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Drainage of thin films beneath parabolic and spherical caps
Author(s) -
Jeelani Shaik A. K.,
Hartland Stanley
Publication year - 1986
Publication title -
the canadian journal of chemical engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.404
H-Index - 67
eISSN - 1939-019X
pISSN - 0008-4034
DOI - 10.1002/cjce.5450640103
Subject(s) - spheres , spherical cap , radius , curvature , plane (geometry) , geometry , physics , radius of curvature , mathematical analysis , surface (topology) , position (finance) , enhanced data rates for gsm evolution , spherical shell , horizontal plane , zero (linguistics) , mathematics , mean curvature , materials science , mean curvature flow , shell (structure) , telecommunications , linguistics , computer science , economics , composite material , philosophy , computer security , astronomy , finance
The 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:\documentclass[article]\pagestyle[empty]\begin[document]$$ t = (3\pi n^2 \mu a^2 /2\beta ^2 f)\ln (h_o /h)(a) $$\end[document]For finite spherical caps with edge radius r f the variation is much more complicated and also involves the parameter S = βr 2 f /2ah o . Fortunately, the gradient\documentclass[article]\pagestyle[empty]\begin[document]$$ - d(\ln h(/dt = (3\pi n^2 \mu a^2 /2\beta ^2 f)(b) $$\end[document]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.