March i8, 1886] 



NA TURE 



477 



empirical formula has been deduced for the viscosity of olive 

 oil at all temperatures between 60" and 120° F.^ 



Besides the effect on m the temperature might, owing to the 

 different expansion of brass and iron, produce a sensible effect 

 on tlie small difference a in the radii of the brass and journal, 

 i.e. on the mean thickness of the film. E was taken for the co- 

 efficient of this effect, and since, owing to the elasticity of the 

 material, the radius would probably alter sUghtly with tlie load, 

 III was taken as a coefficient for this effect, whence a is given 

 by an equation - in terms of a^^, its value with no load and a 

 'temperature zero. 



Substituting these values in the equations, the values of the 

 pressure and friction deduced from the equations are functions 

 of the temperature, which may be tlien as-umed, so as to bring 

 the calculated results into accord with the experimental. 



There was, however, another method of arriving, if not at the 

 actual temperatures, at a law connecting them with the frictions, 

 loads, and velocities. For the rise in temperature was caused 

 by the work spent in overcoming friction, while the heat thus 

 generated had to be carried or conducted away from the oil 

 film. Consideration of this work and the means of escape gave 

 another equation between the rise of temperature, the friction, 

 and velocity." 



The values of the constants in this equation can only be 

 roughly surmised from these experiments, without determining 

 them h-j substituting the experimental values of/, U, and T, as 

 previously determined, but it %\as tlien found that the experi- 

 ments with the lower loads gave remarkably consistent values 

 for A, B, E, III, and a„, which was also treated as arbitrary. In 

 proceeding to the higher loads for which the values of c were 

 greater, the agreement between the calculated and experimental 

 results was not so close, and the divergence increased as c in- 

 creased. On careful examination, however, it appeared that 

 this discordance would be removed if the experimental frictions 

 were all reduced 20 per cent. This implied that 20 per cent, of 

 the actual friction arose from sources which did not affect the 

 pressure of the fdm of oil ; such a source would be tiie friction 

 of the ends of the brass against flanges on the shaft commonly 

 used to keep the brass in its place, or by any irregularity in the 

 longitudinal section of the journal or brass. A coefficient, k, has 

 therefore been iutroduced into the theory, which includes both 

 the effect of necking and of irregularity in longitudinal section. 

 Giving n the value i'25, the calculated results came into accord- 

 ance with all Mr. Tower's results for olive oil, the difference 

 being such as might well be attributed to experimental inaccuracy, 

 and this both as regards the frictions measured with one brass, 

 No. I, and the distribution of the pressure round the journal 

 with another, No. 2. 



Not only does the theory thus afford an explanation of the 

 very novel phenomena of the pressure in the oil film, but it 

 also shows, what does not appear in the experiments, how the 

 various circumstances under which the experiments have been 

 made affect the results. 



Two circumstances in particular which are brought out as 

 principal circumstances by the theory seem to have hitherto 

 entirely escaped notice, even that of Mr. Tower. 



One of these is a, the difference in the radii of the journal and 

 of the brass or bearing. It is well known that the fitting 

 between the journal and its bearing produces a great effect on 

 1 he carrying power of the journal, but this fitting is supposed to 

 be rather a matter of smoothness of surface than a degree of 

 difference in radii. 'Ihe raduis of the bearing must always 

 be as much larger than that of the journal as is necessary to 

 secure an easy fit, but more than this does not seem to have 

 been suggested. 



It now appears from this theory that if viscosity were constant 

 the friction would be inversely proportional to the difference in 

 the radii of the bearing and journal, and this although the arc 

 of contact is less than the semi-circumference ; an 1 taking tem- 

 perature into account it appears from the comparison of the 



' An inch being unit of length, a pound unit of fori 

 le, for olive oil 



fj. = o-O00O4737i-"°°==" 



: of 



' a = (ir,i -I- iiih)c'''''' . . . 



/= ( A -f- ? JT -I- EAT-. . 



A -f- ET represents the rate at which the mechanical equivalent of heat is 

 :_j .._r. ^c . — . j> represents the rate at which it is 



.("7) 

 ■ {120) 



theoretical frictions with the experiment on brass No. i that the 

 difference in the radii at 70° F. was 



a = o"ooo77 (inch), 

 and comparing the theoretical pressures with those measured 

 with brass No. 2, 



a = o"ooo84 (inch), 



or the difference was 9 per cent, greater in the case of brass 

 No. 2. 



Another circumstance brought out by this theory, and re- 

 marked on both by Lord Rayleigh and the author at Montreal, 

 but not before suspected is, that the point of nearest approach of 

 the journal to the brass is not by any means in the line of load, 

 and, what is still more contrary (o common supposition, it is on 

 the oj^^ side of this line. 



This point H moves as the ratio of load to velocity increases ; 

 when this ratio is zero, the point H coincides with o, then as the 

 load increases it moves away to the left, till it reaches a maxi- 

 mum distance - = <!>„, being nearly - - . The load is still 



small, smaller than anything in Mr. Tower's experiments, even 

 with the highest velocities. For further increase of load, H 



returns towards o, or - 



I increases. With the largest loads 



carried away per unit of temperatu 

 conducted away. 



and smallest velocities to which the theory has been applied 

 this angle is about 40°. With a fairly loaded journal well lubri- 

 cated it would thus seem that the point of nearest approach of 

 brass to journal, i.e. the centre of wear, would be about the 

 middle of the offside of the brass. 



This circumstance, the reason of which is rendered perfectly 

 clear by the conditions of equilibrium, at once explains a 

 singular phenomenon, incidentally pointed out by Mr. Tower, 

 viz. that the journal having been run in one direction for some 

 time, and carrying its load without heating, on being reversed 

 began to heat again, and this after many repetitions always 

 heating on reversal, although eventually this tendency nearly 

 disappeared. Mr. Tower's suggested explanation appears to the 

 author as too hypothetical to be satisfactory, even in default of 

 any other ; and particularly as this is an effect which would 

 necessarily follow in accordance with the theory, so long as there 

 is wear. For the centre of wear, being on the offside of the line 

 of loads, this wear will tend to preserve or diminish the radius 

 of the brass on the off side, and enlarge it on the on side, a 

 change which will, if anything, improve the condition for pro- 

 ducing oil pressure while running in this direction, but which 

 will damage the condition on which the production of pressure 

 in the film depends when the journal is reversed and the late off" 

 side becomes the new im side. That with a well-worn surface 

 there should be sufficient wear to produce this result with such 

 slight amounts of using as those in Mr. Tower's experiments 

 before reversal seems doubtful, but supposing the brass new 

 and the surface more or less unequal, the wear for some time 

 would be considerable, even after the initial tendency to heat 

 had disappeared. Hence it is not surprising that the effect 

 should have eventually seemed to disappear. 



The circumstances which determine the greatest load which 

 a bearing will carry with complete lubrication, i.e. with the oil 

 film continuous between brass and journal throughout the entire 

 arc, are definitely shown in the theory, so long as the brass has 

 a circular section. 



The theory shows that the ultimate limit to the load will b; 

 the same with the oil-bath and with partial lubrication as Mr. 

 Tower found it to be. 



The eifect of the limited length of bearings, and the escape of 

 the oil at the ends, is also apparent in the equations. 



Although in the main the present investigation has been 

 directed to the circumstances of Mr. Tower's experiments, 

 namely, a cylindrical journal revolving in a cylindrical brass, 

 the main object has been to establish a general and comp ete 

 theory based on the hydrodynamical equations for viscous fluids. 

 Hence it has been thought necessary to proceed from the 

 general equations, and to deduce the equations of lubrication in a 

 general form, from which the particular form for application has 

 been obtained. It has been found necessary also to consider 

 somewhat generally the characters of fluid friction and viscosity. 



The verification of the equations for viscous fluids under such 

 extreme circumstances affords a severe test of the truth and com- 



^ On and t?^ sides are used by Mr. Tower to express respectively the sides 

 of approach and recession, as B and A, Fig. i, the arrow indicating the 

 direction of motion. 



