Boundary Conditions of a Viscous Flow Between Surfaces With Rolling and Sliding Motion

The fluid motion between surfaces with different radii of curvature and velocities is studied, assuming that the viscous fluid is carried by the solid surfaces but does not fill up the whole space. The boundary conditions at the inlet are examined in connection with those at the outlet of the fluid film. It is shown that only a part of the fluid carried by the surfaces, depending on the velocities and the initial rates of flow ratio, penetrates into the contact zone. Thus an interpretation of the flow field is proposed, differering from the usually assumed shape of the streamlines, by assuming the existence of a counterflow at the inlet. By using some physical conditions in various representative situations, as well as an equilibrium condition for the vortex flow, the real quantity of fluid and the entry and exit points are determined. Thereafter, the film extent, pressure distribution, load-carrying capacity, and minimum film thickness are obtained. Tables are given with the characteristic angles of the fluid film as functions of the minimum film thickness-radius of curvature ratio. The calculated values are in a satisfactory agreement with the experiments of other authors, especially when using the Prandtl-Hopkins conditions at the outlet.