Pendent drop profiles and related capillary phenomena

Abstract

Certain capillary phenomena depend for their interpretation on the configurations of menisci between immiscible fluids. The reduced coordinates ($X$,$Z$) of the curves (meniscus profiles) which on revolution about the axis of symmetry give pendent drops or emergent bubbles, have been computed by a little-used method based on the Laplace equation. The independent variable is the reduced distance $S$ along the curve of the profile, and the shape factor is the reduced hydrostatic head $H$ of fluid at the origin (base of pendent drop). In this way the profiles and drop volumes obtained for $10^{-2}<H<5$ can be realistically related to drops hanging from solid surfaces or to emergent bubbles such as those carrying solid particles in flotation, and certain computational problems inherent in other methods are avoided. The characteristics of pendent drops on the basis of profiles are examined. The maximum sizes which drops hanging from the ends of tubes of finite diameter, and from a horizontal plane of infinite extent can attain are defined, and predictions are compared where possible with the limited experimental data. Analysis is carried out for bulging drops which might spread up the walls of dropping tubes, for situations where the contact angle $\theta $ (on the tip material) is $0<\theta 90$ angstrom. Experimental data are analysed where possible. Problems arise with discontinuities in the solid (edges), and with the possibility of a boundary layer of fluid on the solid. The drop-weight (-volume) method of interfacial tension measurement is examined, especially because the usual explanations are erroneous, and the so-called Tate's law does not represent Tate's own findings and conclusions. The customary correcting factors of Harkins & Brown were obtained for four liquid/vapour interfaces, yet they have been used for liquid/liquid as well as liquid/vapour interfacial tensions of a range of fluids of widely varying densities and viscosities. For the ratio of dropping tube radius to capillary constant $r/a\gtrsim 1.6$, the correction curve is subject to uncertainty. In part it may be represented by a hyperbola (given in figure 6). Data are required covering a range of fluids and experimental conditions. Other aspects of liquids hanging from and detaching from the ends of tubes, including the works of Gross and Sentis, are examined. The possibility that axial flow occurs during spontaneous changes in meniscus configurations leads to suggestions for experimentation, including the use of fluids of comparable density to obtain surfaces of constant curvature.