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General Editor:

S. GAUKROGER, University of Sydney

Editorial Advisory Board:

STEVEN FRENCH, University of LeedsDAVID PAPINEAU, King’s College London

NICHOLAS RASMUSSEN, University of New South WalesJOHN SCHUSTER, University of New South Wales

RICHARD YEO, Griffith University

VOLUME 22

STUDIES IN HISTORY AND PHILOSOPHY OF SCIENCE

RACHEL ANKENY, University of Adelaide

THE GENESIS

1640–1780OF FLUID MECHANICS

JULIÁN SIMÓN CALERO

By

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or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise,

Original title:

La génesis de la Mecánica de los Fluidos (1640–1780) UNED, Madrid, 1996

Translation: Veronica H. A. Watson

© 2008 Springer Science+Business Media B.V.

Library of Congress Control Number: 2008922901

Acknowledgments

In my acknowledgements to the English edition, I would like to mention three persons who helped me with the original Spanish edition: Javier Simón Calero, my brother, for his comments; Eloy Rada García, for his efforts in publishing the work; and very specially to Manuel Sellés García, who was a decisive influence in the gestation of the entire work, in particular, he supervised with an eagle eye the more technical aspects of the book.

In this new edition, I would like to include three new persons. The first is Veronica Watson, generous author of the translation of the Spanish text; she reworked all the quotations, starting from their original source languages. I ac-knowledge that my text was difficult, sometimes baroque and sometimes highly condensed; the result improves the original text considerably, in many occasions sweetening it and rendering it much clearer. Veronica took advantage of the different nuances of tone between the two languages, obliging me to rethink numerous paragraphs. The second person is Stephen Gaukroger, who was the first person to believe that the work was worth translating into English; he advised me on publishers, and in the end it was he himself who has edited the book. Stephen completed Veronica’s work, eliminating academic pedantry from the text, giving it an additional vivacity. The third person is Larrie Ferreiro, who, from the moment he knew it in its original version, has been its best and most assiduous reader to judge from his public comments, and to whom I am very grateful for his favourable views. For this new edition, he analysed the text in great detail, suggested improvements and made me complement and clear up obscure points. These three persons made me work considerably more, each in his own field. Although it is unnecessary, I cannot resist saying that I did it with great enjoyment.

My profound gratitude to all, although I think that truly the ones who should be most grateful are the readers, as once the book arrives in their hands, my task concludes, and they become the proprietors of the text.

v

Contents

ACKNOWLEDGMENTS ................................................................................................v

ABBREVIATIONS..........................................................................................................xi

NOTE ............................................................................................................................. xiii

INTRODUCTION.............................................................................................................1 PROLEGOMENON .............................................................................................................1 THE GENESIS OF FLUID DYNAMICS: A SUMMARY.............................................................5

PART I

THE PROBLEM OF RESISTANCE............................................................................45

CHAPTER 1: THE FORERUNNERS OF IMPACT THEORY: HUYGENS AND MARIOTTE......................................................................................55

MEASUREMENT OF RESISTANCE....................................................................................55 HUYGENS AND HIS EXPERIMENTS .................................................................................57 MARIOTTE AND THE THEORY OF JETS............................................................................64 EXPERIMENT OF A JET AGAINST A PLATE.......................................................................71

CHAPTER 2: IMPACT THEORY: FORMULATION AND FORMALISATION ..............................................................................................73

THE CONTRIBUTION OF NEWTON ..................................................................................73 THE CONCEPT OF FLUIDS IN THE PRINCIPIA ...................................................................76 MOTION OF PENDULUMS ...............................................................................................81 RESISTANCE IN AERIFORM FLUIDS.................................................................................89 AQUIFORM FLUIDS (I)....................................................................................................95 AQUIFORM FLUIDS (II) ..................................................................................................99 EXPERIMENTAL RESULTS ............................................................................................107 FINAL COMMENTS ON THE PRINCIPIA ..........................................................................111 THE FORMALISATION OF IMPACT THEORY...................................................................113 THE APPROACH OF JAKOB BERNOULLI........................................................................114 THE SOLID OF MINIMUM RESISTANCE..........................................................................122 THE SITUATION IN 1714...............................................................................................123

vii

CONTENTS

CHAPTER 3: THE EVOLUTION OF THE PROBLEM OF RESISTANCE ........................................................................................................125

PROBLEMS OF IMPACT THEORY ...................................................................................125 JOHANN BERNOULLI AND THE ‘COMMUNICATION OF MOTION’...................................128 THEORIES AND EXPERIMENTS OF DANIEL BERNOULLI................................................132 JET AGAINST A PLATE: DANIEL BERNOULLI’S CLARIFICATIONS..................................137 THE NAVAL WORK OF PIERRE BOUGUER.....................................................................143 EULER’S SCIENTIA NAVALIS ..........................................................................................145 ROBINS’ NEW PRINCIPLES OF GUNNERY ......................................................................149 EULER’S STREAMLINE MODEL.....................................................................................158 D’ALEMBERT’S PARADOX...........................................................................................167 OTHER WORKS OF EULER ............................................................................................175

CHAPTER 4: EXPERIMENTS ON RESISTANCE..............................................185

THE EXPERIMENTERS ..................................................................................................185 THE EXPERIMENTS OF BORDA .....................................................................................187 SQUARE PLATES ..........................................................................................................190 PRISMS ........................................................................................................................192 PYRAMIDS ...................................................................................................................192 SPHERES AND HEMISPHERES........................................................................................193 TOWING OF FLOATING CUBES......................................................................................193

CHAPMAN’S EXPERIMENTS..........................................................................................197 THE EXPERIMENTS OF BOSSUT ....................................................................................202 PROPORTIONALITY WITH THE SQUARE OF THE VELOCITY ...........................................207 PROPORTIONALITY WITH THE SURFACE.......................................................................208

CHAPTER 5: FLUID-DRIVEN MACHINES AND NAVAL THEORIES.........215

FLUID MACHINES.........................................................................................................215 THE FUNCTIONING PRINCIPLES OF THE MACHINES ......................................................217 PARENT’S STUDIES ......................................................................................................221 PITOT’S WORKS ...........................................................................................................223 THE STUDIES OF BOSSUT AND OTHERS........................................................................226 SMEATON’S EXPERIMENTS ..........................................................................................228 WINDMILLS IN EULER .................................................................................................237 NAVAL APPLICATIONS.................................................................................................242 THE FORCES IN THE HULL ............................................................................................246 FORCE IN SAILS............................................................................................................254

viii

THE THEORY OF JORGE JUAN ......................................................................................182

ROTATING ARM IN WATER...........................................................................................194

PROPORTIONALITY WITH THE SQUARE OF THE SINE ....................................................209

CONTENTS

PART II

THE PROBLEM OF DISCHARGE...........................................................................267

CHAPTER 6: DISCHARGE FROM VESSELS AND TANKS............................271

TORRICELLI’S LAW .....................................................................................................271 THE WORK OF THE PARIS ACADEMY ...........................................................................273 DOMINICO GUGLIELMINI.............................................................................................275 MARIOTTE’S EXPERIMENTS.........................................................................................278 DISCHARGE IN NEWTON..............................................................................................282 GIOVANNI POLENI .......................................................................................................283

CHAPTER 7: THE HYDRODYNAMICA AND THE HYDRAULICA .................293

DANIEL AND JOHANN BERNOULLI ..............................................................................293 THE BASIC PRINCIPLES OF THE HYDRODYNAMICA ........................................................298 MOTION THROUGH TUBES IN THE HYDRODYNAMICA....................................................304 HYDRAULICO-STATICS ................................................................................................314 THE DISCHARGE OF ELASTIC FLUIDS ...........................................................................320 JOHANN BERNOULLI’S HYDRAULICA ...........................................................................326 D’ALEMBERT’S ACCOUNT OF MOTION IN TUBES.........................................................338 BORDA’S WORKS.........................................................................................................346 BY WAY OF SUMMARY…. ...........................................................................................352

CHAPTER 8:

THE GRAND THEORISATION.........................................................................................355 THE SHAPE OF THE EARTH ...........................................................................................358 CLAIRAUT’S EQUILIBRIUM CONDITIONS......................................................................360 GENERALISATIONS ......................................................................................................368 D’ALEMBERT ..............................................................................................................371 PRINCIPLES OF DYNAMICS AND HYDROSTATICS..........................................................374 BODIES IN FLOWING CURRENTS...................................................................................378 SOLUTION USING LATERAL FORCES.............................................................................381 SOLUTION BY CONSTANCY OF VOLUME.......................................................................386

CONCLUSION: D’ALEMBERT’S CONTRIBUTIONS .........................................................399

CHAPTER 9:

LEONHARD EULER ......................................................................................................401 PRINCIPIA MOTUS FLUIDORUM ....................................................................................402 GENERAL PRINCIPLES OF THE STATE OF EQUILIBRIUM OF THE FLUIDS ........................419 GENERAL PRINCIPLES OF THE MOTION OF FLUIDS .......................................................426

ix

THEORETICAL CONSTRUCTIONS (I): CLAIRAUT AND D’ALEMBERT ...................................................................................................355

BODIES MOVING IN A STATIONARY FLUID ...................................................................395

THEORETICAL CONSTRUCTIONS (II): EULER....................401

CONTENTS

SEQUEL TO THE RESEARCHES ON THE MOTIONS OF FLUIDS .........................................435 EULER’S CONTRIBUTION .............................................................................................444 LAGRANGE’S PAPER ON FLUIDS...................................................................................445

CHAPTER 10: APPLICATION OF FLUID MECHANICS TO PUMPS AND TURBINES ..........................................................................................................453

THE HYDRAULIC PUMP ................................................................................................454 PITOT’S THEORY OF PUMPS..........................................................................................456 PUMPS IN EULER’S WORKS..........................................................................................462 BORDA’S STUDIES .......................................................................................................466 DANIEL BERNOULLI ON JET PROPULSION ....................................................................469 SEGNER’S TURBINE .....................................................................................................472 EULER’S ANALYSES.....................................................................................................475 PRESSURE IN THE TUBE................................................................................................477

SEGNER-EULER TURBINE ............................................................................................481 PITOT’S TUBE ..............................................................................................................490 RECAPITULATION: HYDRAULIC MACHINES..................................................................492

APPENDIX....................................................................................................................493

UNITS OF MEASURE .....................................................................................................493

BIBLIOGRAPHY .........................................................................................................497

I PRIMARY SOURCES ..............................................................................................497 II. PERIODICAL PUBLICATIONS, COLLECTIONS AND COMPLETED WORKS..................505 III. SECONDARY SOURCES .........................................................................................506

DRAMATIS PERSONÆ..............................................................................................511

x

TURNING TORQUE .......................................................................................................480 TANK HEIGHT..............................................................................................................481

INDEX............................................................................................................................513

Acta Erud. Acta Eruditorum of Leipzig

Mém. Acad. Berlin Histoire de l’Académie des Sciences et Belles Lettres of Berlin

Nouv. Mém. Acad. Berlin Nouvelles Mémoires Histoire de l’Académie des Sciences et Belles Lettres of Berlin

Mém. Acad. Paris Mémoires de l’Académie Royale des Sciences de Paris Commentarii academiæ scientarum

Novi comm. acad petrop. Novi commentarii academiæ scien-

Phil. Trans.

Royal Society of London

Jour. Sav. Journal de Sçavans

xi

Abbreviations

Comm. acad. petrop. petropolitanæ of St. Petertsburg

tarum petropolitanæ of St. Petertsburg

Philosophical Transations of the

xiii

All measurements and sizes are expressed in the original units, followed by their conversion to the International System. The appendix lists the conversion factors used. Mathematical formulas and notation have been transcribed to present-day symbols, adding the constants required to render the formulas dimensionally consistent. This particularly affects the acceleration of gravity expressed by its customary symbol g, and consideration of density by its absolute value, that is to say mass/volume.

Note

1

Prolegomenon

In what follows we have set out to analyse the genesis of fluid mechanics as a modern scientific discipline. Two prior and interrelated questions must first be raised, however: what do we understand as the basic core of fluid mechanics, and when was this core developed?

As seen today, fluid mechanics is a science of great complexity and diver-sity. Yet, the field derived from a common source, and evolved into a succession of disciplines, each one with an increasing degree of specialization. This hier-archy is headed by fluid dynamics and fluid statics, understood as applied to liquids and gases, which in turn gave rise to subsonic, transonic and supersonic disciplines. Although they all sprang from the same common theoretical basis, each discipline has specific problems determined by the nature of the fluid, the dominating phenomena, the motion undergone and the boundary conditions. Looking back, we find that the basic hypotheses that mathematically regulate fluid mechanics, the structure and formulation of its fundamental equations, and

teenth and first half of the eighteenth centuries, to the extent that the process can be considered in many respects complete by the end of the 1750s. Key concepts in fluid mechanics—such as turbulence, boundary layer, discontinuity surfaces, viscosity and thermodynamic processes—were introduced in the course of the nineteenth and twentieth centuries.

We choose the decade of the 1750s as the jumping-off point for fluid mecha-nics, because a specific body of theory was built up by then, which (together with some experimental studies) provided the groundwork—despite a number of

reached at the end of a century-long process in which the basic concepts ges-tated, were born, and evolved progressively with different theories and assump-tions. It was only with the appearance of Isaac Newton’s Philosophiæ naturalis principia mathematica in 1687, that the scientific nature of the discipline made its first appearance, for this work analyses for the first time the dynamics of

Introduction

the problems of its applications were all forged in the second half of the seven-

residual problems—for the modern scientific theory of fluids. This point was

2 THE GENESIS OF FLUID MECHANICS, 1640–1780

fluids, basing itself on laws of nature of more general characteristics. Specific-ally, Newton introduces two basic premises: one concerning the definition of the constituents of the fluid, the other dealing with its behavior with respect to the general laws of dynamics as set out in the Principia. While it is true that before the appearance of this work, a large number of studies, some very worthy, existed on this subject, none of them provided the overall theoretical body of understanding supplied by the Principia. Nevertheless, despite a number of attempts over a period of 70 years, it was not until 1755 that Leonhard Euler was able to offer a definitive treatment.

While this process opened with Newton and closed with Euler, there were considerable differences in their fundamental assumptions about the nature of fluids. For Newton, a fluid like air was an aggregate of particles that respond individually to the laws of mechanics, in such a way that the force generated by the impact of a current against an object consisted of the sum of the effects of each individual impact. For Euler, by contrast, a fluid was a continuum, ideally separated into elemental domains capable of supporting forces and internal pressures, whose space–time evolution is regulated by the laws of dynamics, expressed by a set of differential equations. These two radically different concepts were established in 1687 and 1755, respectively, and they define the termini of the genesis of fluid mechanics. Before 1685, the concepts of fluid mechanics were still based on a simplistic (to our eyes) view of the physical universe; but after 1755 those concepts were replaced by the construction we use today.

The treatment of the development of fluid mechanics will deal with the dynamic aspects of fluids, liquids or gases subjected to forces, and limited by boundary conditions, as much as in their applications. Although static analyses are not dealt with as such, some aspects of fluid statics are looked at where this is needed to provide a more comprehensive picture. For example, in his 1755 work, Euler united static and dynamic accounts in the same theory, so in this case we deal explicitly to the former. On the other hand, we understand dyna-mics as the attempt to explain and reduce the motion of fluids in terms of the forces applied and the boundary conditions, and so do not deal with hydraulics, understood as the practical use of energy intrinsic to the movement of water.

Although Newton is treated as the founder of the discipline, we have not neglected the contribution of several pre-Newtonian works, tracing the develop-ment of the discipline from Evangelista Torricelli’s work in 1644, through Christiaan Huygens and Edmé Mariotte. Among the post-Newtonians, the Bernoulli family (Jakob, Johann and Daniel), Benjamin Robins, Alexis Claude Clairaut, Jean Le Rond d’Alembert and Leonhard Euler play crucial roles, espe-cially Euler’s 1755 work which brings the development to completion. In the 20 years after this, we witness a burst of activity in applied and experimental fields,

INTRODUCTION 3 in the work of Jean Charles Borda, Charles Bossut, Pierre Bouguer and Jorge Juan y Santacilia, so that the developments we are concerned with effectively come to a head only in the 1770s.

Some clarifying points are worth developing before we begin. In particular, we need to say something about fluids from the point of view of theory, experi-ments and applications; we need to examine the distinction between hydraulics and hydrodynamics; and we need to raise the question of the relation between mechanics in general and fluid mechanics.

Although it is common to think of science as a theoretical–experimental enterprise, problems arise when the question of applied science or of the applica-tion of science is introduced. For our part, we see a connection between both aspects, which we have called ‘applications’. We understand these to be the attempt to use available knowledge, be it theoretical or experimental, for model-ling, explaining and reducing scientifically the behavior of machines, be they real or imaginary devices, but without these studies implying or intending their subsequent construction, which is a matter for technology.1 In a strict definition of science, one might be tempted to exclude these applications. But the advan-tages obtained in the theoretical analysis of any machine are a source of knowl-edge, obliging us to adjust the theories to a specific reality, something which is neither easy nor immediate. This is not the same as experiment, which has the great advantage of granting us freedom for creating an ideal space in order to render a specific effect. We treat the applications as forming part the science itself, on a different plane from experiments, but not constituting a separate group. Applications follow on from theory and experiments, and are continued by the actual construction of machines. This comes under the heading of tech-nology, which is another subject, implying contributions of a different nature. Applications have another important aspect: they are the expression of a social need and through them society acts on science. They form another, additional nexus connecting the conceptual and social aspects of science.

The relation between hydrodynamics and hydraulics comes under the ambit of interrelations between practical and theoretical worlds, if the former is under-stood as coming under the aegis of the theoretical world and the latter under the practical. We must remember that hydraulics flourished from antiquity as an 1 The problem of the technology is the production of technical entities, which is not exactly the same as engineering, although both partly coincide. The first step of the process starts by under-standing the ideal object to be materialized. This is done according to the scientific knowledge available, which is sometimes very scanty or difficult to apply. In the second step this knowledge, together with materials provided by nature, allows these objects to be constructed, in a wide sense of the word, giving rise to the appearance of the technological sciences. In spite of the differences between science and technology, there is something both share: the scientific method. Therefore, the applications have to be treated scientifically.


artisan activity, and remained so in the seventeenth and eighteenth centuries,2 its aim being to optimise the use, handling and distribution of water in watermills, waterwheels, canals, etc. It is natural that hydraulics posed problems which the sages of the time tried to study. However, hydraulics as then understood was not, nor did it become, a science in the modern definition. It remained a dignified artisan activity, which is what it had been since ancient times, with perhaps a few technological improvements. However, some confusion exists as to the terminology of the time, and until the term hydrodynamics appeared, the term hydraulics was often used in its place. D’Alembert tried to clarify the scope of both terms in the Encyclopédie Méthodique,3 but the word hydraulics continued to be used frequently in engineering circles. Nevertheless, while in its debt in some respects, hydrodynamics does not proceed from hydraulics.

As regards mechanics, its evolution during those years deserves comment. The main aim of mechanics was the establishment of its general laws, allowing its problems to be reduced to mathematics. Several attempts were made to do this. One was Newton’s, with the establishment of the laws or axioms of motion, which introduced forces and the concept of mass. Newtonian mechanics assumed that these masses were points, or able to be reduced to points, and that the forces acted between these mass points. Previously, Huygens had considered the underlying principle to be the conservation of what he called live force (vis viva). This was the sum of the products of the masses multiplied by the square of the velocity of all the particles of the system. Conservation was limited to certain conditions, such as elastic collisions and processes under the action of gravity. In contrast to Newton’s laws, which deal with the interaction between individual particles, live force was extended to the entire fluid mass; which, to-gether with the advance of infinitesimal calculus, made the live force theory very suitable for handling large groups of particles, the credit for this development going to Daniel Bernoulli. D’Alembert offered a different proposal, eliminating 2 The most important work on this subject Architecture Hydraulique de Bernard Forest de Belidor. The first edition appeared in 1737 and was published continuously until the nineteenth century. 3 In the volume III of the Encyclopédie Méthodique (1785), under the term Hydraulique it is said: ‘The part of Mechanics which contemplates the motion of fluids and which shows how water is channelled and how to raise them, as much for making the waters form a jet as for other uses. … The hydaulique deals not only with the water tubes and raising waters, but also the machines required to this end, but even more with the general laws of motion of fluid bodies. However, for quite a few years, mathematicians gave the name hydrodynamique to the general science of motion of fluids, and reserved the name hydraulique for the disciplines which, in particular, considered the motions of water: that is to say the art of channelling water, of raising it and handing it for the different requirements of daily life’. In the Hydrostatique talks of ‘the part of mechanics which considers the equilibrium of fluid bodies, together with the bodies which are immersed in them. … The hydrostatique is often confused with the hydraulique, as the subject matter is similar, and several authors hardly separate them at all’.

INTRODUCTION 5 the forces, and introducing conservation of the momentum in impact, along with a practical rule similar to virtual velocities. Subsequently, Maupertuis established the principle of minimum action. Whatever the principle was, the aim was the same: to obtain the laws of motion.

In this connection, a little more needs to be said on the relation between fluid mechanics and mechanics understood in a general sense. If, as we have noted, hydraulics followed its own path, fluid mechanics emerged as part of mechanics.4 This is palpable in Newton, who dedicated Book II of his Principia to fluids, and the Book I to solids, but in both cases with the same title. This way of proceeding is also clear in Mariotte, who proposes some models of what we have called mechanics of jets. The mathematicians of the eighteenth century contributed greatly to mechanics, and they treated fluids as one of three types of bodies: solids, deformable bodies and fluids. The difficulty in treating fluids mathematically was enormous, and the help of differential analysis was absolutely decisive. Moreover, the process of mathematization in Euler and d’Alembert requires the definition of fluids as a continuum, in spite of the fact that they imagined them to be physically constituted by an aggregate of particles. It is interesting to note that in the second half of the 1700s, fluid mechanics, or rather the rational mechanics of fluids, was at the vanguard of theoretical mechanics.

The genesis of fluid dynamics: a summary

Let us begin with a brief summary of the entire process of the genesis and evolu-tion of fluid mechanics in order to aid comprehension, and to place the rest of the work in context. It is a core thesis that this entire evolution took place along two main lines of activity: one dedicated to the effects that fluid current exer-cises upon a body immersed in it, and the other to dealing with how fluids discharge themselves through tubes or reservoirs. We have called the first ‘the problem of resistance’ and the second the ‘problem of discharge’. Almost all the authors of the time treated these subjects separately, and set them in one or other 4 In the Preface of the Principia, Newton says: ‘In this sense rational mechanics will be the sci-ence, expressed in exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required for any motions whatever. The ancients studied this part of mechanics in terms of the five powers that relate to the manual arts. … But since we are concerned with natural philosophy rather than manual arts, and are writing about natural rather than manual power, we concentrate on aspects of gravity, levity, elastic forces, resistance of fluids, and forces of this sort, whether attractive or impulsive. And therefore our present work sets forth mathematical principles of natural philosophy. For the basic problem of philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces’. [p. 382]


of these two contexts, sometimes even in the same book. Both enjoyed consider-able independence, and a set of theories existed in each one, with its experiments and of course with its applications. Their methodology and evolution are not strictly comparable, as there are considerable differences between them, which makes them all the more interesting. Both respond to two sets of different pre-occupations: in the case of resistance, to navigation problems and to machines driven by fluids such as mills and waterwheels; in the case of discharge, to water distributions and to jet reaction machines. The first was more practical and of more immediate interest; the second more theoretical, but conceptually superior, as the subsequent history bears out.

In the main body of this work, we have separated both parts, dedicating the first part to the problem of resistance, and the second to that of discharge, and in each of these we have dealt with the theoretical, experimental and application aspects separately. This way of the presenting the facts emphasizes the coherence of each the two lines, but it breaks the general chronology of the work, so in this Introduction we shall describe the evolution of fluid mechanics as a whole, follow-ing a single chronological line, while making reference to the resistance or the discharge in each case, according to its type.

We establish the starting point of the development, and more specifically of the line denominated ‘the discharge problem’ in Torricelli, who, in 1644, announced the law known today by his name, and which says that the efflux velocity of a liquid contained in a recepticle through an orifice is that which a heavy body would acquire when falling from a height equal to the depth at which the orifice is found.5 The law appeared in the work De motu gravium (On the motion of heavy bodies), which deals with the motion of projectiles, and was a result of his ballistic studies. He tells us that the clue that led him to this law is the commonly observed fact that when a reservoir discharges through a vertical tube acting as a spout, the water reaches a height very close to the level of the reservoir (Fig. I-1a). He observed that the jet ended up converted into droplets in its highest part, and he attributed this to its losing some height. It is interesting to note the empirical origin of his law, which is not so obvious, as the efflux of water is a different phenomenon from the fall of the heavy body that serves him as a reference. Should the orifice be in the lateral wall, he indicates that the trajectory followed would be a parabola, similar to the one a projectile would describe (Fig. I-1b). What Torricelli does, is to assimilate the water to a body in free fall. That is to say, he converts the liquids into an aggregate of solid bodies.

5 Mathematically, it is expressed as ghve 2= . That is, the efflux velocity is proportional to the square root of the depth.

INTRODUCTION 7 The law, apart from being the first one to refer to fluids in motion,6 is of consider-able interest for various reasons. First, it identifies fluids with solid bodies, in spite of their having such a different appearance and behaviour. Second, he pro-poses a mathematical relation for the first time. Third, he opens up an experi-mental field that would be a focal point for the next 50 years. In addition, it was the starting point of the discharge problem.

Fig. I-1. Torricelli springs

As regards the paternity of this law, Torricelli, who was in Italy when he

wrote the aforementioned book, says that it was already previously known by Benedetto Castelli, his master. However, later authors, such as Daniel Bernoulli or Giovanni Poleni, say that some 3 years before (i.e., around 1641), Castelli had supposed that the outlet velocity was proportional to the depth, instead of its square root, as Torricelli had stated. It is difficult to know which account is correct; the fact is that the law was attributed to Torricelli by his contemporaries.

The apparatus used for the experimental verification of this law was based on collecting water flowing out in the discharge during a certain measurable time interval, while the reservoir was kept full. Once the surface of the orifice was known, it was easy to calculate the outlet velocity of the water. A variant consisted in not filling the vessel during the process, and measuring the time it took to discharge completely, as by calculation this time was found to be double that in which an equal quantity of water would be discharged if the level of the vessel was maintained. Whichever procedure was used, the reality was that this apparently clear and simple experiment provided disparate results, a fact that caused enough headaches to stimulate an in-depth study of the phenomenon.

6 The hydrostatics laws are older. The first one is due to Archimedes in the third century BC, and others were found by Simon Stevin (1548–1620) and Blaise Pascal, who was a contemporary of Torricelli.

Huygens and Mariotte in the newly founded Academy of Sciences of Paris,


Huygens performed experiments in Paris in 1668, and his results confirmed Torricelli’s law as opposed to the supposition of Castelli.7 However, in the fol-lowing year he went back on what he said, and questioned the previous results. What happened was that in the experiment less water was collected than pre-dicted, and it would appear that the same thing happened in the various attempts made by him and other experimenters. We now know that the discharge process is not purely kinematic, as Torricelli’s law supposes, but dynamic, and all the water held in the recepticle intervenes in the phenomena; moreover, details such as the form of the vessel, the type of spout and the relation between the surface of the orifice and the surface of the reservoir play an important part. This was then unknown, and was only discovered with great difficulty. According to the sentiment of the times, these anomalies could be interpreted in two ways: either this velocity was not proportional to the square root of the depth; or the height, when considering the fall, was not that of the reservoir itself but a fraction of this. The experiments pointed towards this last alternative, which gave rise to the initiation of a process for revising the law, admitting the mathematical form of

Huygens’ work on motion in fluids was not just limited to discharge: he also devoted considerable effort to the effect undergone by bodies moving inside a fluid. Huygens, like Torricelli, was also a student of ballistics, and one of his preoccupations was to introduce the effect of the resistance of air into the motion of projectiles. If the trajectory followed by projectiles in the supposed absence of a resistant force was a parabola, when the resistance is included, the trajectory deviates from this geometric shape, and in addition, the determination of its path involves a considerable mathematical complication. Initially, it appears that Huygens estimated that the resistance encountered by the projectile was propor-tional to the velocity, but, after the experiments he made in 1669, he deduced that this was not so. Instead it was proportional to the square of the velocity. He introduced this law for the first time, and, with some nuances, it is still consid-ered to hold. The ‘problem of resistance’ also started at this point: as we shall see, it is different in nature from that of the discharge, although it also originated in ballistics. An initial comparison between the two shows that the problem of resistance was dynamic in character, while discharge was rather more kinematic,

7 Cf. Jean-Baptiste du Hamel, Regiæ Scientarum Academia Historia Parisiis, 1698.

and Guglielmini and Poleni in Italy, are the only ones known to have pub-lished results, although many more must have tried.

geometry of the apparatus influenced the results, the discrepancies among thethe square root, but changing the proportionality constant. Moreover, as the

measurements obtained by all and sundry were significant.

INTRODUCTION 9 even in spite of the fact that the concept of force was not yet clear.8 In order to measure the resistance, Huygens assimilated this to the weight of a column of fluid whose base was the frontal surface of the body in question. For the refer-ence height of the column, he took the height from which the moving body would be dropped, in order for it to acquire a velocity equal to its motion; thus it was that the resistance was measured as being equivalent to one or two, or sev-eral times this height. We have called this height the ‘kinetic height’, and, as we shall see, it will play a fundamental role in the study of fluids. This height is the same as the depth in Torricelli’s law.9

Fig. I-2. Huygens experiments

a method which has been perpetuated up to the present in the towing tanks that began to be built in the middle of the eighteenth century. The third, which also makes use of the motion of the object in a fixed medium, consisted of small sleds transporting a plate attached by counterweights, in such a way that the

8 To express it, they used terms such as impulse, impressions and thrusts and it would not be until Newton that the concept of force in a modern sense would enter physics, although it took some time to be admitted. However, the problem of force in Newton has to be dealt with cautious; see Westfall, Force in Newton’s Physics, especially Chapter 7. 9 Its mathematical expression was hc = v2/2g. Let us observe its identity with the Torricelli’s law.

Huygens carried out experiments on resistance following three different procedures as detailed in Fig. I-2. The first consisted in measuring the force pro-duced by a jet against a plate, which was an extension of the discharge pheno-menon. The second was to measure the towing force of a small ship in a pond;


plate fell when the force produced by the moving air was superior to the value adjusted by the counterweights. As regards the last two methods, we can only say that his idea was clear, but that the capability of the instruments available to him was very rudimentary, although the methods were actually ahead of the times. Concerning the first of the three procedures, nowadays we know that the phenomenon of the impact of a jet against a plate differs from the resistance of a body submerged in a fluid. However, Huygens assimilated both in a single phe-nomenon, an idea that was a source of problems until Daniel Bernoulli separated them definitively in 1736. The jet experiment contributed to this identification, due to its closeness to the discharge. Diagrammatically, the matter was very simple: if the plate against which the jet strikes were to close the fluid outlet completely, the force on it would be the product of the static pressure times the outlet surface, i.e., the weight of a column of water whose height was the depth and area of the outlet. However, in the measurements made during the discharge, they found that the force was greater, and they could not find an explanation for this fact.

Huygens did not confine his experiments to jets of water, but dealt also with discharges of air jets on plates, using an interesting apparatus that later served Mariotte as a standard. Huygens ended by giving specific values to the resistance of a plate moving in air or water, and the fact that his values were high, according to our present knowledge, does not detract one jot from his merit, given his rudimentary measuring apparatus.

We have seen that Huygens approached the two major problems of fluids, resistance and discharge with different methods. A little later on we shall speak of Mariotte, who probably co-operated in the experiments with Huygens, and followed a very similar approach. However, for the sake of chronology, we shall go on to Domenico Guglielmini, who in 1683 carried out some experiments that are worth mentioning, and which were published in 1690 in his work Aquarum fluentium mensura (The measurement of the motion of the waters).

Guglielmini is considered as belonging to the so-called Italian School of Hydraulics, a denomination that we owe to René Dugas,10 who traces its origins back to Galileo in the sixteenth century. The existence of the school was due to different factors, above all the Italian scientific flowering in the Renaissance, and the peculiar importance to Italy of solving practical problems of floods, lagoons and marshes. Guglielmini approached the problems of the motion of water in canals and its discharge from reservoirs. In order to study them he used an apparatus consisting of a barrow that discharged itself by successive regularly 10 Carlo Maccagni also calls in that way in his article ‘Galileo, Castelli, Torricelli and others. The Italian School of Hydrodynamics in the 16th and 17th centuries’. Hydraulics and Hydraulic Research. A Historical Review.

INTRODUCTION 11 spaced lateral orifices (Fig. I-3). The test procedure consisted in allowing the water to pass through an orifice, while maintaining the others sealed, and meas-uring the water flowing out in a specific time interval. Comparing the results obtained for the successive holes, he deduced the perfect proportionality of the outlet velocity with the square root of the height, as predicted by the Torricelli’s law. However, he did not base himself on the former measurements for the abso-lute velocities, but made a new reference measurement whose result he took as standard, and which curiously enough did not coincide with any of the previous ones. Using this standard value and the square root law, Guglielmini presented a table of velocities as a function of the height. If we apply Torricelli’s law to these, the height to be considered in the fall would be a quarter of the depth of water in the reservoir, a fact that Daniel Bernoulli made note of years later in his Hydrodynamica.

Fig. I-3. Guglielmini’s bucket

In these very same years, Mariotte carried out several studies and experi-

ments which he published in his Traité de mouvement des eaux (Treatise of movement of the waters) which appeared in 1686, 2 years after his death. As the title of the work indicates, it deals with several problems: sources, winds, etc., and among these those of discharge and resistance are to be found. In general, Mariotte views his work as a practical treatise, thus including numerous meas-urements and application rules. In particular, his experiments on the outlet of fluids were fairly well systematized. He successively analysed the law of flows, and later he measured them, calculating the effect of the outlet area. Like Guglielmini, he not only found that the square root law was satisfied, but also that the height did not correspond to the depth, a fact that, as we have said, was repeated in almost all the experiments.


He carried out several experiments regarding the measurement of resistance whose results were cited for a century. But apart from the experiments, Mariotte felt the need for a basic theory in order to handle the effects of the resistant forces. For this he assimilated these forces to the impact of what he called a ‘jet’ against a body, and he drew up five rules, defining this ‘mechanics of jets’. He imagined one jet as a cylinder of fluid moving like a solid; and given that he supposed the fluid to be formed by an infinity of small corpuscles, the jet would be a set of particles moving together. The behaviour of a jet before an impact against a plate would differ from that of a solid body, but in essence it would be the sum of the individual impacts against the obstacle. From this he deduces, as Huygens had likewise done, that the effect of the impact of a jet against an obstacle is proportional to the square of its velocity. The argument he brings to bear would become classic: if the fluid goes twice as fast, then double the number of particles will impact at the same time, and with double the effect of each one, so that the total effect will be four times greater.

Fig. I-4. Force on a plate in a flow

The interest of his theoretical proposals is complemented by the experi-ments. Even though his jets are a theorization of the phenomenon of the impact of the jet against a plate, he made the experiments with a plate totally submerged in a current, actually a river (Fig. I-4). Huygens, we remember, moved the models in fluids at rest. Mariotte did the opposite, these tests being the first of this type we know of. He interpreted the results in terms of his mechanics of jets, obtaining some standard values which he took as reference. If his tests are

INTRODUCTION 13 interpreted under the form of the modern coefficient of resistance CD,11 we arrive at a value very close to 1, which is the equivalent of the weight of a column of liquid whose height is equal to what we call the kinetic height. Apart from the tests on the plate submerged in the current of a river, he also proposed another one for plates in air flows, although he presents the apparatus by itself without any reference whatsoever to a measurement. On comparing the values of resis-tance in the air and water, he established that the effects were proportional to the densities, another rule which also would persist. On the other hand, in the tests in water, this impinges on the plate perpendicularly, while in the case of air, it does so at an angle of 45°, thus breaking up the impinging force into two components.

Mariotte interprets the phenomena as percussions, and in this sense this appraisal is the first appearance of the ‘impact theory’ which we shall presently consider. The question why Mariotte did not follow the experimental method of Huygens is not easy to answer; but we conjecture he was influenced by the fact that the results of the experiment of the jet against a plate were quite uncertain, because he was familiar with water mills and experimented with them on the river Seine, as he mentions in his Traité. To sum up, Mariotte’s contribution rightly merits the qualification of precursor that Dugas awards him.

To recapitulate, during these years, on the one hand we find that experi-ments with discharges are only partially reconciled with their theoretical predic-tions, and on the other hand that there appears to be a proportionality of the resistance with the square of the impinging velocity, and with the density of the medium. Besides (and this is an important matter) the need was already felt to establish a theoretical basis enabling these phenomena to be explained, a need that Mariotte had anticipated. It is in this context that the first edition of Newton’s Principia saw the light. While Books I and III, respectively titled ‘The Motion of Bodies’ and ‘The System of the World’, are well known, Book II is not. It is also named ‘The Motion of Bodies’ although it deals with fluids, while the first deals with rigid bodies. The coincidence in the titles indicates that Newton put both solids and fluids under the same umbrella. However, Book I is structured as a closed set, characterised by an attempt to derive all motions from a few axioms or laws of motion. This is not the case in Book II, where he has to use several different additional hypotheses, which at the very least complement the others. In this respect, insufficiently justified suppositions abound, together with some small fudges and implausible constructions. In spite of this, Book II was of capital

11 The definition of this coefficient is CD = F/(½ρv2S), where F is the resistance force, ρ the fluid density, v the velocity and S the frontal surface. The meaning of this coefficient, as the times sense, was equivalent to the number of kinetic height the cylinder of equal weight that the resistance force would have. (cf. later Chapter 1, note 3).


importance throughout the eighteenth century, and almost all authors referred to it, some in order to follow it, others to refute it.

Book II deals with several different problems and the most significant one refers to the motion of projectiles. He dedicates the first sections to the dynamics of projectiles in dense media, that is, to bodies whose movement is resisted by forces depending on the velocity, be it in simple proportion, squared or a combi-nation of both. Assuming the existence of this resistance and its mathematical form, the resulting problem is similar in nature to that dealt with in Book I, of which these sections are but an extension. We deduce from this that Newton arrives at the problem of resistance due to his interest in ballistics, as Huygens also did, and unlike Mariotte, who arrived at it via hydraulic machines.

Newton proposed a definition of a fluid as a body whose parts yield to any force impressed on it. However, he encounters difficulties when he tries to explain the different sources generating resistance. In terms of our present understand-ing, Newton considered the existence of a resistance produced by the forces of inertia, which he calls ‘inertia of matter’, and another resistance produced by viscous forces. His idea concerning the latter was a lot less clear, and he called it ‘lubricity, tenacity or fluidity’. Although he established the definition of visco-sity as a shear force proportional to the variation of the velocity, the range of viscous phenomena and their mechanisms lay outside his horizon. This explains why he gave them various names according to the aspects they presented.

Fig. I-5. Pendulums in resistant media

Regarding resistance, and with the aim of determining its mathematical

form by experiment, Newton proposed using pendulums oscillating in fluids (Fig. I-5). The idea was simple. The pendulum would maintain the amplitude of oscillation constant if there were no resistant forces. If these forces did exist, the amplitudes would be reduced with time, progressively diminishing until the pendulum came to rest, so that Newton could infer the mathematical laws of

INTRODUCTION 15 resistance from the observation and measurement of these diminished motions. Therefore, he introduced an extensive study of the motion of pendulums in resis-tant media, and arrived at the point of establishing a relation between the ratio of these reductions and the mathematical expression of the velocity in this law. Once the theory of the instrument had been established, he performed a set of experiments with pendulums of various sizes, making them oscillate in air, water and mercury. However, the conclusions he obtained did not indicate that the resistance was proportional to the square of the velocity, as was expected, and in consequence Newton tried to approach the problem by polynomials with differ-ent powers of velocity. What appeared more obvious to him was the proportion-ality of resistance to the cross section of the oscillating mass, and to a lesser degree to the density of the fluid. From the comments he makes, one can infer that the results of the experiment did not satisfy his expectations, but that in spite of these dubious results, he continued to think that the resistance was propor-tional to the square of velocity.

After the work with the pendulums, which comes within the framework of experimental physics, Newton directly tackled the case of bodies in fluid currents or vice versa, that is, moving bodies in a fluid at rest. He dedicated the most memorable sections of Book II to this problem. He begins with a differentiation of behavior among liquids or uncompressible or non-elastic fluids, as opposed to air, which he qualified as being compressible or elastic. Although he considered both classes to be aggregates of particles, he established a difference between them: in the air the particles are separated from each other and repel each other due to forces which he called ‘centrifugal’. These are inversely proportional to the separation existing among the particles; for liquids all particles are in con-

It is in the study of the motion of bodies in air that we consider fluid mechanics as becoming a truly modern scientific discipline, as it is there that Newton, in order to explain the phenomenon, introduces three basic hypothesis: one on the constitution of air, a second on how moving air and a body in its midst interact, and the third on the existence of a law regulating these inter-actions. The latter is the second law of dynamics. As we have said, he took a set of individual particles submitted to repelling forces for the air model, and for the interaction he adopted the mechanical model of impact. However, although the

in the air are easily individualized, while this is not true for those of a liquid. tact, and are only united by the forces of lubricity. That is to say, the particles

The theories in each case are different, we ought to say qualitatively different. But there is more; in the second edition of the Principia, published in 1713, Newton rewrote the section referring to liquids almost completely, although he maintained the same methodology as in the first edition, which also obliges us to treat the topic in two different moments.


theory can be constructed with these three elements, the difficulty of calculation was almost insurmountable, as the motion of one particle of air affected all others to a certain extent, given the repulsion among them. In order to solve this difficulty, he simplified the air model by eliminating these repulsive forces, thereby rendering the particles immobile. He called the fluid resulting from this operation a ‘rare medium’, which, although it did not have the properties that he attributed to air, would behave in a very similar way to air. In this new medium, the phenomenon would be modeled as the motion of all the particles, completely independent from each other, that would rebound elastically in the impact against the body, or would remain immobile if the impact was not elastic. In either case, they would transmit a certain momentum to the body, whose varia-tion with time would be the resistant force.

Fig. I-6. Impact theory

This model, which we have called the ‘impact theory’, was already sketched by Mariotte, although it acquires its authentic meaning in Newton. He supposes (Fig. I-6) that there is a deflection in the impact, depending on the angle of incidence. The corollary of the supposition is the existence of shadow zones where the fluid cannot impact. As we shall see, this theory will remain in force in fluid dynamics for more than a century. It will be criticized and doubted, but because of the lack of a better theory it will prove essential due to its instru-mental function.

Newton applied this theory to a plate and a sphere. In the case of an elastic rebound, he obtained the values CD = 4 and CD = 2, respectively12 as resistance coefficients which were reduced by half when dealing with a non-elastic impact. 12 We quoted numerical values because the calculation of the actual magnitudes of these coeffi-cients was one of the major milestones in the evolution of the problem of the resistance.

INTRODUCTION 17 In addition, he made a short incursion into the case of a solid of minimum resistance.

When dealing with liquids, neither this model nor the impact theory was of use to him, as the particles are not only innumerable, but also they rub together, and when they impact with the object they transmit this action to the ones behind in a domino effect. Faced with this difficulty, Newton was forced to explain the phenomena using an alternative method. The model he used was the vertical discharge of fluid through an orifice, in whose current he had placed a resistant body (Fig. I-7). The method he followed in order to find the force acting on this body is based initially on finding the velocity of the discharging fluid as a func-tion of the depth of the water in the reservoir. Afterwards, as a second step, he calculates the force upon the body also as a function of this height. Knowing both, and after eliminating the height parameter, the resistance is deduced as a function of the velocity. Obviously, this approach does not correspond with the impact theory, as there is no particle rebound. Likewise, we see that Newton made use of the discharge problem as an ancillary element, and although he used this methodology in the two first editions of the Principia, as we shall see, the solution he arrived at was different in each one.

a) b)

Fig. I-7. Discharge with a body inside the current

In the first edition he offers an argument from which he concludes that the

discharge velocity corresponds to a fall height that is half that of the depth; that is to say the contrary of Torricelli’s law. This was to some extent in tune with experimental results of the day, although it contradicted the experimental evidence of the vertical ascent of water in spring jets. The question was under the spotlight, and Newton must have convinced himself shortly afterwards of his error, as in 1690, in his personal copy of the Principia he made some notes from which we deduce that he had changed his mind. As regards the force exercised by the current on the sphere placed in it, he supposes it to be equal to the weight


of a column of water whose height was that of the reservoir and had the same cross section as the sphere. If the submerged body were to have another geomet-rical form, but the same cross section, the weight of the column would be the same; that is to say, the resistance for a liquid movement is independent of the shape of the body, which did not occur for a body in air. The result he obtains for a sphere in a liquid is that the resistance coefficient CD = 2, which is equal to that of the sphere in the air.

Although Newton deals with the discharge quite briefly, we find it has two opposing aspects. On the one hand was Torricelli’s law, corroborated by the fact

that when the reservoir is discharged by a lower orifice, and Torricelli’s formula is introduced, the height of the discharge will oscillate around half the depth of the reservoir. The situation was contradictory, and at first Newton opted for the latter solution, which was included in the first edition of the Principia. Later on, as we have said, with the correction he wrote in his personal copy, the new coef-ficient of resistance was reduced to CD = 1, that is, half the former, which fur-thermore partially agreed with the experimental values found by Mariotte. The solution to this contradiction will be given by Newton in the second edition.

To recap, we must underline two aspects of Newton: the new approach to the formalisation of the impact theory using the ‘rare medium’ instead of air, and the use of discharge phenomenon as an instrumental element for resolving the question in the case of liquids, where impact theory, strictly speaking, is not used. There is therefore a crossing over of the two major lines of work, resistance and discharge, although this crossover is only partial.

We now briefly leave Newton, but we will return to him when we consider the second edition of the Principia. In the two decades following Principia’s publication (1690–1710), a group of mathematicians—geometricians as they were known at the time—and another group of hydraulic mechanics entered the stage. The preoccupations of the former swung between what we could call pure applications, such as the discussions of the minimum resistance solid, and other more practical aspects such as applications to naval theory. Compared to them, the mechanics focused on the study of hydraulically driven machines, that is to say waterwheels with paddles.

The characteristic common to the mathematicians was the acceptance and extension of the impact theory even to liquids, without anyone questioning of its physical reality. Among them Jakob Bernoulli stood out, having used differential calculus to analyse the effects of liquid flow on two-dimensional shapes of ships. His conclusions went partly unnoticed, but for many years they were not sur-passed, as he obtained the maximum advantage from an analysis of this type.

water in the reservoir. On the other hand, the results of the experiments indicated that vertical fountain discharges almost reach the level of the surface of the

INTRODUCTION 19 At the same time he extended his studies to sails, which, inflated by the wind, formed so-called sailing curves (velarias), related to the catenaries, which were also in fashion then. As regards the solid of minimum resistance, Newton offered a solution in the Principia, although almost without any justification. Its obscure nature caused Nicolas Fatio de Duillier, Guillaume Antoine de l’Hôpital and Johann Bernoulli to enter into the fray with new solutions in the final years of the century. All the solutions were purely mathematical; therefore we must consider the contributions they made to the study of fluids as being purely theo-retical with little or no practical application. Proof of this is that they limited themselves to relative solutions between one shape and another, without includ-ing any values of the resistant forces.

Analysts of hydraulic machines, like Philippe de la Hire and Antoine Parent, constitute the opposite extreme. In 1704 and 1705, they studied the behavior of fluid-driven machines. They based their experiments on impact theory, with a pro-portionality coefficient derived from Mariotte’s experiments, rounding it up to the nearest unit. To complement their reasoning, they introduced considerations concerning the mechanical efficiency of mechanical systems.

The application of fluid mechanics to ships and to hydraulic machines res-ponded to important social needs. It is not necessary to go into detail regarding the importance of the navy at this time, but by the eighteenth century, maritime power had become one of the defining elements of a nation’s military strength, and its most relevant exponent was the ship of the line. The ship of the line, at its most elemental, is a machine which balances the effects of the wind in the sails with that of the water on the hull. It is the epitome of the ‘fluid machine’. As regards other machines, it is nothing new to say that windmills and waterwheels had been almost the only sources of mechanical power from ancient times. The growing urbanization of the eighteenth century required a method of distributing water, that is to say, a source of driving power and a raising device, which were the waterwheel and the water-raising pump.

Thus, we arrive at 1713, a year during which we find ourselves once again with Newton, and his second edition of the Principia. No new theory had appeared since the first edition, only, as we have seen, some mathematical and practical applications. Newton’s new contributions were important, as he practi-cally rewrote the part referring to motion of liquids. We remember that, where we had left him, his findings contradicted Torricelli’s law, even when they com-plied with the discharge experiments. In order to deal with this, he modified the discharge process that had served as the basis of his method, introducing a theo-retical construction, the cataract (Fig. I-8), with which he intended to reconcile both positions. In this type of discharge, a contraction of the stream is produced, which implies that the actual cross section of the outlets is smaller than the transit


section was approximately half that of the outlet orifice. This allowed him to adjust things in order to say that the outlet velocity followed Torricelli’s law, but that due to this contraction the outgoing mass was reduced to half. Bringing the theory into line with the experiments in this way, he had to assume that the dis-charge took place as it was passing through a funnel of ice that became liquid without interfering with the falling water. The explanation was most unlikely, and was widely criticised. As regards the force exercised by the current on the body located in its midst, he placed another cataract on this body, concluding that the resulting force corresponded to their resistant coefficient of CD = 0.5, whether the body was a disk or a sphere, and that it was a quarter of what he had obtained in the first edition. To all this he added a new collection of experimen-tal results, consisting in spheres falling into buckets of water, where he played with different sizes and densities, concluding that the theoretical predictions were verified. He even included among his experiments glass spheres dropped from the dome of St. Paul’s Cathedral. Further experiments were carried out by himself and others by Jean Théophile Desaguliers.

Fig. I-8. Newton’s cataract

Bernoulli applied impact theory in order to calculate both the forces of water against the hull, and the wind in the sails. The work was the end result of a long

mechanics of naval ship maneuvering. For Bernoulli, the shape of the hull was

orifice. Newton measured this contraction, and found that the contracted cross

The Essay d’une nouvelle théorie de la manœuvre des Vaisseaux (Essay on

scientist Christiaan Huygens and the Swiss scientist Jakob Bernoulli on the

a new theory of ship manoeuvering) appeared in 1713. This treatise by Johann

debate between the French naval officer Bernard Renau d’Elizagaray, the Dutch

INTRODUCTION 21 responsible for the drift when the ship advanced, and he calculated it based on impact theory as applied to the position of the sail.

Another author of the time was the Italian scientist and engineer Giovanni Poleni, and we mention two of his works in particular, the De motu aquæ mixto (On the motion of mixed waters) and the De castellis per quæ derivantur flu-viorum aquæ habentibus latera convergentia (On constructions with convergent openings, through which the waters of rivers are discharged). In both works Poleni tackles the discharge problem. In the first, he makes measurements of some models of ‘mixed’ waters, which he means to include ‘live and dead’, which in turn recalls ‘live forces’ (vis viva) and ‘dead’ (i.e., static) forces. The method in evi-dence in De motu appears to follow a rather artisan line with not much future. In the second work, he goes back again to verify Torricelli’s law. In order to do this, he performs discharge experiments through orifices with spouts of varying shapes, simultaneously measuring the contraction of the outlet vein. He obtains confirmation of the proportionality of the efflux velocity with the square of the height, although he does not manage to explain the cause of the contractions. Poleni is possibly the last of the great experimenters of the Italian school.

By 1720, it was common knowledge that discharge theory still suffered from a considerable conceptual weakness, while impact theory, supporting the problem of resistance, was well established by then. Nevertheless, the latter had a drawback: even though the model was accepted, it was not clear which values to use for the coefficients of proportionality. There were the values obtained by Newton, some theoretical as in the case of air, others justified experimentally such as for water, although these were almost double what Mariotte had meas-

the experiments made by Mariotte were made with flat plates, whereas Newton used spheres, and he had established that a sphere in air has half the resistance of a plate. Besides, the mathematicians had supposed that impact theory could be applied as much to air as to liquids, therefore both sets of results could be con-sidered compatible.

discharge and resistance, and at the end of the decade he had sketched out the basic ideas of his Hydrodynamica, a work that would appear in 1738. Others made lesser contributions including Desaguliers, who went back to the idea of spring-driven pendulums oscillating in water and mercury.

In 1724, we find Johann engaged in a work essentially about mechanics,

annual Rouillé prize, and which contains a part dedicated to analysis of the

‘Discours sur le loix de la communication du mouvement’ (‘Discourse on the law

The decades of 1720 and 1730 were marked by the contributions of the Bernoullis, father Johann and son Daniel. Daniel contributed to the problems of

of communication of motion’) which he presented for the Paris Academy’s

ured. Nevertheless, it is not surprising that these were taken to be congruent, as


motion of solids in fluids. We note yet again that the contributions on fluids were frequently intertwined with those on solid mechanics. The work we quote

tion of any shape whatsoever, upon which he calculates the absolute forces,

In 1727, Daniel Bernoulli, who was living in St. Petersburg at the time, submitted the work ‘Disertatio de actione fluidorum in corpora solida et motu solidorum in fluidis’ (‘Dissertation on the effect of fluids in solid bodies and the motion of solid bodies in fluids’) on the analyses of bodies submerged in cur-rents, to the Commentarii academiæ imperialis scientarum petropolitanæ (Mem-oirs of the St. Petersburg Academy), where he contributed a new theoretical construction in order to justify Newton’s results. This construction consisted of a

that impacted in successive layers on the plate, acting as a set of continuous percussions. The analysis of this model gave the resistance coefficient of value

Dwith the experiment on the discharge of a vertical jet against a plate, which showed that CD =1. Faced with this dilemma, Daniel decided on the latter result, justifying this by experiment. Besides, when this corrected result was applied to a sphere using impact theory, it gave a CD = 0.5, which again coincided with the value found by Newton in the descent of spheres in water tanks. It goes without saying that any coincidence with Newton’s results was, at the time, a guarantee of acceptance. He ended the work with the application of impact theory to the

phenomenon. In the following decade, the British artillery engineer Benjamin Robins will come back to this problem.

There is another work of Daniel in the same 1727 volume of the Commen-tarii petropolitanæ, which is dedicated to the motion of water through ducts. The

confirms the separation of the phenomena of resistance and discharge. This monograph marks a change in the treatment of movement through ducts, and it already contains ideas that appear later in the Hydrodynamica. The fundamental

is specifically Newtonian, and it extends the impact theory to bodies of revolu-

des vaisseaux. What can be considered as new is his application of impact theory to the problems of ballistics, which he outlined in the Essay, and complementedin the later work, ‘Problema Ballisticum’, at the same time making an incursioninto the case of a pendulum in a resisting medium.

something he had not done in the Essay d’une nouvelle théorie de la manœuvre

ballistics of a cannonball fired vertically, in which he comments on the different

paper, entitled ‘Theoria nova de motu aquarum per canales quoscunque fluen-

C = 4, the same found by Newton in the motion in air. He compared this result

flight times as a function of the projectile weight. His intentions are sound, but

plate inside a current flow, which he assumed was formed by elastic particles

he lacks the data (the projectile velocity at the muzzle) needed to understand the

tium’ (‘New theory about the water motion flowing through whatever channels’)

INTRODUCTION 23

Fig. I-9. Motion through ducts

basis of the new approach consists of two principles: the conservation of the live forces and motion in parallel sections (Fig. I-9). For the first, he supposes that the set of particles comprising the liquid collide with each other and with the walls, in an elastic manner, therefore justifying the conservation of the total live force. However, in order to study motion of fluids in ducts, a second hypothesis is required: that the velocity be the same in all points of any section perpendicu-lar to the axis, thus causing this section to advance like a plane. Given that the velocity is inversely proportional to the section, when this is known, the problem is reduced to a one-dimensional case, and the calculation of the total live force of fluid volume is reduced to a simple integral. In the specific case when it is applied to a reservoir in a vertical position with an exit orifice in the bottom part, the outlet velocity would be calculated by assuming that the live force of the whole reservoir is kept constant. The entire mass of the fluid would intervene in the process, and the final formula arrived at is no longer Torricelli’s, as it coin-cides with his only in the case where the cross section of the outlet orifice is negligible compared with the cross section of the reservoir and the upper surface of the water is not very close to the bottom. Although these ideas are greatly perfected in the Hydrodynamica, we can appreciate that there is a qualitative leap with respect to the previous approaches, which may be resumed by consider-ing the motion of the fluid mass as a global phenomenon, not individualized in particles.

Daniel presents another work dealing with discharges: ‘Experimenta coram societate instituta in confirmationem theoriae pressionum quaas latera canalis ab aqua tranfluente sustinet’ (‘Experiments performed before the Academy in order to confirm the lateral pressure in fluid currents’) which appeared in the 1729 volume of Commentari petropolitanæ. This is the first attempt to relate pressure


and velocity, a prelude to the formulation of the theorem later named for Bernoulli, and whose emblematic character all that work in the abstruse world of fluid mechanics knows.

Fig. I-10. Daniel Bernoulli’s experiment

At this point, a few additional comments on pressure might be helpful. The

concept of pressure as a force that the fluid exercises on the walls belongs to the world of hydrostatics. Pascal extended it to the entire fluid by a very well-known principle taking his name. However, up to the date which concerns us, the con-cept of pressure remained as a property linked to hydrostatics, independent of motion, until Daniel Bernoulli demonstrated experimentally that this supposition was not true, and that the pressure on the walls of a discharge tube (for the moment only on the walls) depends on the velocity of the water in the duct (Fig. I-10). To this end, Daniel used the water height manometer, the apparatus shown in the figure, which consists of a vertical tube through which the water ascends according to the pressure in its base. If the outlet orifice of the discharge is blocked, and therefore there is no movement, the level of water in the tube reaches the height of the fluid in the reservoir. However, when circulation is allowed, the pressure will be lowered, which is evident from the drop in the tube level, and it will go down as the outlet flow increases. We repeat, the experi-ment has several remarkable points: it shows the relation between pressure and velocity, contributes a measuring instrument such as the water manometer, and presents an experimental result which will promote the theoretical development required to explain it.

All these explanations are given in the Hydrodynamica, a work which was almost complete when Daniel left St. Petersburg in 1733, although its publica-tion was delayed until 1738. But before we examine the Hydrodynamica, let us recall once again the case of the impact of the jet against the plate. This phenomenon had been mistakenly interpreted as being equivalent to the effect of

INTRODUCTION 25 the current on a body submerged in the same current, although this equivalence was strictly true according to the model of the impact theory. Daniel undid this supposition, identifying it as a different phenomenon (Fig. I-11) in which there is no rebound of the fluid, but where it is deflected towards the sides. The application of Newtonian mechanics, already well developed as a mathematical instrument, allowed him to estimate the expected forces, which he confirmed experimentally and quite precisely with a well-designed apparatus. The results appeared in the 1736 volume of the Commentarii petropolitanae. Three years later, another announcement appeared in the Commentarii with a new set of experiments, this time made by Georg Wolffang Krafft.

Fig. I-11. Jet against flat plate

In the decade of the 1730s, the figure of Henri Pitot also appears. Although he was principally a mathematician, he dedicated a good part of his activity to the analysis of machines. Among his contributions we find the work La théorie de la manœuvre des vaisseax reduite en pratique (The theory of the manoeuvering of ships reduced to practise), which is an extension with more practical applica-tions of the work with the same title by Johann Bernoulli; and three monographs on water pumps in the Mémoires de l’Academie Royal des Sciences of Paris in which he studied the application of the theory of live forces. He is especially remembered for an instrument for measuring the velocity of a current, which is described in the 1732 volume of these Memoirs, and which is still known as Pitot’s tube. Apart from this invention, which continues to be in use, he made no theoretical or experimental contributions. However, his work concerning the application of fluid mechanics theory was a different matter: although water pumps were known from ancient times, until Pitot came on the scene there were no studies explaining how they functioned.


We return to Bernoulli père et fils, Johann and Daniel. For unknown rea-sons, although it seems to have been due to fit of jealousy, the appearance of Hydrodynamica by Daniel aroused the ire of his father Johann. He prepared the Hydraulica, which came to light in its complete form in 1742, although accord-ing to Johann, it was written in 1732, that is, before his son had delivered his version. Today it has been shown that this was not true, and that the father pla-giarized his son regarding the theory that bears his name. Nevertheless, Johann’s work is of higher quality, as it tackles the problem in a more general manner, and has a more solid Newtonian basis than the works concerning the conserva-tion of live forces.

As Daniel relates, the title of Hydrodynamica was dictated by a combina-tion of the two branches of fluid mechanics: hydrostatics and hydraulics, which were separate up to then and which, he argued, should go together. Nevertheless, we note that in the text Daniel uses the word ‘hydraulic-static’. The subject mat-ter of the work is very wide, and touches on the entire spectrum of the fluid problems of era. The contents can be divided into several areas, with differences of form and content between them, such as fluid discharge, oscillations of fluids in tubes, the passage of liquid between reservoirs and ‘hydraulic-statics’. To all these types of motions, which are due only to the force of gravity, he adds those produced by external forces like pumps, as well as the forces generated by the reaction of jets and the motion of liquids in reservoirs that are themselves in motion. All these areas deal with liquids, except one dedicated to a mass of air enclosed in a reservoir, which he analyses introducing a version of what we would nowadays call the kinetic theory of gases. The Hydrodynamica is a wide-ranging, dense and complex book whose premises are the principles of conserva-tion of live forces and of motion through parallel sections, which Daniel presents in a more elaborate form in this work. In this respect, he introduces what he calls ‘potential ascent’ and real descent instead of the live forces. His definition of the first term is the height to which the moving body would ascend if its velocity were to be directed instantaneously upwards, while the real descent would be the amount by which the centre of gravity of the system descends in reality. That the loss of one must be the gain of the other, is easy to demonstrate, and can be seen intuitively. The application of this theory enables discharge problems to be solved. But Daniel is not content with theoretical solutions, proposing after each chapt

The Genesis of Fluid Mechanics, 1640–1780fluid mechanics, the structure and formulation of its...

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