The double-helical branch structure of fixed contact line liquid bridge equilibria

Abstract

Attempts to stabilize the fixed contact line cylindrical liquid bridge have generally implicitly assumed that it was related to stable equilibria in a continuous manner. An examination of the branch structure of equilibria demonstrates that for longer liquid bridges, the cylinder becomes separate from normally stable equilibria and hence likely cannot be stabilized by a continuous perturbation. All axisymmetric equilibria for a fixed contact line liquid bridge of fixed moderate length in the absence of applied forces (zero gravity, no spin) are found to lie on a single semi-infinite bridged double helix. This helix breaks repeatedly as the length of the liquid bridge is increased, separating the stable truncated sphere state from the generally desirable cylindrical state. This structural change appears to explain why several approaches to stabilization of long cylindrical liquid bridges beyond the classical Plateau–Rayleigh limit have been largely unsuccessful over the past few decades. The first breakage of the helix, when liquid bridge length equals 9.0973 times radius, corresponds roughly to the maximum length of cylindrical liquid bridge achieved experimentally via a perturbative applied force (using acoustic radiation pressure). An examination of the deformations of the helix under gravity reveals that for moderate length liquid bridges at small Bond number, all axisymmetric equilibria lie on a single unbroken branch which forms a single (unbridged) semi-infinite double helix. Combined effects of gravity and length are investigated as well, resulting in multiple disconnected loops of equilibria.