A magnetic field coupling lattice Boltzmann model and its application on the merging process of multiple-ferrofluid-droplet system

Highlights

• A magnetic field coupling LB model is proposed for magnetic multiphase flows.

• The magnetic effect is physically described by a surface force derived from Lorentz force law.

• The elongation of ferrofluid droplet is well consistent with the experiment.

• The merging process of the multiple-ferrofluid-droplet system is investigated.

Abstract

The present study concerns the dynamics of ferrofluid droplets in a Newtonian fluid. The incompressible Navier-Stokes (NS) equations are applied for the dynamics of the immiscible magnetic and Newtonian fluids while the Cahn-Hilliard (C-H) equation is adopted to describe the behavior of their interface. A magnetic field coupling lattice Boltzmann (LB) model is developed to solve both the NS equations and the C-H equation. Specially, a mass-correcting term is introduced into the C-H equation, which strongly enforces the mass conservation. The magnetic field is evaluated by a Poisson equation solver with a self-correcting procedure for the static Maxwell equations. The magnetic dipole force is transformed into the magnetic surface force by a rigorous mathematical procedure, which can physically describe the magnetic effect on the interface. Moreover, the magnetic force after this treatment becomes easy-to-implement, which can be directly incorporated into the external force term of the LB model. The capability of the present combination method to simulate the magnetic multiphase flows is demonstrated by three typical numerical examples, i.e., Laplace law for a stationary droplet, a stationary cylinder under an external uniform magnetic field, the deformation of a single ferrofluid droplet. Further, the merging process of multiple-ferrofluid-droplet system in organic oil is investigated. It is found that the elongation of ferrofluid droplet in the direction of magnetic field shows positive impact on the merging process when the orientation of ferrofluid droplets is parallel to the external magnetic field, while negative impact on the merging process when their orientation is perpendicular to the external magnetic field.

Introduction

Multiphase flows, especially immiscible functional multiphase flows, have raised concerns not only for its marvelous interfacial behaviors in various industrial applications, but also for its underlying mechanisms [[1], [2]–3]. Since the pioneering work of Taylor [4], the deformation and coalescence of droplets in complex multiphase flows have been extensively investigated. Due to the complicated interfacial interactions between different phases, the accurate prediction of immiscible two-phase flow by the numerical simulation is still a challenging problem. From the computational point of view, the numerical methods can be divided into two broad categories, i.e., interface tracking and interface capturing techniques. On the other hand, in physical standpoint for immiscible two-phase flows, there are two categories of algorithm to describe the interfacial behaviors: sharp-interface method [5] and diffuse-interface method [6]. The diffuse-interface method was initially developed by Lord Rayleigh [7] and Van der Waals [8], who proposed the gradient interfacial theories based on thermodynamic principles. Thus, this method implies a non-zero thickness interface between binary fluids.

Recently, the simulation of multiphase flows based on lattice Boltzmann (LB) method [9,10] and Cahn-Hillard (C-H) equation [11] has received a lot of attention. The complex macroscale interfacial behaviors, including interfacial displacement, coalescence, separation etc., can be physically considered as a nature consequence of the inter-molecular interactions which is represented as a series of forces in macroscope, such as adhesion, cohesion and bulk effects. From this point, the LB method based on the mesoscopic kinetic theory has the well-known merits on the algorithmic simplicity, parallelization and easy implementation of complex boundary, and a unique superiority on simulation of mesoscopic scale which distinguishes from the traditional numerical methods. Due to these advantages, a variety of LB models has been successfully developed into promising scheme for multiphase flows, especially one pioneering work, i.e., the well-known Shan-Chen pseudopotential LB model [12]. Later, He et al. [13] developed an LB model incorporating molecular interactions for the simulation of Rayleigh-Taylor instability, which is also called the free-energy LB model. Further, the LB models for capturing the interfacial behaviors have been developed and modified by many talented scientists, such as Zheng et al. [14], Huang et al. [15], Wang et al. [16], Yang et al. [17], Yuan et al. [18], Matin et al. [19], Karami et al. [20], Niu et al. [21] and so on. The original Zheng's LB model [14] adopted double distribution functions strategy which is widely used. One of the distribution functions is used to capture the interfacial behavior which can recover the C-H equation, the other one for the hydrodynamics of two immiscible fluids which can recover incompressible Navier-Stokes (N-S) equations. However, the poor numerical stability and the mass diffusion problem is unsolved in that LB model, which entails some combined numerical methods to overcome these drawbacks. Yuan et al. [18] combined adaptive mesh refinement method and volume of fluid method into LB model to improve the numerical stability on the simulation of multiphase flows. Matin et al. [19] applied collision and external force term locally for a finite element formulation. Those kinds of combined method have a great progress of the numerical stability and inherit the kinetic nature, but the numerical solver is complex which lose some unique advantages of LB method.

Even though a lot of theoretical and numerical investigations on multiphase LB models exist, it is still a challenging issue to precisely simulate the magnetic field coupling multiphase flows, due to the complex interfacial behaviors and the magnetic interaction between the magnetic and nonmagnetic materials. Starting from the work of Séro-Guillaume et al. [22,23] which theoretically analyzed the potential flow of an inviscid ferrofluid droplet as a Hamiltonian system, the numerical simulation of magnetic multiphase flows by the macroscopic model based on finite-element method [24], finite differences method [25], volume-of-fluid (VOF) method [26] and level-set method [27] have been extensively performed. Afkhami et al. [24] numerically investigated droplet shapes and velocity fields under different magnetic Laplace numbers and droplet diameters by a finite-element method. Habera et al. [25] used a finite difference method and a level-set method to simulate the ferrofluid droplet deformation under the effect of gravity and magnetic fields. Yamasaki and Yamaguchi [26] analyzed the bubble deformation under an external magnetic field by a numerical model consists of a coupled level-set and VOF method. Hassan et al. [27] simulated the ferrofluid droplet deformation in shear flow under a magnetic field, and the effects of magnetic field direction, magnetic Bond number and capillarity number were studied systematically. On the other hand, the efficient mesoscopic models for the simulation of magnetic multiphase flows is still lacking. To simulate the magnetic effect, Niu et al. [28] proposed an LB model consisting of triple LB equations to simulate the temperature-sensitive ferrofluids. The model [28] can be used to simulate the single-phase thermal flow but without the interaction between magnetic and nonmagnetic materials. Ghaderi et al. [29] numerically investigated a falling ferrofluid droplet under uniform magnetic field by the Shan-Chen interparticle potential LB model [12] without interfacial tracking, and the magnetic interaction between magnetic materials and nonmagnetic materials was not involved. Hu et al. [30] extended a complicated multi-relaxation phase-field-based LB model with 27 relaxation parameters to simulate the ferrofluid droplet behaviors. Thus, it is necessary to develop a magnetic field coupling multiphase model for the simulation of magnetic multiphase flow under the external magnetic field, which is ease-to-implement while maintains good accuracy and efficiency.

Each numerical method for the simulation of magnetic field coupling magnetic multiphase flows has its own unique features. The C-H equation, a popular mathematic method to capture the interfacial behaviors, has been widely used in the simulation of multiphase flows, although it suffers from a serious mass diffusion problem. However, the interfacial behaviors in magnetic multiphase flow are more complex and the mass diffusion problem becomes more severe in numerical simulation. Hu et al. [30] reported that the computational mass is continuously diffused with the computational time in their magnetic multiphase flow simulation.

Thus, the present study aims to solve the two challenges mentioned above, i.e., minimizing the mass diffusion and effectively and accurately implementing the magnetic forces in the numerical simulation. Specially, the multiphase LB method proposed by Zheng et al. [14] is extended by introducing a magnetic field coupling LB model with a mass-correcting term for the simulation of magnetic multiphase flows. In detail, the fluid flow is simulated by the incompressible Navier-Stokes equations with the nonlinear Langevin magnetization law. The interfacial behavior is captured by the C-H equation with mass correction term which can strongly enforce the mass conservation. Both the governing equations are solved by using the LB method. As the interfacial behaviors of two-phases flows are naturally captured by the LB model of the modified C-H equation from both sides of the interface, the present numerical model is more rigorous. To involve the interaction between magnetic and nonmagnetic materials, a Poisson equation solver with a self-correcting procedure for the static Maxwell equations is employed to evaluate the magnetic field. Further, the magnetic dipole force is transformed into the magnetic surface force by a rigorous mathematical procedure, which can physically describe the magnetic effect on the interface. After this treatment, the effect of the external magnetic field becomes easy-to-implement, which can be directly incorporated into the external force term of the LB model. Moreover, the physical magnetic boundary conditions can be directly implemented by using the macroscopic variables without transforming to the LB distribution functions.

The remainder of this article is organized as follows. In Section 2, the Poisson equation solver with a self-correcting procedure for solving magnetic field is first represented. The governing equations for two-phase flow recovered by the general multiphase LB equation are then introduced. And, a mass-conserving LB model for magnetic multiphase flows based on the modified Cahn-Hilliard equation is finally presented. In Section 3, several typical physical problems, such as Laplace law for a stationary droplet, a stationary cylinder under an external uniform magnetic field, the deformation of a single ferrofluid droplet, the merging process of multiple-ferrofluid-droplet system in organic oil, are simulated to test the accuracy and numerical stability of the present model. Finally, the conclusions are drawn in Section 4.