A theoretical mesh-free scheme to model viscous drop interactions: a particle-based method

Alejandro Acevedo-Malav 0 0 Centro Multidisciplinario de Ciencias, Instituto Venezolano de Investigaciones Cientficas (IVIC) , Mrida 5101, Venezuela Here a Lagrangian mesh-free formalism is presented to simulate the coalescence process between three unequal-sized liquid drops in the three-dimensional space. The surface tension forces acting on the surface of the drops cause the formation of a circular flat section when the droplets collide. The effect of polydispersity on the collision dynamics is simulated using a set of droplets with radius around 30 m. It is important to see that the inhomogeneous distribution of the droplets size results in very important changes on the drops dynamics. The smoothed particle hydrodynamics scheme proposed here can be used to model situations where a continuum phase is included in the problem. The velocity vector fields are computed for each situation, and it can be seen that in the zone of contact between the droplets, there is an increment of the velocity value. This is due to the pressure distribution inside the drops. - Introduction In the literature, many studies have been proposed for the numerical simulation of the coalescence and break up of droplets. The authors propose different methods to approach the dynamics of liquid drops by a numerical integration of the Navier-Stokes equations. These examine the motion of droplets and the dynamics that it follows in time and study the liquid bridge that arises when two drops collide. The effects of parameters such as Reynolds number, impact velocity, drop size ratio, and internal circulation are investigated, and different regimes for droplets collisions are simulated. In some cases, those calculations yield results corresponding to the four regimes of binary collisions: bouncing, coalescence, reflexive separation, and stretching separation. These numerical simulations suggest that the collisions that lead to rebound between the drops are governed by macroscopic dynamics. In these simulations the mechanism of formation of satellite drops was also studied, confirming that the principal cause of the formation of satellite drops is the end pinching, while the capillary wave instabilities are the dominant feature in cases where a large value of the parameter impact is employed. In this work the smoothed particle hydrodynamics (SPH) method is applied to simulate for the first time in three-dimensional space the hydrodynamic coalescence collision of three liquid drops in a vacuum environment. This method is employed in order to obtain the approximate numerical solutions of the equations of fluid dynamics by replacing the fluid with a set of particles. These particles may be interpreted as corresponding to the interpolation points from which properties of the fluid can be determined. Each SPH particle can be considered as a system of smaller particles. The SPH method is particularly useful when the fluid motion produces big deformations and a large velocity of the whole fluid. A brief review of previous studies Rekvig and Frenkel [1] reported a molecular simulation study of the mechanism of droplets covered with a surfactant monolayer coalesce. The authors proposed a model system where the rate-limiting step in coalescence is the rupture of the surfactant film. For this numerical study, one made use of the dissipative particle dynamics method using a coarse-grained description of the oil, water, and surfactant molecules. The authors found that the rupture rate is highest when the surfactant has a negative natural curvature, lowest when it has a zero natural curvature, and lying in between when it has a positive natural curvature. Gokhale et al. [2] studied the coalescence of two condensing drops and the shape evolution of the coalesced drops. Image analyzing interferometry is used to study the coalescence of two drops of 2-propanol, and the shape evolution after the coalescence is found to be driven by the capillary forces inside the drop. Foote [3] proposed a method to study the dynamics of liquid drops by a numerical integration of the NavierStokes equations. This author examined the motion of droplets with the application to the raindrop problem. The study was restricted to the collision of equal-sized drops along their line of centers. Numerical solutions were developed to study the rebound of water droplets in air. It is found that except for a small viscous effect, the Weber number of the drops determines the dynamics of the collision and the bounce time. Decent et al. [4] studied the formation of a liquid bridge during the coalescence of droplets. In this paper, the authors considered a mathematical model where the pressure singularity is removed at the instant of the impact for the coalescence of two viscous liquid volumes in an inviscid gas or in a vacuum environment. The formation of the liquid bridge is examined for two cases: (a) two infinitely long liquid cylinders, and (b) two coalescing spheres. In both cases the numerical solutions are calculated for the velocity and pressure fields, and the removal of the pressure singularity is confirmed. Mohamed-Kassim and Longmire [5] conducted particle image velocimetry (PIV) experiments to study the coalescence of single drops through planar liquid/liquid interfaces. Sequences of velocity vector fields were obtained with a high-speed video camera and the subsequent PIV analysis. Two ambient liquids with different viscosities but similar densities were examined. After rupture, the free edge of the thin film receded rapidly, allowing the drop fluid to sink into the bulk liquid below. The vorticity generated in the collapsing fluid developed into a vortex ring, straddling the upper drop surface. The inertia of the collapse deflected the interface downward before it rebounded upward. During this time, the vortex core split so that part of its initial vorticity moved inside the drop fluid while part of it remained in the ambient fluid above it. The velocity of the receding free edge was smaller for higher ambient viscosity, and the pinching of the upper drop surface caused by the shrinking capillary ring wave was stronger when the ambient viscosity was lower. This resulted in a higher maximum collapse speed and higher vorticity values in the dominant vortex ring. Qian and Law [6] proposed an experimental investigation of binary collision of drops with emphasis on the transition between different regimes, which may be obtained as an outcome of the collision between droplets. In this study the authors analyzed the results using photographic images, which show the evolution of the dynamics exhibited for different values of the Weber number. As a result of the experiment reported by Qian and Law [6], five different regimes governing the collision between droplets are proposed: (a) coalescence after a small deformation, (b) bouncing, (c) coalescence after substantial deformation, (d) coalescence followed by separation for head-on colli (...truncated)