Some Early Attempts at Theory Formation in Fluid Mechanics

SIMON INCE

Department of Hydrology and Water Resources, University of Arizona, Tucson, Arizona

Despite similarities in the axiomatic structure of theories in m~t~ematics and natural sciences, theirobjectives are different. Mathematics is devoid of factual or. empIrIcal content a.nd need not, and doesnot, assert anything about the physical universe. The theo~les In the na!ural SCIences, n~ matter howwell packaged in complex and sophisticated mathematIcs '. ar~ subject. to observa!lonal and~orexperimental verification. Some attempts ~t theory formatIon In the hls~ory. of fluId mechanIcsillustrate the axiomatic structure of the theorIes and show the reasons fo.r theIr fatlure o~ succes~. T~ebasic theoretical structure of physical hydrology, resting on the foundatIons ~f Ne~tonlan physIcs, ISdeterministic. Mathematical random-process theory is a very powerful technIque In the managementof many hydrological issues.

INTRODUCTION

The goal of any branch of knowledge is to obtain a theorywithin the framework of which the body of knowledge can beinterpreted, explained, forecast, and events predicted quan-titatively with a high degree of probability. The fact thatmathematics is often referred to as the queen of sciences isdue to the absolute certainty of its results.

For a long time the idea was prevalent in wide circles thatif mathematics could be used as the language of naturalscience, or empirical knowledge formulated in mathematicalterms and patterned according to the methodology of mathe-matics, the certainty of the results of empirical science couldalso be guaranteed. Western man's confidence in mathemat-ics was so great that, in the nineteenth century, it wasbelieved that the universe could be described by an infinitesystem of self-solving differential equations.

Then, in the twentieth century, Bertrand Russell shookthe foundations of the entire system of sciences by callingmathematics "the only science where one never knows whatone is talking about, nor whether what is said is true." Theshock treatment liberated the patient from his delusions,which meant that, philosophically, the provisional nature ofempirical science was recognized and differentiated frommathematics. It was realized that no proposition in empiricalscience-no matter how well-packaged in sophisticatedmathematics-can attain the certainty of mathematics.

What is the nature of this mathematical certainty, andwhat is its significance in relation to the structure of physicaltheory?

Mathematical theory consists in the logical deduction of aproposition from other propositions previously established.This procedure presupposes an arbitrary origin where somepropositions are accepted without proof; they are the axiomsor postulates. Once the postulates for a theory have beenlaid down, each further proposition of the theory must beproven exclusively by logical deduction from the postulates;in this process no appeal is allowed to self-evidence, to thecharacteristics of the physical universe, or to our experi-ences concerning the behavior of rigid bodies in physicalspace, etc. The purely deductive character of mathematicalproof forms the basis of mathematical certainly. In mathe-

Copyright 1987 by the American Geophysical Union.

matics, no assertion is made that the axioms have physical orfactual content. For this reason a mathematical derivation ordeduction is absolutely certain because it is devoid offactualor empirical content.

Historically speaking, however, Euclidean geometry, forexample, had its origin in the generalization and systematiza-tion of empirical discoveries in connection with the measure-ment of areas and volumes, the practice of surveying and thedevelopment of astronomy. Thus understood, geometry hasfactual content and may be called physical geometry.

The physical interpretation transforms a given pure geo-metrical theory into a physical theory of the structure ofphysical space. Whether this theory is correct in interpretingnature, is not the concern of mathematics but of empiricalscience. It can only be proven by suitable experimentationand observation. It might seem that an easy way to test thevalidity of a theory is to test the axioms or postulates.However, in all theories of natural sciences, it is neithernecessary nor in general possible, to submit the basic axiomsto direct experimental tests. The testing of any scientifictheory has to proceed indirectly by testing the deducedtheorems. If enough relevant experimental evidence is foundsupporting the theorems and therefore the theory, it acquiresa certain degree of reliability and may be accepted "untilfurther notice." But, however great the degree of confirma-tion, the possibility always exists that new disconfirmingevidence will be found. Herein lies the provisional characterof all theories in empirical science. The great importance ofmathematics for the empirical sciences lies in the fact thatwhile it does not assert anything about empirical fact, itprovides an indispensable and efficient machinery for deduc-ing, from abstract concepts, such as the laws of Newtonianmechanics, concrete consequences which can be tested.

It has become customary to call the sciences which have ahigh degree of confirmation "exact sciences or hard sci-ences"; other sciences with lower degrees of confirmation"soft sciences." In fact we should only distinguish naturalsciences and social sciences, both empirical, and mathemat-ics. Only mathematics affords us abstract certainty, while allsciences-no matter how much sophisticated mathematicsthey contain in their theoretical structure-are subject to

.experimental verification.Now, what happens if repeated experimental evidence

disconfirms the theory? Do we have to reject the theory?

35

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36 THEORY FORMATION IN FLUID MECHANICS

Not necessarily. We can introduce an additional postulate ora corrolary which brings the theory in line with the experi-mental data. This process can be continued until the theoryis so loaded with additional hypotheses that it becomes verycumbersome. In that case, a search is warranted for a theorywith a new set of axioms, which is more compact, andtherefore more elegant.

What happens to the old theories? Should they be relegat-ed to the junkheap of history? Certainly not. Beside formingan integral part of mankind's intellectual heritage and there-fore worthy of conservation and study, many of the super-seded theories maintain their validity and usefulness withinwell defined albeit narrower boundaries, and within theselimitations, can indeed be utilized very efficiently. Newtoni-an mechanics is a good example. The theory of mechanics offluids, which is ultimately based on the laws of Newtonianmechanics, provides an interesting example of the checkeredhistory of theory formation in this field.

SOME THEORIES OF FLUID MECHANICS

Aristotle's (384-322 B.C.) concept of "horror vacuii" or"nature's abhorrence of a vacuum" could be considered asone of the earliest theories of fluid motion. On the basis ofthis postulate, we are able to explain why liquid will not flowout of an- inverted can with a single small hole punched intoit; for if the liquid discharged from the hole, a vacuum wouldbe created at the upper end of the can, which would becontrary to the postulate that nature will not tolerate avacuum. On the basis of this theory we can predict that thesame thing will occur during future experiments; we can alsopredict that if a second small hole is punched into the can,liquid will pour out of it, since now air can rush in andprevent the formation of a vacuum. This is a perfectlyrespectable and adequate theory for the phenomenon athand. The short-coming of the theory is that it is qualitative-ly and quantitatively very limited and incapable of explainingand/or predicting many of the other observed phenomena offluid flow.

Another early attempt to formulate a theory of hydrostat-ics was by Archimedes (287-212 B.C.). [For this and allother references, unless specifically mentioned, see Rouseand Ince, 1957]. Judging by the deductive method employedand the validity of the results achieved, Archimedean hydro-statics is an "amazing monument in the history of fluidmechanics. He based his theory upon two postulates:

Postulate 1: We pose in principle that the nature offluids is suchthat its parts being uniformly placed and continuous, that whichis less pressed is displaced by the one which is pressed themore, and that each part is always pressed by the whole weightof the column perpendicularly above it, unless this fluid isenclosed some place or is compressed by something else.Postulate 2: Let it be granted that bodies which are forcedupwards in a fluid are forced upwards along a perpendicularwhich passes through their center of gravity.

Based on these two postulates, Archimedes derived, bypurely mathematical deduction, many of the propositions ofhydrostatics and equilibrium offloating bodies. Some twentycenturies later Lagrange commented on this: "This work isone of the most splendid monuments attesting to the geniusof Archimedes, for it contains a theory of stability of floatingbodies to which the moderns have added very little."Despite the limited scope of Archimedes' hydrostatic the-

ory, the results obtained are impressive. Even more note-worthy, however, is the method employed by Archimedes.It is a perfect example of the method of theory formationdiscussed in the introduction. There is nothing self-evidentor obvious about the axioms, especially when viewed in thegeneral philosophical and scientific atmosphere of the thirdcentury B.C. The theorems deduced by purely geometricalconsiderations, however, can be verified-even thoughthere is no evidence that Archimedes did so himself-andbear out the validity of the postulates.

The next major contribution to hydrostatic theory was byStevin (1548-1620). Stevin followed the method of Archime-des by setting down postulates and deriving propositions ortheorems by logical deduction. His exposition is a goodexample of the axiomatic method, based on two postulates.

Postulate VI: That the upper surface of the water (what isordinarily called the fleur d'eau) be plane and level, that is tosay, parallel with the horizon.

The other postulate, given In terms of a theorem andproposition, reads:

Any designated body of water maintains whatever position isdesired in water, (because if it were not so) . . . . . this waterwould be in perpetual motion, which is absurd.

As a corollary to the second postulate, Stevin stated that"a solid body parigrave to water holds itself in such positionand place as is desired." Broadly interpreted, this impliesthat if any part of the water is replaced by a rigid body uf ~;1~same density, the forces exerted on it by the rest of the waterwill remain unchanged.

On the basis of the two postulates, Stevin demonstrated,for example, that: "The bottom of a mass of water, parallelto the horizon, supports a weight, equal to the weight of acolumn of water of which the base is the aforesaid bottomand the height a line perpendicular to the horizon, betweenthe bottom and the surface."

Thus, the so-called hydrostatic paradox was shown not bebe a paradox at all, but a lawful proposition in a new theoryof hydrostatics. In evaluating this theory of hydrostatics,one must st~ll agree with Dijksterhuis that a very essentialconcept was missing from his reasoning: "Stevin does notknow the idea of hydrostatic pressure acting at a point of theliquid equally in all directions; the consequence is thatStevin does not succeed in deducing the various subjectstreated by him (Archimedes' principle, hydrostatic paradox,pressure upon an inclined wall) from a single point of view tobe brought into relation with static conditions." It would beinteresting, however, to perform a thought experiment todemonstrate how close Stevin came to this concept ofpressure [Conant, 1951].

Consider the postulate of the impossibility of perpetualmotion. In Figure 1 is shown a vessel containing a liquid,with a tube running through the sides of the vessel asindicated. If the pressure at A in one direction were greaterthan in the other, then the liquid would flow around the tube;but this would be perpetual motion, which is impossible bypostulation. Hence the pressure in both directions must bethe same. It is this concept of pressure which eluded Stevin.

Pascal (1623-1662) reformulated many of Stevin's argu-ments and demonstrated the principle of the instantaneoustransmissibility of pressure. In the case of the hydraulicpress, which he called "a machine for multiplying forces,"

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THEORY FORMATION IN FLUID MECHANICS 37

Fig. 1. Demonstration of the concept of pressure based on theimpossibility of perpetual motion [Conant, 1951]

he introduced the notion of pressure as force per unit area,but there still was no clear statement about the concept ofpressure acting equally in all directions. It remained forEuler (1707-1783) to formulate this concept in a clear formand thus to lay the foundations of a more ample and usefultheory of hydrostatics. How fruitful a concept this is canperhaps best be judged from the fact that Euler used it alsoas one of the axioms for a theory of hydrodynamics. In aseries of three papers, published in the Memoires de L'Aca-demie des Sciences de Berlin, 1755, Euler formulated a newtheory of fluid mechanics.

The historical importance of these papers is not only thatthey created a new theory of hydrostatics and hydrodynam-ics, but in doing so introduced a new principle in mechanics,the continuum concept of matter. In many cases, the historyof scientific thought is too intertwined to allow settingdefinite dates for scientific breakthroughs, except by tradi-tion or for convenience. Euler was influenced, as he himselfhas admitt~d, by the ideas of pressure and continuum putforward by John Bernoulli (1667-1748), who first suggestedthe method of calculating the force acting on an infinitesimalelement. "A true and genuine method," Euler called it.Around the same time, d'Alembert was tangling with ideasof internal pressure but was unable to free himself from theatomistic view of matter.

By 1749, Euler I had reached the conclusion that furtheradvances in mechanics were only possible by giving up theatomistic view of matter. Instead, he put forward the con-cept of a continuous medium, defined a particle to mean apoint in that continuum, and postulated an internal pressureat a point with equal intensity in all directions. On this onepostulate of pressure Euler erected, by logical deduction, acompletely new, extremely broad and very simple theory ofhydrostatics.

Now the question arises whether this theory is any betterthan its predecessors and if so, why? Furthermore, is thistheory the final formulation of fluid statics? The answer tothe first question is easy; it is better because it is simpler,more general, and it works. Up to now we have found no

observation or experiment which contradicts the theory, sountil further notice, we accept it as a useful theory.

There is nothing to indicate, however, that this is theultimate of all possible theories of hydrostatics. It is entirelywithin the realm of possibility that some day a better theorywill be found.

It is customary to indicate the origin of theoretical fluiddynamics with the publication, in 1738, of Daniel Bernoulli's"Hydrodynamica." Leaving aside, for the time being, thelegitimacy of this assumption, it is certain that until thebeginning of the 18th century there was no broad theory offluid motion. Daniel Bernoulli's "Hydrodynamica" was alandmark in the history of fluid mechanics, not only becauseit coined an imaginative and descriptive new name, but alsobecause it attempted to build a wider theory based on thepostulate of the conservation of the total energy of a body offluid. The criticisms of the disciples of Euler are all valid:The "Hydrodynamica" was obscure, complicated and limit-ed in scope, treating mainly some of the problems of appliedhydraulics. Nevertheless, it was a theory as defined in theintroduction of this paper.

When the scientific climate in a given period in history isfertile for the development of new conceptual schemes,cross-fertilization of ideas does frequently lead to the suc-cessful formulation of new theories. The first half of the 18thcentury was ripe for a theory of hydrodynamics. The effortsof Daniel Bernoulli, d'Alembert, Clairaut, John Bernoulliand others to establish a broader and better theory remainedby and large fruitless; but the sparks of insight emanatingfrom their work kindled Euler's imagination. In 1755, Eu-ler's efforts culminated in the publication of a classicalhydrodynamics of ideal fluids, which, in form, content andnotation, remains essentially unchanged until today.

In Euler's hydrodynamics, the fundamental postulate wasagain the equality of pressure in all directions at a point in acontinuum. In addition, he introduced Newton's momentumprinciple which he had earlier adapted to apply to aninfinitesimal element in a rectangular coordinate system,fixed in space. The third postulate in Euler's theory of fluidmotion was the principle of conservation of mass, known inhydrodynamics as the continuity equation. The history ofthis principle is more difficult to follow, but it is beyondqyestion that the first quantitative statement of the continu-ity equation is to be found in the writings of Leonardo daVinci. It is written in simple language, about the flow in ariver, expressed in forms of proportions and applicable toincompressible fluids: "A river in each part of its length in anequal time gives passage to an equal quantity of water,whatever the width, the depth, the slope, the roughness, thetortuosity. "

In this form the continuity equation was used by thehydraulicians for practical purposes; d'Alembert and JohnBernoulli recognized the principle of continuity as a funda-mental notion in hydrodynamics, but credit for the clear andelegant formulation of the principle of mass conservation in acontinuum must go to Euler. Euler treated the problem in itsfull generality by extending his analysis to embrace com-pressible fluids.

Now, once again, it is pertinent to ask why this theory isbetter than, say, Bernoulli's. It is better 'because it worksbetter; that is, it explains and predicts a larger range ofphenomena with a greater degree of reliability and withgreater economy of intellectual effort.

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38 THEORY FORMATION IN FLUID MECHANICS

Is this the final theory of hydrodynamics? Obviously not,since already there have been occasions when disconfirmingevidence put into question the validity of the theory.D'Alembert's paradox-that the net thrust on a sphere in aflow field was calculated to be zero-was a severe blow tothe prestige of Eulerian hydrodynamics, which we now callthe theory of potential flow. To bring it in line with theexperimental results, additional hypotheses had to be intro-duced. One such postulate was the free streamline concept,which, while still working with an ideal fluid, was successfulin overcoming some of the more blatant departures of thetheory from observation. Other important additional postu-lates were the concepts of viscosity and boundary layer, theintroduction of which enlarged considerably the scope andthe limits of validity of hydrodynamic theory. However,Eulerian hydrodynamics is still very effectively used in thetheory of deep water waves and in the theory offlow throughporous media.

Euler's success was largely due to his ability to breakaway from the conventional atomistic thinking of his con-temporaries and to visualize a continuous medium. It wouldbe absurd to maintain that this is the only valid way tohydromechanics. It is quite possible that an even bettertheory can be formulated on atomistic concepts. But, untilthen, and so long as it works, we provisionally believe inwhat we have.

The conceptual framework of physical hydrology rests onthe foundations of Newtonian physics, and, as such, itstheoretical structure is deterministic. Yet we find it neces-sary to use the mathematics of probability and randomfunctions to overcome the uncertainties encountered in itsoperation. Does this mean that something is wrong with theconceptual framework, and that we should now try to move

away from Newtonian physics and seek a better understand-ing by adapting the tenets of modern physics? Would therebe a philosophical necessity and an operational advantage? Ithink not; and I have the feeling that physicists wrestlingwith the problems of the particle zoo would advise us againstsuch a move at this time. Even though the possibility is notprecluded that some day we might not find a better theoreti-cal foundation, for the present and the near future thedeterministic structure is adequate; the imperfections anduncertainties are not conceptual. For the practical andpracticing hydrologist, to whom all hydrological processessuch as rainfall, runoff, infiltration, etc. appear to be ran-dom, I recommend the point of view of Bras and Rodriguez-Iturbe:

Randomness and the applicability of random-process theorymay be inherent in the structure of the process or may resultfrom the lack of knowledge or from the scale of observation.Many arguments, mainly philosophical, exist to refute or justifythe above statement. The techniques and philosophy in thisbook have proved their usefulness to us. The nonbelieverhopefully will be impressed by the power of the varioustechniques and therefore accept them. [Bras and Rodriguez-lturbe, 1984]

REFERENCES

Conant, J. B., Science and Common Sense, 371 pp., Yale Universi-ty Press, New Haven, CT, 1951.

Bras, R. L., and I. Rodriguez-Iturbe, Random Functions andHydrology, 559 pp., Addison-Wesley, Reading, MA, 1984.

Rouse, H., and S. Ince, History of Hydraulics, 264 pp., IowaInstitute of Hydraulic Research, Iowa City, lA, 1957.

S. Ince, Department of Hydrology and Water Resources, Univer-sity of Arizona, Tucson, AZ 85721.

History of Geophysics: Volume 3The History of Hydrology

Some Early Attempts at Theory Formation in Fluid MechanicsIntroductionSome Theories of Fluid MechanicsReferences