Three-component phase-field Lattice Boltzmann method with high density ratio and ability to simulate total spreading states

Highlights

• Ternary phase-field LBM for high density ratios and total spreading situations.

• Stable discretization adopted to improve the density ratio between components.

• Versatility-accuracy of the model validated through wide range of benchmark tests.

• LBM simulations of collision and coalescence of bubble and droplet in the water.

• Drainage-coalescence time for off-center collision is higher than head-on impact.

Abstract

In this paper, a ternary phase-field based LBM, capable of handling high density ratios [O(1000)] and total spreading situations is presented. In order to improve the density ratio between three components, stable discretization incorporating mixed differentiating scheme (average of the biased and central schemes) is adopted, which instead of 8, involves 16 neighboring points to approximate the derivative terms. The multi-component Cahn-Hilliard equations are employed to track phasic evolution of the system. The bulk free energy in these equations is chosen in such a way that it enables the model to cope with total spreading problems, where one of the components is totally wetted by one, or both of the other fluids. Two LB equations are used to capture the interface between three incompressible-immiscible fluids, and another one for simulation of the hydrodynamic flow field. To demonstrate versatility and accuracy of the proposed model, a wide range of benchmark problems and numerical tests, including static and dynamic interaction of a bubble at liquid-liquid interface, ternary phase separation, binary and ternary Rayleigh-Taylor instability (RTI), and finally collision and coalescence between an oil droplet and a gas bubble that are rising in water due to the buoyancy force, are investigated. A good agreement is found between simulation results and available data/theoretical predictions. The simulation results of the bubble-droplet interaction reveals that the film drainage and coalescence times of the off-center collisions are higher compared to the head-on impacts, and these times increase with the horizontal distance separating the bubble and the droplet.

Introduction

Three-phase and three-component flows, not only are omnipresent in everyday human life, but also are of great importance in many industrial, chemical, food and environmental processes. Gas flotation for removal of fine oil droplets from produced and waste waters [1], [2], [3], [4], [5], enhanced oil recovery [6], foamed foods such as ice cream, mayonnaise or dressings [7], [8], paper and pulp industry [9], [10], and microfluidic systems for separation processes, chemical reactions and encapsulation (generation of multiple emulsions) [11], [12], [13], are only a few examples of wide-ranging applications of three-component flows. This type of flows, due to the presence of different components and interfaces often involve interesting and yet complex phenomena that make their study very challenging. Despite the invention of new instruments and techniques, and the advancements of high speed cameras, still it is very difficult to perform precise experiments on them, since they normally involve many different parameters and wide range of time and length scales, from microscopic to meso and macroscopic.

At the other hand, especially with development of high processing power computers, computational fluid dynamic (CFD) has shown that it can produce accurate and reliable results in solving single and multi-phase fluid flow problems. Generally the numerical methods for fluid flow simulations can be divided into two different categories: conventional Navier-Stokes (NS) solvers that are based on continuum theory and more suited for macroscopic studies, and particle based methods that take a microscopic or mesoscopic view to the problem. In the latter case, depending on the method, a particle may represent e.g. an atom or a molecule as in molecular dynamics (MD) simulations, a collection of molecules which is the case for direct simulation Monte-Carlo (DSMC) and dissipative particle dynamics (DPD) methods, or a portion of the macroscopic fluid as it is in lattice Boltzmann method (LBM). The LBM originally stems from lattice gas automata, with the difference that instead of tracking discrete particles, it tracks distribution of the fluid particles [14]. The LBM is based on kinetic theory and can be derived from Boltzmann's equation using Bhatnagar-Gross-Krook (BGK) approximation, it also can be viewed from evolution of particle distribution function, f in the phase space. At the macroscopic scale, the continuous NS equations can be recovered from LBM using Chapman-Enskog (CE) expansion. It is believed that LBM has some advantages over other conventional CFD methods, particularly in dealing with complex boundaries and geometries, ease of implementation and parallelization, and incorporating microscopic interactions [15]. The latter property specifically is very important in multi-phase flow studies, because surface tension, phase separation or coalescence of miscible components in another fluid medium, wetting and spreading of one component over another, and generally topological changes of the interfaces in these flows result from interparticle forces and microscopic interactions. Enhancing and improving capabilities of the LB methods to tackle with various theoretical, practical and industrial problems is still part of an ongoing process. Besides, fluid flows with more than two phases/components have many applications nowadays, particularly the newly found ones in microfluidic and biomedical/bioengineering (such as encapsulation, drug delivery and drug release). Because of its mesoscopic nature, and its ability of bridging between micro and macro scales, LBM specifically can be very helpful in these relatively new applications. It should be mentioned that many thermal LBM models also have been proposed for either single or multiphase flows in different applications, which is out of the scope of the present paper and interested readers are referred to Refs. [16], [17], [18], [19], [20], [21], [22], [23].

Various types of multiphase LB models have been proposed in the literature, including: color-gradient model [24], [25], Shan-Chen (pseudo-potential method) [26], free-energy based models [27], [28], and phase-field (or generally models that use Cahn-Hilliard or similar equations in order to describe interface dynamics) [29], [30], [31], [32], [33]. A more detailed description of these models alongside with their point of strength and drawbacks can be found in Refs. [15], [23]. These models mainly focus on the two-phase case, and relatively little attention has been paid to the construction of LB model for immiscible three-component flows.

The first attempt to model ternary fluid mixtures backs to [34] and [35]. These researchers tried to simulate oil-water-surfactant mixtures, but their approach is only applicable to the cases where one of the components is amphiphile. The first notable multi-phase model capable of handling three different fluid component was proposed by Bao and Schaefer [36]. Their model is based on Shan-Chen pseudo-potential MCMP LBE method and can reach to density ratios of 1000 and 300 for static and dynamic cases, respectively. Their model suffers from relatively high spurious currents, particularly in higher density ratios and also they did not discuss that whether their method is able to recover correct macroscopic NS equations or not.

Leclaire and his coworkers [37] extended two phase color-gradient operator to model N-fluid systems. Their model was able to reach high density ratios for some test cases, such as static contact angle, but it failed to simulate multi-layered Couette's flow with variable density ratios. Although in their next paper [38], they successfully simulated two layered Couette's flow using an improved equilibrium distribution function. A few years later, Fu et al. [39] used the model framework established by Leclaire et al. [37] to study double-emulsion formation in a microfluidic channel. Very recently, Yu and his co-workers [40] have proposed another ternary LBM model based on the color-gradient approach. They derived a new interfacial force for N-phase system and introduced this force to their model via body force scheme. Doing so, they were able to reduce parasitic currents compared to the model developed by Leclaire et al. [37], [38]. The main advantage of these models is probably the ability of covering a wind range of surface tensions, which makes them a good choice for simulation of multiple emulsions and encapsulation problems. In their latest work, Yu et al. [41] also incorporated contact line dynamics in to their model. Very recently a Rothman-Keller type three-component LBM has also been adopted and modified by Xie et al. [42] for simulation of viscoelastic fluid flows. In order to implement viscoelastic effects, Maxwell's constitutive equation has been used in this model.

Shi et al. [43] based on multi-phase LBM flux solver, proposed a model for simulation of three-component flows. They used a three-component Cahn-Hilliard model (originally developed by Boyer et al. [44], [45]) to capture evolution of the fluid-fluid interfaces, but this model is also limited to very low (O (1)) density, viscosity and surface tension ratios. Another important point that should be noted is that their model seems to be limited to the partial wetting cases, since the additional term that allows the model to account for total spreading states was not included in the free energy functional, though they did not discuss the matter on their paper. Besides Zhang and Kwok's [46] mean field multicomponent free energy LBM, Semprebon and his co-workers [47] probably were the first group to developed a free energy based ternary LBM. The main feature of their method is that fluid-fluid interfacial tensions and fluid-solid contact angles can be tuned independently. Liang et al. [48] adopted ternary CH model of Boyer et al. [44], [45] to present a new three-component incompressible LBM based on phase-field theory. This model can simulate partial lens spreading problem and dynamic case of ternary spinodal decomposition with density ratios of around 100 and 20, respectively. Very recently, Liang et al. [49] have also extended the model to simulate the contact line motion of ternary fluids on a solid surface.

Although the proposed models in [[34], [35], [36], [37], [38],[40], [41], [42], [43],[46], [47], [48], [49]] are very valuable considering the fact that they are pioneering works, but they can only be suitable for simulation of three-component flows with small or moderate density ratios. Development of a versatile ternary LB model that can handle high-density ratio three-phase/component flows is still very attractive in LB community, since the density ratio practically reaches to a few hundreds or even 1000 for a realistic gas-liquid-liquid system.

Wei et al. [50] used a Shan-Chen (pseudopotential) ternary model to study meniscus induced motion of bubbles/droplets in a fluid-fluid interface. By adjusting interaction force parameters in the force balance model, and developing an effective density technique, they were able to reach a high density ratio of 1000 between the gas and liquid components for the studied problem. Based on the Cahn-Hilliard equation and Lee and Liu's [32] two-phase LBM, Haghani et al. [51] proposed a LB model for large density ratio ternary flows, where the mixed difference scheme is used to improve the model's instability, but it is not clear that whether the model can recover the correct interface capturing equations. Combining the free energy and entropy concepts, Wöhrwag et al. [52] developed an entropic Lattice Boltzmann model for high density ratio ternary flows. This model is capable of simulating total spreading states and has thermodynamic consistency, but since it uses a liquid-gas equation of state, it is only suitable for cases with one gas phase and several liquids. Also, the liquid components must have equal densities in this model and the implementation of the collision process is very complex. Recently, based on the Allen-Cahn phase-field method, Haghani et al. [53] developed another ternary LB model for high density ratios. The main contribution of this work is the use of three-component Allen-Cahn equation for capturing the interfaces, which can be less dissipative than the Cahn-Hilliard equation (see Refs. [54], [55]). In this type of formulation, the interface normal and curvatures are expressed in terms of equilibrium phase-field profile across the interface.

In this paper, an alternative LB model for large-density-ratio three-component flows from the perspective of the CH phase-field theory is proposed. This model is based on combining the advantages of the Lee and Lin's model [31] with the Liang et al.’s [33], [48] multi-phase LB framework, which is relatively easier to implement, and the developers showed that it recovers the correct NS and continuity equations via CE analysis [48]. This is where consistency of the proposed LBMs with NS/CH equations has not been reported in any of the other above mentioned ternary models. The primary contribution of the present work is to address the two main limitations of the original model of Liang et al. [48], namely the density ratio and inability of coping with total spreading states. In order to improve the stability of the model in higher density ratios, the mixed difference scheme of Lee and Lin [31] is employed in the calculation of the derivatives in the force terms. While, free energy functional developed by Boyer et al. [44] is used in the CH interface tracking equations to allow the model to handle the total spreading/wetting situations. The other contribution of this paper is presenting LBM simulations of collision and coalescence between rising gas bubble and oil droplet in the water, under a constant gravity field (see Section 4.2.). This phenomena is of great importance in practical applications and is considered to be the most determining step in the separation of fine oil drops from wastewaters through gas flotation technology [2]. In the next section, theory and formulation of the model is described, and then it is evaluated through a series of benchmark problems and numerical tests in Sections 3 and 4.