Axisymmetric Liquid Hanging Drops

Abstract

The formation of liquid drops delivered from a circular capillary has found application in drop-volume tensiometers ranging from the early stalagmometer to more recent computer-controlled instruments. Although the phenomenon of drop formation can be observed daily, it is rarely discussed in detail in physics or chemistry teaching courses. The aim of this work is to provide an understanding about the buildup of axially symmetric hanging drops, starting from the Young–Laplace equation and the governing differential equation. Drop shapes that are common for all liquids and capillary dimensions are obtained by numerical integration of the dimensionless differential equation. The growing of drops is discussed in terms of changing geometrical quantities such as drop volume, height, surface area, and contact angle to the capillary. A diagram is presented that shows the existence of three different drop shape types at respective values of capillary size and relative drop volume. Accurate geometrical properties are presented for the drop with maximum height and that with maximum volume and diameter. The historical hypothesis of Lohnstein, who suggested a method to calculate the ratio of the critical hanging volume to the falling volume in the drop separation process, is evaluated. Finally, critical volumes are shown as a function of capillary size and compared with falling and residual volumes that have been calculated using a recommended correction function.