This, however, is not the usual problem that arises. What is generally wanted is to find the thickness of the film where least (h0) and its angular position with respect to direction of load, to resist a definite load with a particular surface velocity. If the surfaces are plane, the general solution involves only one arbitrary constant, the least thickness (h0); since in any particular case the variation of h with x is necessarily fixed, as in this case lubrication affords no automatic adjustment of this slope. When both surfaces are curved in the plane of motion there are at least two arbitrary constants, h0, and φ the angular position of h0 with respect to direction of load; while if the surfaces are both curved in a plane perpendicular to the direction of motion as well as in the plane of motion, there are three arbitrary constants, h0, φ0, z0. The only constraint necessary is to prevent rotation in the plane of motion of one of the surfaces, leaving this surface free to move in any direction and to adjust its position so as to be in equilibrium under the load.
The integrations necessary for the solutions of these problems are practicable—complete or approximate—and have been effected for circumstances which include the chief cases of practical lubrication, the results having been verified by reference to Tower’s experiments. In this way the verified theory is available for guidance outside the limits of experience as well as for determining the limiting conditions. But it is necessary to take into account certain subsidiary theories. These limits depend on the coefficient of viscosity, which diminishes as the temperature increases. The total work in overcoming the resistance is spent in generating heat in the lubricant, the volume of which is very small. Were it not for the escape of heat by conduction through the lubricant and the metal, lubrication would be impossible. Hence a knowledge of the empirical law of the variation of the viscosity of the lubricant with temperature, the coefficients of conduction of heat in the lubricant and in the metal, and the application of the theory of the flow of heat in the particular circumstances, are necessary adjuncts to the theory of lubrication for determining the limits of lubrication. Nor is this all, for the shapes of the solid surfaces vary with the pressure, and more particularly with the temperature.
The theory of lubrication has been applied to the explanation of the slipperiness of ice (Mem. Manchester Lit. and Phil. Soc., 1899).
(O. R.)
[1] Mém. de l’Acad. (1826), 6, p. 389.
[2] Mém. des sav. étrang. l. 40.
[3] Mém. de l’Acad. (1831), 10, p. 345.
[4] B.A. Report (1846).
[5] Cambridge Phil. Trans. (1845 and 1857).