Viscous film flow coating the interior of a vertical tube. Part 1. Gravity-driven flow

Abstract

The gravity-driven flow of a viscous liquid film coating the inside of a tube is studied both theoretically and experimentally. As the film moves downward, small perturbations to the free surface grow due to surface tension effects and can form liquid plugs. A first-principles strongly nonlinear model based on long-wave asymptotics is developed to provide simplified governing equations for the motion of the film flow. Linear stability analysis on the basic solution of the model predicts the speed and wavelength of the most unstable mode, and whether the film is convectively or absolutely unstable. These results are found to be in remarkable agreement with the experiments. The model is also solved numerically to follow the time evolution of instabilities. For relatively thin films, these instabilities saturate as a series of small-amplitude travelling waves, while thicker films lead to solutions whose amplitude becomes large enough for the liquid surface to approach the centre of the tube in finite time, suggesting liquid plug formation. Next, the model's periodic travelling wave solutions are determined by a continuation algorithm using the results from the time evolution code as initial seed. It is found that bifurcation branches for these solutions exist, and the critical turning points where branches merge determine film mean thicknesses beyond which no travelling wave solutions exist. These critical thickness values are in good agreement with those for liquid plug formations determined experimentally and numerically by the time-evolution code.

Stability of viscous film flow coating the interior of a vertical tube with a porous wall

“…Assuming 1 and Re ∼ O(1), we can remove the contributions of the inertial terms. We consider the relative sizes of the parameters for experiments conducted by Camassa et al [16], in which Re ∼ O(10 −2 ) or smaller. Thus, even as ∼ O(10 −1 ), we also take Re → 0, and we can neglect the inertial term.…”

“…Duprat et al [35] have studied the absolute and convective instability of an exterior coating film on a vertical fiber both experimentally and theoretically. The most relevant work on our problem is the study of the AI-CI characteristics of an interior coating film down a vertical tube [16]. Camassa et al [16] have conducted a series of experiments to study the gravity-driven film coating the interior of a vertical tube.…”

“…The most relevant work on our problem is the study of the AI-CI characteristics of an interior coating film down a vertical tube [16]. Camassa et al [16] have conducted a series of experiments to study the gravity-driven film coating the interior of a vertical tube. In their experiments, tubes of three different inner radii were used for the experiments: 5 × 10 −3 , 2.95 × 10 −3 , and 1.7 × 10 −3 m. The liquid used was a silicone oil with dynamic viscosity μ = 12.9 N s/m 2 , density ρ = 970 kg/m 3 , and surface tension σ = 2.15 × 10 −2 N m −1 .…”

“…An implicit Gears method for stiff problems was used for the time advance, and the relative error is set less than 10 −6 . Camassa et al [16] performed numerical simulations on the long-wave model for a thin film coating the interior of the tube. The results showed that for film with a mean thickness smaller than some critical value depending on a, i.e., h 0 < h c (a), these instabilities saturate as a series of small-amplitude traveling waves.…”

“…Ring waves were identified as a mass transport mechanism in air-driven core-annular flows. Camassa et al [16] have studied the linear stability and the nonlinear evolution of a viscous film falling down a vertical tube driven by gravity. The results of linear stability analysis indicate that for any tube size there exists a film thickness large enough to become absolutely unstable.…”