Features Fluids, Fluorescence and a Hat Full of Beetles Mark Mccartney, University of Ulster

features Fluids, Fluorescence and a Hat Full of Beetles Mark McCartney, University of Ulster

George is so fond of lightning. ... He puts his head things’ [1, p. 35]. Or he could be found on holiday on the north under all the waterspouts he can find. ... He flew coast of Ireland, in the sea, trouser legs rolled up, checking if the about, now up, now down, trying to find a better path; maximum angle of wave crests agreed with his calculations. On he quite enjoys dangerous places and looks so happy one occasion as a young man, he had to apologise for not bowing when his neck might be broken. [1, p. 16] to a group of ladies in a Cambridge street because his hat was full of beetles. It seems virtually all aspects of the natural world held t first glance these read as a parent’s indulgent de- a fascination for him. scriptions of a playful and inquisitive child. But they are A not. They are taken from the journal of George Gabriel Stokes’ wife, and are all the more remarkable because they were Early life written on honeymoon. They give a hint of Stokes’ omnivorous George Gabriel Stokes was born on 13 August 1819 into a fam- and lifelong curiosity. ily with a habit of producing Church of Ireland clerics and aca- Similarly, with respect to lightning, his daughter said: demics. His father was rector of Skreen parish church, which ‘He would go any distance to see houses and trees that had was a couple of kilometres from the Atlantic Ocean in County been struck’ [1, p. 32]. Despite his apparent lifelong inter- Sligo. He was the youngest of eight children, two of whom died est in thunderstorms, Lord Kelvin, in his obituary of Stokes, in infancy. His three brothers all entered the church. His early stated that electricity was virtually the only area of natural education was at home and overseen by the parish clerk. In 1832, philosophy to which he did not make a significant contri- aged 13, he was sent to Dublin to live with his uncle and attend bution [3]. The term natural philosophy translates to what school there. Then in 1835, he moved to Bristol College. His we today call physics, and in the 19th century for people mathematical ability was commented on in Skreen, Dublin and such as G.G. Stokes, Lord Kelvin and James Clerk Maxwell Bristol, with his sister Elizabeth claiming that: ‘There is a tradi- that meant being engaged both experimentally and theoreti- tion that he did many of the propositions of Euclid as problems, cally, with the mathematics being driven by the need to solve without having looked at the book’ [1, p. 5]. physical problems.

Figure 2: Stokes’ birthplace: Skreen Rectory, Co Sligo, as it appeared at the beginning of the 20th century [2]

Cambridge In 1837, Stokes entered Pembroke College, Cambridge. As was typical for any student who wanted to appear high on the exam- ination lists, he studied under a private tutor. The tutor, William Hopkins, had an eye for strong students and a reputation for con- Science Photo Library verting their strength into becoming Senior Wrangler (i.e. obtain- ing the highest marks in the Mathematical Tripos examinations). Figure 1: George Gabriel Stokes (1819–1903) A significant number of major figures in 19th century mathematics and natural philosophy passed through Hopkins’ hands, While today Stokes is remembered by undergraduates be- including J.J. Sylvester, Arthur Cayley, William Thomson (later cause his name is attached to mathematical results such as the Lord Kelvin), E.J. Routh, P.G. Tait and James Clerk Maxwell. Navier–Stokes equation and Stokes’ theorem (see box), in his In 1841, Stokes’ studies paid off and he was announced as Se- lifetime he could also be found in his home in Cambridge ex- nior Wrangler, and shortly afterwards also took first place in the perimenting on fluorescence with ‘strange infusions which smelt Smith’s Prize examinations. Such a strong performance meant horribly, horse-chestnut bark and leaves and all sorts of other that a fellowship at Pembroke inevitably followed.

Mathematics TODAY AUGUST 2019 142 StokesStokesStokes went went went on, on, on, in in inOctober October October 1849, 1849, 1849, to to tobe be be elected elected elected the the the 13th 13th 13thFromFromFrom that that that suggestion suggestion suggestion flowed flowed flowed over over over 20 20 20 papers, papers, papers, clustered clustered clustered at at the at the the be- be- be- LucasianLucasianLucasian Professor Professor Professor of of Mathematics.of Mathematics. Mathematics. P.G. P.G. P.G. Tait, Tait, Tait, a a student astudent student at at Cam- at Cam- Cam-ginningginningginning and and and end end end of of hisof his his life life life (1842–50 (1842–50 (1842–50 and and and 1880–98), 1880–98), 1880–98), covering covering covering the the the bridgebridgebridge at at the at the the time, time, time, declared declared declared that that that he he heand and and his his his fellow fellow fellow undergraduates undergraduates undergraduatesnaturenaturenature of of fluidof fluid fluid flow flow flow and and and the the the theory theory theory of of waterof water water waves. waves. waves. Interestingly Interestingly Interestingly hadhadhad not not not even even even heard heard heard of of ofStokes Stokes Stokes before before before his his his appointment. appointment. appointment. The The The sug- sug- sug-thethethe work work work he he he considered considered considered his his his most most most important important important in in thein the the area area area of of offluids fluids fluids gestiongestiongestion that that that Stokes Stokes Stokes was was was a a relative arelative relative unknown unknown unknown is is backed is backed backed up up up by by by the the thewaswaswas an an an1851 1851 1851 paper, paper, paper, ‘On ‘On ‘On the the the effect effect effect of of internalof internal internal friction friction friction of of fluidsof fluids fluids on on on factfactfact that that thatTheTheThe Times Times Timesandandand two two two Cambridge Cambridge Cambridge papers papers papers erroneously erroneously erroneously an- an- an-thethethe motion motion motion of of pendulums’of pendulums’ pendulums’ [5, [5, [5, pp. pp. pp. 1–141] 1–141] 1–141] (see (see (see box). box). box). Certainly Certainly Certainly an an an nouncednouncednounced that that that it it was it was was Stokes’ Stokes’ Stokes’ elder elder elder brother, brother, brother, Reverend Reverend Reverend W.H. W.H. W.H. Stokes, Stokes, Stokes,accurateaccurateaccurate understanding understanding understanding of of pendulaof pendula pendula for for for timekeeping timekeeping timekeeping and and and geodesy geodesy geodesy SeniorSeniorSenior Fellow Fellow Fellow of of ofCaius, Caius, Caius, who who who had had had been been been appointed. appointed. appointed. Stokes Stokes Stokes was was waswaswaswas important important important in in thein the the 19th 19th 19th century, century, century, but but but it it is it is not is not not the the the paper paper paper where where where amusedamusedamused rather rather rather than than than piqued piqued piqued by by bythe the the error. error. error. hehehe writes writes writes down down down what what what we we we now now now call call call the the the Navier–Stokes Navier–Stokes Navier–Stokes equation. equation. equation. TodayTodayToday the the the Lucasian Lucasian Lucasian Chair Chair Chair is is seen is seen seen as as aas a stellar stellara stellar appointment, appointment, appointment, asas- as-ThisThisThis appeared appeared appeared in in 1845 in 1845 1845 in in ‘On in ‘On ‘On the the the theories theories theories of of theof the the internal internal internal friction friction friction sociatingsociatingsociating the the the incumbent incumbent incumbent with with with previous previous previous holders holders holders such such such as as asNewton, Newton, Newton,ofofof fluids fluids fluids in in motion,in motion, motion, and and and of of ofthe the the equilibrium equilibrium equilibrium and and and motion motion motion of of ofelas- elas- elas- Dirac,Dirac,Dirac, Hawking Hawking Hawking and and and Stokes Stokes Stokes himself. himself. himself. However, However, However, it it was it was was not not not quite quite quitetictictic solids’ solids’ solids’ [4, [4, [4, pp. pp. pp. 75–129]. 75–129]. 75–129]. Stokes Stokes Stokes was was was not not not the the the first first first to to write to write write the the the seenseenseen that that that way way way in in his in his his day. day. day. A A Lucasian A Lucasian Lucasian Professor Professor Professor from from from 20 20 20 years years years be- be- be-equationequationequation down. down. down. Claude-Louis Claude-Louis Claude-Louis Navier Navier Navier had had had done done done so so inso in 1822, in 1822, 1822, and and and in in in foreforefore Stokes, Stokes, Stokes, George George George Biddell Biddell Biddell Airy, Airy, Airy, complained complained complained that that that he he hehad had had to to take to take takebetweenbetweenbetween then then then and and and 1845 1845 1845 so so hadso had had Cauchy, Cauchy, Cauchy, Poisson Poisson Poisson and and and Saint-Venant. Saint-Venant. Saint-Venant. aa one-thirda one-third one-third cut cut cut in in payin pay pay to to taketo take take the the the role, role, role, and and and matching matching matching his his his action action actionStokesStokesStokes notes notes notes that that that on on on finishing finishing finishing the the the paper paper paper [4, [4, [4, p.77]: p.77]: p.77]: toto histo his his complaint complaint complaint stayed stayed stayed in in the in the the job job job for for for scarcely scarcely scarcely more more more than than than a a year. a year. year. II afterwards afterwardsI afterwards found found found that that that Poisson Poisson Poisson had had had written written written a a memoir memoira memoir WhileWhileWhile in in post, in post, post, Airy Airy Airy cunningly cunningly cunningly used used used his his his position position position to to arrange to arrange arrange an an anin- in- in- ononon the the the same same same subject, subject, subject, and and and on on onreferring referring referring to to it to it I itI found found I found that that that creasecreasecrease in in thein the the salary salary salary of of theof the the Plumian Plumian Plumian Professor Professor Professor of of Astronomyof Astronomy Astronomy and and and hehehe had had had arrived arrived arrived at at theat the the same same same equations. equations. equations. The The The method method method ExperimentalExperimentalExperimental Philosophy Philosophy Philosophy at at Cambridge, at Cambridge, Cambridge, before before before switching switching switching chairs. chairs. chairs. whichwhichwhich he he he employed employed employed was was was however however however so so so different different different from from from EarlyEarlyEarly in in inhis his his time time time as as asLucasian Lucasian Lucasian Professor, Professor, Professor, Stokes Stokes Stokes augmented augmented augmented his his his mineminemine that that that I I feel Ifeel feel justified justified justified in in inlaying laying laying the the the latter latter latter before before before inadequateinadequateinadequate salary salary salary by by by lecturing lecturing lecturing at at the at the the Royal Royal Royal School School School of of ofMines Mines Mines in in in [the[the[the Cambridge Cambridge Cambridge Philosophical Philosophical Philosophical Society]. Society]. Society]. London,London,London, but but but unlike unlike unlike Airy, Airy, Airy, he he heremained remained remained in in the in the the chair chair chair until until until his his his death. death. death. HeHeHe then then then adds adds adds in in a in a footnote footnotea footnote that that that Navier Navier Navier has has has also also also written written written them them them down down down ‘but‘but‘but his his his principles principles principles differ differ differ from from from mine mine mine still still still more more more than than than do do do Poisson’s’ Poisson’s’ Poisson’s’ ResearchResearchResearch [4,[4,[4, p.77]. p.77]. p.77]. Navier Navier Navier had had had built built built his his his model model model on on onideas ideas ideas of of molecular of molecular molecular forces, forces, forces, whereaswhereaswhereas Stokes Stokes Stokes preferred preferred preferred to to toavoid avoid avoid any any any molecular molecular molecular speculations. speculations. speculations. InIn 1901,In 1901, 1901, near near near the the the end end end of of hisof his his life, life, life, Stokes Stokes Stokes recalled recalled recalled that that that after after after gradu- gradu- gradu-TheTheThe work work work by by byStokes Stokes Stokes and and and his his his predecessors predecessors predecessors was was was little little little enough enough enough known known known ationationation in in 1841 in 1841 1841 [1, [1, [1, p. p. 8], p. 8], 8], toto beto be berediscovered rediscovered rediscovered again again again by by by Helmholtz Helmholtz Helmholtz in in 1859, in 1859, 1859, and and and you you you can can can read read read textbookstextbookstextbooks up up up to to tothe the the mid-20th mid-20th mid-20th century century century which, which, which, while while while giving giving giving due due due II thought thoughtI thought I I would wouldI would try try try my my my hand hand hand at at original at original original research; research; research; creditcreditcredit to to Navier to Navier Navier and and and Stokes, Stokes, Stokes, do do do not not not name name name the the the equation equation equation after after after them. them. them. andandand following following following a a suggestion asuggestion suggestion made made made to to meto me me by by by Mr Mr Mr Hop- Hop- Hop- WhileWhileWhile in in incontemporary contemporary contemporary science science science the the the name name name of of ofStokes Stokes Stokes is is bestis best best kinskinskins while while while reading reading reading for for for my my my degree, degree, degree, I I took tookI took up up up the the the sub- sub- sub- knownknownknown for for for fluid fluid fluid dynamics, dynamics, dynamics, in in his in his his lifetime lifetime lifetime he he healso also also had had had a a high ahigh high repu- repu- repu- jectjectject of of ofHydrodynamics, Hydrodynamics, Hydrodynamics, then then then at at a at a rather a rather rather low low low ebb ebb ebb in in in tationtationtation in in optics. in optics. optics. In In this In this this his his his contributions contributions contributions ranged ranged ranged across across across the the the whole whole whole thethethe general general general reading reading reading of of theof the the place place place [i.e. [i.e. [i.e. Cambridge]. Cambridge]. Cambridge]. subject,subject,subject, encompassing encompassing encompassing theory, theory, theory, experiment experiment experiment and and and instrument instrument instrument design. design. design.

ThreeThreeThree cases cases cases of of of Stokes Stokes Stokes

Stokes’sStokes’sStokes’s name name name is is associated is associated associated with with with a a number anumber number of of phenomenaof phenomena phenomena ThisThisThis result result result appears appears appears in in ain a 141-page a 141-page 141-page paper paper paper entitled entitled entitled ‘On ‘On ‘On the the the andandand equations. equations. equations. Here Here Here are are are three three three of of theof the the best best best known. known. known. effecteffecteffect of of internal of internal internal friction friction friction of of fluids of fluids fluids on on onthe the the motion motion motion of of pendu- of pendu- pendu- lums’lums’lums’ [5, [5, [5, pp. pp. pp. 1–141]. 1–141]. 1–141]. Stokes Stokes Stokes points points points out out out that that that the the the result result result ex- ex- ex- plainsplainsplains why why why very very very small small small particles, particles, particles, including including including those those those that that that make make make Stokes’Stokes’Stokes’ theorem theorem theorem upupup clouds, clouds, clouds, are are are suspended suspended suspended in in air.in air. air. At At At the the the beginning beginning beginning of of ofthe the the 20th20th20th century, century, century, the the the result result result was was was key key key to to enablingto enabling enabling Robert Robert Robert Mil- Mil- Mil- ForForFor a a vector avector vector field field fieldFFFandandand surface surface surfaceSSboundedSboundedbounded by by by a a simple asimple simple likanlikanlikan to to indirectly to indirectly indirectly measure measure measure the the the size size size of of oilof oil oil drops drops drops in in his in his his fa- fa- fa- curvecurvecurveΓΓ: :Γ: mousmousmous experiment experiment experiment to to determine to determine determine the the the charge charge charge on on on the the the electron. electron. electron. FF.Fd. dl.ld==l = FF.Fd. dS.Sd.S. . ∇×∇×∇× ΓΓ Γ SS S TheTheThe Navier–Stokes Navier–Stokes Navier–Stokes equation equation equation Actually,Actually,Actually, Stokes Stokes Stokes did did did not not not come come come up up upwith with with this this this theorem! theorem! theorem! It It was It was was ForForFor an an anincompressible incompressible incompressible fluid, fluid, fluid, the the the velocity velocity velocity field field fieldvvofvof aof a fluid, afluid, fluid, hishishis long-term long-term long-term correspondent correspondent correspondent William William William Thomson. Thomson. Thomson. Thomson Thomson Thomson wherewherewhereρρisρis theis the the fluid fluid fluid density, density, density,ppispis theis the the pressure, pressure, pressure,ηηisηis theis the the dy- dy- dy- putputput it it in it in a in a letter aletter letter to to Stokes to Stokes Stokes in in 1850, in 1850, 1850, and and and Stokes Stokes Stokes put put put its its itsproof proof proof namicnamicnamic viscosity viscosity viscosity and and andffarefareare external external external forces, forces, forces, is is given is given given by by by asas aas a question aquestion question in in the in the the Smith’s Smith’s Smith’s Prize Prize Prize examinations examinations examinations in in 1854. in 1854. 1854. ∂∂vv∂v 11 1 +(+(+(vvv )v)v)+v++ pp pηη η2v2v2vff=0f=0=0. . . ∂t∂t∂t ·∇·∇·∇ ρρ∇∇ρ∇−−−∇∇∇ −−− Stokes’Stokes’Stokes’ law law law TheTheThe ubiquity ubiquity ubiquity of of fluids of fluids fluids makes makes makes the the the Navier–Stokes Navier–Stokes Navier–Stokes equation equation equation a a a TheTheThe drag drag dragDDDononon a a small asmall small sphere sphere sphere of of ofradius radius radiusRR,R, moving moving, moving with with with cornerstonecornerstonecornerstone of of modernof modern modern physical physical physical science science science and and and engineering. engineering. engineering. speedspeedspeedvvthroughvthroughthrough a a fluid afluid fluid with with with viscosity viscosity viscosityµµisµis givenis given given by by by ItsItsIts nonlinearity nonlinearity nonlinearity makes makes makes it it hard it hard hard to to solve,to solve, solve, rich rich rich in in dynamicsin dynamics dynamics andandand the the the subject subject subject of of oneof one one of of theof the the Clay Clay Clay Mathematics Mathematics Mathematics Institute’s Institute’s Institute’s DD=6D=6=6πµRv.πµRv.πµRv. MillenniumMillenniumMillennium Prize Prize Prize Problems. Problems. Problems.

Mathematics TODAY AUGUST 2019 143 Stokes was an excellent and elegant experimentalist. J.J. Thom- son stated: ‘It has been said that if you give Stokes the sun and three-quarters of an hour there is not an experiment in optics which he cannot perform’ [6, p. 370]. Those aspects of his work which were entangled with that most Victorian snare, the ether, have not aged well, but other studies have stood the test of time. Thus, in his work on aspects of interference and diffraction, he was able to bring new depths of mathematical and physical insight. However, one of his most significant contributions contained no mathematics whatsoever. In 1845, John Herschel noted the blue glow produced within a thin layer near the surface of a quinine solution. Under the then current understanding that monochromatic light was immutable, this phenomenon was difficult to explain. Stokes realised that if he took the dramatic step of assuming monochromatic light was mutable then the phenomenon was easily explained. He per- formed a series of experiments to test the hypothesis, and tracked the matter down to the absorption of ultraviolet. He then went on to find a similar effect in ‘solutions made directly from various parts of vegetables’ [5, p. 262] (hence the foul smelling infusions mentioned earlier). In a footnote to the 140-page paper where he Science Photo Library laid out his results, Stokes stated: ‘I am almost inclined to coin a word and call the appearance fluorescence’ [5, p. 289]. The work Tonic water fluorescing under ultraviolet radiation (right) compared won him the Royal Society’s Rumford medal in 1852. with visible light (left). The fluorescence is due to quinine. While the vast majority of Stokes’ over 130 papers are on fluids and optics, he did publish seven papers on more directly mathematical work. In a paper published in 1847 he considered the headstone. But it hardly matters, his name is cut into equations nature of the convergence of a Fourier series near a finite discon- that will last longer than any granite. tinuity, and in the same paper introduced the idea of uniform convergence. He also made use of what we now call the Riemann– Further reading Lebesgue lemma seven years before Riemann. In 1850 Stokes considered the integral A readable and warm recollection of Stokes, written by his daughter, can be found in the first 90 pages of [1]. D.B. Wilson provides

∞ π 3 a fine set of perspectives on Stokes’ life and work in [8]. [9] gives W (m)= cos (x mx) dx a collection of chapters on many aspects of Stokes’ life and work 0 2 − written by a range of experts. For those wishing to delve more whose zeros Airy had shown corresponded to dark bands in the deeply into Stokes’ work the five volumes of his Mathematical theory of caustics. Airy had been able to find zeros close to the and Physical Papers are freely available at www.archive.org. origin, but Stokes, ‘after many trials’ [7, p. 330], produced an asymptotic analysis which allowed the position of larger zeros to be found efficiently. In this work he was again anticipating Rie- R mann, this time by a decade, by using the saddle point method 1 Larmor, J. (ed.) (1907) Memoir and Scientific Correspondence of the for integrals in the complex plane. This work led on to what we Late George Gabriel Stokes, Bart, Vols. 1 and 2, reprinted 2010, Cam- now call Stokes’ phenomenon, the varying asymptotic behaviour bridge University Press, Cambridge. of functions in different sectors of the complex plane; a key idea 2 Greer, J. (1924) The Windings of the Moy with Skreen and Tireragh, in the modern research areas of super- and hyper-asymptotics. Alex Thom, Dublin. 3 Thomson, W. (1911) Mathematical and Physical Papers, Cambridge University Press, Cambridge, vol. 6, p. 339. Conclusion 4 Stokes, G.G. (1880) Mathematical and Physical Papers, Vol. 1, Cam- There was much more to Stokes than the science and mathematics bridge University Press, Cambridge. he has left behind. He served for more than 30 years as Secretary 5 Stokes, G.G. (1901) Mathematical and Physical Papers, Vol. 3, Cam- of the Royal Society. As such he devoted much time to editing its bridge University Press, Cambridge. Philosophical Transactions and acted as one of the gatekeepers of 6 Thomson, J.J. (1899) The Stokes jubilee, Camb. Rev., vol. 20, p. 370– Victorian scientific standards. A committed Christian, he thought 371. deeply about his faith and its relation to science, writing at book 7 Stokes, G.G. (1883) Mathematical and Physical Papers, Vol. 2, Cam- length on theological issues. He acted as MP for Cambridge Uni- bridge University Press, Cambridge. versity, and in the last months of his life he served as Master of 8 Wilson, D.B. (1987) Kelvin and Stokes: A Comparative Study in Vic- Pembroke; and it was from Pembroke that his coffin was finally torian Physics, Adam Hilger, Bristol. taken, processing round the Chapel Court, on to the University 9 McCartney, M., Whitaker, A. and Wood, A. (eds) (2019) George Church, and then to the Mill Road Cemetery. Strangely, if you Gabriel Stokes: Life, Science and Faith, Oxford University Press, go looking for his grave today, you will not find his name on any Oxford.

Mathematics TODAY AUGUST 2019 144