Stokes’ law, viscometry, and the Stokes falling sphere clock royalsocietypublishing.org/journal/rsta Julyan H. E. Cartwright1,2

1Instituto Andaluz de Ciencias de la Tierra, CSIC–Universidad de Granada, 18100 Armilla, Granada, Spain Discussion 2Instituto Carlos I de Física Teórica y Computacional, Universidad de Cite this article: Cartwright JHE. 2020 Stokes’ Granada, 18071 Granada, Spain law, viscometry, and the Stokes falling sphere JHEC, 0000-0001-7392-0957 clock. Phil.Trans.R.Soc.A378: 20200214. http://dx.doi.org/10.1098/rsta.2020.0214 Clocks run through the history of physics. Galileo conceived of using the pendulum as a timing device on watching a hanging lamp swing in Pisa Accepted:17June2020 cathedral; Huygens invented the pendulum clock; and Einstein thought about clock synchronization in One contribution of 13 to a theme issue ‘Stokes his Gedankenexperiment that led to relativity. Stokes at 200 (part 2)’. derived his law in the course of investigations to determine the effect of a fluid medium on the swing Subject Areas: of a pendulum. I sketch the work that has come out of this, Stokes drag, one of his most famous results. And fluid mechanics to celebrate the 200th anniversary of George Gabriel Stokes’ birth I propose using the time of fall of a sphere Keywords: through a fluid for a sculptural clock—a public kinetic Stokes drag, viscometer, falling sphere clock artwork that will tell the time. This article is part of the theme issue ‘Stokes at 200 Author for correspondence: (part 2)’. Julyan H. E. Cartwright e-mail: [email protected]

In 1851, Stokes derived [1] the drag force on a spherical body in a fluid, now called Stokes’ law, = πμ v Fd 6 R , where R is the radius of the sphere, v is the flow velocity and the constant μ is the fluid viscosity. Stokes’ paper was ‘On the effect of the internal friction of fluids on the motion of pendulums’. In 1851, such research was very important for an accurate determination of g, the acceleration due to gravity, which could be made with a pendulum [2,3]. However, as Stokes [1] stated,

The present paper contains one or two applications of the theory of internal friction to problems which are of some interest, but which do not relate to pendulums. The resistance to a sphere moving uniformly in

2020 The Author(s) Published by the Royal Society. All rights reserved. a fluid may be obtained as a limiting case of the resistance to a ball pendulum, provided the 2 circumstances be such that the square of the velocity may be neglected. royalsocietypublishing.org/journal/rsta ...... Those circumstances are slow viscous flow—what we would now call Stokes flow—and low Reynolds numbers. Stokes [1] begins his paper

The great importance of the results obtained by means of the pendulum has induced philosophers to devote so much attention to the subject, and to perform the experiments with such a scrupulous regard to accuracy in every particular, that pendulum observations may justly be ranked among those most distinguished by modern exactness.

He goes on to relate that his aim is to apply the Navier–Stokes equations to the pendulum [1]:

Having afterwards occupied myself with the theory of the friction of fluids, and arrived Phil.Trans.R.Soc.A at general equations of motion, the same in essential points as those which had been previously obtained in a totally different manner by others, of which, however, I was not at the time aware, I was desirous of applying, if possible, these equations to the calculation of the motion of some kind of pendulum. The difficulty of the problem is of course materially increased by the introduction of internal friction, but as I felt great confidence

in the essential parts of the theory, I thought that labour would not be ill-bestowed on the 378

subject. I first tried a long cylinder, because the solution of the problem appeared likely to 20200214 : be simpler than in the case of a sphere. But after having proceeded a good way towards the result, I was stopped by a difficulty relating to the determination of the arbitrary constants, which appeared as the coefficients of certain infinite series by which the integral of a certain differential equation was expressed. Having failed in the case of a cylinder, I tried a sphere, and presently found that the corresponding differential equation admitted of integration in finite terms, so that the solution of the problem could be completely effected.

The difficulties that Stokes encountered with the cylinder arise from what is today sometimes called the Stokes paradox, or Oseen paradox [4]: the flow perturbation from the body does not decay in two dimensions [5]. The cylinder problem was solved by Lamb in 1911 [6], 60 years after Stokes’ paper, based on Oseen’s work of 1910 [7]. Eames & Klettner [5] provide a nice derivation. The extra forces that come into play in unsteady flow about a sphere, which lead to what is now called the Maxey–Riley equation, are discussed in a companion paper in this theme issue [8]. As Eames & Klettner [5] also comment, Stokes’ drag law contributed to the determination of two fundamental constants—Avogadro’s number and the electron charge—and concomitantly to three Nobel Prizes in physics and one in chemistry. Einstein’s 1921 Nobel Prize was ‘for services to theoretical physics’, and one piece of work from his wonder year of 1905 was on the theory of Brownian motion, in which he used Stokes’ law, giving the so-called Stokes–Einstein equation. Likewise, Perrin’s experimental work on Brownian motion won him the 1926 Nobel Prize in physics; in the same year, Svedberg won the Nobel Prize in chemistry for the development of the ultracentrifuge, which could be used to determine the molecular weights and size—the Stokes radius—of colloids with Stokes’ law. Lastly, there is Millikan’s oil drop experiment; this gives the charge on the electron, for which Millikan won the Nobel Prize in physics in 1923. Millikan even puts Stokes into the title of his paper [9]. During the twentieth century, viscometers based on Stokes drag, such as the instrument of Höppler [10] that is shown in figure 1, were used for precision measurements of viscosity [11–13]. (The old centimetre–gram–second (CGS) unit of kinematic viscosity was the stoke or stokes: − − 1St= 1cm2 s 1 = 0.0001 m2 s 1.) The same physics is seen in the desk toys that are sometimes called liquid timers. Unfortunately, they are generally cheap and poorly made, and they stop working quickly. A basic one has a less dense solid rising up through an orifice into a more dense liquid (figure 2a). Of course, the solid can also be more dense than the liquid. The falling sphere viscometer is based upon this principle [11–13]. One can also have drops, rather than solid 3 royalsocietypublishing.org/journal/rsta ...... Phil.Trans.R.Soc.A 378

Figure 1. AntiquefallingsphereviscometerofHöppler[10],whichwasusedtomeasuretheviscosityofaliquidbasedonStokes 20200214 : drag. (Online version in colour.)

spheres, of a more dense liquid falling in a less dense liquid (figure 2b). Again, one can have the reverse: a less dense fluid rising within another, either as drops or as gas bubbles. The bubble viscometer is based upon this principle [14]. There are also those where the fall or rise is not vertical but constrained to be at an angle. One can have balls or drops move down ramps, which do not show the simplest Stokes drag (the rolling ball viscometer [15] is a modification of the falling sphere viscometer), and one can have a set-up where the drop moves down the ramp and then falls freely over another distance; or the reverse: free fall and then the drops are collected at the bottom in a ramp (figure 2c). And there are some such toys where the falling drops move a ‘water wheel’ (figure 2d). Thus, these desk toys illustrate various principles on which a viscometer based upon Stokes drag can operate. Stokes had pendulums in mind when thinking about his drag law. Of course, the things that pendulums are good for are clocks; in other words, for telling the time [16,17]. A crucial difference between fluids and granular media is the reason why sand timers or clocks are more prevalent than water clocks in the history of horology. The flow rate at the nozzle of a silo or a sand timer is constant whatever the height of sand above [18,19], so sand clocks are linear, while that is not true for a fluid [20,21]. However, a clepsydra or water clock, but one using the fall time of a sphere—determined by its Stokes velocity—to mark time, does not seem to have been proposed. A Stokesian water clock avoids the nonlinearity of traditional clepsydrae [22]. Although there are some clever modern water clocks, it is strange that Stokes’ law does not seem to have been used for a clock. A clock based upon the physics of Stokes drag might consist of a set of long transparent tubes, like viscometers, through which spheres would be falling—or rising—at rates tuned by the density difference and the viscosity to mark seconds, minutes, hours, days and so on. If one could manage even slower fall times of a month or a year with reasonable precision and accuracy, then so much the better. At the bottom of each free fall, there might be a wheel to move a counter for each event—to indicate, respectively, seconds, minutes, hours, for example—and perhaps a spiral or zig-zag ramp at the top or the bottom where the spheres or drops would collect before being transported back to the top in some manner. A further idea is to make the clock even more (a) 4 royalsocietypublishing.org/journal/rsta ......

(b) Phil.Trans.R.Soc.A

(d)

(c) 378 20200214 :

Figure 2. Desk toys show viscometer principles involving Stokes flow and Stokes drag:a ( ) light solids rise through a denser liquid; (b) heavy liquid drops fall through a less dense liquid; (c) heavy liquid drops descend on ramps; (d) heavy drops drive a wheel as they fall. (Online version in colour.)

like a sand timer. One could have a simple Stokes timer: a sphere falling down a tube, with a time scale on the surface of the container. Then turn it upside down, like a sand timer, when the sphere has completed its trajectory. Inverting the whole tube when the sphere hits the bottom might work rather well for a clock as a kinetic sculpture. One should think about the most elegant way to move things from bottom to top, to keep the clock wound, as it were. What about temperature differences if the clock is to be installed outside? Maintaining the mechanism at a reasonably constant temperature would be one solution. Whether that would mean one would prefer to have it indoors or venture to heat up the fluid in winter would need to be decided. An alternative might be to create a moving time scale which would adapt to the temperature and viscosity variations. To have sufficient precision and accuracy, one solution would be to have one mechanism guiding another, which is often called a phase-locked loop in engineering. The Clock of the Long Now, a mechanical clock being designed and built to tell time for 10 000 years [23], has had to deal with this problem, and the proposal is to use a solar signal, that is, heat from the noonday Sun would be used to synchronize it. While that clock is a much more exigent engineering problem than the present one, the fundamental idea of the Stokes mechanism receiving a nudge from time to time from a signal, either electronic or from celestial mechanics, such as the Sun at noon, is 5 just as valid for the Stokes clock. Since Stokes had a hand in developing the Campbell-Stokes royalsocietypublishing.org/journal/rsta sunshine recorder, sometimes called a Stokes sphere, a mechanism making use of it would be ...... most appropriate. Would one use electronics or try to keep everything mechanical? There are all sorts of interesting problems to resolve in designing and building such a clock. The present discussion is not intended to provide a set of blueprints, but an idea that might be worked on. I find it most appropriate to use Stokes’ result for a clock that would use spheres falling through fluids to tell the time. Today, having our timekeeping running on atomic clocks probably lets us be a little relaxed about how accurate and precise the Stokes clock would be as a timekeeper. A Stokes clock would surely make an effective piece of scientific art. It may be thought of as a kinetic sculpture [24]; kinetic art is a movement that has, like Stokes himself, strong links to Cambridge [25]. There is a considerable tradition of clocks as kinetic art, for instance in the works of sculptor Rowland Emett [26] and of clockmaker Martin Burgess [27]. It is also appropriate to think of balls falling as a clock because, historically, a different sort of falling ball clock was Phil.Trans.R.Soc.A important. So-called time balls were a common way to indicate and so synchronize time in Stokes’ day [28,29]; for instance, one was installed at the Royal Observatory at Greenwich in 1833 and was dropped—and is still dropped—each day at 1 o’clock in the afternoon so that people could synchronize their clocks with the official time. A Stokes falling sphere clock would provide a fitting arts–sciences memorial to George Gabriel Stokes. 378 Data accessibility. This article has no additional data. Competing interests. I declare that I have no competing interest. 20200214 : Funding. No funding has been received for this article. Acknowledgements. I dedicate this paper to my father, Albert Edward Cartwright (1910–2002), who introduced me to the wonderful world of clocks.

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