Film-Lubrication between Spherical Surfaces: with an Application to the Theory of the Four-Ball Lubricant Testing Instrument

Abstract

The paper is concerned mainly with the flow of viscous liquid between the outer surfaces of two equal spheres placed in an infinite body of the liquid so that the ``minimum separation'' of the surfaces (the distance apart at the points of closest approach) is small compared with the radius; one sphere is held at rest, the other rotates at constant speed about an axis which does not pass through the center of the fixed sphere. The assumption is made that the surfaces are separated by a continuous film of liquid to which the classical theory of hydrodynamical lubrication can be applied—that is, ``boundary lubrication'' and actual contact between the surfaces are excluded; the partial differential equation (the Reynolds equation) for the pressure distribution in this film is set up, and it is shown that with a suitable choice of boundary conditions a solution can be found in very simple form. From this solution are deduced the total force exerted by the liquid on the moving sphere and the torque, about the axis of rotation, of the forces acting on the fixed sphere. The results are applied to the particular case of the four-ball lubricant testing instrument, in which a sphere is made to rotate about a vertical axis, under axial load, in the central space formed by a set of three equal spheres held stationary so as to touch one another with their centers in a horizontal plane; the whole set of spheres is immersed in the lubricant to be tested. The relations finally obtained express the load on the moving sphere and the torque on the set of fixed spheres in terms of the radius of the spheres, the velocity of rotation, and the viscosity of the liquid; they involve as a parameter the minimum separation of the moving sphere from the fixed spheres, and elimination of this gives the relation between the measurable quantities, load, and torque, which could provide a basis for the use of the instrument as a viscometer. It is found that the torque is a very slowly varying function of the load, being approximately proportional to the logarithm of the load. Further, if it is assumed that the hydrodynamical theory of lubrication is valid only for films whose thickness is greater than some agreed minimum, then the expression for the load on the moving sphere, into which the minimum separation enters, provides a criterion for the range of conditions (of which the load is much the most important) within which the theory does in fact apply to the instrument. Numerical examples are considered, and it is shown that, with allowance for wide departures from the standard size and speed of operation of the instrument, the maximum load which can be supported by the moving sphere in the hydrodynamical regime cannot be expected to exceed a few hundred grams weight; for loads of greater order the standard continuous-film theory cannot hold, and the forces on the spheres are no longer determined by the viscosity of the liquid.