Forced dynamics of a short viscous liquid bridge

Abstract

The dynamics of an axisymmetric liquid bridge of a fluid of density ρ, viscosity µ and surface tension σ held between two co-axial disks of equal radius e is studied when one disk is slowly moved with a velocity U(t). The analysis is performed using a one-dimensional model for thin bridges. We consider attached boundary conditions (the contact line is fixed to the boundary of the disk), neglect gravity and limit our analysis to short bridges such that there exists a stable equilibrium shape. This equilibrium is a Delaunay curve characterized by the two geometric parameters 0 = V 0 /(πe 3 ) and S = (πe 2 )/V 0 , where V 0 is the volume of fluid and the length of the bridge. Our objective is to analyse the departure of the dynamical solution from the static Delaunay shape as a function of 0 , S, the Ohnesorge number Oh = µ/ √ ρσ e, and the instantaneous velocity U(t) and acceleration ∂ t U (non-dimensionalized using e and σ /ρ) of the disk. Using a perturbation theory for small velocity and acceleration, we show that (i) a non-homogeneous velocity field proportional to U(t) and independent of Oh is present within the bridge; (ii) the area correction to the equilibrium shape can be written as U 2 A i + U Oh A v + ∂ t UA a where A i , A v and A a are functions of 0 and S only. The characteristics of the velocity field and the shape corrections are analysed in detail. For the case of a cylinder (S = 1), explicit expressions are derived and used to provide some insight into the break-up that the deformation would induce. The asymptotic results are validated and tested by direct numerical simulations when the velocity is constant and when it oscillates. For the constant velocity case, we demonstrate that the theory provides a very good estimate of the dynamics for a large range of parameters. However, a systematic departure is observed for very small Oh due to the persistence of free eigenmodes excited during the transient. These same eigenmodes also limit the applicability of the theory to oscillating bridges with large oscillating periods. Finally, the perturbation theory is applied to the cylindrical solution of Frankel & Weihs (J. Fluid Mech., vol. 155 (1985), pp. 289-307) obtained for constant velocity U when the contact lines are allowed to move. We show that it can be used to compute the correction associated with acceleration. Finally, the effect of gravity is discussed and shown to modify the equilibrium shape but not the main results obtained from the perturbation theory.

Self-similar impulsive capillary waves on a ligament

“…While the stress penetration into a solid bar allows solving for its stability, and ultimate breakup 4 , the dispersive dynamics of capillary waves on a liquid ligament helps understanding the remnant liquid mass attached to a rod quickly removed from a pool 5 . Impacts distort liquid surfaces into a variety of non-trivial shapes [6][7][8] , in particular when they are mediated by capillary waves 9,10 ; Here, we provide an original solution for the short time response of a liquid ligament of which one extremity is either pulled, or pushed, and in all cases suddenly accelerated axially.…”

Statics and dynamics of a viscous ligament drawn out of a pure-liquid bath

“…2006; van Hoeve et al. 2010; Vincent, Duchemin & Le Dizès 2014 a ; Martínez-Calvo et al. 2018) and films (Champougny et al.…”

The break-up of free films pulled out of a pure liquid bath

“…The above system is closed by the unchanged conservation equation (7). Note that we did not truncate at O(ε 2 ) the expression (6) for the mean curvature, despite the higher-order correction at the denominator, as done for liquid bridges by [25] for example. This keeps the model compatible with the solution of the static meniscus near the bath, where the slope is not small anymore, and allows describing the entire film shape, from the bath up to the fiber, without having to use a matching procedure between the film region and the static meniscus.…”