Maximal liquid bridges between horizontal cylinders

TL;DR

This paper analytically studies the shape and maximum trapping capacity of liquid bridges between horizontal cylinders, revealing simple relationships for optimal configurations when cylinders are small relative to the capillary length.

Contribution

It provides analytical solutions to the nonlinear Laplace-Young equation for liquid bridges and identifies parameters that maximize trapping capacity for small cylinders.

Findings

Maximum trapping occurs when the center-to-center distance is about twice the capillary length.

Maximum cross-sectional area scales linearly with small separations.

Linearized solutions confirm relationships for meniscus slope and trapping capacity.

Abstract

We investigate two-dimensional liquid bridges trapped between pairs of identical horizontal cylinders. The cylinders support forces due to surface tension and hydrostatic pressure which balance the weight of the liquid. The shape of the liquid bridge is determined by analytically solving the nonlinear Laplace-Young equation. Parameters that maximize the trapping capacity (defined as the cross-sectional area of the liquid bridge) are then determined. The results show that these parameters can be approximated with simple relationships when the radius of the cylinders is small compared to the capillary length. For such small cylinders, liquid bridges with the largest cross sectional area occur when the centre-to-centre distance between the cylinders is approximately twice the capillary length. The maximum trapping capacity for a pair of cylinders at a given separation is linearly related to the separation when it is small compared to the capillary length. The meniscus slope angle of the largest liquid bridge produced in this regime is also a linear function of the separation. We additionally derive approximate solutions for the profile of a liquid bridge making use of the linearized Laplace-Young equation. These solutions analytically verify the above relationships obtained for the maximization of the trapping capacity.