The stability of axisymmetric menisci

Abstract

The conditions that govern the equilibrium and stability of a meniscus have been obtained from the first and second derivatives of the energy of the meniscus when it undergoes axisymmetric deformation. The energy of forming a meniscus is defined in thermodynamic terms and methods are given for calculating the free energy of a mensicus in the perturbed and unperturbed state. The stable, critically stable and unstable equilibrium states of a meniscus are all defined in terms of an energy profile, that is, the variation of energy with degree of perturbation. The variational problem of defining parameters for a critically stable meniscus is solved graphically by using a three-dimensional cluster of energy profiles, and it is shown that certain properties of the meniscus, notably volume or pressure, reach limiting values at critical conditions. Four types of stability are considered for each of three forms of axisymmetric menisci. The stability types are those limited by volume or pressure, in conjunction with limitation by the size of the supporting solid surface or the angle of contact. The three forms of menisci are pendant drops, sessile drops and rod-in-free-surface menisci. Detailed stability criteria are given for each of the twelve different combinations of stability type and meniscus form. The stability criteria of this study are all derived by numerical interpolation methods applied to the tables of equilibrium meniscus shapes - they are thus theoretical. Where possible they have been compared with experiment and with other studies, and are found to predict critically stable states with an accuracy greater than that likely to be found in the normal course of experiments.

Hydrodynamical models for the chaotic dripping faucet

“…These pendant drops are shown in Fig. (1) for different faucet radii R. However, the only stable stationary drop corresponds to the branch starting at the origin and ending at the first turning point (Padday and Pitt 1973) which denotes the critical drop volumes V c for which the weight of the drop is just balanced by the capillary forces. The corresponding shapes have a waist where the dynamic instability associatedwith eventual pinch off first starts.…”

“…For a faucet of radius R that is sufficiently small, drops with a volume V < V c are stable and axisymmetric (Padday and Pitt 1973) which case we will restrict ourselves to. The shape of such a pendant drop is determined by minimising the sum of its gravitational and surface energy subject to the constraint of constant volume.…”

Drainage of liquid from a small circular hole in a vertical wall

“…For example, water ceased to flow from a PP container with D = 1.7 mm at h f = 13.5 mm. In this case, Equation (8) over predicted h f to be 17.3 mm.…”

“…To flow into or through these materials, a so-called Laplace pressure that arises from the surface tension and curvature of the liquid interface often must be overcome. Discussions of the basic equations for Laplace pressure can be found in textbooks [3][4][5] and in the scientific literature [6][7][8]. For example, Padday and colleagues have theoretically examined the shape of pendant and sessile drops attached to various horizontal substrates [7,8].…”

The periodicity of droplets emanating from interconnected orifices

“…One of the earliest investigations was reported in 1864 by Tate, a pharmacist, who performed experiments to quantify the effects of tube diameter and other parameters on the weight of the dispensed droplet. Additional, related studies were reported by Rayleigh (1899), Harkins and Brown (1919), Padday and Pitt (1973), and Wilson (1988). Manfr6 (1966) presented a model of and experimental results for droplet formation driven by a hydrostatic head and reviewed previous, related works.…”