708 Mr. E. Buckingham : Notes on 



9. Second example of the occurrence of a Universal Constant; 

 Heating of a Bearing. 



To show that, as regards dimensional reasoning, there 

 is nothing exceptional about the gravitation constant, it is 

 well to consider a problem which involves another universal 

 constant, namely Joule's equivalent. Let us therefore 

 consider a plain cylindrical bearing, of given design, which 

 supports a journal running at constant speed and under a 

 constant load, the bearing being thoroughly lubricated. 

 Since power is being dissipated in the film of lubricant, 

 the temperature of the journal rises above that of its sur- 

 roundings. After a sufficient time, a steady state is reached 

 and the excess of temperature remains constant. Let us 

 inquire what this rise of temperature depends on, and 

 how. 



To make the problem reasonably simple we may suppose 

 that there is no longitudinal conduction along the shaft, 

 and that the outside of the bearing is kept at a fixed 

 temperature. When the shape and size ot' the bearing- 

 are given, the heat-flow outward, from the oil film where 

 the dissipation occurs, now depends only on the temperature 

 difference and the thermal conductivity of the metal. TTe 

 shall also suppose that the behaviour of the oil depends only 

 on its viscosity, — which is not strictly true but nearly so 

 under ordinary conditions. 



Let D be the diameter of the journal ; a the speed, in 

 r.p.m. ; /jl the viscosity of the oil at its steady temperature ; 

 X the thermal conductivity of the bearing ; and A the rise 

 of temperature. 



Let us first suppose that heat is measured in work units. 

 The initial equation is 



F (A, D 5 <r, fi, X) = 0, . . . . (33) 



and the dimensions we need are [Al = [0], [D] = [Zl, 

 [<r] = p-i], [/*] = [ml-H- 1 ], [\] = [mlt-W- 1 ]. Equa- 

 tion (33) contains 5 quantities requiring 4 fundamental 

 units, and upon applying the IT theorem we have 



/ (dv>) 



0, (34) 



A-n5£c, (35) 



in which the dimensionless shape factor N is fixed by the 



