The papers on this aspect of his work, which are all to be found in Volume I of the Collected Works, deal with matters concerned with comets; the solar corona and the aurora; terrestrial magnetism; the electrical properties of clouds; the bursting of trees struck by lightning; the destruction of sound by fog; the refraction of sound by the atmosphere; the action of rain to calm the sea; the action of oil on water in preventing wind waves; surface tension and capillary action.
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Paper 16 (1874) and Paper 22 (1876) are likewise concerned with the refraction of sound by the atmosphere. In these Reynolds considers the effects due to the difference of wind velocity near the surface of the ground and at a height above it. This causes sound to be lifted when the waves move into the wind and to fall when waves move with the wind. He goes on to examine the effect of the variation of temperature in the atmosphere and explains that this causes sound waves to rise. Experiments are reported in which he studied these various effects but the papers are particularly noteworthy for the keenness of Reynolds' observations and revealing of his interest in outdoor pursuits. Thus:
`It has often astonished me, however, when shooting, that a wind which did not appear to me to make the least difference to the direction in which I could hear small sounds distinctly, should yet be sufficient to cover one's approach to partridges, and more particularly to rabbits, even until one was within a few feet of them - a fact which shows how much more effectively the wind obstructs sound near the ground than even a few feet above it.'
In a presentation to the Manchester Literary and Philosophical Society, `On the action of rain to calm the sea' (Paper 15, 1875) Reynolds showed by experiment that vortex rings produced by droplets of rain cause water to be carried well below the surface in appreciable amounts leading to the damping of wave motion (see Figure 2).
Papers 29 (1875) and 30 (1877) deal with the formation of raindrops, hailstones and snowflakes. Reynolds points out that hailstones are formed by the aggregation of small frozen particles resulting from coalescence with more rapidly descending larger particles. To prove it he ingeniously produced artificial hailstones by chilling a flow of air laden with tiny droplets of water through the use of an ether spray (see Figure 3).
In a paper read to the British Association in 1880 `On the effect of oil in destroying waves on the surface of water' (Paper 38), Reynolds attributes this to the surface tension varying inversely with the thickness of the oil film as the wind flows over it. This introduces tangential stiffness which prevents the surface taking up the motion of the water beneath. The effect is a dynamic one; instead of wave motion occurring in the water, eddies are formed below the surface. At the centennial British Association meeting held at York the following year Reynolds developed his ideas `On surface-tension and capillary action' still further (Paper 39).
On October 4th 1881 Reynolds made a short presentation to the Manchester Literary and Philosophical Society `On the floating of drops on the surface of water depending only on the purity of the surface' (Paper 40). In this Reynolds reported experiments using powder in the form of flowers of sulphur to determine the circumstances under which such drops are suspended.
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Papers 12 (1874) and 23 (1876) deal with forces exerted by the communication of heat to a surface immersed in a rarefied gas. Such forces are attributed to molecular influences and used to afford an explanation of the operation of the Crookes light mill (Figure 4), again using the kinetic theory model. The ideas involved led Reynolds to invent a simple photometer. Experiments on a light mill performed in collaboration with a colleague at Owens College, Arthur Schuster, demonstrated conclusively that the force which turns the vanes is not directly due to thermal radiation.
Reynolds' interest in the Crookes light mill was perhaps the initial stimulus for his thoughts on the so-called `transpiration' of gases. Graham had applied this term to the passage of gases through capillary tubes. Reynolds re-applied it to describe the motion of gases through minute channels including porous plugs and apertures in thin plates as well as capillary tubes. The results of his 1879 investigation are presented in one of the longest and most original of his papers entitled `On Certain Dimensional Properties of Matter in the Gaseous State' (Paper 33). In this, he showed by theory and experiment (using the apparatus shown in Figures 5a and 5b) that not only would a difference of pressure cause a gas to flow from one side of a porous plate to the other, but so also would a difference of temperature, even when initially the pressures on the two sides were equal. To this phenomenon he gave the name `thermal transpiration'.
In the same paper Reynolds also demonstrated that the extremely low pressure of the gas in the light mill was necessary because of the comparatively large size of the vanes and that similar results ought to be obtainable with smaller vanes at higher pressure. This he proved by experiments on fibres of silk and a `spider line' using the apparatus in Figure 6. He showed that, provided the pressure in the vessel containing the fibre was not more than approximately one-quarter of an atmosphere, the fibre moved towards an external source of heat.
Reynolds considered his investigation as a whole to have very profound implications, affording a proof that gas is not a continuous medium but possesses a `dimensional structure'. His description and explanation introduced the dependence of the density of the gas in relation to the size of the passages in the porous medium or the vanes of the light mill, and so involved what he called the `dimensional properties of gases'. This he believed could have more than philosophical importance:
`The actions only become considerable within extremely small spaces, but then the work of construction in the animal and vegetable world, and the work of destruction in the mineral world, are carried on within such spaces. The varying action of the sun must be to cause alternate inspiration and expiration of air, promoting continual change of air within the interstices of the soil as well as within the tissue of plants. What may be the effects of such changes we do not know, but the changes go on; and we may fairly assume that in the processes of nature the dimensional properties of gas play no unimportant part.'
Advancement of the kinetic theory of gases was a notable feature of the science of the 1870s and one to which Reynolds made a significant contribution alongside more established figures such as Maxwell. A short note by Reynolds on thermal transpiration (Paper 34), written in response to a criticism of his ideas by Maxwell and communicated to the Royal Society by its Secretary Sir George Stokes in 1879, provides an intriguing insight into the very formal manner in which the scientific establishment operated.
Reynolds' work on the surface forces due to heating and on thermal transpiration provided valuable experimental support for the developing kinetic theory (i.e. that `heat' is a manifestation of the molecules of which a gas is composed). Widespread acceptance of the theory by the scientific community did not come until much later. Reynolds' experiments and theoretical contributions in the period 1874 to 1879 were many years in advance of the time. It is of interest that one of Reynolds' students in his last years at Manchester, Sidney Chapman, was later to make further important advances in this field.
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He applied the term `dilatancy' to the property possessed by a mass of granular material to alter its volume in accordance with a change in the arrangement of its grains. His illustration of this in terms of the piling of spheres in two different ways is shown in Figure 8: the pile of spheres in cubical formation occupies a volume greater than that of the same number and size of spheres when piled in the second way.
Reynolds went on to illustrate this by characteristically simple means. If an india- rubber bottle with a glass neck is filled with water and the bag is then squeezed, water will be forced up the neck. But if the bottle is full of granular material and water, the effect of squeezing, up to a point, is to draw water down from the neck into the bag, because the grains have adopted an arrangement in which the volume of the interstices has been increased. In Reynolds' words, `the result, but for the knowledge of dilatancy, would appear paradoxical, not to say magical... Sand presents many striking phenomena well known but not hitherto explained, which are now seen to be simply evidence of dilatancy'. A familiar phenomenon explained by Reynolds was that observed when a foot is planted on firm moist sand on the sea- shore: an area around the foot appears to become dry, however, when the foot is raised, the sand beneath is found to be abnormally wet. The pressure of the foot has increased the volume of the interstices between the grains of sand below it and water has been drawn in to occupy the additional voids.
Reynolds stated that the recognition of this property of dilatancy would, from a practical point of view, place the theory of earth-pressures on a true foundation, but that `the greatest results are likely to follow in philosophy, and it was with a view to these results that the investigation was undertaken'. He goes on to declare that
`the recognition of this property of dilatancy places a hitherto unrecognized mechanical contrivance at the command of those who would explain the fundamental arrangement of the Universe, and one which, so far as I have been able to look into it, seems to promise great things, besides possessing the inherent advantage of extreme simplicity.'
These were the thoughts in his mind in 1885. Indeed, the title of his discourse to the Royal Institution in February, 1886 (Paper 51), on the same property of granular material, contained the significant words `possibly connected with gravitation'. Seven years previously, a paragraph concluding his paper on dimensional properties of matter in the gaseous state had shown that his thoughts were turning towards the possibility of solving the riddle of the luminiferous ether. However, it was not until February 1902, that his memoir `On the Sub- Mechanics of the Universe' was communicated to the Royal Society.
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The first step in Reynolds' discovery of this fundamental parameter appears to have been his observation that `the tendency of water to eddy becomes much greater as the temperature rises'. It occurred to him that this might be related to the well-known fact that the viscosity of water diminishes as the temperature rises, and moreover, that the physical property, kinematic viscosity, `is a quantity of the nature of the product of a distance and a velocity'.
He proceeded to consider the equations of motion and to establish that the forces per unit mass are of two distinct types, inertial and viscous, and further that the ratio of these is related to the dimensionless group D Um/n, in which Um is the mean velocity of the flow, D is the tube diameter and n the kinematic viscosity. In his paper he states:
`This is a definite relation of the exact kind for which I was in search. Of course without integration the equations only gave the relation without showing at all in what way the motion might depend upon it. It seemed, however, to be certain, if the eddies were due to one particular cause, that integration would show the birth of eddies to depend on some definite value of D Um/n .'
The story of his experiments using colour bands in glass tubes is well known. His final apparatus, so effectively portrayed by the well known illustration shown in Figure 9, consisted of a glass-sided tank, 6 feet long, 18 inches deep and 18 inches wide. Inside it was a glass tube with `a trumpet mouth of varnished wood, great care being taken to make the surface of the wood continuous with that of the glass'. On the right-hand side, the tube was connected to an iron pipe equipped with a valve which could be controlled by means of a long lever. On the left-hand side is the device for introducing a streak of dye into the trumpet, while a float and dial wagister the waterer the water-level in the tank and hence the volume being discharged through the glass tube. The experiments, made in 1880, consisted of filling the tank with water, allowing several hours for conditions to become steady, then opening the valve, at first only slightly.
Referring to Figure 10 the colour-band established itself as a beautifully steady streak (10a) but a point was reached on increasing the flow along the tube by opening the valve still further, when `the colour band would all at once mix up with the surrounding water, and fill the rest of the tube with a mass of coloured water' (10b). `On viewing the tube by the light of an electric spark, the mass of colour resolved itself into a mass of more or less distinct curls, showing eddies.' (10c).
Reynolds proceeded to measure the critical velocity for onset of eddies using three tubes of different diameter and in each case varying the water temperature. To a first approximation, the Reynolds Numbers based on these critical values of velocity were found to be the same (about 13000) for each of the tubes and for all water temperatures. He then set out to find the critical condition for an eddying flow to change into non-turbulent flow, referring to this as the `inferior limit'. To do this, he allowed water to flow in a disturbed state from the mains through a length of pipe and measured the pressure-drop over a five-foot distance near the outlet (see Figure 11).
Starting with low flows and gradually increasing them, he found that at a certain point the fluid levels in the differential manometer connected to the pressure-holes began to fluctuate: this coincided with the change in the character of the flow and provided visual evidence of the attainment of the critical velocity which he later determined by plotting the mean velocity against the pressure-gradient. Two sizes of pipe were tested. The result was to demonstrate that the critical velocities for the two pipes were in fact so related as to imply the same critical value of the Reynolds Number (about 2000).
In this renowned contribution to the development of fluid mechanics, Reynolds not only evolved the number to which his name was later attached and determined the critical value below which flow in a pipe is always stable and laminar, but also provided a detailed picture of the resistance to flow in pipes. In addition, he took the further step of showing that, for given conditions of surface roughness, the friction coefficient is a unique function of the Reynolds Number. The following year (1884) in his Presidential Address to the British Association in Montreal, Lord Rayleigh paid this tribute:
... Professor Reynolds has traced with much success the passage from the one state of things to the other, and has proved the applicability under these complicated conditions of the general laws of dynamical similarity as adapted to viscous fluids by Professor Stokes. In spite of the difficulties which beset both the theoretical and experimental treatment, we may hope to attain before long to a better understanding of a subject which is certainly second to none in scientific as well as practical interest.
Stokes himself, in his capacity as President of the Royal Society, also singled out this exceptional paper in his statement of 30th November 1888 on the occasion of the presentation of a Royal Medal to Osborne Reynolds `for his investigations in mathematical and experimental physics, and on the application of scientific theory to engineering.'
In a subsequent shorter contribution entitled `On the two manners of motion of water', Paper 48 (1883), Reynolds compared the characteristics of flow in converging and diverging channels, pointing out that, whereas in the former the conditions are favourable for producing steady flow, in the latter the flow is likely to be turbulent and unsteady. This he contrasted with flow in parallel channels where below a certain flow rate steady streamline conditions prevail and above that turbulence is encountered.
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`....must be of a general and important kind, such as the unexplained laws of the resistance of fluid motions, the laws of universal dissipation of energy and the second law of thermo-dynamics'It was the culmination of twenty-five years of research and came as a result of Reynolds conducting `a more rigorous examination and definition of the geometrical basis on which the analytical method of distinguishing between molar-motions and heat-motions in the kinetic theory of matter is founded; and of the application of the same method of analysis, thus definitely founded, to distinguish between the mean-molar-motions and relative-molar-motions, where, as in the case of steady-mean (turbulent) flow along a pipe, the more rigorous definition of the geometrical basis shows the method to be strictly applicable.' Its origins date back to Reynolds' interests in the properties of gases at the outset of his career.
Section I of the paper is a lengthy introduction which sketches the background to the work to be reported, states the objectives, outlines the approach adopted and summarises the findings. He begins by tracing the development between 1822 and 1845 of the equations governing fluid motion, the Navier-Stokes equations. He refers to the comparisons reported by Stokes in 1857 between theoretical solutions of the equations and certain experimental observations, which seemingly proved the assumptions made in the formulation of the equations. These were restricted to the drag on slowly moving objects of small size and the resistance to the flow of liquid at low rates through long tubes of small bore. These he contrasts with examples where theoretical results were found to be directly at variance with common experience in the case of the motion of larger bodies at higher velocity and the discharge of fluid through large tubes at greater flow rates. He points to the fact that Stokes was aware that the discrepancies resulted from the presence of eddies which rendered the actual motion other than that to which the theoretical solutions referred.
Reynolds then goes on to discuss his own contribution in 1883 in identifying the dimensionless parameter, the Reynolds Number, which governs whether the flow in tubes will be direct (laminar) or unsteady (turbulent), and establishing by experiment the value of `the inferior limit', above which transition can occur. He asserts that:
`These experimental results completely removed the discrepancy previously noticed, showing that whatever may be the cause, in those cases in which the experimental results do not accord with those obtained by the singular solution of the equations, the actual motions of the water are different. But in this there is only a partial explanation, for there remains the mechanical or physical significance of the existence of the criterion to be explained'Reynolds then flatly states `my object in this paper is to show that the theoretical existence of an inferior limit to the criterion follows from the equations of motion,' before continuing, `I also show that the limit to the criterion obtained by this method of analysis, and by integrating the equations of motion in space, appears as a geometrical limit to the possible simultaneous distribution of certain quantities in space, and in no wise depends on the physical significance of these quantities.'
Expressed in modern terms, Reynolds sought to write the components of velocity in a turbulent flow in terms of mean and fluctuating quantities and to perform averaging of the momentum equations. This showed that these equations contained additional terms which could be thought of as apparent stresses due to turbulence. He then derived equations for the kinetic energy of the mean motion and the kinetic energy of the turbulent motion and noticed that they contained terms, the turbulent energy production terms, which represent the total exchange of energy between the mean motion and the kinetic energy of the turbulence.
To explain the occurrence of transition in channel flows Reynolds examined the conditions under which the turbulence energy could be sustained. Using the turbulence energy equation and considering a control volume for which the turbulent diffusion of turbulent energy would integrate to zero, he arrived at a condition for `the inferior limit' based on the idea that the total turbulence production must equal the total turbulence dissipation. He analyzed the particular case of flow between parallel walls driven by a pressure gradient. Using an analytical function to describe a small disturbance superimposed on a fully developed laminar flow he evaluated the total turbulence production and the total turbulence dissipation. The result was that they were in balance at a particular value of the Reynolds number of 517, based on the bulk mean velocity and the distance between the walls.
It is clear that Reynolds' historic paper contained the foundations of modern turbulence modelling. The concept of turbulent stress, the role of the turbulent production terms in the exchange of energy between the mean motion and the turbulence and the dissipation of turbulence are matters which remain of central importance in the subject of turbulence. In essence he conceived the idea of the energy cascade in turbulent flows. One can take his equations for the kinetic energy of the mean motion and for turbulence energy and with little modification derive the corresponding equations currently in use. A century later the basic ideas contained in Reynolds' paper are still utilised in almost all our numerical predictions of practical turbulent flows, at least in situations close to industrial applications.
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This showed that it was possible for a journal to drag oil between itself and the bearing, causing a rise of pressure sufficient to support the shaft. Reynolds realized that the maintenance of a film of oil between the shaft and its bearing might be explained by hydrodynamics on the assumption that the centre of the rotating shaft shifted away from the centre of the bearing in such a direction as to make the film of oil thicker on the ingoing than on the outgoing side.
Excited by Tower's results and after a preliminary reference to them at the 1884 British Association meeting, he pursued the subject with such energy that his famous contribution `On the Theory of Lubrication...' appeared in the Philosophical Transactions of the Royal Society in 1886 (Paper 52). In this very lengthy and detailed paper, Reynolds not only formulated and integrated the hydrodynamic equations but also, by allowing for the variation of viscosity with temperature, obtained close agreement with the observed pressures.
The paper provides a very good example of Reynolds' approach of discussing the physical or mechanical picture of things before proceeding with the mathematics. Under the heading `General View of the Action of Lubrication', he evolves the basic concepts involved by first considering two plane surfaces. In Case 1 (Figure 12) they are parallel to one another but AB is moving to the left with a velocity U and `pumping' fluid between itself and the stationary surface CD. The sloping lines show how the velocity varies between U at AB and zero at CD. The pressure is constant between D and C although there are tangential viscous stresses on the two surfaces.
Next, he considers the same plates without tangential motion with the upper one being forced downwards to squeeze out the fluid. A pressure distribution is then created, reaching its highest value at the centre (Case 2, Figure 12). He then combines the two: tangential motion and squeezing of the surfaces together (Case 3, Figure 12). In this instance, the distribution of pressure resembles that of Case 2, while the mean viscous stress on CD is similar to that of Case 1.
In order to account for the case of lubricated surfaces which are not approaching one another, but which nevertheless are capable of sustaining a transverse load, it only remains to visualise Case 4, Figure 12, where one surface is inclined to the other. At the section P1 Q1, there is the same uniform distribution of velocities as if the surfaces were parallel to one another and at a distance P1Q1 apart. But to either side of P1Q1, the velocities must be modified to preserve continuity and so they adopt a shape similar to those of Case 3. Correspondingly, the pressure follows the general shape of the curve shown in Figure 12, with its maximum value at the section P1Q1. `This', Reynolds concludes, `is the explanation of continuous lubrication. The pressure of the intervening film of fluid would cause a force tending to separate the surfaces.'
Ever practical, Reynolds then considers the question of a cylindrical surface, developing the detailed mathematics and comparing the implications with Tower's observations:
...The result of the whole research is to point to a conclusion (important in Natural Philosophy) that not only in cases of intentional lubrication, but wherever hard surfaces under pressure slide over each other without abrasion, they are separated by a film of some foreign matter, whether perceivable or not. And that the question as to whether this action can be continuous or not, turns on whether the action tends to preserve the matter between the surfaces at the points of pressure, as in the apparently unique case of the revolving journal, or tends to sweep it to one side, as is the result of all backwards and forwards rubbing with continuous pressure...An interesting postscript to this giant amongst Reynolds' papers is provided by Paper 67 (1899), the final one which he read to the Manchester Literary and Philosophical Society. This is entitled `On the slipperiness of ice'. In it he proposes that an explanation of the phenomenon is afforded by the ideas on lubrication contained in Paper 52.
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`Now I will ask you to look at these balloons. They are familiar objects enough, and yet they are most sensitive anemometers, more sensitive than anything else in this room; but even they do not show any motion; each of them forms an internal bounding surface of the air. I send an aerial messenger to them, and a small but energetic motion is seen by which it acknowledges the message, and the same message travels through the rest, as if a ghost touched them. It is a wave that moves them. You do not feel it, and, but for the surfaces of the air formed by the balloons, would have no notion of its existence.'
He then repeated the experiment but added smoke to the vortex-generating box:
`I will now fulfil my promise to reveal the silent messenger I sent to those balloons. The messenger appears in the form of a large smoke ring, which is a vortex ring in air rendered visible by smoke instead of colour ...
These are, if I may say so, the phenomenal instances of internal motion of fluids. Phenomenal in their simplicity, they are of intense interest, like the pendulum, as furnishing the clue to the more complex. It is by the light we gather from their study that we can hope to interpret the parallel of the vortex wrapped up in the wave, as applied to the wind of heaven, and the grand phenomenon of the clouds, as well as those things which directly concern us, such as the resistance of ships.'
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Paper 2 `The tails of comets, the solar corona, and the aurora, considered as electrical phenomena'. Manchester Literary and Philosophical Society, Memoirs, Series 3, Vol. 5, Session 1870-71. Back
Paper 3A `On cometary phenomena'. Manchester Literary and Philosophical Society, Memoirs, Series 3, Vol. 5, Session 1871-72. Back
Paper 3B `On an electrical corona resembling the solar corona'. Manchester Literary and Philosophical Society, Memoirs, Series 3, Vol. 5, Session 1871-72. Back
Paper 4 `On the electro-dynamic effect which the induction of statical electricity causes in a moving body. - This induction on the part of the sun a probable cause of terrestrial magnetism'. Manchester Literary and Philosophical Society, Memoirs, Series 3, Vol. 5, Session 1871-72. Back
Paper 5 `On the electrical properties of clouds and the phenomena of thunder storms'. Manchester Literary and Philosophical Society, Proceedings, Vol. 12, Session 1872-73. Back
Paper 7 `On the bursting of trees and objects struck by lightning'. Manchester Literary and Philosophical Society, Proceedings, Vol. 13, Session 1873-4.Back
Paper 7A `On the destruction of sound by fog and the inertness of a heterogeneous fluid'. Manchester Literary and Philosophical Society, Proceedings, Vol. 13, Session 1873-4.Back
Paper 11 `On the forces caused by evaporation from, and condensation at, a surface'. Royal Society, Proceedings, No. 153, 1874. Back
Paper 12 `On the surface-forces caused by the communication of heat'. Philosophical Magazine, November 1874.Back
Paper 15 `On the action of rain to calm the sea'. Manchester Literary and Philosophical Society, Proceedings, Vol. 14, Session 1874-5. Back
Paper 16 `On the refraction of sound by the atmosphere'. Royal Society, Proceedings, No. 155, 1874.Back
Paper 22 `On the refraction of sound by the atmosphere'. Royal Society, Phil. Trans., Vol. 166, Pt. 1. Back
Paper 23 `On the forces caused by the communication of heat between a surface and a gas; and on a new photometer'. Royal Society, Phil. Trans., Vol. 166, Pt. 2.Back
Paper 27 `On the rate of progression of groups of waves and the rate at which energy is transmitted by waves'. Nature, Vol. 16, Aug. 23, 1877.Back
Paper 29 `On the manner in which raindrops and hailstones are formed'. Manchester Literary and Philosophical Society, Memoirs, Series 3, Vol. 6, Session 1876- 77. Back
Paper 30 `On the formation of hailstones, raindrops, and snowflakes'. Manchester Literary and Philosophical Society, Memoirs, Series 3, Vol. 6, Session 1877- 78.Back
Paper 31 `On the internal cohesion of liquids and the suspension of a column of mercury to a height more than double that of the barometer'. Manchester Literary and Philosophical Society, Memoirs, Series 3, Vol. 7, Session 1877-78. Back
Paper 33 `On certain dimensional properties of matter in the gaseous state'. Royal Society, Phil. Trans., Pt. 2, 1879.Back
Paper 34 `Note on thermal transpiration'. (In a letter to Professor Stokes, Sec. R.S. Communicated by Professor G.G. Stokes.) Royal Society, Proceedings, No. 203, 1880.Back
Paper 35 `Some further experiments on the cohesion of water and mercury'. Manchester Literary and Philosophical Society, Proceedings, Vol. 20, Session 1880-81.Back
Paper 38 `On the effect of oil in destroying waves on the surface of water'. British Association Report, 1880.Back
Paper 39 `On surface-tension and capillary action'. British Association Report, 1881. Back
Paper 40 `On the floating of drops on the surface of water depending only on the purity of the surface'. Manchester Literary and Philosophical Society, Proceedings, Vol. 21, Session 1881-82. Back
Paper 44 `An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels'. Royal Society, Phil. Trans., 1883.Back
Paper 48 `On the two manners of motion of water'. Royal Institution, Proceedings, March 1884. Back
Paper 50 `On the dilatancy of media composed of rigid particles in contact. With experimental illustrations'. Philosophical Magazine, December, 1885. Back
Paper 51 `Experiments showing dilatancy, a property of granular material, possibly connected with gravitation'. Royal Institution, Proceedings, February 12, 1886. Back
Paper 52 `On the theory of lubrication and its application to Mr. Beauchamp Tower's experiments, including an experimental determination of the viscosity of olive oil'. Royal Society, Phil. Trans., Pt. 1, 1886.Back
Paper 61 `Study of fluid motion by means of coloured bands'. Royal Institution, Proceedings, June 2, 1893. Back
Paper 62 `On the dynamical theory of incompressible viscous fluids and the determination of the criterion'. Royal Society, Phil. Trans., 1895.Back
Paper 67 `On the slipperiness of ice'. Manchester Literary and Philosophical Society, Memoirs and Proceedings, Vol. 43, Session 1898-9.Back
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