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BREWER AND P. Y. ALEXANDER. TAYLOR AND FRANCIS, Red Lion Court, Fleet Stree E.C. 4. All applications for space to be made to Mr. H. A. COLLINS, 32 Birdhurst Road, Croydon. All THE LONDON, EDINBURGH, AND DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.] JANUARY 1918. I. Notes on the Theory of Lubrication. ODERN views respecting mechanical lubrication are experiments of B. Tower†, conducted upon journal bearings. He insisted upon the importance of a complete film of oil between the opposed solid surfaces, and he showed how in this case the maintenance of the film may be attained by the dragging action of the surfaces themselves, playing the part of a pump. To this end it is necessary that the layer should be thicker on the ingoing than on the outgoing side "‡, which involves a slight displacement of the centre of the journal from that of. the bearing. The theory was afterwards developed by O. Reynolds, whose important memoir § includes most of what is now known upon the subject. In a later paper Sommerfeld has improved considerably upon the mathematics, especially in the case where the bearing completely envelops the journal, and his exposition is much to be recommended to those who wish to follow the details of the investigation. Reference may also be made to Harrison T, who includes the consideration of compressible lubricants (air). * Communicated by the Author. † Proc. Inst. Mech. Eng. 1883, 1884. British Association Address at Montreal, 1884; Rayleigh's Scientific Papers, vol. ii. p. 344. Phil. Trans. vol. 177. p. 157 (1886). Zeitschr. f. Math. t. 50. p. 97 (1904). ¶ Camb. Trans. vol. xxii. p. 39 (1913). Phil. Mag. S. 6. Vol. 35. No. 205. Jan. 1918. B In all these investigations the question is treated as twodimensional. For instance, in the case of the journal the width-axial dimension-of the bearing must be large in comparison with the arc of contact, a condition not usually fulfilled in practice. But Michell* has succeeded in solving the problem for a plane rectangular block, moving at a slight inclination over another plane surface, free from this limitation, and he has developed a system of pivoted bearings with valuable practical results. It is of interest to consider more generally than hitherto the case of two dimensions. In the present paper attention is given more especially to the case where one of the opposed surfaces is plane, but the second not necessarily so. As an alternative to an inclined plane surface, consideration is given to a broken surface consisting of two parts, each of which is parallel to the first plane surface but at a different distance from it. It appears that this is the form which must be approached if we wish the total pressure supported to be a maximum, when the length of the bearing and the closest approach are prescribed. In these questions we may anticipate that our calculations correspond pretty closely with what actually happens,-more than can be said of some branches of hydrodynamics. In forming the necessary equation it is best, following Sommerfeld, to begin with the simplest possible case. The layer of fluid is contained between two parallel planes at y=0 and at y=h. The motion is everywhere parallel to æ, so that the velocity-component u alone occurs, v and w being everywhere zero. Moreover u is a function of Moreover u is a function of y only. The tangential traction acting across an element of area represented by dr is μ(du/dy)dx, where μ is the viscosity, so that the element of volume (dx dy) is subject to the force μ(d3u/dy2) dx dy. Since there is no acceleration, this force is balanced by that due to the pressure, viz. -(dp/dx) dx dy, and thus dp dx u = d2u =μ dy3 (1) In this equation p is independent of y, since there is in this direction neither motion nor components of traction, and (1), which may also be derived directly from the general hydrodynamical equations, is immediately integrable. We have 1 dpy2+A+By, · 2μ dx (2) where A and B are constants of integration. We now Zeitschr. f. Math. t. 52. p. 123 (1905). • suppose that when y=0, u=-U, and that when y=h, u=0. Thus u = y2 — hy dp — ( 1 — %) U. 2μ da h hU Sudy=-124 de --Q. 12μ d 2 where Q is a constant, so that dp 6μU h3 The whole flow of liquid, regarded as incompressible, between O and h is 2Q (3-28). U dp 6μU = When y=0, we get from (3) and (5) 4h-3H h2 (4) If we suppose the passage to be absolutely blocked at a place where x is negatively great, we are to make Q=0 and (4) gives the rise of pressure as a decreases algebraically. But for the present purpose Q is to be taken finite. Denoting 2Q/U by H, we write (4) du μ =μU (h-H). (3) (5) (6) which represents the tangential traction exercised by the liquid upon the moving plane. It may be remarked that in the case of a simple shearing motion Q=hU, making H=h, and accordingly dp/dx=0, du/dy=U/h. Our equations allow for a different value of Q and a pressure variable with x. So far we have regarded h as absolutely constant. But it is evident that Reynolds' equation (5) remains approximately applicable to the lubrication problem in two dimensions even when his variable, though always very small, provided that the changes are not too sudden, a being measured circumferentially and y normally to the opposed surfaces. If the whole changes of direction are large, as in the journal-bearing with a large arc of contact, complication arises in the reckoning of the resultant forces operative upon the solid parts concerned; but this does not interfere with the applicability of (5) when his suitably expressed as a function of x. In the present paper we confine ourselves to the case |