Liquid Films Falling Down a Vertical Fiber: Modeling, Simulations and Experiments

Abstract

We provide a new framework for analyzing the flow of an axisymmetric liquid film flowing down a vertical fiber, applicable to fiber coating flows and those in similar geometries in heat exchangers, water treatment, and desalination processes. The problem considered is that of a viscous liquid film falling under the influence of gravity and surface tension on a solid cylindrical fiber. Our approach is different from existing ones in that we derive our mathematical model by using a control-volume approach to express the conservation of mass and axial momentum in simple and intuitively appealing forms, resulting in a pair of equations that are reminiscent of the Saint-Venant shallow-water equations. Two versions of the model are obtained, one assuming a plug-flow velocity profile with a linear drag force expression, and the other using the fully-developed laminar velocity profile for a locally uniform film to approximate the drag. These can, respectively, model high- and low-Reynolds number regimes of flow. Linear stability analyses and fully nonlinear numerical simulations are presented that show the emergence of traveling wave solutions representing chains of identical droplets falling down the fiber. Physical experiments with safflower oil on a fishing line are also undertaken and match the theoretical predictions from the laminar flow model well when machine learning methods are used to estimate the parameters.