The Problem of Discharge

The Problem of Discharge We have called the second major line of research into theoretical fluid mechanics in the seventeenth and eighteenth centuries the ‘discharge problem’ because its origin lay in the study of the discharge from vessels through orifices. To be more exact, two types of studies fall under this rubric: the first is the discharge we have just mentioned, the second is the motion of fluids through pipes. The second derives directly from the first, and it ranges from the rationalisation of fluid mechanics to deriving its fundamental equations. Unlike the resistance problem, rather different from the discharge problem in many ways, its social relevance is not so obvious or urgent—even in spite of the problems of water supply and distribution to towns—as the problem is more or less within the compass of experimentation. However, its theoretical interest was in fact richer and further reaching than the case of resistance, the most significant milestone being Euler’s equations, which have shone out from the middle of the eighteenth century right up to the present day. Compared to the conceptual simplicity of the impact model, the problem of discharge led first to the discovery of the relation between pressure and velocity, thus linking two hitherto separate phenomena, and later on they led to the introduction of the concept of internal pressure, a major milestone that allowed differential calculus to be applied to a fluid. Thus the dynamics of the three major categories of bodies, rigid, flexible and fluid, were unified under the same umbrella. It is true that the intervention of differential calculus was deci- sive, but in its turn the entire set of queries and questions constituted a stimulus for the advance of calculus. Authors like Bernoulli, a name that includes an entire family (though sometimes this is forgotten), Euler, and d’Alembert are asso- ciated as much with the mechanics or dynamics of fluids as with mathematics. Moreover, each of these complained of the difficulty in advancing further, due to the limitations of the mathematical instruments at their disposal. We must not forget that it was at that time that the frontier of theoretical mechanics passed through fluid dynamics, and note that some of the important problems in areas such as meteorology and aerodynamics, that nowadays require the integration of the equations presented by Euler and later completed by Navier and Stokes in the nineteenth century, are examples of this. 267 268 THE GENESIS OF FLUID MECHANICS, 1640–1780 The beginnings of the discharge problem go back to the decade of the 1640s, when Evangelista Torricelli and his master Benedetto Castelli studied the outlet velocity of a jet through an orifice in a tank. In 1644, Torricelli published the first known law concerning the subject, which carries his name, and indi- cated that this velocity was equal to that acquired by a heavy body falling from a height equal to the depth of the outlet orifice. The experimental verification of this law appeared to be very simple: to open a spigot of a known exit area and to measure the volume of liquid that flows out in a certain time, and from this data to infer the outlet flow velocity of the liquid. From recorded comments, it seems that a fair number of researchers must have devoted themselves to this task, the most outstanding being the Academy of Sciences of Paris, and the so-called Italian School of Hydraulics. The former sponsored the study of this problem in the first years following its foundation in 1666. Figures like Huygens, Mariotte and other lesser-known ones collaborated in this task. As regards the Italian School,1 a lot less is known, and this is perhaps an opportune moment to say something about it. Apart from belonging, not to one but to the multitude of states that constituted the ‘Italy’ of the era, a characteristic feature of the ‘Italian school’ was its preoccupation with currents of water and experimental hydraulics, fields with great practical interest due to the profusion of floods and over- flows that took place in the Italian lakes and rivers. Galileo Galilei can be taken as the first point of reference, followed by Benedetto Castelli, Evangelista Torricelli, Domenico Gugielmini, Giovanni Poleni, Bernandino Zendrini and many more. The greater part of the work of these authors was collected in the Raccolta d’autori che trattano del moto dell’acque (Collection of authors dealing with motion in water)2 which collected together a wide range of activities, the majority of which were experimental, carried out in Italy during the second half of the seventeenth century and the first third of the eighteenth. The Raccolta constitutes an opus magnus, not only for its intrinsic interest, but also because it shows the level that hydraulics studies in Italy had attained. The decline of this School at the beginning of the eighteenth century coincides with the ascent of 1 We took this denomination from Rouse, who names it thus, and justly so, in his History of Hydraulics, p. 113. A work on the same theme exists written by Carlo Maccagni, ‘Galileo, Castelli, Torricelli and others. The Italian School of hydraulics in the 16th and 17th centuries’. 2 Edited for the first time in Florence in 1723, and the second between 1765 and 1771. This second edition, which consisted of nine volumes began with the treaty on Des corps flottants of Archimedes and the discourse Intorno alle cose che stano in sull’acqua, o che in quella si mouvono of Galileo, followed by authors of the time, some of whom were not Italians, such as Mariotte and Couplet. However, some important works of the time are missing in the Raccolta, such as De Castellis of Poleni. THE PROBLEM OF DISCHARGE 269 the French School, which as we have said began its experiments precisely with discharge in vessels. In a certain manner, by the end of the seventeenth century the French School took over the relay baton. Let us begin with the experiments verifying Torricelli’s Law. The identifi- cation of the outflow velocity with the fall of a heavy body is mathematically expressed by the law v2 = 2gh which implies two hypotheses: that the velocity is proportional to the square root of the height, and that the proportionality constant is the root of 2g. The first experiments verified the first hypothesis, but not the second, which was interpreted as follows: the height equivalent to the fall was not the depth, but rather only a fraction of this, and furthermore, the results pointed towards this fraction being only half. Therefore, and considering the belief—we repeat, the belief—that nature seeks whole numbers, the conclusion was that the height of the equivalent fall was half the depth. Even Newton fell into this error, as the first edition of the Principia shows, and he rectified it in the second edition by introducing the phenomena of the stream contraction, about which we have already spoken. The reality of the problem lies precisely there: in the formation of the outlet jet, as the field of velocity in the neighbour- hood of the orifice is quite complex, and in which the geometry of the outlet spout, should it exist, plays a determinant role, along with other properties of fluid not well known at the time. In this contradictory situation, Daniel Bernoulli tried to study motion through pipes. He considered the fluid as a group of corpuscles that advanced, subject to the principle of conservation of lives forces, in a form that we might identify today as the conservation of mechanical energy of the fluid. The novelty of this approach was that it invoked a regulating principle of a general nature involving the entire fluid. Bernoulli’s ideas followed the main lines of dynamics: the reduction of problems to laws of a general nature. Newton introduced what we know today as the laws of dynamics, years before Huygens had introduced the conservation of lives forces, and later d’Alembert was to do this with his principle of dynamics. Both Bernoulli and Newton imagined the fluid to be a grouping of corpuscles, but contrary to Newton, who needed to know the indi- vidual interactions in order to resolve the problem, Daniel applied a law covering the entire fluid in an overall fashion, with the single hypotheses that the internal interactions were elastic. Bernoulli’s ideas appeared for the first time in 1727 in the Commentarii Petropolitanae, but where they attained their maximum develop- ment was in another of the memorable works of the eighteenth century: the Hydro- dynamica (1738), in which he not only obtained the basic formula of motion through ducts, with specific particularisation in the case of outlet through an orifice, but he also dealt with the relation between velocity and pressure, which is nowadays known as Bernoulli’s Theorem. 270 THE GENESIS OF FLUID MECHANICS, 1640–1780 The Hydrodynamica was answered by another no less important work: the Hydraulica written by his father Johann Bernoulli, which appeared in 1742. The history of these two works is worth telling. The basic difference between them is that Johann took Newtonian principles, already transformed into his differential formulation, as the regulating ones, and he introduced the concept of internal pressure. Using both of these, he divides the fluid into differential elements capable of being analysed individually, which he employs to reformulate the analysis carried out by his son, in particular what we call today ‘Bernoulli’s Theorem’, which in the process comes very close to its current version. Actually, we are of the opinion that the theorem ought to be called ‘the Bernoullis’ theorem’, as it is due as much to the father as to the son.

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