Drainage of thin films beneath parabolic and spherical caps

Abstract

AbstractThe variation in film thickness h with time t for the approach of an infinite sphere to a plane horizontal surface (β = 1) or of two infinite spheres (β = 2) is given by: For finite spherical caps with edge radius rf the variation is much more complicated and also involves the parameter S = βr2f/2aho. Fortunately, the gradient is the same in both cases, providing t is large enough (the critical value of t increases with decreasing S). A similar result is obtained if the spherical cap is approximated by a parabolic cap with apex curvature 1/a equal to that of the sphere. In both cases the variation in dynamic pressure close to the centre of the draining film is identical and independent of the radial position where the dynamic pressure falls to zero when the film thickness is small.MacKay and Mason (1961) measured the film thickness beneath a sphere of finite size approaching a horizontal plane and experimentally verified Equation (b). This does not however, as they assumed, prove the correctness of Equation (a), which only applies to infinite spheres. The more complicated equations describing the approach of finite spheres and parabolic caps are presented in this paper.