singular Perturbation Theory, to the Method of Multiple Scale Analysis, to the Inverse Scattering Transform, and of other advanced mathematical techniques. Later, these techniques spread over other fields of Mathematics and Physics.

For a long time the solutions proposed for the flows of a fluid seemed very far from reality. Indeed, the interpretation of many of the most common phenomena of our everyday life involving fluids is still very poor. While Physics evolved in fields very far from the possibility of a direct observation, but rich of important applications. The field of Fluid Dynamics was essentially abandoned by physicists because of its difficulties. The development of the discipline relied on the important civil and military applications of engineering. The advent of computers revived interest in this topic, not only in its numerical aspects, but also in its theoretical and experimental components.

Today, Fluid Dynamics appears as one of the most advanced fields of Physics, characterized by a boundless number of problems linked to each other in various ways. The role of a problem inside this complex network is difficult to establish, but essential for its full understanding. Unfortunately, the complication of this matter leads very often to the presentation of the discipline as a set of uncorrelated phenomena, very far from each other, thus generating a misunderstanding not only of the discipline, but also of its true importance.

In particular, the meteo-oceanographic system appears to be as the most complicated system present in physics as stated, for example, at the beginning of their book