THE VISCOSITY COEFFICIENT OF AIR WITH AN INQUIRY INTO THE EFFECT OF THE RONTGEN RAY THEREON BY FREDERICK G. REYNOLDS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF SCIENCE IN THE FACULTY OF SCIENCE, NEW YORK UNIVERSITY Salem TME SALEM PRESS Co., SALBAV, MASS. 1904 THE VISCOSITY COEFFICIENT OF AIR WITH AN INQUIRY INTO THE EFFECT OF THE RONTGEN RAY THEREON BY FREDERICK G. REYNOLDS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF SCIENCE IN THE FACULTY OF SCIENCE, NEW YORK UNIVERSITY Salem press: THE SALEM PRESS Co., SALEM. MASS. 1904 -: : '. "/I" : ;\ ' * * : " * : * *"* "" THE VISCOSITY COEFFICIENT OF AIR WITH AN INQUIRY INTO THE EFFECT OF THE RONTGEN RAY THEREON BY FREDERICK G. REYNOLDS ONE of the chief features of the gaseous form of matter is the distinguishing simplicity of its laws. The molecules of the medium in this state seem to act in a less complicated manner than when in a liquid or solid state. The investigation of any property of matter in the form of a gas might therefore be more simple than a similar investigation on any other form of matter, and the results attending such an effort might serve to throw some light on these other and more complicated forms. The particular phenomenon in question has claimed the attention of investigators for many years and while the general truths may be said to be settled, there have always existed many differences of detail in which one may find a justification for further effort in the same field. By the viscosity of a gas is meant the resistance which it offers to a continuous change of form depending on the rate at which that change is effected. The coefficient of viscosity may perhaps be best understood by considering a stratum of the gas infinite in extent. If this stratum were enclosed by two parallel planes at a distance d centimetres from each other and if one of these planes were to move parallel to the other with a velocity of v centimetres per second, keeping always in the same direction, till the gas embraced takes up its final velocity thus imparted, this velocity would decrease uniformly as we pass from the moving plane towards the other. If again the strata in contact with the two planes have the same velocity as the planes themselves, the strata will decrease in velocity centimetres per second for every centimetre we depart from the moving plane. The friction between any two contiguous (3) REYNOLDS strata will then be the same as between either plane and the stratum in contact with it. If this friction be denoted by a tangential force F on every square centimetre of surface then F = U -3 where U is the coefficient of viscosity of the gas. The measurement of the internal friction or viscosity of a gas is the determination of the value of this coefficient. The property which enables one layer to drag along another might in liquids be looked upon as a cohesion acting in opposition to the motion but in the case of a gas we must- look for another explanation. In the light of the Kinetic Theory we look upon the gas as manifesting itself by the very rapid motion of its molecules. When two layers are moving therefore with unequal velocities, the molecules from the faster moving layer penetrate the slower moving one with their faster forward velocity and vice versa, so a measure of the friction would be the momentum which is thus carried over. The heat motion of the molecules is so great (being at Cent, a mean velocity of 447 metres per second), that it is not affected by the comparatively slow forward motion of the molecules. It would naturally follow that this friction would increase with an increase of the temperature for the speed of the molecules increases with the temperature, being proportional to the square root of the absolute temperature. That the friction is inde- pendent of the density seems paradoxical at first, but the theory is at least justified when we consider that the transfer is made by the molecules and is greater as their number and activity are greater, hence as the density is greater ; but at the same time the transfer takes place in layers whose distance apart may be traversed by the molecules and as the density increases, this distance is restricted so that it is possible that the combined result of the two causes should produce a con- stant effect. Internal Friction as a property of fluid media is spoken of as early as the year 1687 by Newton 1 who makes use of the 1 Phil. Nat. Princ. Math. 1687, Lib. II, Sect. 9. THE VISCOSITY COEFFICIENT OF AIR 5 following language : " Attritus vel resistentia quee oritur ex defectu lubricitatis," and he advanced the theory that the friction between two neighboring layers of a fluid does not depend upon the pressure, that it is proportional to the difference of the velocities of the layers and that it is propor- tional to the size of" the surface. The methods of approaching and handling the investiga- tions into the viscosity of fluids, both theoretical and practical, may be divided into four general classes : I. By their transpiration through capillary tubes, in which phase of the investigation Poiseuille 1 and Graham 2 were the pioneers, the former as to liquids and the latter as to gases. II. By the swinging of pendulums, a method followed by Baily 3 , Bessel 4 and Du Buats all of which is dis- cussed in a paper by Stokes 6 " On the Internal Friction of Fluids on the Motion of Pendulums." III. By the torsional vibrations of an immersed disc, first adopted by Coulomb. 7 IV. By the torsional vibrations of a vessel filled with the fluid in question which might rather be considered a development of the preceding method. 8 Each method seems to have its practical and theoretical advantages and disadvantages and the discordant results at- tending the early efforts in each have since been brought more closely into harmony through criticism and refinements of process. In speaking of the transpiration and torsion methods Meyer remarks of the discrepancies "als begrundet in den Annahmen, welche zur Gewinnung brauchbarer Formeln 1 Mem. de Savants Strangers, 1846, IX, p. 433. 2 Phil. Trans. 1846, CXXXVI, p. 573; 1849, CXXXIX, p. 349. a Phil. Trans. 1832. 4 Berlin Acacl. 1826. 5 Principes d'Hydraulique, 1786. 6 Cambridge Phil. Trans. 1850, IX, pt. 2. 7 Mem. de 1'Institut National, III, p. 246. s Helmholtz and Pietrowski, Sitzungsber. der kk. Akad. April, 1860. 6 REYNOLDS in die theoretische Entwickelung eingefiihrt wiirden ; die dabei eintretenden Vernachlassigungen waren in dem Falle schwin- gender Scheiben derart, dass der Coefficient zu gross, in dem Falle des Ausflusses durch Capillarrohren derart dass erzu klein erhalten werden miisste ; die wahren Werthe wiirden also zwischen den Resultaten diesen beiden -Methoden liegen." Which method comes nearer to the truth is not decided, but the possible objection to the transpiration experiments lies in the difficulty of measurement of the diameters of the tubes and also in the fact that on account of the smallness of the bore one cannot be certain that the action between the molecules of the gas and of the substance of the tube does not affect the result. The pendulum method seems to be considered capable of great accuracy, but in the theory applicable toitBaily made the error of considering that the viscosity is dependent on the density. A correction of his results on the basis of a correct hypothesis has been promised but does not seem to have been made. The difficulties or objections connected with the method of a vibra- ting disc seem to be less than those in the transpiration method and it may be for this reason that it has been the method gener- ally selected by the later investigators ; but the trouble here lies in the necessity of accounting properly for the effect of the motion of the medium near the edge of the discs. As a result of Maxwell's 1 analysis and deductions given in 1860, both he and Meyer 2 started to demonstrate practically the truth of the same, using as a basis of their investigations the method adopted by Coulomb ; but Meyer altered the earlier method in so far as to substitute three oscillating discs with a common axis in- stead of the one used by Coulomb. These plates were so ar- ranged that it was possible to separate them, thereby exposing six surfaces or to place them together exposing only the two. In this way the friction on the exposed parts of the hanging system such as the mirror, etc., as well as the internal friction of the wire itself, could be eliminated by considering the differ- 1 Phil. Mag. 1860, (4) XIX, p. 31; Phil. Trans. 1866, CLVI, p. 249. 2 Pogg. Ann. 1865, CXXV, p, 177. THE VISCOSITY COEFFICIENT OF AIR 7 ence of the effects obtained when the plates were apart and to- gether. Maxwell improved on this by inserting four fixed plates between the moving ones, which had the effect of caus- ing the friction of the gas to exert a greater resistance and therefore one more easily measured, an advantage which was immediately recognized by Meyer 1 himself. A further ad- vantage of Maxwell's method lies in the fact that his deductions give the viscosity coefficient directly, whereas Meyer's deduc- tions lead to the square -root which would cause the effect of any error to be increased. As a consequence Maxwell's results showed a better agreement in the effort to verify the independ- ence of the coefficient and the pressure within the range of his observations, viz. : from one to ^L atmospheres. In 1887, as a result of a correction suggested by Koenig, 2 an improvement was made by Meyer in his method and he calculated his results again on the improved theory 3 . This independence of the viscosity and the pressure cannot be expected to hold in the limiting cases, as would follow from a consideration of the theory. The density and the free path enter we have seen as factors, but when the former approaches the latter approaches co which would cause this phenomenon to fall outside of a development where only finite densities and free paths are considered. Investigations on this point at low pressures were taken up by Kundt and Warburg 4 , who followed Maxwell's methods and found that below a pressure of -$\ atmospheres there was a marked diminution in the logarithmic decrement from which the viscosity coefficient is calculated. They however found an ex- planation of this in a phenomenon which they called Slipping (or External Friction), by which is meant the sliding of the medium along the bounding surface as distinguished from the 1 Pogg. Ann. 1871, CXLIII, p. 14. 2 Wied. Ann. 1887, XXXII, p. 193. a Wied. Ann. 1887, XXXII, p. 642; Sitzuugsber. d. Miinchener Akad. 1887 XVII, p. 343. 4 Monatsber. d. Berl. Akad. 1875, p. 160; Pogg. Ann. 1875, CLV, pp. 337, 525. 8 REYNOLDS sliding of one layer of the medium along another layer ; and they explained this falling oft' as not caused by a decrease of the coefficient of viscosity, but by an increase of the Slipping at the lower pressure. Even a highly 'polished surface must, in comparison with the molecules of the gas, be considered rough, and under ordinary pressures a sliding along this, especially since the forward motion is very slow, is practically impossible, but this would not be the case in a very rare medium. By the term Coefficient of Slip they designated the ratio of the internal to the external friction, with the result that, between the pres- sures of .ti and 20 millimetres of mercury, the value of the Coefficient of Slip was found to be inversely proportional to the density of the gas and very nearly equal to the free path of the molecules. Therefore the external friction is directly pro- portional to the density. Warburg 1 experimented further on the external friction by the transpiration method and obtained smaller values for the Coefficient of Slip but Breitenbach 2 later obtained results nearer the earlier ones. The correctness of this explanation of Kundt and Warburg for the sudden falling off of the logarithmic decrement was later proven by Crookes 3 on a theory developed by Stokes, 4 wherein he used his vacuum tubes with vertically suspended strips of mica thereby excluding the possibility of any external friction. The results of these investigations show the inde- pendence of viscosity and pressure down to pressures so low as to be no longer accurately measured. However, at much higher vacuums there is a break in the constancy. At extremely high pressures the law does not hold, but this might also be expected from the assumptions in the theory. Starting with the hypothesis that the particles traverse straight paths between their successive encounters with each other, the curved paths traversed during the period of actual encounter would be negligibly small, at ordinary densities or at low 1 Pogg. Ann. 1876, CLIX, p. 399. Wied. Ann. 1899, LXVII, p. 826. 3 Phil. Trans. 1881, CLXXII, p. 387. 4 Phil. Trans. 1881, CLXXII, p. 435. THE VISCOSITY COEFFICIENT OF AIR 9 / densities ; but this would not be the case when the density be- comes very great. The law considers the molecules unchange- able and the Dissociation of the molecules under this extreme condition may contribute to this variation. This would be particularly true in the consideration of vapors. Warburg and Babo 1 have shown in the case of carbonic acid that the coefficient of viscosity increases with the density at pressures from 30 to 120 atmospheres, which law seems to hold for other gases as well. In 1866 Meyer 2 published a theory wherein, assuming the independence of pressure upon the friction, he gave a law governing the speed of flow of gases through capillary tubes which corresponded with the law found by Poiseuille for liquids and which, by considering the amount of gas transpired in a given time, leads to the value of the coefficient of friction. His calculations from the results of Graham's observations proved the correctness of his assumption. He obtains for the coeffi- cient of viscosity of the air at Cent, the values .000171, .000170, .000174 and he also gives the coefficients for nine- teen other gases. Obermayer, 3 Puluj 4 and Mutel, following this method, obtain for air the values .000167, .000180, .000172 respectively. As to the fact of the increase of the viscosity with the in- crease of the temperature there can be no doubt, but as to the rate of such change there is not such an agreement among the different investigators. Maxwell first put it as increasing directly as the absolute temperature or proportional to (1 + ad) where 6 is the temperature from the freezing point and a is the coefficient of expansion. Later experiments show that, for air at least, it does not increase so rapidly. Efforts were made to represent the change by a factor of the form (1 + ad) 11 where n was first given as f ; but later Barus, 5 adopting the transpiration 1 Wied. Ann. 1882, XVII, p. 390; Sitz. d. Berl. Akad. 1882, p. 509. 2 Pogg. Ann. 1866, CXXVII, pp. 253, 353. 3 Carl's Rep. 1876, XII, p. 13. 4 Wien. Ber. 1874, LXIX(2), p. 287; LXX(2), p. 243. 5 Bull, of thell. S. Geol. Surv. No. 54, Washington 1889. Wied. Ann. 1889, XXXVI, p. 358. 10 REYNOLDS method and with a range of temperature from to 1300, gave to n the value f for air and hydrogen. Different values were found by others for other gases and Wiedeman 1 found that n became smaller with the increase of temperature, a fact also found by Holman 2 . Schumann 3 however chose to represent the change by a formula consisting of two factors (1 + 7#) 2 y/l + aO from the consideration that the viscosity is dependent upon two things, the molecular free path and the molecular speed; and while this satisfied his own observations founded upon Maxwell's method it did not agree with those of Barus. Sutherland, 4 tak- ing into consideration the apparent sphere of action of the molecules as distinguished from the real sphere of action, changed the first factor and represented the change by the equation, . + e) being the absolute temperature, a the coefficient of expansion and c a constant depending on the attractive forces exerted be- tween the molecules when near together. With this he found results agreeing well with Barns and Holman. Vapors as distinguished from gases proper have also been the subject of observations to determine if they follow the laws of gases as to viscosity. In these researches L. Meyer 5 and Koch 6 have followed the transpiration method, while Puluj 7 and Schumann 8 have used the oscillation method with the result that, at ordinary pressures, like true gases the coefficient of viscosity is independent of the pressure. However the formulae given for true gases do not here represent exactly the change with the 1 Arch. d. Sc. Phys. et Nat. 1876, LVI, p. 273. 2 Proc. Amer. Acacl. Boston 1877, XII, p. 1; Phil. Mag. (5), III, p. 81; XXI, p. 199. 3 Wied. Ann. 1884, XXIII, p. 353. 4 Phil. Mag. 1893 (5) 202. p. 507. 5 Wied. Ann. 1879, VII, p. 497; 1881, XIII, p. 1; 1882, XVI, pp. 369, 394. 6 Wied. Ann. 1883, XIX. p. 857. 7 Wien. Ber. 1878, LXXVIII (2), p. 279; Carl's Repert. 1878, XIV, p. 573. 8 Wied. Ann. 1884, XXIII, p. 353. THE VISCOSITY COEFFICIENT OF AIR 11 temperature and it seems that the viscosity changes in a much larger ratio. Attempts to justify this disagreement have been made wherein the causes which make a perfect gas differ from the laws of vapors are used for a basis of argument. Of these perhaps the principal one which might be mentioned is the phenomenon of Dissociation. The question of the effect of Electricity and Magnetism on the viscosity of a gas has not as yet been very satisfactorily an- swered, but that these cause no effect would appear from the results of Pagliani 1 and Noack 2 . Koenig 3 oscillated crystal spheres in a magnetic field, but the effect on the coefficient of viscosity was not apparently considered. Quincke 4 also inves- tigated the viscosity of fluids in the electric field, showing an increase without any apparent change in the period of swing of the spheres. To these results, however, objection was made by Boltzman 5 . The results of the efforts of investigators to de- termine the Coefficient of Viscosity of the Air, with the methods used by them may be seen by the table on page 12. As a consequence of the corrections in theory and through the refinements of process the results of the different methods, while they varied widely at the start, have been brought into closer harmony. Since no new method differing in principle from these has been given, the only available way of testing the correctness of the results is in a variation of the methods at hand. The method of torsional vibrations seems to permit of a wider range of variation and therefore to be best suited to such a purpose. The objection to the use of discs, however, has been mentioned as lying in the allowance to be made for the action at the edge of the disc ; but the improvement of method in substituting several in place of the one partly obviated this. This objection would not of course hold if a spherical surface were substituted and for this reason as well 1 Ac. Torino, 1885, XX, p. 615; 1887, XXII, p. 1. 2 Wied. Ann. 1886, XXVII, p. 289. 3 Wied. Ann. 1887, XXXI, p. 273. 4 Wied. Ann. 1897, LXII, p. 1. 5 Wied. Ann. 1897, LX, p. 399. REYNOLDS as that its symmetry renders possible a rigorous theoretical development, the sphere or spherical shell seems to answer the requirements best. Investigator. Method. Coef. C.G.S. Temp. Stokes Meyer 1 1 1 Baily's Pendulum Oscillation Bessel's Exper. Girault's < ' .000104 .000353 .000275 .000384 44 44 44 .000360 18 44 2 44 Oscillation (2) 44 .000333 .000323 8.3 21.5 44 .000366 34.4 44 3 44 Transpiration .000168 .000174 Maxwell 4 Oscillation .000200 18 Puluj 5 Von Obermayer 6 44 7 Transpiration 4 I ( ( .000179 .000171 .000168 Schumann 8 Oscillation .000168 Schneebeli 9 Tomlinson 10 4 4 Transpiration Oscillation (cylinder) 44 .000171 .000179 .000176 12.02 10.64 4 4 . 44 .000177 14.63 4 4 4 4 .000178 11.69 4 4 " (spheres) ,000176 9.97 DESCRIPTION OF APPARATUS A vertical glass tube 92.71 cm. long and 6.85cm. in diam- eter has its lower end fitted air tight into a brass collar 7 cm. 1 Crelle's Jour. 59, p. 229. 2 Fogg. Ann. 1871, CXLIII, p. 14. 3 Pogg. Ann. 1873, CXLVIII, pp. 37, 203. 4 Phil. Trans. 1866, 156, p. 249. 5 Wien, Sitz. Abth. 2, 1874, LXIX, p. 278. Wien. Sitz. Abtb. 2, 1874, LXX, p. 243. 6 Carl's Rep. 1876, XII, p. 15. 7 Wien. Sitz. 1875, LXXI, p. 281 ; 1876, LXXIII, p. 433. 8 Wiecl. Ann. 1884, XXIII, p. 353. 9 Arch, des Sci. Phys. et Nat. Geneve, 1885 (3) XIV, p. 197. 10 Phil. Trans. (2) 1886, CLXXVII, p. 768. THE VISCOSITY COEFFICIENT OF AIR 13 high which screws into a brass circular bed-plate 28cm. in diameter. The lower surface of the plate is ground and polished so that the ground edge of a bell jar when placed against it may form an air-tight connection. The upper end of the glass tube is capped with a brass cylinder in the form of an inverted cup whose upper surface is 3.49cm. thick and is pierced by a conical shaped opening into which fits the cone of a torsion head, which being carefully ground is capable of easy motion but at the same time is air-tight. The brass collar at the base is fitted with a plane glass window which is adjusted air-tight over a part of the glass tube which has been cut away. In this way the observer may see clearly within that part of the cylinder since the rays of light are not interfered with by the cylindrical surface of the tube. The bed-plate rests upon a tripod fitted with adjusting screws in its legs so that the glass cylinder may be made vertical. By an arrangement of clamps acting as a wedge, a bell jar may be attached to the tripod and held tightly against the under surface of the bed-plate. In the first and second sets of experiments a hollow brass spherical shell was used and careful measurements of its diameter were made for the purpose of determining any variations from a truly spherical form with the result that the diameter used as an axis was found to be very slightly smaller than were the diameters perpendicular to it, but the difference was so small as to cause no appreciable error in considering the surface spherical. The dimensions were therefore taken as follows: external diameter 12.68cm., weight 235.7 grammes and moment of inertia (calculated) 6316.0647. This sphere hung within the bell jar and was attached to the suspending wire by means of a brass rod 28.8 centimetres long whose lower end screwed into the sphere and whose upper end was fitted with a cross pin in the shape of a T which fitted into a hook-shaped saddle attached to the wire. The wire was of German silver .0254 cm. in diameter and 79.3cm. long and was used throughout the entire set of experiments. A small plane mirror was attached to the brass connecting rod, and this was 14 REYNOLDS capable of a perpendicular adjustment so that the mirror could be brought directly opposite the opening in the base of the apparatus. The whole suspended apparatus was thus protected from any extraneous currents of air and a very small turn to the torsion head would suffice to set the system into motion which could be seen through the agency of the mirror. The readings were made by means of a tangent scale and telescope directly in front of the opening opposite the mirror and from 95 cm. to 100 cm. distant therefrom. The temperature of the gas was observed from a thermometer placed within the bell jar and easily read from without. The length of time taken for a swing from one position of rest to the next was obtained by means of a break-second chronometer with an electrical attachment to a recording apparatus from which yj^ seconds could be estimated. The barometer from which the readings were taken hung near the apparatus. A preliminary set of experiments was made and the results were calculated for the purpose of determining : 1. The best working limits of the apparatus as used. 2. Whether either the logarithmic decrement or the period of one swing depends upon the amplitude of the same. 3. Whether within the range of future experiments there is any change corresponding to the varying barometric pressure. 4. Any peculiarities liable to occur and therefore to be avoided. PRELIMINARY SET The method of proceeding in this instance was as follows : The sphere was first attached to the brass rod and the latter was then placed with its end in the saddle ; the bell jar would then be brought up and with as little jar as possible would be clamped into position. Then by menus of the torsion head, which would be moved first in one direction and then in the reverse, an oscillating movement would be given to the sphere. In many cases a slight pendulum motion would also be given, but with a little practice it was found that this could be entirely stopped by a slight pressure of the hand on the outside of the THE VISCOSITY COEFFICIENT OF AIR 15 apparatus. The oscillations were allowed to continue some- times at the start for an hour, whereby over two hundred would have occurred, the object of this being to accustom the sus- pending wire to its load and to the torsional vibrations so that when the readings proper were taken the damping effect of the internal friction of the wire itself would be reduced to as small and as constant a quantity as possible. The necessity of this precaution was shown by the work of Tomlinson 1 where he found that rest and change of temperature as well as any sudden shock would raise temporarily the internal friction of the wire. After this the sphere would be given a fresh swing and when the amplitude of the'swing was the desired one, the temperature, barometric pressure and time of start would be noted as well as the starting points on the scale. The readings on the scale were always taken in groups of ten successive swings and at the end of every two or three groups a pressure on the electrical key would record the time. This would con- tinue till two hundred readings had been taken. The method of getting from this the logarithmic decrement is as follows : Suppose %, a. 2 , a 3 , a 4 , a 5 , a 6 and 6 1? 5 2 , 5 3 , > 4 , b 5 are eleven con- secutive readings from the left and right of the scale. The ten corresponding arcs from rest to rest are therefore a l + b^ b + a 2 , 2 + 6 2 , b. 2 + a s , 3 + 6 3 , 6 3 + a 4 , a 4 + &4 ^ + 5 , a 5 -f 6 5 , 6 5 + a 6 . The mean of the first and last, second and ninth, third and eighth, etc., would then be taken and if these did not vary much, as was generally the case, the mean of these five would be taken and the logarithm of the corresponding number of seconds noted. This would also be done with the second and succeeding sets of ten readings and the mean of the ten differences between the first and eleventh, second and twelfth, etc., logarithms would when divided by 100 be taken as the logarithmic decrement. These differences would vary slightly but no particular law of variation was noticeable, the variation being sometimes in one direction and sometimes in the other. The accompanying set shows more particularly the method : 1 "The Influence of Stress and Strain on the Physical Properties of Matter." Phil. Trans. 1886(2) CLXXVII, p. 801. 16 REYNOLDS log A n\ 110.33 22.666 4.355375 103.65 21.303 4.328441 97.40 20.027 4.301616 91.60 18.841 4.275104 86.19 17.693 4.247802 81.12 16.695 4.222587 76.33 15.714 4.196286 71.85 14.795 4.170115 67.62 13.926 4.143825 63.73 13.127 4.118165 59.96 . 12.353 4.091772 .263603 56.41 11.623 4.065318 .263123 53.06 10.935 4.038818 .262798 50.05 10.314 4.013426 .261678 47.06 9.700 3.986772 .261030 44.28 9.127 3.960328 .262259 41.75 8.607 3.934852 .261434 39.12 8.065 3.906604 .263511 36.81 7.589 3.880185 .263640 34.67 7.148 3.854185 .263980 In which /^represents the respective means of ten consecu- tive amplitudes as noted on the scale. A the amplitudes reduced to seconds. n\ a multiple of the logarithmic decrement found by sub- tracting 11 from 1, 12 from 2 etc. The mean of the ten differences n\ gives .2627056 and therefore the average logarithmic decrement for one swing is taken as .002627056. The results of such observations, taken at different times under varied conditions of barometric pres- sure and temperature, are shown by the table on page 17, from which it appears : 1 The small difference of barometric pressure has no effect on the logarithmic decrement. 2 The period of oscillation is unaffected by the amplitude of swing or by the temperature. 3 The logarithmic decrement is unaffected by the ampli- tude of swing. THE VISCOSITY COEFFICIENT OF AIR 17 Temp. Barom. Period Amplitude Log. Dec. 12.65 30.21 15.98 121.1 36.9 .00257493 15.25 30.2 15.91 129.5 199.5 .00260613 16.40 29.95 16. 113.7 33.7 .00262706 16.67 30.15 15.97 168.7 - 50.2 .00262837 17.85 30.08 15.98 164.4 84.2 .00263851 18.15 30.06 15.99 74.1 - 38.1 .00264170 18.22 29.81 15.97 200.5 199.5 .00264190 18.30 30.18 15.94 205.1 60. .00265430 4 The temperature is the only element which affects the logarithmic decrement. This was further emphasized in two cases where during the observation the temperature changed grad- ually to the extent of one degree. In one case, where the tem- perature increased, the value of the logarithmic decrement increased steadily instead of varying in one way and the other around the mean value ; and in the other case, where the tem- perature gradually fell, there was a steady falling off hi the value of the logarithmic decrement. This sensitiveness to the changes in temperature rendered it advisable in the future observations to confine them to a shorter period of time so that there would be less liability that a change in temperature would occur, and therefore in the fol- lowing sets the observations were confined to 100 complete vibrations. SECOND SET With the swinging apparatus as described it was of course impossible to know how large a part of the logarithmic decre- ment was due to the internal friction of the wire itself and to the action of the air on the mirror. A change therefore was necessary so that this effect could either be eliminated or ac- counted for. This was accomplished by an addition to the apparatus as before described. A hollow brass tube 14.818 cm. long was pierced by a hole in the plane of its mid-section so as to permit the hanging brass rod to pass through it and to which it was fastened so as to hang 8.6 cm. from the lower end. 18 REYNOLDS Great care was necessary in the arrangement of this, so that when finished it should hang in a perfectly horizontal position and at right angles to the suspending rod in order not to de- flect the mirror. Two solid brass cylindrical weights were now constructed which fitted snugly into the ends of the hollow cyl- inder and each weight was carefully cut down till it was half the weight of the spherical shell (i. e. 117.85) when the length of each was found to be 4.842 cm. The moment of inertia of the spherical shell had been calculated to be 6316.0647 and it was desired to place each cylinder at such a distance from -the axis as to have its moment of inertia equal to 3158.03235. The moment 01 inertia of each around an axis through its centre of inertia is 229.9353 and therefore 2928.09705 is the moment about the original axis of the whole mass considered as placed at the centre of inertia. If the distance of the centre of inertia from the axis be denoted by x, then MX* = 2928.09705 from which x = 4.988 cm. and, since it was also necessary to have the ends of the weights flush with the ends of the enveloping cylinder, this determined the proper length of the cylinder 14.818 cm. The aim in this was to reproduce the weight and moment of inertia of the spherical shell in the weight and moment of inertia of the two cylindrical weights. The method of procedure in this set was as follows : The cylinders would be removed and the sphere attached to the brass rod and the apparatus then left for several hours, usually over night. At the end of this time, by means of the torsion head, the sphere would be oscillated for some time to accustom the wire to the oscillations, and then the vibrations would be started again for the readings, care always being taken to avoid any pendulum m'otion. In this case 100 com- plete swings from rest to rest were taken and the time thermometer and barometer noted as before. The sphere would then be detached, care being taken not to jar the wire, THE VISCOSITY COEFFICIENT OF AIR 19 and the cylinders would then be inserted into place, the preparatory oscillations given and then 100 more readings from the scale with temperature and pressure would be noted. The two swinging systems being the same in every respect save that of the extra surface exposed by the spherical shell, the difference between the decrements when the sphere was swing- ing and when the counter weights were inserted was that due to the action of the spherical surface alone. The method was varied by taking alternately the sphere reading first and then the readings with the weights first, and the whole extended over a range of eight degrees in temperature during a series of twenty-seven observations. A slight discrepancy in the period of swing was noticed when the sphere was attached and when the cylinders were in place which led to a further investigation to reconcile the results. The cylinders were removed and circular paper caps were attached to the horizontal hollow cylinder and the time and decrement noted, and compared with the time and decrement when the caps were removed exposing the interior of the cylinder. With the use of the caps the times were found to agree and a slight correction was found necessary to the logarithmic decrement of the sphere as found originally. The corrected results are inserted in the following table : Time Temp. \i-\2 18.965 16.58 .002672 18.99 16.91 .002679 19.02 17.56 .002691 18.98 18.27 .002717 19.005 19.63 .002760 18.955 20.72 .002808 18.97 21. .002819 18.972 21.78 .002851 18.977 22.36 .002873 18.99 23.29 .002904 19. 24.61 .002924 20 REYNOLDS KirchofF in his "Mechanik" 1 derives a formula involving the coefficient of viscosity, the logarithmic decrement, the time of swing and certain other known elements as follows : In which rj coefficient of viscosity. ft = density of the medium. R radius of the swinging sphere. X = the logarithmic decrement. T = the swinging time in the fluid. T = the swinging time free from influence of friction. K = moment of inertia of the system, which in the case of a spherical shell = %Ma?. The relation between the observed Tand T is given by : Using for the density of the air the value .001293 in the above formula the value of the coefficient of viscosity at the temperature 20.72 wag calculated to be : .00018697 THIRD SERIES The object of this series was to see if, with an entirely different swinging system, results could be obtained which would approximate closely the results obtained with the spher- ical shell. Such agreement if obtained would justify the mathe- matical theories giving rise to the formula used. For this purpose two hollow brass cylinders were pro- cured and after careful tests were found to depart but the i Vierte Aufl. 26 Vorlesung, p. 383. THE VISCOSITY COEFFICIENT OF AIR 21 slightest from exact uniformity of measurement. The dimen- sions were as follows : Cylinder Length Weight Inside Diameter Outside Diameter A 30.4cm. 327.8 gr. 5.00 5.08 B 30.4 cm. 316.3 gr. 4.70 4.78 A thin circular brass cap was soldered to one end of the smaller cylinder in the centre of which was fastened a perpen- dicular brass rod 16 centimetres long so that it coincided ex- actly with the axis of the cylinder. In a manner similar to the one used on the sphere, this rod was fitted at the top extremity with a cross-pin to rest in the hanging hook on the wire. The wire was the same one as used before, its length being this time 80.1 cm. By means of two small felt collars, one on the lower end of the smaller cylinder and the other on the upper end of the larger one, the larger when placed over the smaller was held snugly in place and the whole, when placed on the sus- pending wire, hung in a perpendicular position with the axis of the two cylinders coinciding with the wire. This telescopic arrangement of the cylinders permitted the outer one to be drawn out over the inner one or pushed back so as to cover it and in this way it was possible to eliminate the effect of the friction of the wire and the action of the air on the mirror. Two marks 25.4 cm. apart were made on the inner cylinder and the exten- sion was limited to this amount. In the same manner as before the readings were made from the scale and the temperature, time and pressure were noted. The observations were always made in pairs ; one set with the cylinders collapsed and the other with the cylinders extended and alternately, one set and then the other being taken first. The results of 34 sets with temperature varying about six degrees were of a very satisfac- tory character. The time of swing averaged 13-09 seconds, any one differing from this by less than .05 seconds. 22 REYNOLDS Temp. Cylinders Extended Temp. Cylinders Collapsed 21.4 .001504 21.25 .000953 22.11 .001507 22.73 .000954 22.75 .001509 23.35 .000956 23.6 .001518 24.3 .000960 24.35 .001522 25.52 .000961 25.21 .001526 25.61 .000962 25.95 .001529 25.7 .000964 26.38 .001531 26.73 .000966 27.1 .001533 27.4 .000969 The differences between the two corresponding logarith- mic decrements vary from .000551 to .000564 and these differ- ences were taken for the logarithmic decrement corresponding to a cylinder of the internal and external diameters of the smaller cylinder and 25.4 cm. in length. From the mathematical deductions of Stokes the logarith- mic decrement due to the action on the walls of the cylinder will be given by : M/JLT v/2 0.375/- 1 - v/2 0.4922/- -f etc.) where In which A, = the logarithmic decrement. M mass of air confined in a cylinder of same length and mean radius of the one con- sidered. yu. the coefficient of viscosity. r = the observed time. I = the moment of inertia of the cylinder con- sidered. p the density of the air considered as about half saturated with moisture. a = the mean radius of cylinder. THE VISCOSITY COEFFICIENT OF AIR 23 In applying the formula a first approximation was made for the value of p and this used in a second approximation. A third was found unnecessary and the value of //. resulting from an observed time 13.09 seconds, a logarithmic decrement .000551 and at a temperature 21.33 was calculated to be : .00018711 This value agrees very closely with the value found by the preceding method and both agree well with the later results of other investigators as may be seen by the table. FOURTH SERIES Attention has been called by Thomson 1 to the facility with which a gas can be changed from a conductor to a non-con- ductor by the application and removal of Rontgen rays ; and in particular he caused the gas so exposed to be swept past a wire charged with electricity and in this way part of the charge would be carried to an electrometer. When the ray was not acting on the gas no charge was carried over by the gas. The gas it was found loses this conductivity when a current of electricity passes through it and also when forced through a plug of glass wool, which latter fact would indicate that the structure in virtue of which the gas conducts is of a coarse character, since it does not survive the passage through the fine pores of the glass wool. The study of any other property of a gas when in the state into which it is thrown by the Rontgen ray might therefore lead to further interesting and valuable results. The object of this phase of the investigation is to ascertain if there is any difference in the coefficient of vis- cosity of air when it is affected by the influence of the Rontgen ray. In this set the brass spherical shell used before was hung in position and the bulb from an X-ray apparatus was brought 1 Phil. Mag. 1896 (5), 208, p. 392. "Discharge of Electricity through Gases." 24 REYNOLDS within 15 cm. to the surface of the sphere. It was found nec- essary to have the coil in an adjoining room in order to keep the floor and air free from the vibrations of the apparatus. Furthermore in order to shield the air surrounding the sphere from the electrostatic influences of the bulb, the latter was sur- rounded by a metal gauze screen which was connected to the earth. The observations were taken in pairs, alternately with the current on and off" and attention was first directed to the period of swing. In the following table where the average time of ten consecutive swings is given, at different temperatures and barometric pressures, no evidence of any uniform law of "change is apparent. Temp. Bar. Without Ray With Ray Amplitude 15 29.6 15.985 16. 80. 70. 44 4 4 15.97 16. 19. -13.5 tl it 15.95 16.94 16.6 29.77 15.98 15.98 35. 30. 17.75 15.94 15.96 70. 64. 17.75 15.965 15.94 64. 56. 15.97 15.96 84. 771.5 At a later time further sets were taken with a slight change in the swinging apparatus and at higher temperatures and the logarithmic decrement as well as the time were noted for com- parison. The method of procedure consisted in taking the readings of thirty complete oscillations with the current off", then thirty more with the current on, and a final thirty with the cur- rent on again ; the interval between each set of thirty being the same, usually one complete oscillation. The average of the first and third sets was taken and compared with the second set with the following results : THE VISCOSITY COEFFICIENT OF AIR 25 Time off Time on Current off Current on Amplitude 15.68 15.7 .0025498 .0025428 112. 94.9 15.7 15.73 .0025681 .0025647 56. _ 47.6 15.7 15.7 .0026147 .0026042 38.4 32.4 .0026893 .0026223 84.771.5 15.7 15.73 .0025472 .0025471 120.6 102.3 .0026653 .0026273 151.3 126. 15.7 15.7 .0025697 .0025648 100.8 85.4 15.7 15.7 .0025791 .0025729 70.6 59.8 In this set the periods agree very closely but it might be noted that the variation favors usually a slight increase in the period of swing when the air is affected by the Rontgen ray but the difference is so small that this might easily be caused by errors in observation. In the case of the logarithmic decre- ment the difference is also very slight but points to a decrease when the gas is under the influence of the Rontgen ray. 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