Origins of concentric cylinders viscometry*Prasannarao Dontula,a) Christopher W. Macosko,b) and L. E. Scriven

Department of Chemical Engineering and Materials Science,University of Minnesota, Minneapolis, Minnesota 55455

(Received 7 March 2005; final revision received 27 April 2005d

Synopsis

The history of the concentric cylinders apparatus for measuring the shear viscosity of liquids, andits attribution to Maurice Couette, have been explored. Examination of the Nineteenth Centuryliterature has revealed that the concept goes back to Stokes and later Margules, the design andexecution of the apparatus, apparently independently, to Perry, Couette, Mallock, and Schwedoff.Mallocks and Schwedoffs measurements were the most accurate and were within 1% of theviscosities derived from Poiseuilles measurements on the basis of no slip at the tube walls andcylinder surfaces. Measurement of fluid viscosity was closely linked to the adoption of the no-slipboundary condition at solid-fluid interfaces. 2005 The Society of Rheology.fDOI: 10.1122/1.1940640g

I. INTRODUCTIONConcentric cylinders are widely employed to measure shear viscosities of complex

liquids, i.e., liquids whose microstructures require more than microseconds, and typi-cally milliseconds or longer, to equilibrate locally within a flow. Familiar examples arepolymer solutions, certain surfactant solutions, concentrated suspensions of colloidal par-ticles, and many composite colloidal and polymeric materials in chemical, food, andother technologies.

Liquid contained in a narrow annulus between two coaxial cylinders, one or both ofwhich rotate, experiences a nearly uniform shear rate. In the simplest situation, onecylinder is stationary and the other is set in motion with either a constant velocity orconstant torque. To arrive at the viscosity, the torque on the stationary element or theangular velocity of the moving element is measured. The ratio of the torque to the angularvelocity is by a definition the shear viscosity of the liquid up to a constant.

m = 2pR2tY 4pFS a2R2a2 1DsVo VidG , s1d

where R is the radius of the inner cylinder, a is the ratio of the radius of the outer cylinderto that of the inner cylinder, Vo and Vi are, respectively, the angular velocities of theouter and inner cylinders, and t is the shear stress per unit length of the inner cylinder.

*This paper is based on a portion of C. W. Macoskos 2004 Bingham medal address, 15 February 2005,Lubbock, TX.

adPresent address: General Electric, Global R&D, Bangalore, India.bdAuthor to whom correspondence should be addressed; electronic mail: macosko@umn.edu 2005 by The Society of Rheology, Inc.807J. Rheol. 49s4d, 807-818 July/August s2005d 0148-6055/2005/49s4d/807/12/$25.00

808 DONTULA, MACOSKO, AND SCRIVENToday, this flow is widely referred to as Couette flow, and the method Couetterheometry, after Maurice Marie Alfred Couette, who is regarded to have first constructedsuch an apparatus and made accurate measurements fCouette s1888, 1890dg. Two authori-tative reviews in the 1930s fDryden et al. s1932d, Graetz and Stckl s1931dg establishedbeyond doubt that at least two other researchers independently designed and built similardevices at around the same time, if not earlier. However, neither of these reviews iscomplete. Nor is Donnellys s1991d recent account of the history of concentric cylinderflow. Here we recount and evaluate the Nineteenth Century work of Mallock, Schwedoff,and Perry, as well as of Couette, with the concentric cylinder apparatus.

II. NO-SLIP BOUNDARY COUNDITIONA little background is germane. The story of the determination of fluid viscosity is

closely linked with the adoption of the no-slip boundary condition at a solid-liquid inter-face. Navier s1823d, in his celebrated memoir on the laws of fluid motion, introduced andexplained two coefficients of the internal resistance to slipping, one within the liquid andthe other between the solid and the liquid fNavier s1823d, p. 416g. Moreover, he did notexplicitly use the word viscosity to denote the first. He also derived an expression forthe fluid velocity in a circular tube by assuming perfect slip at the wall. sThe flow ratewould then vary with the cube of the tube radius, which he did not specifically mention.dIn 1839, Hagen reported experiments in which the liquid flow rate in circular tubes variedwith slightly more than the fourth power of the tube radius fHagen s1839dg. Sutera andSkalak s1993d established that in the same year as Hagen, Poiseuille submitted his ex-perimental measurements on tube flow of liquids to the French Academy. In those mea-surements he clearly demonstrated that at sufficiently low liquid velocities, the flow ratethrough circular tubes varies with the fourth power of the tube radius. His result contra-dicted theoretical treatments available in 1839, and it was probably the reason it was notaccepted until he repeated the experiments with additional liquids in the presence of adesignated committee! By 1847, his experiments and measurements of liquid viscositieswere fully communicated fcf. Sutera and Skalak s1993dg and thereafter became the stan-dards to match. However, Poiseuille did not himself calculate the absolute viscosity ofliquids, for that would have been tantamount to adopting the no-slip boundary conditionat the tube wall.

Stokes s1845d in his memoir was probably the first to argue that the liquid next to asolid wall is at rest. But he found that his formulas swhich he does not reportd did notagree with the experiments of Bossut and Dubuat from 70 years earlier fcf. Stokess1845d, p. 96g, which, it is today clear, fell in the turbulent regime of tube flow. However,two pages later he later mentions fcf. Stokes s1845d, p. 98g:

Dubuat found by experiment that when the mean velocity of water flowing througha pipe is less than one inch in a second, the water near the inner surface of the pipeis at rest. If these experiments may be trusted, the conditions to be satisfied in thecase of small velocities are those which first occurred to me, and which are in-cluded in those just given by supposing n=.

Stokes n is now known as the slip coefficient sor its inversed. He then derived anexpression for the velocity profile of flow in a tilted pipe of radius a fcf. Stokes s1845d,

pg. 105g:

809ORIGINS OF CONCENTRIC CYLINDERS VISCOMETRYw =gr sin a

4msa2 r2d + U , s2d

where U was the velocity close to the pipe surface. Unfortunately, Stokes was not awareof Poiseuilles more accurate measurements, or the no-slip boundary condition wouldhave been adopted earlier. Hence, he suggested that the flow between coaxial cylinders inrelative angular motion with respect to each other be used to elucidate the friction inliquids and also the boundary condition at a solid wall fStokes s1845d, p. 104g. hManyyears later, Margules apparently independently suggested using the same flow to measurethe coefficients of friction and gliding in 1881 fMargules s1881dgj. One year later, inhis 1846 report to the British Association, Stokes interpreted Coulombs oscillating-diskexperiments as supporting his arguments for no slip at a solid wall, but he did notadvocate this boundary condition unequivocally fStokes s1846dg. Five years later, Stokess1851d verified the no-slip boundary condition at solid-fluid boundaries in his celebratedstudy on the motion of pendulums fStokes s1851d, see pp. 145, 103d.

It was not until the late 1850s and early 1860s that an exact expression of the liquidvelocity and flowrate in laminar flow in a circular tube was publishedindependently bythree researchers. Jacobson fs1860d, p. 91g derived the formula, was probably the first tocall it Poiseuilles law, and verified it with an extensive set of his own experiments. Hecredited the derivation to Neumann. In March 1860, Hagenbach published the same resultfHagenbach s1860d, pp. 394739g, and commented on Naviers error in a footnote. In1863, Mathieu derived the same formula fMathieu s1863dg. Each of the earlier adopted noslip at the solid-liquid boundary. However, when in April 1860 Helmholtz published theformula for tube flow, he included a slip coefficient because he had compared his resultswith some inaccurate early experiments. This issue of slip appears not to have beenresolved in that era, because even Lamb, in the first edition of his treatise Hydrodynamicsin 1879 fLamb s1879dg, reproduced Helmholtzs formula for tube flow, and mentionedthat when slip at the walls was neglected, the resulting formula agreed with Poiseuillesexperimental results, in particular the fourth-power dependence of flow rate on tubediameter.

It was in this context that Couette, Mallock, Schwedoff, and Perry published theresults of their experiments with concentric cylinder devices in the next dozen years, andadopted the no-slip condition to extract the absolute viscosity. Interestingly, when Thorpeand Rodger delivered their famous Bakerian Lecture in 1894 on the relation between thechemical nature of liquids and their viscosities, they did not so much as mention the issueof slip or the concentric cylinder device fThorpe and Rodger s1894dg. And from hissecond edition s1895d onward, Lamb held it possible that there is no slip at solid bound-aries in all ordinary cases fLamb s1895dg He specifically cited not only Poiseuillesresults, but also Whethams s1890a, 1890bd conclusion from glass, silvered, and coppertubes slightly less than a millimeter in diameter that no slip occurs, at any rate withsolids that are wetted by the liquid. Poiseuilles tube diameters went down to 14 mm,and provided a much more severe test.g.

Today no-slip is accepted as the wall boundary condition for the flow of all smallmolecule fluids that wet the boundaries. For example in a recent review of microfluidicflows, fStone et al. s2004d, p. 388g conclude that the no-slip boundary conditionremains an excellent approximation for flows at scales above tens of nanometers. Mea-surements have been reported, however, of apparent viscosities of certain polymersolutions flowing through cylindrical pores in membranes, of four different diameters

between 2.5 and 11 mm, lower than viscosities measured in larger tubes and gaps at verylow rates of shear fChauveteau s1982d, also cf. Aubert and Tirrell s1982dg.

810 DONTULA, MACOSKO, AND SCRIVENIII. COUETTES CONTRIBUTIONCouette s1890d studied both pressure gradient-driven flow in tubes and the rotational

flow in an elaborate concentric cylinder apparatus. Piau et al.s s1994d biography is themost complete account available of Couettes personal life and scientific career. Couettestheoretical correction for pressure loss at the tube entrance was more accurate thanHagenbachs s1860d correction, from which it differed by a numerical multiplier sbutCouettes formula was identical to Neumanns formula reported 30 years earlier by Ja-cobson, of which Couette was apparently unawared. He also developed an experimentaltechnique by which these corrections were totally eliminated. Couettes concentric cyl-inder apparatus was inspired by Margules s1881d concept and lies work with concen-tric spheres flie s1882dg ssee Fig. 1d. His measurements with the concentric cylinderdevice in which the inner cylinder was stationary were not accurate. His measured valuesof shear viscosity of water were higher than Poiseuilles values by 15% fCouette s1890d,p. 460g. For instance, on page 460 of his 1890 publication he observed

Pour complter la vrification, jai calcul le coefficient de frottement intrieur de leau 16,7, en remplacant, dans la formule s13d, P /Npar la valeur moyenne que nous venons de trouver. Jai obtenu ainsi la valeur

= 0,01255

FIG. 1. History of the concentric cylinder apparatus to measure fluid viscosity. Arrows connect those workscited by other authors. Perry, Couette, Mallock, and Schwedoff all built their first concentric cylinder deviceindependent of each other. Perry built the first device; Couette and Mallock reported the first shear viscositymeasurements; Schwedoffs viscosity measurements were 0.2% lower than Poiseuilles reported standards;Mallock was the first to build an apparatus in which either cylinder could be turned, and reported that the flowis unstable when the inner cylinder rotates she was always above the critical Taylor numberd.au lieu que les expriences de Poiseuille, qui mritent toute confiance, donnent,pour la mme temprature,

811ORIGINS OF CONCENTRIC CYLINDERS VISCOMETRY = 0,01096

La valeur que jai trouve est done trop forte denviron 15 pour 100. Ce rsultat meparat d aux imperfections de lappareil,

He attributed the evident errors to the guard rings with which he tried to eliminate endeffects fg and g8 in Fig. 2sddg, and to the eccentricity of the cylinders. Eccentricitybetween coaxial cylinders does raise the measured torque scf. Sommerfelds lubricationtheoryd and consequently the apparent viscosity, but in order to explain Couettes dis-crepancies, the eccentricity in his apparatus would have to be in excess of 25% of theradial gap between the cylinders. Furthermore, Couette tabulated results of his experi-ments on water over 7 months, and his values were between 12.5% and 22.9% larger thanPoiseuilles values fCouette s1890d, p. 461g. These errors were most likely due to hisguard rings. Couette then calibrated his instrument with Poiseuilles values of waterviscosity, and thereupon estimated the viscosity of air. His estimate was 10% lower thanJames C. Maxwells and Oscar E. Meyers values, and 1% higher than O. Schumannsvalues, all with the oscillating-disk method fCouette s1890d pp. 467846g. Thus, hismeasurements of air viscosity with the concentric cylinder apparatus were not absolute asfDonnelly s1991d, p. 37g claims.

It is curious that Couette used the word viscosity sparingly; he preferred to usecoefficient of friction in liquids sincluding in the title of his 1890 articled, even thoughacross the English Channel the word viscosity had replaced coefficient of friction atleast 25 years earlier fKelvin s1865d, p. 290g.

IV. MALLOCKS CONTRIBUTIONUnremarked by Thorpe and Rodger s1894d, Mallock, working under the aegis of

Rayleigh, had independently constructed a concentric cylinder device of which the innercylinder was stationary, measured the viscosity of water over a range of temperatures,and obtained results that were close to Poiseuilles reported values fMallock s1888d, p.131g. They were more accurate than Couettes measurements, which were published thesame year. In 1896, he recomputed the viscosities after G. G. Stokes pointed out that hehad not used quite the right formula; his recomputed viscosities were very close toPoiseuilles values. hWe recomputed viscosities from Mallocks s1888d data using theformula he gave fMallock s1896dg and found them to be within 1% of viscosity valueslisted by Bingham s1922d, which in turn were 0.34% lower than those computed fromPoiseuilles measurements at 15 Cj. Towards the end of his paper, Mallock pointed outthe significance of his own work

The chief interest of these experiments, beyond that attaching to an independentdetermination of m by a new method, lies in the comparatively high velocities atwhich viscous forces remain the principal cause of resistance.

He concluded his short paper withMany experiments were made on the viscosity of fluids other than water, but as Ifind that the results do not differ materially from those of Poiseuille it is unneces-sary to give them here.

Mallocks arrangement to reduce end effects was ingenious and convenient, as much asCouettes guard rings were elaborate. The bottom of the inner cylinder trapped an airbubble that separated the inner cylinder from the liquid beneath it fFig. 2scdg. This designreduced the extra torque from the liquid sheared below the level of the inner cylinder.This design is now standard in the ARES rheometer sTA Instruments, New Castle, DEd

and other commercial rheometers, though Mallocks contribution has been forgotten andMercier s1932d is generally credited with the invention fsee Mooney and Ewart s1934dg.

812 DONTULA, MACOSKO, AND SCRIVENMallock was a careful experimenter; the outer cylinder must have been transparent shedid not explaind because he observed and commented on a minute secondary flow set upbetween the stationary inner cylinder and the rotating outer cylinder by the bottom of the

FIG. 2. Concentric cylinder apparatus used to measure liquid viscosities and designed by sad Perry s1883d, sbdSchwedoff s1889d, scd Mallock s1888d, and sdd Couette s1890d.latter fMallock s1888d, p. 128gIt was found that at all these speeds the force tending to turn the inner cylinder

813ORIGINS OF CONCENTRIC CYLINDERS VISCOMETRYB could be represented by the sum of two terms, one varying as the velocity and theother as the square of the velocity; the latter being small compared to the former,even at the highest speed.The cause of the square term seems to be that, owing to the action of the bottom ofthe revolving cylinder, a circulation is set up in the fluid in the annulus, the flowbeing up the side of the revolving cylinder and down the side of the stationary one,the result being that the fluid having the velocity due to a position near the outercylinder is by this circulation continuously carried near the inner one, thus makingthe variation in velocity in the neighborhood of the latter greater than it wouldotherwise be. As far as could be observed there was no trace of eddies with axesparallel to that of the two cylinders. The proportion between the two terms dependson the ratio between the length of the cylinders, and the breadth of the annulus, thesquare term become smaller and smaller compared to the other as the ratio in-creases. Professor J. Thomson fThomson s1876d, s1877dg has pointed out that acirculation having a very similar origin must take place in a stream when flowinground a bend.

Rayleigh communicated this paper to the Royal Society. Mallock then built a new versionof his apparatus and studied the stability of the flows with either the inner or the outercylinder rotating and with a variety of end conditions; Kelvin communicated this paper tothe Royal Society fKelvin s1896dg. Mallocks results went largely unnoticed by thoseresearching viscosity measurements, at least up to the time of Taylors s1923d landmarkexperiments and theoretical analysis. Even Leroux s1925d, who measured water viscositywith a greatly improved version of Couettes device, made no mention of Mallockswork. Hatschek s1913d, who measured the viscosity of suspensions, was among thosewho earlier had overlooked Mallock.

V. SCHWEDOFFS CONTRIBUTIONIn 1889, Schwedoff reported measurements of the elastic shear modulus of gelatin by

tracing the liquid relaxation after an imposed deformation with a concentric cylinderapparatus fSchwedoff s1889dg. In 1890, he reported the viscosity of water, castor oil,glycerin and a gelatin solution with a concentric cylinder device whose outer cylinderwas made of glass fFig. 2sbdg fSchwedoff s1890dg. By that time he was aware of Cou-ettes work fcf. Piau et al. s1994dg, and so he made clear that he had designed and builthis apparatus independently fSchwedoff s1890d, p. 38g

Je me suis arrt la mthode des deux cylindres concentriques, dont lintervalleest rempli du liquide tudier. Propose dabord par M. Margules pour ltude dela viscosit des liquides, cette mthode avait prov depuis des perfectionnementsessentiels entre les mains de M. Couette s1888d, qui eut lheureuse ide dintroduiredans son appareil duex cylindres de garde, pour simplifier la thorie delexprience. Par hasard, cette mthode se rapproche beaucoup de celle que javaisadopte pour mes recherches sur la rigidit des liquides, sans avoir pourtant con-naissance des essais de M. Couette dans cette direction. Seulement, au lieu descylindres de garde, je me suis servi du procd delimination du fond que jai dcritdans la premire Partie de ces recherches. sSchwedoffs emphasisd

Schwedoff accounted for the extra torque from the liquid below the inner cylinder andsubtracted it; this is the process of bottom elimination he refers to in the earlier pas-sage. His measurement of water viscosity was 0.2% lower than Poiseuilles value

fSchwedoff s1890d, p. 44g, more accurate than either Mallocks or Couettes measure-ments with the concentric cylinder apparatus. In the same work, he determined that the

814 DONTULA, MACOSKO, AND SCRIVENviscosity of a gelatin solution depended on shear rate. Schwedoff cited neither Mallocksnor Perrys work. Neither Graetz and Stckl s1931d nor Donnelly s1991d cite Schwedoffswork, but fColeman et al. s1966d pp. 8889g cite both his works, and also comment thatSchwedoffs measurements on the gelatin solution were probably the first report of ashear-thinning liquid viscosity in the steady flow between two coaxial cylinders free fromturbulence or secondary flow.

VI. PERRYS CONTRIBUTIONPerrys contribution to the development of the concentric cylinder apparatus is the

least known. Perhaps that is because he was a teacher sof engineeringd first, and onlysecondarily a researcher. Once a student of James Thomson and an Honorary Assistant ofKelvin at Glasgow, he later taught engineering at the Imperial College of Engineering inJapan s18751878d, at the City and Guilds of London Technical College in Finsburys18821896d, and at the Royal College of Science in Kensington s18961913d beforeretiring scf. J. C. Poggendorff Biographisch-literarisches Handworterbuch der exactenWissenschaften 34, and obituary notice in 1926 in Proc. Roy. Soc. London A 111, iiid.His campaign to improve engineering education methods, and his efforts to bridge thegap between mathematicians and engineers were famous scf. his obituary noticed. Perryalso contributed prolifically to the scientific literature while he was teaching san exami-nation of the Catalogue of Scientific Papers compiled by the Royal Society reveals 105papers coauthored by Perry between 1875 and 1900d. Perrys book Practical Mechanics,an attempt to put before non-mathematical readers a method of studying mechanicssPerrys emphasisd, contains a drawing of a concentric cylinder apparatus similar to thedouble-cup, a current method to measure the viscosity of low viscosity liquids fMa-cosko s1994dg. He also described the method he used to measure liquid viscosities fPerrys1883d, Sec. 242, pp. 248249, also see Fig. 2sadg

Fig. 143 shows a hollow cylindric body, F, supported so that it cannot movesidewise, and yet so that its only resistance to turning is due to the twist it wouldgive the the suspension wire, A. CC is water or another liquid filling the annularspace between the cylindric surfaces DD and EE, and wetting both sides of F.When the vessel DD EE is rotated, the water moving past the surfaces of F tend tomake F turn around, and this frictional torque is resisted by the twist which is givento the wire. The amount of twist in the wire gives us, then, a measurement of theviscosity of liquids, and investigations may be made under very different condi-tions.

He published his work on liquid friction in 1893. He began it by statingA piece of apparatus such as is used in this investigation was designed and partlyconstructed in Japan in 1876; it is described in my book on Practical Mechanicss1883d. The specimen actually used by us was constructed at the Finsbury Techni-cal College in 1882, and occasionally used since that time, but no complete sets ofobservations were attempted till October 1891.

Perry was also concerned about the extra torque from the liquid sheared below the innercylinder, i.e., end effects fPerry s1893d, p. 442g, and his double gap design reduced it.Perry also claimed to have measured viscosities of several liquids, but reported only theabrupt fall in viscosity of sperm oil above 40 C. His paper sone of the reviewers ofwhich was Reynoldsd contained no references to Mallock, Couette, or Schwedoff.

Perrys work went unnoticed until Drew s1901d, working in Professor Michelsons

laboratory at the University of Chicago, cited his work when he reported measurementson viscosity of water. Interestingly, Drew did not cite Couette or Mallock or Schwedoff.

815ORIGINS OF CONCENTRIC CYLINDERS VISCOMETRYThe concentric cylinder apparatus he used, he explained, was designed and built by hispredecessor Johonnott, under the direction of Professor Michelson. Subsequently Gurney,supervised by Professor Michelson and Professor Millikan, reported further work onwater and cited Couette, but not Mallock, Schwedoff, or Perry fGurney s1908dg.

VII. CONCLUSIONSThe question of whether a fluid slips at a solid wall, and the determination of fluid

viscosity are inseparable. The first issue was slip. Stokes illuminating research on vis-cous drag on spheres and slender cylinders fStokes s1851dg was offset by Piotrowskissand Girardsd data, which Whetham s1890a, 1890bd finally showed were erroneous, andHelmholtzs analysis of them in terms of slip fHelmholtz s1860dg. Poiseuilles data s1846dand Jacobsons sor Neumannsd analysis of them s1860d might otherwise have settled theissue; they did establish what became a primary technique for measuring fluid viscosity.Even as Lamb s1879d prolonged the ambiguity about slip by giving credence to theHelmholtzPiotrowski paper, the experimenters who were developing viscosity measure-ment all adopted the no-slip hypothesis, as had Maxwell as early as 1866. And Maxwells1866d, though fully aware of Helmholtz and Piotrowski, on the basis of his own andother experiments, stated

I have no doubt that. there is no slipping..The leading alternative technique to Poiseuilles flow emerged in the concentric cylinderapparatus frequently called the Couette rheometer after Maurice Couette, who is widelycredited with devising the method and making the first accurate measurements. Dryden etal. s1932d, Graetz and Stckl s1931d, and Donnelly s1991d all show that at least twoothers developed the same apparatus independently. But none of these reviews tell thewhole story. Here we have re-examined the origins of concentric cylinder rheometry andits attribution to Couette.

Stokes and later, Margules s1881d, separately proposed that the coefficients of fric-tion and gliding sboth seemed unaware that the no-slip boundary condition at the solid-liquid interface had been conclusively established by Poiseuilles experiments with tubeflowd could be measured with a concentric cylinder apparatus. Margules solved for theflow allowing for some slip. Perry built the first concentric cylinder device to measureliquid viscosity in 1882, but he did not publish measurements until 1893. slie built aconcentric sphere device in 1882 that was inconsequential except for catching Couettesnotice.d Couettes measurements of water viscosity s1888, 1890d were between 10% and23% higher than Poiseuilles reported values. His measurements of air viscosity were notabsolute because he calibrated his apparatus with Poiseuilles measurements on water.Mallocks s1888d measurements of water viscosity were more accurate and within 1% ofthose derived from Poiseuilles measurements. The next year, s1889d, Schwedoff reportedmeasurements of the elastic modulus of a gelatin solution with concentric cylinders of hisown design. In 1890, he noted that his device was similar to Couettes, and reported theviscosity of water measured with his apparatus that was only 0.2% lower than Poiseuillesvalues. He also reported the first instance of a shear rate-dependent viscosity with mea-surements on a gelatin solution, foreshadowing widespread applications of CouettesMallockSchwedoffPerryd rheometers to rheological characterization of non-Newtonianliquids as surveyed by Coleman et al. s1966d, fPiau et al. s1994dg. A little later Mallocks1896d, working under the eyes of Lord Rayleigh and his friend Lord Kelvin, built the

first apparatus with which the inner cylinder could be rotated and found, inter alia, thatthe flow between concentric cylinders is unstable when the inner cylinder rotates. The

816 DONTULA, MACOSKO, AND SCRIVENensuing history of flow stability, from Taylor s1923d onward, is summarized by Donnellys1991d.

It is interesting to note that the original concentric cylinder designers used four dif-ferent approaches to eliminate end effects fsee Dontula s1999d for further discussiong.Three of these are still the methods used today; only Couettes guard rings did notsurvive. The falling weights which drove rotation in the first designs soon gave way toelectric motors. Sensing of torque by mirrors reflecting onto meter sticks was replaced bycapacitors but the torsion springs of the original designs remained for over 80 years. Onlysince the 1970s have springs been replaced by strain gauges and then rebalance trans-ducers and constant torque drag cup motors fMacosko s1994dg.

ACKNOWLEDGMENTSThe authors are indebted to E. G. Arlinghaus for her translations of the passages in

French and to Dr. I. Iliopoulos for his help in translating the works in German. Thisresearch was supported by the members of IPRIME, the Industrial Partnership for Re-search in Interfacial and Materials Engineering at the University of Minnesota.

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