OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY VOLUME XXII. No. III. pp. 39-54. THE HYDRODYNAMICAL THEORY OF LUBRICATION WITH SPECIAL REFERENCE TO AIR AS A LUBRICANT BY W. J. HARRISON, M.A. FELLOW OF CLARE COLLEGE, CAMBRIDGE CAMBRIDGE: AT THE UNIVERSITY PRESS M.DCCCC.XIII. ADVERTISEMENT THE Society as a body is not to be considered responsible for any facts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors. THE SOCIETY takes this opportunity of expressing its grateful acknowledgments to the SYNDICS of the University Press for their liberality in taking upon themselves the expense of printing this Volume of the Transactions. III. The Hydrodynamical Theory of Lubrication with Special Reference to Air as a Lubricant. By W. J. HARRISON, M.A., Fellow of Clare College, Cambridge: [Received 11 May 1913. Read 19 May 1913.] THE theory of the lubrication of surfaces moving relatively to one another and separated by a thin film of oil or other lubricant is one of considerable practical interest. The cognate problems are essentially hydrodynamical in their character, and have this interest, that they are among the few problems in the motion of viscous fluids which can be solved approximately for the case of large velocities. The theoretical work of Osborne Reynolds* and of Pétroff† is of extreme complexity, partly because Reynolds considered the case of an incomplete cylindrical bearing, and Pétroff introduced further complications into its form. It must be admitted that the forms of the bearing considered by these investigators are those which occur most frequently in practice, but their analysis is so complex and methods of approximation so laborious, that even the mathematician may fail to grasp the essential character of the results obtained, or to be expected. This fact alone is a valid reason for treating a simpler form of bearing. All results become simple, and the theory of lubrication can be elegantly illustrated by considering the case of a complete cylindrical bearing. It was not till after I had completed my investigations in this case that I came across the very elaborate treatment by A. Sommerfeldt of the same problem. Our resulting formulae are identical. But the present treatment of the problem being somewhat different from Sommerfeld's, shorter and in one or two points more direct, will perhaps appeal more directly to experimenters. A question is raised in the course of this paper as to the validity of experiments which have hitherto been made to determine the moment exerted by the traction of the lubricant on the journal. This point is of importance, as by means of this moment the nominal coefficient of friction of the journal is obtained. In the latter part of this paper the method is extended to take account of lubrication by means of an elastic viscous fluid such as air. It was stated as long ago as 1885 by Hirn§, that under suitable circumstances air is the most perfect lubricant. In 1897 a series of very beautiful experiments was carried out by Prof. Kingsbury|| on the lubrication of a cylindrical journal by air. The results he obtained, which are apparently accurate to a fair degree, exhibit in certain details wide variation from those to be looked for in the case of lubrication by an incompressible liquid. I have not succeeded in obtaining an explicit solution of the differential equation determining the pressure in the film of air, but I have integrated it numerically by Runge's method, using the data of Kingsbury's experiments. The degree of approximation of theory to experiment is quite satisfactory. I have integrated the differential equation in the case of plane surfaces, and give some results below which exhibit more clearly the very marked effects of the compressibility of the air on the magnitude and distribution of the pressure. But apart from the new results obtained, this paper will serve the useful purpose of recalling attention to Sommerfeld's work. A subsequent paper by A. G. B. Michell* is also worth attention and will be referred to below. It might be in place to remark here that I have obtained some results and have work in hand treating of cases in which the influence of variable speed and variable load on the lubrication of a cylindrical bearing is taken into account. Case of Incompressible Liquid. In proceeding to determine the equations which give the motion and the pressure of a film of liquid separating two surfaces moving relatively to one another, it is to be observed that the inertia terms can be neglected as well as the effect of gravity, since forces depending on these terms are negligible compared with the internal stresses arising from the rapid shearing of the liquid. Again, on account of the thinness of the film its curvature can be neglected, and therefore the same equations hold whether the surfaces are plane or cylindrical. Sommerfeld has transformed the equation V1 = 0, which is satisfied by the stream function, from Cartesian coordinates (x, y) to polar coordinates (r, 0). He proceeds to use essentially the same method of approximation as employed by Osborne Reynolds. The only result of this transformation is to introduce relatively unimportant terms, as will be seen below. The coordinate a will be measured along the moving surface in the direction of motion, the coordinate y normal to this surface. The motion is steady and will be assumed to be two-dimensional. If u, v be the component velocities at any point in the liquid, p the pressure, the equations of motion are where is the coefficient of viscosity, and the equation of continuity is .(1), .(2), .(3). |