Reduction-consistent Cahn–Hilliard theory based lattice Boltzmann equation method for N immiscible incompressible fluids

Highlights

• Reduction-consistent lattice Boltzmann equation method.

• Droplets dynamics.

• $N$ ($N\ge 2$) immiscible incompressible fluids.

Abstract

When some fluid components are absent from N (N $\ge$ 2) immiscible fluids, the reduction-consistent property should be guaranteed. In phase-field theory, the evolution of fluid–fluid interface in N immiscible fluids can be captured by a reduction-consistent Cahn–Hilliard equation (CHE), which has a variable dependent mobility. However, it is difficult for lattice Boltzmann equation (LBE) method to solve this kind of CHE with variable mobility. To eliminate this issue, in this paper, a reduction-consistent LBE is proposed for N immiscible fluids. In the model, the reduction-consistent formulation of fluid–fluid interface force is reformulated into a chemical potential form, which can be implemented by a force term in LBE, while a source term treatment is used to achieve the reduction-consistent property for CHE. Numerical simulations of spreading of a liquid lens, spinodal decomposition, and dynamic interaction of drops are carried out to validate present LBE, and the results show the accuracy and capability of present phase-field based LBE for $N$ ($N\ge 2$) immiscible fluids.

Introduction

Transport phenomena of interface dynamics with complex topological change in multiphase flow system are of great interest in nature and engineering. Due to the complex topological evolution of fluid–fluid interface, the efficient and accurate interface tracking technique for multiphase flow modeling is still a challenging task. In the literature, the Cahn–Hilliard equation (CHE) was widely applied to capture the interface evolution of phase separation, and the CHE-based numerical models gained great success in two-phase system [1]. In the CHE theory, the driven force was proportional to the gradient of chemical potential, which could be derived by the system free energy, and the theory showed that fluid–fluid interfaces were modeled by a narrow finite transition layer, which was related to the gradient parameter in the system free energy. When CHE was coupled with fluid dynamics, the fluid properties such as the density and viscosity varied continuously through the transition layer.

In practice, the multiphase fluid system with more than two immiscible fluids is frequently observed such as the oil–water–air or oil–water–oil ternary fluids system, and the interface topological evolution is more complex than that in two-phase system. When one wants to investigate the mechanism of more than two immiscible fluids numerically, one should design a suitable interface capturing technique. Recently, some attempts are devoted to this direction within the framework of CHE theory together with reduction-consistent issue for more than two immiscible fluids [2], [3], [4]. In Refs. [2], [3], the free energy density function and the general CHE were investigated for $N$ ($N\ge 2$) immiscible fluids, and the reduction property of free energy was discussed in detail. In Ref. [4], the hydrodynamic effect and reduction-consistent properties of free energy density function and governing equations were investigated, and numerical results showed the predictions were consistent with the theoretical analysis. On the other hand, the lattice Boltzmann equation (LBE) method as one of the diffuse interface methods [5] has been successfully applied to the multiphase system [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. Recently, the CHE-based LBEs are further developed to ternary or more immiscible fluids system [23], [24], [25], [26], [27], [28], [29], [30]. Nevertheless, most of them focused on ternary fluids system, due to the unusable phase specific decomposition [2], [31], [32], the extension of the phase-field theory [2] based LBE to more than ternary immiscible fluids seems to be unfeasible, especially, the reduction-consistent issue is not clearly addressed in aforementioned CHE-based LBEs.

In this paper, we aim to design a reduction-consistent CHE-based LBE for N (N $\ge$ 2) immiscible incompressible fluids. In the model, the incompressible Navier–Stokes equations (NSE) is solved by LBE, while a simple LBE is proposed to solve the reduction-consistent CHE with a source term for fluid–fluid interface capturing in N (N $\ge$ 2) immiscible fluids, which can achieve the reduction-consistent property. The outline of this paper is given as follows: In Section 2, the phase-field theory of N (N $\ge$ 2) immiscible fluids are briefly overviewed, and the CHE-based LBE solvers are presented in Section 3; a series of benchmark simulations are conducted to validate present LBE models in Section 4, and a brief summary is included in Section 5.