Model and experiments of a drop impinging on an immersed wall

Abstract

We present a model describing the rebound of a drop impinging on a rigid plane wall immersed in water. This model is based on the resolution of the drop equation of motion in an unbounded fluid in which an additional pressure force is introduced accounting for the wall effect on the drop motion. This force is computed from a film drainage simulation model during the approach of the deformable particle to the wall. Results of the model have been compared with experimental trajectories of drops impinging vertically at terminal rise velocity against a horizontal wall immersed in water at rest. These trajectories have been obtained with the help of an image processing technique. A wide range of experimental conditions has been studied (drop diameter, interfacial tension, drop viscosity, and density). In most of the cases, the model predicts the experimental trajectories within a very good accuracy (height of bouncing, deformation, number of rebounds) even in the case of a significant deformation. The numerical results show that the rebound of a deformable inclusion against a wall in water is essentially governed by the balance between the added mass force and the film pressure force exerted on the drop during the impact. The model has also been successfully tested in the case of an impinging bubble at high particle Reynolds number, based on experimental data taken from Tsao and Koch [Phys. Fluids 9, 44 (1997)].

A note on the modelling of the bouncing of spherical drops or solid spheres on a wall in viscous fluid

“…The time of contact with the wall is the half period of this oscillator: Fig. 4 shows the comparison between the predictions of relation (5) and the experimental results of Okurama et al (2003) for drops in air, Klaseboer et al (2000) and Legendre et al (2005) for toluene drops in water, Klaseboer et al (2000) for cyclohexane drops and silicone oil drops in water. In the latter case, the contact time is deduced from the trajectories of the drops obtained using a high-speed camera.…”

“…For drops, the contact time is controlled by the drop's surface tension during the bouncing as shown by Klaseboer et al (2000), Richard and Quéré (2000), Okurama et al (2003) and Legendre et al (2005). It is the time necessary to deform the surface (kinetic energy transferred into surface energy) and to restore the spherical shape (surface energy transferred into kinetic energy).…”

“…In the latter case, the contact time is deduced from the trajectories of the drops obtained using a high-speed camera. The experiments of Klaseboer et al (2000) were performed using a rate of 50 images per second which is not enough to deduce the contact time with a satisfactory precision. The values reported in Fig.…”

“…For drops, the bouncing on a wall is characterized by significant deformation and contact time with the wall, both increasing with the particle diameter and the impact velocity (Richard and Quéré, 2000;Klaseboer et al, 2000;Legendre et al, 2005). The bouncing is also controlled by the drainage of a film formed between the particle and the wall.…”

Slip velocity and drag law in a liquid‐liquid homogeneous dispersed flow

“…The validity of the Schiller and Nauman correlation for single drops rising in an unbounded medium has been already observed for a broader variety of liquid-liquid systems in a previous work Ž . Klaseboer et al, 2001 . This correlation is therefore supposed to be accurately predicting the terminal velocity of drops in the range of measured diameters.…”