The equilibrium and stability of axisymmetric pendent drops

Abstract

The stability of the equilibrium of axisymmetric drops suspended from a horizontal circular orifice is studied mathematically in this paper. For axisymmetric perturbations it has been shown by Pitts (1976) that limit point instability occurs at the positions of maximum volume, or of maxi­mum internal pressure at the point of support, depending on whether the drop is held at constant volume, or at constant pressure head respectively. Here a criterion is given for asymmetric instabilities. It is shown that bifur­cation of the equilibrium into an asymmetric mode, with azimuth wave-number m = 1, will occur when the profile of the drop becomes horizontal at the point of support. For a drop grown from an initial horizontal plane interface calculations show that when the orifice radius a is greater than 3.219... in units of the capillary length, onset of this instability will precede the axisymmetric instability. When a reaches the value 3.812... the m = 1 instability sets in at the plane interface, and we recover the earlier results of Plateau (1873) and Maxwell (1875) on the instability of horizontal plane interfaces. Higher order instabilities are briefly discussed, and it is shown that modes m = 2, 3, 4,... will not precede the m = 1 mode for a drop suspended in this way.