On the stability of a cylindrical liquid bridge

Abstract

The problem of the stability of a liquid bridge stretched between parallel plates with a wetting contact angle of 90◦ is revisited. A closed form expression is derived for the height of the bridge, in terms of its volume, within which a cylindrical and various types of unduloidal equilibrium configurations can exist. For a given volume of the liquid and specified height of the bridge, the lateral surface of a uniform cylindrical bridge is smaller than the surface area of any unduloidal equilibrium shape. The lateral surface of the unduloidal shapes increases with the increase in the number of their inflection points. The force required to keep the bridge in equilibrium is evaluated in each case. All unduloidal equilibrium configurations are unstable, the only stable configuration being that of a cylindrical bridge whose height is less than one-half of its circumference. A lower bound estimate is also derived based on a simple energy consideration. The stretching force required for the equilibrium at the onset of instability is compared with its upper bound estimate. For a given height of the bridge, the force required to keep a cylindrical bridge in equilibrium is greater than the force required for equilibrium of any unduloidal configuration of the same height. The opposite is true for the capillary pressure.