248 P N E U M A T 1 S than in the other. Rut this difference will not prevent the individual molecules diffusing across the interface between the layers. Diffusion will go on freely. The result will be that the .slower moving layer will on the whole gain momentum in the direction of its motion and the faster moving layer lose momentum. Thus, dift usion tends to the equalization of momentum between two contiguous regions, and the rate at which this takes place across unit area is the measure of the viscosity. Maxwell has proved l that the viscosity so measured is independent of the density of the gas when the temperature is constant ; whereas the relation between the viscosity and temperature depends upon the particular mode of action between the molecules when they approach each other. The above definition of viscosity is not one which can be used in experimental determinations, since we cannot take account of the individual molecules of a gas. The coefficient of viscosity must be defined in terms of directly measurable quantities. Viscosity Maxwell has defined viscosity in these words : the viscosity of defined a substance is measured by the tangential force on unit of area of physi- either of two horizontal planes at the unit of distance apart, one of cally. which is fixed, while the other moves with unit of velocity, the space between being filled with the viscous substance. This is the dynamical definition. When the effects of viscosity on the internal motions of a fluid itself are being considered it is often more convenient to use the kinematical definition. It is given in terms of /t, the coefficient of viscosity, by the equation where p is the density of the substance, and v the kinematic viscosity. Experi- The viscosity of fluids has been determined experimentally in mention three distinct ways by flow of the fluid through tubes, by motion viscasity. in the fluid of pendulums or vibrating disks, and by the oscilla tions of spheres filled with the fluid. The last was employed by Helmholtz and Von Piotrowski in their investigation of liquids, - but it is not applicable to the case of gases. Experiments on the flow through capillary tubes have been carried out by Poiseuille 3 for liquids, and by Graham, 4 Meyer, 5 Springmiihl, 5 andPuluj 6 for gases. This, tlie transpiration method, is the most effective for comparing viscosities, which are directly proportional to the times of trans piration of the respective gases. There is, however, a little uncertainty as to the effect of the capillary tube, so that, for measuring absolutely the viscosity for any one gas, the method is not so trustworthy as the second method. Here we may use either pendulums swinging through small arcs or disks oscillating in their own plane under the action of torsion. In both the measurement depends upon the rate at which the amplitude of oscillation diminishes. Stokes, who first satisfactorily discussed the true nature of viscosity, tested the theory by a discussion 7 of the pendulum experiments of Dubuat (1786), Ressel (1826), and Raily (1832). From Rally s results he calculated 000104 (in metric units) as the coefficient of viscosity of air. Meyer 8 similarly deduced from Ressel s and Girault s 9 experiments the values 000275 and 000384. It is not in the least surprising that these are all pretty wide of the true value, seeing that the experimenters had not the special problem of finding the viscosity before them. Meyer, to whom we owe a very complete series of valuable memoirs on the subject, has more recently experimented 10 with three different-sized pendulums. The values deduced for the viscosity were 000232, -000233, and 000184. The last number, given by the shortest pendulum, Meyer considers to be the best. Maxwell, 11 Meyer, 12 and Kundt and Warburg 13 have experimented with oscillating disks. The methods of Maxwell and Meyer were so far similar that each used an arrangement of three horizontal circular disks, fixed centrally to the same vertical axis, and sus pended by a torsion wire inside a receiver. The pressure and tem perature of the air or gas inside could be adjusted to any desired values within certain limits. In Maxwell s apparatus, which Meyer adopted in his later researches, the moving disks vibrated between parallel fixed disks, which were perforated in the centre so as to allow the vertical suspended axis to pass freely through them. Each disk thus oscillated in its own plane between two parallel fixed surfaces. After the disks were set in position, and the air in the receiver brought to the desired temperature and pressure, the suspended disks were set in oscillation. This was effected magneti cally, a small magnet fixed to the end of the suspended axis being acted upon by an external magnet suitably adjusted. Each disk, in its oscillations, dragged after it the layer of air in immediate con tact with it ; and in virtue of viscosity this oscillation was trans- 1 Phil. May., I860, and Phil. Trans^ 1867. 2 Kitzungsber. der Winner Akad., I860. 3 Mim. dft Savants Etrangers, 1846. < Phil. Trans., 184G, 1849. 5 Poggendarft Ann., cxlviii.. 1873. B Kitzungtber. d. Wittier Akad., Ixix., 1878. 1 Cam. Phil. Trans. ,ix., 1850. Poggendorff s Ann., Cxxv., 1SC5. > Mem. de CAcad., etc., de Caen, I860. " Poggendorff s Ann., cxlii., 1871. " Phil. Trans., clvi.. 180(5. - Poggendorff"* Ann., cxciii., 1871, and cxlviii., 1873. s Poyyendorff t Ann., civ. clvi., 1875. mitted with diminishing amplitude from layer to layer until the fixed disks were reached. In thus setting and sustaining in motion a mass of gas, the disk was doing work ; and, if left to itself and to the action of the torsion suspension, it oscillated with gradually diminishing range until it came to rest. The viscosity of the air was not the only retarding influence. The torsion wire had also a coefficient of viscosity ; and then there was a possible resistance due to the slipping of the fluid at the surfaces of the disks. Those vari ous effects were discriminated by suitable modifications. Thus by placing the oscillating disks in contact with each other, and setting tsvo of the fixed disks at measured distances above and below, Maxwell reduced the number of surfaces in contact with the fluid, and so increased the relative importance of the effect due to the wire s viscosity. Again, by diminishing the distances between the fixed and oscillating disks, he made the conditions more favourable to the effect (if any) due to the slipping. This latter effect was found to be so small as to be almost within the errors of observation ; consequently Maxwell felt himself warranted in. calculating the coefficient of viscosity on the assumption that there was no slipping. Maxwell s final result in metric (C. G. S. ) units for the coefficient of viscosity of dry air is /i== -0001878(1 + -003650), where 6 is the temperature in degrees Centigrade. Meyer s result is M = -000190(1+ -00250). Maxwell found the effect of pressure to be inappreciable down to a pressure of 12 mm., and thus verified the deduction from theory. Kundt and Warburg, in their experiments, used only one disk, which oscillated under the influence of a bifilar suspension between two fixed disks. They carried the pressure down to as low as 6 mm. At 20 mm. pressure the viscosity was the same as at the atmospheric pressure ; but at lower pressures a slight diminution began to show itself. According to Crookes s later researches, this diminution becomes more and more marked at tin- higher exhaustions. The manner in which the viscosity then diminishes coincides remarkably with the manner in which the free path increases. It could not be expected that in such modified circumstances Maxwell s law would continue to apply. When the gas becomes so far rarefied that the mean free path of a mole cule is not small compared to the space in which the gas is confined, the motion of the molecules cannot be treated statistically. Hence the deductions from a theory based upon the statistic method will no longer hold good. Maxwell, Kundt and Warburg, and Crookes investigated by the disk method the viscosities of other gases, the values for which are compared below with the transpiration times of the same gases through capillary tubes. Maxwell also found that damp air, at 100 mm. pressure, and over water at about 20 C. , was one-sixtieth less viscous than dry air at the same temperature. Kundt and Warburg found for water vapour, at 21 C. and 16 mm. pressure, the value ju- 0000975, a little more than half that of air. The results obtained by Meyer and Springmiihl and by Pnluj from their transpiration experiments agree well with those already giveji. In such experiments, however, the slipping of the gas over the solid surface has in certain circumstances a measurable effect. This slipping is measured by a certain coefficient, called the Gleitungs-Coejjicient by Helmholtz and Von Piotrowski. When this coefficient becomes appreciable, the giis in contact with the solid surface, instead of being at rest relatively to that surface, will be gliding over it with a finite velocity v. The circumstances of the motion will be very nearly the same if we remove a layer of the solid surface and replace it by fluid, the new surface of fluid in contact with the new solid surface being at rest. The thickness which must be so removed is the measure of the coefficient of slipping. Kundt and Warburg, 14 in their experiments with glass tubes, found this coefficient for dry air at about 20 C. to be 8/p centimetres, where;; is the pressure in dynes per square centimetre, which is nearly the same as in millionths of an atmosphere. The value for hydrogen on glass is 15/p. Hence at ordinary pressures and moderate exhaustions this coefficient is very small, becoming appreciable only at low pressures. The relation between viscosity and temperature is indicated at once by Maxwell s and Meyer s formulae given above. According to Maxwell, the viscosity is proportional to the absolute tempera ture. If in the kinetic theory the forces between the molecules arc disiegarded, that is, if the molecules are assumed to rebound after collision like elastic spheres, the relation deduced is that the vis cosity varies as the square root of the absolute temperature. Hence the mutual molecular forces must be taken into account. Maxwell s experimental law would require any two molecules to repel each other with a force varying inversely as the fifth power of the distance. According to Meyer, however, the viscosity varies
H Pnggendorff t Ann., 187K.