# Escape from the Jail of Shape; Dimensionality and Engineering Science

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## Abstract

In V. Sackville-West’s Poem, “The Land” the poet deals with craftsmen. The noem reads in part:

All craftsmen share a knowledge. They have held

Reality down fluttering to a bench;

Cut wood to their own purposes; compelled

The growth of pattern with the patient shuttle,

Drained acres to a trench.

Control is theirs. They have ignored the subtle

Release of spirit from the jail of shape.

They have been concerned with prison, not escape.^{1}

## Keywords

Heat Transfer Dimensionless Parameter Engineering Science Dimensional Analysis Fluid Resistance
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## Notes

- 1.V. Sackville-West, ‘The Land — Summer’, in
*Collected Poems*(London: Webb and Bower, 1989), p. 69. Reprinted courtesy of Curtis Brown and John Farquharson. I would like to thank Professor Rutherford Arris for calling my attention to this poem and for his inspiration and help over many years.Google Scholar - 2.On the diameter of tubes in relation to their mechanical properties, see Charles MacGregor, ‘Mechanical Properties of Materials’, in Theodore Baumeister
*et al.*(eds.),*Marks’ Standard Handbook for Mechanical Engineers*(New York: McGraw-Hill, 1978), Sec. 5, pp. 49–50. For a cylindrical tube the circumferential or “hoop” stress,*S*, is equal to*S = pr/t*where*p*is the internal pressure,*r*the radius and*t*the plate thickness.Google Scholar - 3.On “specific speed” see W.G. Whippen, ‘Hydraulic Turbines’, in Baumeister,
*Marks’ Standard Handbook for Mechanical Engineers*, Sec. 9, pp. 137–138. The discovery of specific speed will be discussed below. One of the most important keys to rational design of water wheels was to avoid “angle of incidence” losses due to shock at entrance. To eliminate shock it is necessary to have “tangent entry” of the water, which, in turn, required that the designer knew the speed of the rotor.Google Scholar - 4.Robert L. Daugherty and Joseph B. Franzini,
*Fluid Mechanics with Engineering Applications*(New York: McGraw-Hill, Seventh Edition, 1977), pp. 445–446.Google Scholar - 5.J.J. Cornish, ‘Aeronautics’, in Baumeister,
*Marks’ Standard Handbook for Mechanical Engineers*, Sec. 11, pp. 60–64, 66–71. In the most obvious case, in laminar flow the boundary layer flows as if made up of many thin layers moving smoothly parallel to one another; in a turbulent boundary layer the flow is no longer parallel; there are irregular motions (such as eddies) normal to the surface of the layer.Google Scholar - 6.There is a modern tendency to restrict the meaning of “dimensional analysis” to the contents of the Vaschy—Buckingham π theorem, but a much broader definition is used here. I have been much influenced by Stephen J. Kline,
*Similitude and Approximation Theory*(New York: McGraw-Hill, 1965), who uses “dimensional analysis” in a broad sense. He analyzes four distinct types of dimensional analysis (pp. 262–215). These are (1) the π theorem (discussed below), (2) the use of governing equations, that is, normalizing the governing differential equations to make them dimensionally homogeneous. This method was pioneered by Lord Rayleigh; it is favored by physicists, and (3) the “method of similitude”, (which is largely Kline’s own creation). To this one should add the method of “configurational analysis” which is built upon Kline’s work, but which emphasizes the use of group theory for the analysis of dimensional issues. (For configurational analysis, see H.A. Becker,*Dimensionless Parameters*(New York: John Wiley, 1976), Vol. 3, pp. 10–11. To these I add historical methods of analyzing dimensions such as that of Fourier, who invented the first formalism of dimensional analysis. In addition, I follow Kline in including the derivation of model laws within dimensional analysis (pp. 2–7). Model laws are often called rules or principles of similitude (or similarity). Other works dealing with dimensional analysis that I found particularly helpful include Percy W. Bridgman,*Dimensional Analysis*(Cambridge, Mass: Harvard University Press, 1921); W.J. Duncan,*Physical Similarity and Dimensional Analysis*(London: Edward Arnold, 1953); H.E. Huntley,*Dimensional Analysis*(London: McDonald, 1953); Henry L. Langhaar,*Dimensional Analysis and Theory of Models*(New York: John Wiley, 1951); and L.I. Sedov,*Similarity and Dimensional Methods in Mechanics*(New York: Academic Press, 1959). On modelling theory I found Langhaar especially useful. Several of these contain useful bibliographies, notably Becker and Kline. Kline has comments on major contributions to the literature which can be found in the text . On the history of dimensional analysis I am deeply in the debt of Enzo O. Macagno, ‘Historico-Critical Review of Dimensional Analysis’,*J. Franklin Inst.*232(1971), pp. 393–394. Macagno’s study is a critical and informative history of dimensional analysis; Macagno follows the more common practice of restricting the meaning of “dimensional analysis” to the Vaschy—Buckingham theorem and closely related matters.Google Scholar - 7.Philo of Byzantium quoted in Morris R. Cohen and I.E. Drabkin,
*A Sourcebook in Greek Science*(Cambridge, Mass.: Harvard University Press, 1958), p. 318.Google Scholar - 8.On parameter variation see Walter G. Vincenti, ‘The Air-Propeller Tests of W. F. Durand and E. P. Lesley: A Case Study in Technology Methodology’,
*Technology and Culture*20 (October, 1979), p. 714.CrossRefGoogle Scholar - 9.The traditional phrase “Research and Development” reflects the now outdated “applied science” theory of technology, which has now been replaced by the “interactive” model. There is evidence suggesting that it is more common for research to follow the appearance of a novel engineering (or design) idea. For example, one of the largest and best of the innovations studies was done by the well-known economists Sumner Myers and D. G. Marquis, who found that successful innovations in the industries studied did not originate with research in 95 percent of the cases, but came about in implementing a technological idea. (See Sumner Myers and D.G. Marquis,
*Successful Industrial Innovations*(Washington D. C.: The National Science Foundation, NSF 69–17, 1969) especially Table 19, p. 46.)Google Scholar - 9a.On the interactive model see Barry Barnes, ‘The Science—Technology Relationship: A Model and a Query’,
*Social Studies of Science*12 (February, 1982), pp. 166–171 and Edwin T. Layton, Jr., ‘Mirror Image Twins: The Communities of Science and Technology in 19thCentury America’,*Technology and Culture*12 (October, 1971), pp. 562–580. Studies of technology as knowledge include Edwin T. Layton, Jr., ‘Technology as Knowledge’,*Technology and Culture*15 (January, 1974), p. 41; Walter G. Vincenti,*What Engineers Know and How they Know It, Analytical Studies from Aeronautical History*(Baltimore: Johns Hopkins, 1990).CrossRefGoogle Scholar - 10.Marcus Vitruvius Pollo,
*The Architecture of Marcus Vitruvius Pollo in Ten Books*, transl. Joseph Gwilt (London: Priestley and Weale, 1826), p. 342.Google Scholar - 11.Charles Hutton,
*A Course of Mathematics*, 2 vols. (New York: Samuel Campbell and others, American edition edited by Robert Adrian based on 5th and 6th London editions, 1818). Volume II is an applied mechanics for engineers which used a simplified Newtonian notation. “Hutton” seems to have been very influential in the English speaking countries. It went through a total of 29 editions, of which nine were American, between 1798 and 1860.*(National Union Catalog of pre-1956 Imprints*262, pp. 184–186.) Though still used in English speaking countries as late as 1860, Hutton’s work was made obsolete by the French tradition which employed Leibnitz’s notation and shifted mechanics from a geometric to an analytical formalism. For a broad survey of mechanics which stresses the European and French perspectives, see René Degas,*Histoire de la Mécanique*(Neuchatel: Editions du Griffon, 1950).Google Scholar - 12.Isaac Newton,
*Mathematical Principles of Natural Philosophy and His System of the World*, 2 vols., transl. Florian Cajori (Berkeley: University of California Press, 1962), Vol.II, pp. 327–328. See also Proposition 33, Theorem 26 including the several corollaries (pp. 328–331) (hereafter: Newton,*Principia).*Google Scholar - 13.Newton, Principia, Book I, section 12 deals with spherical bodies, section 13 extends the analysis to non-spherical bodies, pp. 193–225. See particularly Proposition 88, Theorem 45, “If the attractive forces of the equal particles of any body be as the distance of the places from the particles, the force of the whole body will tend to its center of gravity; and will be the same with the force of a globe, consisting of similar and equal matter, and having its center in the center of gravity.” (I, p. 216).Google Scholar
- 14.Newton’s ontology also included his “geometrization of space”, and other things. These issues have long been debated. Though now somewhat outdated, historically important works in understanding Newton’s ontology (or metaphysics) included Edwin A. Burtt,
*The Metaphysical Foundations of Modern Science: A Historical and Critical Essay*(New York: Harcourt Brace, 1925) and Alexandre Koyré,*Etudes Galiléennes*(Paris: Hermann, 1939) and the same author’s,*Metaphysics and Measurements: Essays in Scientific Revolution*(London: Clapham and Hall, 1968).Google Scholar - 15.Newton,
*Principia*, I, pp. 328–336. Newton’s awareness of the limits, and possibilities, of his ontology is remarkable and evident in these propositions of fluid resistance. Newton did deal with some of the effects of the mutual interactions of the particles of a fluid, in Proposition 33, pp. 328–329, where he shows that the centripetal and centrifugal forces by which the particles of the system act on each other, can be reduced to simple considerations of the force and matter as with a simple system by using the similarity principles developed in Proposition 32. Thus Newton refuted Descartes, while not solving the problems of Fluid Mechanics with which modern engineers must deal. The fact that Newton’s physics was associated with a particular ontology does not mean that it was invalid except with that ontology. Thus it is possible (with later analytical tools) to expand Newton’s*F= MA*to get the Navier—Stokes equation, and by considering the ratios of two such expressions for two different systems to derive the key dimensional concepts of dynamic and kinematic similarity. This is done, for example, by Max Jakob,*Heat Transfer*, 2 vols. (New York: John Wiley, 1949, 1955), Vol. I, pp. 16–19, 429–430. It would be a-historical to expect Newton to have done this; the fact that it can be done is a tribute to the foundations which Newton provided for physical science. The Navier—Stokes theorem will be discussed below.Google Scholar - 16.Theodore von Kármán, ‘Isaac Newton and Aerodynamics’,
*J. Aeronautical Sci.*9 (December, 1942), pp. 521–522, 548.CrossRefGoogle Scholar - 17.Charles de Borda, ‘Expériences sur la résistance des fluides’,
*Mémoires de l’Académie Royale des Sciences*(1767), pp. 495–503. Borda’s tests were repeated and extended using a towing tank (so that the model did not move through disturbed water) and the towing tank became a significant advance in model testing methodology. Some other conflicts between theory and experiment in the 18th century are summarized in Terry S. Reynolds,*Stronger Than a Hundred Men. A History of the Vertical Water Wheel*(Baltimore: Johns Hopkins, 1983), pp. 204–223, 237–241.Google Scholar - 18.Froude’s relationship to Ferdinand Reech and the story of the law of similarity and the dimensionless parameter named after Froude is complicated and will not be discussed in this paper. As Rouse and Ince note, Froude did not originate the similarity rule (it was discovered by Reech), and Froude never used the “Froude Number” . On the other hand he did anticipate boundary layer analysis. See Rouse and Ince,
*History of Hydraulics*, pp. 154–155, 182–187.Google Scholar - 19.John Smeaton, ‘An Experimental Inquiry Concerning the Natural Powers of Water and Wind to Turn Mills, and Other Machines, Depending upon Circular Motion’,
*Phil. Trans. Royal Soc.*51(1, 1759–1760), pp. 100–101. Smeaton’s role in the experimental tradition in engineering is discussed in Terry S. Reynolds,*Stronger than a Hundred Men*, pp. 218–233.Google Scholar - 20.Reynolds,
*Stronger than a Hundred Men*, pp. 223–226, 280–284.Google Scholar - 21.Quoted in Edwin T. Layton, Jr., ‘American Ideologies of Science and Engineering’,
*Technology and Culture*17 (October, 1976), p. 693.Google Scholar - 22.Edwin T. Layton, Jr., ‘Scientific Technology: the Hydraulic Turbine and the Origins of American Industrial Research’,
*Technology and Culture*20 (January, 1979), pp. 72–78, 81.CrossRefGoogle Scholar - 22a.James B. Francis,
*Lowell Hydraulic Experiments*(New York: D. Van Nostrand, second edition, 1868).Google Scholar - 23.Francis,
*Lowell Hydraulic Experiments*, p. 52; the experimental derivation of the Francis weir formula is on pp. 69–135.Google Scholar - 24.Francis,
*Lowell Hydraulic Experiments*, p. 52 and Table 4, pp. 53–54. The properties which Francis measured turned out to be dependent upon dimension, as will be noted below in discussing specific speed.Google Scholar - 25.One can infer from Newton’s Proposition 32 basic geometric, time, and force similarity conditions, where 1 and 2 stand for similar systems, and
*S, T*and*F are*any lengths, time intervals, and forces occurring in the similar systems, the relations S_{1}/S2 = a, T_{1}/T_{2}=*b*, and T_{1}/T_{2}=*c*, where*a, b*, and*c*are factors of proportionality. A common alternative for this system of units is to consider length, time and force instead of mass.Google Scholar - 26.Peter Jeffrey Booker,
*A History of Engineering Drawing*(London: Chatto & Windus, 1963), pp. 18–36, 86–106.Google Scholar - 27.Stephen Timoshenko,
*History of the Strength of Materials*(New York: McGraw-Hill, 1953), pp. 190–197, 283–288. Both of these are systems of analyzing the stresses within a framework by graphical means. Culmann, a mathematician by training, became convinced that structural calculations could be best done graphically. See Carl Culmann,*Die graphische Statik*(Zurich: Meyer und Zellers Verlag, 1869).Google Scholar - 28.Julius Weisbach,
*A Manual of the Mechanics of Engineering*(New York: John Wiley, Trans. A. Jay Du Bois, 1880), II, pp. xliii—xlv, 222–226, 321–326, 355–358, 397–400.Google Scholar - 29.D.S.L. Cardwell,
*From Watt to Clausius*(Ithaca: Cornell University Press, 1971), pp. 80–83.Google Scholar - 30.Francis,
*Lowell Hydraulic Experiments*, pp. 36–43.Google Scholar - 31.Larry Owens, ‘Vannevar Bush and the Differential Analyzer: Text and Context of an Early Computer’,
*Technology and Culture*27 (January, 1986), pp. 88–89.CrossRefGoogle Scholar - 31a.John Perry,
*Calculus for Engineers*(London, 1897).Google Scholar - 32.Philip Franklin, ‘Graphical Representation of Functions’, in Baumeister,
*Marks’ Standard Handbook for Mechanical Engineers*, Sec. 2, pp. 54–62.Google Scholar - 33.There is an interesting discussion of graphical statics and the link between this science and design in David Billington,
*Robert Maillart’s Bridges. The Art of Engineering*(Princeton: Princeton University Press, 1979), pp. 5–7Google Scholar - 34.As noted above, I am much indebted to Kline and follow his very broad definition of dimensional analysis. He categorizes dimensional analysis as a part of “fractional analysis” (e.g., analysis which provides useful but not complete information) and sees the latter as a fundamental methodology in engineering. See Kline,
*Similitude and Approximation Theory*, pp. v–vii, 2–7, 87–219.Google Scholar - 35.I have used Alexander Freeman’s English translation: Joseph Fourier,
*The Analytical Theory of Heat*(New York: Dover reprint of the 1878 English edition, 1955).Google Scholar - 36.
*Ibid.*, p. 6.Google Scholar - 37.
*Ibid.*, 2. This is true for heat transfer where rates (that is time) are critical; but the later science of thermodynamics can be considered in terms of equilibrium, without concern for the rates at which the transfer of heat takes place.Google Scholar - 38.
*Ibid.*, pp. 23–26.Google Scholar - 39.Macagno, ‘Historico-Critical Review of Dimensional Analysis’, pp. 393–394. The further development of heat transfer added a good deal of complexity to Fourier’s theory. It would be surprising if the first investigator in a new science discovered everything of relevance and importance. Fourier’s theory was primarily useful in conduction; though he made beginnings in convective and radiative heat transfer, these latter theories were less satisfactory. Modern heat transfer uses the concept of thermal diffusivity in the study of heat conduction, and especially the Fourier number, a dimensionless parameter needed to correlate information in unsteady heat conduction. (E.R.G. Eckert and Robert M. Drake,
*Heat and Mass Transfer*(New York: McGraw-Hill, 1959), p. 77).Google Scholar - 40.Fourier,
*Theory*, p. 127. See also pp. 126–130. The variables Fourier admitted were x (length),*t*(time), v (temperature),*c*(heat capacity), and two heat transfer coefficients (in Fourier’s nomenclature), “specific conductibility”,*K*, and “surface conductibility”,*h*(p. 130). See also Jakob,*Heat Transfer*, Vol. I, pp. 3–5. ‘Google Scholar - 41.Macagno, Historico-Critical Review of Dimensional Analysis , pp. 391–402. See also H.E. Huntley,
*Dimensional Analysis*(London: McDonald, 1952), contains a brief history (pp. 33–44). For another historical study see Alton C. Chick, ‘The Principle of Similitude’, in John R. Freeman (ed.),*Hydraulic Laboratory Practice*(New York: the American Society of Mechanical Engineers, 1929), pp. 796–797.Google Scholar - 42.Max Jakob,
*Heat Transfer*, Vol. I, pp. 3–5.Google Scholar - 43.
*Ibid., I*, p. 5.Google Scholar - 44.
*Ibid.*, I, pp. 716–726.Google Scholar - 45.Eckert and Drake,
*Heat and Mass Transfer*, pp. x—xiv. The appendices of properties are located on pp. 493–521.Google Scholar - 46.Edwin T. Layton, Jr. and Richard J. Goldstein, ‘Interviews with Ernst R. G. Eckert’, (five videotaped interviews, Walter Library, University of Minnesota, Minneapolis, Minnesota).Google Scholar
- 47.Eckert and Drake,
*Heat and Mass Transfer*, pp. x—xiv.Google Scholar - 48.Newton,
*Principia*, Book II. See also Rouse and Ince,*History of Hydraulics*, pp. 83–86.Google Scholar - 49.George Gabriel Stokes, ‘On the Effect of the Internal Friction of Fluids on the Motion of Pendulums’, in George Gabriel Stokes,
*Mathematical and Physical Papers, 5*vols. (Cambridge: Cambridge University Press, 1880–1905), Vol. III, pp. 1–141, especially pp. 7–17. The broader historical context of the Navier-Stokes equation is sketched by Rouse and Ince,*History of Hydraulics*, pp. 193–200. Dominique Noir has argued that the Stokes law of similitude for the drag of a sphere in a resisting fluid can be derived from Aristotle’s physics. See Dominique Noir, ‘La première loi de similitude de la mécanique des fluides’,*Revue de l’Histoire des Sciences et de leurs Applications*25 (1972), pp. 271–274.Google Scholar - 50.Hermann von Helmholtz, ‘Über ein Theorem, geometrisch ähnliche Bewegungen flüssiger Körper betreffend, nebst Anwendung auf das Problem, Luftballons zu lenken’,
*Monatsber. Kön. Preuss. Akad. Wiss. Berlin*18B (1873), pp. 501–514. This is a remarkable foreshadowing of dimensional analysis in which he anticipated the later similarity considerations associated with the Froude, Mach, and Reynolds criteria; there is no evidence, however, that he anticipated the existence or the role of dimensionless parameters. For the more modern modelling of the Helmholtz’s problem see Edgar Buckingham, ‘Model Experiments and the Forms of Empirical Equations’,*Trans. Am. Soc. Mech. Engineers*37 (1915), pp. 273–277.Google Scholar - 51.Lord Rayleigh (John William Strutt), ‘The Principle of Similitude’,
*Nature*95 (1915), pp. 66–68, 644 is a very remarkable virtuoso performance. Kline has a remarkably clear and insightful chapter in which he analyzes the comparatives advantages and drawbacks of Rayleigh’s method with that of Buckingham and others. (See Kline,*Similitude and Approximation Theory*, pp. 262–215.) For a succinct statement, see also Murdock, ‘Mechanics of Fluids’ in Baumeister,*Marks’ Standard Handbook for Mechanical Engineers*, Sec. 3, p. 50.CrossRefGoogle Scholar - 52.Rayleigh’s use of dimensional analysis and his use of the inverse Reynolds number is discussed in Macagno, ‘Historico-Critical Review of Dimensional Analysis’, pp. 395–396.Google Scholar
- 53.Lord Rayleigh (John William Strutt), ‘On the Question of the Stability of Fluids’,
*Scientific Papers*, 6 vols. (New York: Dover Reprint of 1869–1919, 1964), Vol. IV, p. 575.Google Scholar - 54.Jakob,
*Heat Transfer*, pp. 429–431. Reynolds is not completely clear on his reliance upon Stokes in this paper.Google Scholar - 55.Osborne Reynolds, ‘An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous and the Law of Resistance in Parallel Channels’,
*Papers on Mechanical and Physical Subjects*, 3 vols. (Cambridge: Cambridge Univ. Press, 1900–1903), Vol. II, pp. 54–55. See also Rouse and Ince,*History of Hydraulics*, pp. 206–211, which puts Reynolds’ work in perspective.Google Scholar - 56.Reynolds, ‘On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion’,
*Papers*, II, p. 543.Google Scholar - 57.
*Ibid.*, pp. 51–105.Google Scholar - 58.Macagno, ‘Historico-Critical Review of Dimensional Analysis’ , pp. 395–396. Macagno points out that Fourier used a dimensionless term, and that he was aware of its dimensionless character (p. 394). Though Reynolds was very important, the concept of the dimensionless parameter and its role in engineering was the work of many investigators, and only gradually crystallized in the period 1890 to 1920.Google Scholar
- 59.Reynolds, ‘On Certain Laws Relating to the Régime of Rivers and Estuaries and on the Possibility of Experiments on a Small Scale’,
*in Papers*, II, pp. 326–335. See also his three committee reports on scale-modeling in hydraulics,*ibid.*, pp. 351–380.Google Scholar - 60.Reynolds, ‘On the Theory of Lubrication and its Application to Mr. Beauchamp Tower’s Experiments . . .’ ,
*ibid.*, pp. 228–310.Google Scholar - 61.Reynolds, ‘On the Dynamic Theory of Incompressible Fluids and the Determination of the Criterion’, and ‘On the Extent and Action of the Heating Surface of Steam Boiler’,
*ibid.*, pp. 81–85. This analogy linked heat with mass transfer; it is discussed in Ernst Eckert, ‘Heat Transfer’, in D.M. McDowell and J.D. Jackson (eds.),*Osborne Reynolds and Engineering Science Today*(Manchester: Manchester University Press, 1970), pp. 160–175. McDowell and Jackson’s anthology gives a good idea of the scope of Reynold’s work.Google Scholar - 62.Rudolph Camerer, ‘Klassification von Turbinen’,
*Z. Ver. Deutsch. Ing.*49 (4 March, 1905), p. 308.Google Scholar - 63.Rudolph Camerer,
*Vorlesungen über Wasserkraftmaschinen*(Leipzig: Wilhelm Englemann, 1914). Camerer was assisted by the fact that American turbine companies produced stock turbines at standard sizes and these were tested at the Holyoke Testing flume at constant head, so that entire families of geometrically similar turbine could be compared in their speed and their “swallowing ability” (capacity) at constant head (p. 295). For the capacity coefficient of turbines, see Denis G. Shepherd,*Principles of Turbomachinery*(New York: Macmillan, 1956), pp. 31–32.Google Scholar - 64.Daugherty and Franzini,
*Fluid Mechanics with Engineering Applications*, pp. 445–446, 468–469, 490–493.Google Scholar - 65.John H. Lienhard, ‘Ernst Kraft Wilhelm Nusselt’,
*Dictionary of Scientific Biography*, s.v. See also John H. Lienhard, ‘Notes on the Origins and Evolution of the Subject of Heat Transfer’,*Mechanical Engineering*105 (June, 1983), pp. 20–27. See also Elizabeth Jacob, ‘Max Jakob; July 20, 1879—January 4, 1955’, and Virginia Dawson, ‘From Braunschweig to Ohio: Ernst Eckert and Government Heat Transfer Research’, and Ernst R.G. Eckert, ‘Ernst Schmidt — As I Remember Him’, in Edwin T. Layton and John H. Lienhard, eds.*History of Heat Transfer*(New York: American Society of Mechanical Engineers, 1988), pp. 87–116, 125–137, 139–146.Google Scholar - 66.John H. Lienhard, ‘Nusselt’, and ‘Notes on the Origins and Evolution of Heat Transfer’. Wilhelm Nusselt, ‘Das Grundgesetz des Wärmeüberganges’,
*Gesundheits-Ingenieur*38 (October, 1915), pp. 477–482. Nusselt’s correlation of the Nusselt, Prandtl, and Grashof was reviewed by Jakob,*Heat Transfer*, I, pp. 481–488. On the overall impact of Nusselt’s similarity methods and the Nusselt number on heat transfer, and Nusselt’s other publications of importance see (in addition to Lienhard), Jakob,*Heat Transfer”*I, pp. 496–499, 522, 527–528, 744. Ernst R. G. Eckert thought Nusselt’s paper so fundamental that he wrote that its publication in 1915 marked “the birth year of a science of heat transfer.” (Eckert and Drake,*Heat and Mass Transfer*, p. 229).Google Scholar - 67.Edgar Buckingham, ‘On Physically Similar Systems: Illustrations of the Use of Dimensional Equations’,
*Phys. Rev*4 (1914), pp. 345–376.CrossRefGoogle Scholar - 68.Aimé Vaschy, ‘Sur les lois de similitude en physique’,
*Ann. Télégraphiques*19 (1892), pp. 25–28 and the same author’s application of his principle in ‘Sur les lois de similitude de électricité’,*ibid.*, pp. 189–211. Macagno has useful remarks on Vaschy for which I am, again, in his debt; he points out that Vaschy did not make it completely explicit that the terms he dealt with were dimensionless, though he did note that his terms were independent of the system of units employed (a characteristic of dimensionless numhers).Google Scholar - 69.Aimé Vaschy, ‘Sur les lois de similitude en physique’, p. 345. Vaschy’s statement is very compressed; his statement of his theorem takes less than a page, and he moves on to illustrations and applications at the bottom of the first page. At the outset of the theorem he states that , “The most general law of similitude in mechanics and physics results in the following theorem.” He then states that a function of
*n*physical quantities*(a1, a*_{2}*...,an)*can be reduced to another of*(n—p)*parameters of the form f(x_{1}, x_{2}...,*xn(*_{p}) in which the parameters x1, x_{2}*xn*_{p}are monomial (”single term”) functions of a1, a_{2},.... Vaschy does not state that this is the most general or fundamental form for physical equations, only that it is possible. Vaschy, in his first statement, makes no statement suggesting that the x parameters are dimensionless. Later on in the paper, however, he does say that these terms are independent of the units in which these terms are expressed. This is today a definition of a dimensionless parameter, but perhaps in 1892 these useful entities were not so well known and an explicit statement might have helped some readers. In any case Vaschy and his fellow French telegraph engineers do not appear to have made wide use of the theorem. One problem appears to have been Vaschy’s extreme brevity. 70 Buckingham, ‘On Physically Similar Systems’,*J. Wash. Acad. Sci.*4, pp. 345–350 (Sec. 2, pp. 345–346). As Langhaar has shown, the mere fact of dimensional homogeneity leads directly to the π theorem. (Langhaar,*Dimensional Analysis*, pp. 55–58). In some early works the π terms were called variables, but the term parameter is more correct. The dimensionless parameters are composed of terms which, in the non-dimensional form, are correctly called variables. But the process of dimensional analysis does more than just group together “variables” into groups in which the units of dimension cancel out. ” In his paper ‘Model Experiments and the Forms of Empirical Equations’, p. 265, Buckingham presented a short version of his theorem which is strikingly similar to Vaschy’s earlier statement of the theorem.Google Scholar - 72.Buckingham’s paper ‘Model Experiments and the Forms of Empirical Equations’, pp. 273–296, presented to the American Society of Mechanical Engineers in 1915 was a particularly striking and influential example of Buckingham’s concern to popularize his work among engineering. See also Buckingham, ‘Physically Similar Systems’, pp. 347–353 and his ‘The Principle of Similitude’,
*Nature*96 (December 9. 1915). pn. 396–397, and his ‘Notes on the Dimensional theory of Wind Tunnel Experiments’,*Smithsonian Miscellaneous Collections*62 (4, 1914), pp. 15–26.Google Scholar - 73.Langhaar,
*Dimensional Analysis and Theory of Models*, p. 64. Buckingham discussed model experimentation in his ‘On Physically Similar Systems’, pp. 369–372, and with many practical examples in his ASME paper, ‘Model Experiments and the Forms of Empirical Equations’, pp. 273–296.Google Scholar - 74.Kline.
*Similitude and Approximation Theory.*Google Scholar - 75.For remarkable examples of the power of Kline’s methods see his Chapter 4, pp. 68–199.Google Scholar
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*Similitude and Approximation Theory*, pp. 74–75.Google Scholar - 77.Becker,
*Dimensionless Parameters*, p. 6.Google Scholar - 78.A. Sarlemijn and Peter Kroes, ‘Technological Analogies and Their Logical Nature’, in P.T. Dublin (ed.),
*Philosophy and Technology 4: Technology and Contemporary Life*(Dordrecht, Holland: Reidel, 1988), pp. 237–255.CrossRefGoogle Scholar - 79.Peter Kroes, ‘Structural Analogies between Physical Systems’,
*Brit. J. Phil. Sci.*40 (1989), pp. 145–154. Structural analogies include both physical models and dimensional analysis applied to governing equations. As Kroes points out, these structural analogies between physical systems (or the equations describing them) are not the same as the nomic isomorphism of laws.CrossRefGoogle Scholar - 80.
*Ibid.*, p. 153.Google Scholar - 81.Ernst R.G. Eckert and R.J. Goldstein (eds.),
*Measurement Techniques in Heat Transfer*(London: The Advisory Group for Aerospace Research and Development, NATO, 1970). This book does not include the Mach-Zehnder interferometer, which, however, is discussed by the editors in numerous publications.Google Scholar - 82.On the Mach-Zehnder interferometer see W. Kinder, ‘Theorie des Mach-Zehnder Interferometer und Beschreibung eines Gerätes mit Einspielgeleinstellung’,
*Optik*I (1946), pp. 413–448. On the role of Ernst R. G. Eckert in this development, see Virginia P. Dawson, ‘From Braunschweig to Ohio: Ernst Eckert and Government Heat Transfer Research’, in Layton and Lienhard,*History of Heat Transfer*, pp. 125–134.Google Scholar - 83.Günter Küppers, ‘On the Relation between Technology and Science — Goals of Knowledge and Dynamics of Theories. The Examples of Combustion Technology, Thermodynamics and Fluid Mechanics’, in Wolfgang Krohn, Edwin T. Layton, Jr., and Peter Weingart (eds.),
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