Fluid mechanics calculations in physics of droplets – IV: Head-on and off-center numerical collisions of unequal-size drops

Abstract

In this study, the finite volume method is employed to simulate the coalescence collision between water drops immersed in a continuous phase (n-heptane). For that purpose, it is chosen a range of values for the velocity of collisions for the finite volume calculations may yield different possible outcomes of the collision process. It can be seen for head-on collisions that when the velocity of collision is 0.2 m/s and 3.5 m/s, the little drop induces the formation of a hole in the bigger drop, until the surface tension forces to restore the circular form of the resulting drop. For a velocity of collision of 16.0 m/s, the little drop deforms the bigger one, and the system is converted into a thin ligament with the evolution of the dynamics. In this case, a little mass of n-heptane is trapped between the two drops, but at the end of the dynamics it drains to the continuous phase. For off-center collisions, two different values for the velocity of collisions were chosen, and the drops exhibit a lot of waves on the droplets’ surface. The streamlines are calculated for the process of coalescence of drops. These streamlines allow the understanding of the dynamics of the droplets immersed on the n-heptane phase. The effect of the interfacial tension it is showed due to the oscillations that the droplet exhibits. When the coalescence has begun, the streamlines form circular patterns at the zone of contact between the drops which explain the increment of the thickness of the bridge structure of the fluid between the two drops. At the end of the dynamics, when the velocity is of 0.2 m/s, the bigger drop reaches a circular form approximately, but when the velocity is of 3.5 m/s the drop reaches an elongated form.

Introduction

Experimental studies of the binary collision of alkane droplets were carried out by Ashgriz and Givi.^1,2 These authors found that when the weber number is increased, the collision takes the form of a high-energy one and different types of results arise. In these references, the results show that the collision of the droplets can be bouncing, grazing and generating satellite drops. A similar experimental study about binary collision of equal-size alcohol droplets has been carried out by Brenn and Frohn, ³ showing different stages of the collision process. Gokhale et al. ⁴ study the coalescence of two condensing drops and shape evolution of the coalesced drops. Also image-analyzing interferometry has been 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. Ashgriz and Poo ⁵ conducted an experimental study for the binary collision of water droplets for a wide range of weber numbers and impact parameters. These authors identify two types of collisions leading to the drops separation, which can be reflexive separation and stretching separation. It is found that the reflexive separation occurs for head-on collisions, while stretching separation occurs for high values of the impact parameter. These authors study experimentally the border between two types of separation and also collisions leading to coalescence. Menchaca-Rocha et al. ⁶ conducted a study on the coalescence and fragmentation of mercury drops of equal and unequal sizes. In this study, these authors find out the limits for the coalescence measured in terms of the relative velocity and impact parameter. Gotaas et al. ⁷ conducted an experimental study of the effect of viscosity in the collision of drops. Various organic substances are used as liquid phase corresponding to a range of viscosities from 0.9 to 48 MPa/s. The collision weber numbers ranged from 10 to 420, and binary collisions of liquid drops were reported using a modified stroboscopic technique, varying the impact parameter.

Mohamed-Kassim and Longmire ⁸ 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 viscosity but similar density were examined. After rupture, the free edge of the thin film receded rapidly allowing the drop fluid to sink into the bulk liquid below. 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 in such a way that part of its initial vortices moved inside the drop fluid while the other part 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. Eow and Ghadiri ⁹ report a study on the effects of the direction of the applied electric field as well as the geometry of the electrodes. In this study, the angle between the electric field and the centre line of two drops (θ) should be zero for the electrically induced force to attain its maximum attractive value. The maximum induced force is chosen large enough to deform the adjacent surfaces of the drop prior to coalescence. It has been shown experimentally and theoretically that the drop–drop attraction can occur also when θ is less than 54.7° or more than 125.3°. Aarts et al. ¹⁰ propose a study of droplet coalescence in a molecular system with a variable viscosity and a colloid-polymer mixture with an ultralow surface tension. When either the viscosity is large or the surface tension is small enough, it is observed that the opening of the liquid bridge initially proceeds at a constant speed set by the capillary velocity. In the first case studied one finds that the inertial effects become dominant at a Reynolds number of about 1.5 and the neck then grows as the square root of time. In the second case one finds that decreasing the surface tension by a factor of 105 opens the way to a more complete understanding of the hydrodynamics involved. Xing et al. ¹¹ put forward a lattice Boltzmann method-based single-phase free surface model to study the interfacial dynamics of coalescence, droplet formation and detachment phenomena related to surface tension and wetting effects. A perturbation similar to the step one in Gunstensen’s color model is added to the distribution functions of the interface cells in order to incorporate the surface tension into the single-phase model. Implementations of the model are verified simulating the processes of droplet coalescence, droplet formation and detachment from ceiling and from nozzles with different shapes and different wall wetting properties.

Baldessari and Leal ¹² report a comparison of the experimental results for flow-induced drop coalescence with the existing theories. For head-on collisions, the experiments show a plateau in the dependence of drainage time versus capillary number that cannot be explained by either the existing scaling analysis or the existing thin-film theory of the film drainage process. ¹³ These results indicate that the existing theories are incomplete in providing a framework for a comprehensive explanation of the experimental results. In this study, the authors find that a quasi static model in which deformation is localized within the thin film is in general not sufficient to describe the asymptotic approximation of the flow-induced collision to the coalescence of two slightly deformable drops at low capillary number. Wang et al. ¹⁴ studied the effect of glycerol on the coalescence of water drops in stagnant oil phase. In this reference the authors consider the binary coalescence of water drops formed through capillaries at low inlet flow rates in an immiscible stagnant oil phase. The evolution of the coalescence process is shown in this case. Qian and Law ¹⁵ propose 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 analyze 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 ¹⁵ five different regimes governing the collision between droplets are proposed: (i) coalescence after a small deformation, (ii) bouncing, (iii) coalescence after substantial deformation, (iv) coalescence followed by separation for head-on collisions, and (v) coalescence followed by separation for off-center collisions.

Gotaas et al. ¹⁶ propose a method for determining the boundaries between the different outcomes that can be achieved in the collision of liquid drops setting the weber number and varying the impact parameter. Menchaca-Rocha et al. ¹⁷ investigate the collision of two mercury drops at very low kinetic energy using fast digital and analog imaging techniques. In this paper, these authors studied the time evolution of the surface shape as well as an amplified view of the contact region. Zhang et al. ¹⁸ conducted a study on coalescence of unequal-size drops. In this work, the coalescence of a drop with a flat liquid surface pinches off a satellite droplet from its top, whereas the coalescence of two equally sized drops does not appear to produce in this case a satellite drop. These authors find that the critical ratio grows monotonically with the Ohnesorge number and the experimental coalescence of two unequal-size droplets is reported. Wu et al. ¹⁹ report experimental results on the coalescence of two liquid drops driven by surface tension. Using a high-speed imaging system, these authors study the early-time evolution of the liquid bridge that is formed upon the initial contact of two liquid drops in air. It is found that the liquid bridge radius follows the scaling law in the inertial regime. Further experiments show that such scaling law is robust when using fluids of different viscosities and surface tensions. The dimensionless pre-factor is measured to be in the range of 1.03–1.29, which is lower than the pre-factor 1.62 predicted by the numerical simulation of Duchemin et al. ²⁰ for inviscid drop coalescence.

Governing equations

The governing equations can be given by the continuity equation (1) and the momentum equation (2)

$\frac{\partial }{\partial t}(r_{\alpha }{\rho }_{\alpha }{V}_{\alpha })+\nabla ·(r_{\alpha }({\rho }_{\alpha }{V}_{\alpha }\otimes V_{\alpha }))\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=-r_{\alpha }\nabla p_{\alpha }+\nabla ·(r_{\alpha }{\mu }_{\alpha }(\nabla V_{\alpha }+(\nabla V_{\alpha }{)}^T))\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+\sum _{\beta =1}^{N_p}({\Gamma }_{\alpha \beta }^{+}{V}_{\beta }-{\Gamma }_{\beta \alpha }^{+}{V}_{\alpha })+M_{\alpha }$

(2)

where V is the velocity, p is the pressure, µ is the viscosity, ρ is the density, r_α is the volume fraction of the continuous phase α, N_p is the total number of phases, M_α describes the interphase forces acting on phase α due to the presence of other phases, Γ⁺_αβ represents the positive mass flow rate per unit volume from phase β to phase α. The unique term in the right hand of the equality on equation (1) occurs if the interphase mass transfer takes place.

CFX® (ANSYS® 15.0, ANSYS®, Inc. Southpointe 2600 ANSYS Drive Canonsburg, PA 15317 USA), a flow solver based on the finite volume method, was used to solve equations (1) and (2). A rectangular mesh was used for this calculation with 360,000 elements. Inside the CFX module was defined the following constants: ρ (n-heptane) = 683.8 kg/m³, ρ (water) = 1000.0 kg/m³, µ (n-heptane) = 0.000408 × 10⁻³kg/m.s, µ (water) = 1.12 × 10⁻³Pa.s, r_drop1 = 30.0 × 10⁻⁶ m, r_drop2 = 12.0 × 10⁻⁶ m, σ (water/n-heptane) = 49.1 × 10⁻³N/m. Here r_drop1 and r_drop2 are the droplets radius, σ (water/n-heptane) is the interfacial tension of the system water-heptane.

The surface tension model

Surface tension is the force that holds the interface together and this effect is included here according to the method that is described in Acevedo. ²¹ If an interface is cut, the magnitude of the surface tension force per unit length is called the surface tension coefficient. Surface tension has a number of important physical effects, including:

If the interface is curved, it induces a force normal to the interface. For a droplet at rest, this induces a pressure rise within the droplet. The effect of this normal force is to smooth regions of high curvature; it tends to reduce the surface area of droplets.

When the surface tension coefficient is not constant, the surface tension force has a tangential component that tends to move fluid along the interface toward regions of high surface tension coefficient. This is often called Marangoni convection.

When the free surface interface touches a wall, the wall may attract the liquid (wetting wall) or repel the liquid (non-wetting wall). In a static situation, the wall contact angle may be measured. This phenomenon, known as wall adhesion, gives rise to effects such as the meniscus and capillary rise in tubes.

For free surface flows, the option of including surface tension effects exists. When it is enabled, a surface tension coefficient must be specified. If the surface tension coefficient is variable, the Marangoni force is automatically included. If the surface tension model is activated, using double precision is often required to avoid roundoff errors in the curvature calculation.

Head-on collisions: Coalescence and fragmentation of liquid drops

In order to obtain the numerical solution of equations (1) and (2) for head-on collisions of liquid droplets the finite volume method was used. In this case, the value that was chosen for the velocity is of 0.2 m/s which correspond to coalescence of water drops immersed in n-heptane continuous phase.

In Figures 1 and 2 is shown a sequence of snapshots for the collision between two water drops (blue color) immersed on a hydrocarbon continuous phase (white color). These unequal-size droplets approach each other and the system coalesce to form a bigger drop. At t = 2.66e−5s it can be seen the formation of a little interface between the drops that disappear at t = 2.74 e−5 s.

Figure 1.

Sequence of times showing the evolution of the collision between two drops with V_col = 0.2 m/s. The time scale is given in seconds.

Figure 2.

Sequence of times showing the evolution of the collision between two drops with V_col = 0.2 m/s. The time scale is given in seconds.

At t = 3.04e−5 it can be seen the formation of a hole just in the zone that the little drop coalesce with the bigger one. After this snap shot the surface tension forces produce oscillations in the bigger drop and these oscillations are damped with the evolution of time in the dynamics, and the final stage shows a very circular and defined bigger drop.

In order to study the coalescence collision of the water drops immersed in n-heptane phase the finite volume method was used for the following calculations. In this case, the value that was chosen for the velocity is of 3.5 m/s.

It can be seen in Figure 3 that for t = 8.39e−6s an interfacial film is formed. This interface disappears at t = 9.39e−6 s. It is very important to note that for t = 8.05e−6s the process of collision induces the bigger drop to stretch its surface and the drop is deformed.

Figure 3.

Sequence of times showing the evolution of the collision between two drops with V_col = 3.5 m/s. The time scale is given in seconds.

At t = 1.12e−5s the formation of a hole at the zone of contact between the drops can be seen. This hole disappears completely due to surface tension forces operating in the surface of the drop, which follow a very interesting dynamics until reaching a very defined circular form (see Figure 4 at t = 2.29e−5).

Figure 4.

Sequence of times showing the evolution of the collision between two drops with V_col = 3.5 m/s. The time scale is given in seconds.

In order to model the coalescence collision of water drops immersed in a continuous phase (n-heptane), the equations of momentum and continuity are solved numerically with the finite volume method. In this case, the value that was chosen for the velocity of collision is of 16.0 m/s.

In Figures 5 and 6, it can be seen that a very well-defined interface is formed. In this zone, the n-heptane that composes this interfacial film forms a little drop inside the mass of water that is deformed due to inertial forces acting at the surface of the bigger drop when the impact of the little drop take place.

Figure 5.

Sequence of times showing the evolution of the collision between two drops with V_col = 16.0 m/s. The time scale is given in seconds.

With the evolution of the dynamics it can be seen than a very thin ligament is formed. In this case the inertial forces prevailing and the surface tension forces are not enough to restore the drop into a circular form.

Off-center collisions: Coalescence of liquid drops

In order to model off-center collisions of two unequal-size liquid drops some calculations were carried out using the finite volume method. The velocity of collision with values of 0.2 m/s and 3.5 m/s were chosen which correspond to coalescence of water drops immersed in n-heptane continuous phase.

In Figure 7 is shown a sequence of snapshots for the collision between two water drops (blue color) immersed on a hydrocarbon continuous phase (white color). It can be seen that for a velocity of collision of 0.2 m/s, the coalescence of drops is carried out, without the formation of the circular interfacial film. This behavior occurs because the liquid that drains out between the droplets has the time enough to drain and no fluid is trapped at the interface of the drops.

Figure 6.

Sequence of times showing the evolution of the collision between two drops with V_col = 16.0 m/s. The time scale is given in seconds.

Figure 7.

Sequence of times showing the evolution of the collision between two drops with v_col = 0.2 m/s. the time scale is given in seconds.

At t = 3.05e−7s, it can be seen the formation of a bridge structure at the zone of contact between the drops. The thickness of this bridge is incremented with the evolution of the dynamics of the system. For this case, the surface tension forces prevailing over inertial forces and the dynamics of the system water-heptane show multiple oscillations at the surface of the bigger droplet that result of the coalescence process. At t = 1.48e−5s the bigger drop is transformed into a very well defined circular drop.

In Figures 8 and 9 can be seen the internal streamlines for the system of drops at t = 8.88e−7s and t = 1.48e−5 s. When the surfaces of the drops touch between them, the flux around the bridge structure is circular and the streamlines explain the increment of the thickness of the mass of water in that zone. In Figure 9 can be seen a set of circular patterns of the flow inside the bigger drop that mainstains the circular form of the droplet.

Figure 8.

Streamlines for the system water-heptane with V_col = 0.2 m/s at t = 8.88e−7 s.

Figure 9.

Streamlines for the system water-heptane with V_col = 0.2 m/s at t = 1.48e−5 s.

In Figure 10 is shown a sequence of snapshots for the collision between two water drops immersed on a hydrocarbon continuous phase. In this case, the velocity of collision is 3.5 m/s and the coalescence of drops is carried out but with the deformation of the drop at the end of the dynamics.

Figure 10.

Sequence of times showing the evolution of the collision between two drops with V_col = 3.5 m/s. The time scale is given in seconds.

It can be seen that at t = 4.11e−6s the surface of the drops reaches a elongated form due to the inertial forces acting in the system of drops. At this stage, the drops start to form a bridge structure between them and the coalescence process has begun. At t = 4.72e−6, the drops exhibit a very thin interface of n-heptane located at the middle of the bridge. This interface disappears with the evolution of the dynamics and finally the drops reach their elongated form at t = 1.48e−5 s.

In Figures 11 and 12 can be seen the internal streamlines for the system of drops at t = 4.72e−6 s and t = 9.16e−6 s. It can be seen that the streamlines of the fluid flow explains the subsequent increment of the thickness of the bridge between the drops.

Figure 11.

Streamlines for the system water-heptane with V_col = 3.5 m/s at t = 4.72e−6 s.

Figure 12.

Streamlines for the system water-heptane with V_col = 3.5 m/s at t = 9.16e−6 s.

Conclusions

In this paper, an adequate methodology for the study of droplet collisions was presented. The equations of fluid mechanics were resolved using the finite volume method for a system composed of water and one hydrocarbon (n-heptane) that represent a continuous phase. Depending on the velocity of collision it was shown different outcomes such as: coalescence and formation of a thin ligament of water. When the velocity of collision was 0.2 m/s and 3.5 m/s a little interface is formed between the drops and the formation of a hole at the zone of contact between the droplets can be seen. On the other hand, when the velocity of collision is 16.0 m/s a very well-defined interfacial film composed by n-heptane is formed. This mass of n-heptane is trapped in the mass of water and at the end of the dynamics this mass return to the continuous phase and a very thin ligament is formed. On the other hand, the modeling of off-center collisions of droplets was presented. The equations of continuity and momentum were resolved using the finite volume method for a system of two unequal-size drops of water immersed in a hydrocarbon phase. Two values for the velocity of collision were chosen and these values conduce to the coalescence between the droplets. For a collision velocity of 0.2 m/s an approximately circular drop at the end of the calculation can be seen. On the other hand when the velocity of collision was 3.5 m/s a elongated drop is formed at the end of the snap shots. The streamlines are showed for each case. These streamlines explain the direction of the flow inside the drops. Only for a velocity of collision of 3.5 m/s was observed the formation of very thin interface composed by the n-heptane continuous phase.