Improved three-dimensional color-gradient lattice Boltzmann model for immiscible two-phase flows

Reaction-limited evaporation for the color-gradient lattice Boltzmann model

We propose a reaction-limited evaporation model within the color-gradient lattice Boltzmann (LB) multicomponent framework to address the lack of intrinsic evaporation mechanisms. Unlike diffusion-driven approaches, our method directly enforces mass removal at the fluid interface in a reaction-limited manner while maintaining numerical stability. Using the inherent color-gradient magnitude and a single adjustable parameter, evaporation sites are chosen in a computationally efficient way with seamless mass exchange between the components, with no change to the core algorithm. Extensive validation across diverse interface geometries and evaporation flux magnitudes demonstrates high accuracy, with errors below 5% for unit density ratios. For density contrasts, the method remains robust in the limit of smaller evaporation flux magnitudes and density ratios. Our approach extends the applicability of the color-gradient LB model to scenarios involving reaction-limited evaporation, such as droplet evaporation on heated substrates, vacuum evaporation of molten metals, and drying processes in porous media.

Color-gradient-based phase-field equation for multiphase flow

In this paper, the underlying problem with the color-gradient (CG) method in handling density-contrast fluids is explored. It is shown that the CG method is not fluid invariant. Based on nondimensionalizing the CG method, a phase-field interface-capturing model is proposed which tackles the difficulty of handling density-contrast fluids. The proposed formulation is developed for incompressible, immiscible two-fluid flows without phase-change phenomena, and a solver based on the lattice Boltzmann method is proposed. Coupled with an available robust hydrodynamic solver, a binary fluid flow package that handles fluid flows with high density and viscosity contrasts is presented. The macroscopic and lattice Boltzmann equivalents of the formulation, which make the physical interpretation of it easier, are presented. In contrast to existing color-gradient models where the interface-capturing equations are coupled with the hydrodynamic ones and include the surface tension forces, the proposed formulation is in the same spirit as the other phase-field models such as the Cahn-Hilliard and the Allen-Cahn equations and is solely employed to capture the interface advected due to a flow velocity. As such, similarly to other phase-field models, a so-called mobility parameter comes into play. In contrast, the mobility is not related to the density field but a constant coefficient. This leads to a formulation that avoids individual speed of sound for the different fluids. On the lattice Boltzmann solver side, two separate distribution functions are adopted to solve the formulation, and another one is employed to solve the Navier-Stokes equations, yielding a total of three equations. Two series of numerical tests are conducted to validate the accuracy and stability of the model, where we compare simulated results with available analytical and numerical solutions, and good agreement is observed. In the first set the interfacial evolution equations are assessed, while in the second set the hydrodynamic effects are taken into account.

Generalized equilibria for color-gradient lattice Boltzmann model based on higher-order Hermite polynomials: A simplified implementation with central moments

We propose generalized equilibria of a three-dimensional color-gradient lattice Boltzmann model for two-component two-phase flows using higher-order Hermite polynomials. Although the resulting equilibrium distribution function, which includes a sixth-order term on the velocity, is computationally cumbersome, its equilibrium central moments (CMs) are velocity-independent and have a simplified form. Numerical experiments show that our approach, as in Wen et al. [Phys. Rev. E 100, 023301 (2019)2470-004510.1103/PhysRevE.100.023301] who consider terms up to third order, improves the Galilean invariance compared to that of the conventional approach. Dynamic problems can be solved with high accuracy at a density ratio of 10; however, the accuracy is still limited to a density ratio of 1000. For lower density ratios, the generalized equilibria benefit from the CM-based multiple-relaxation-time model, especially at very high Reynolds numbers, significantly improving the numerical stability.

Lightweight lattice Boltzmann

A regularized version of the lattice Boltzmann method for efficient simulation of soft materials is introduced. Unlike standard approaches, this method reconstructs the distribution functions from available hydrodynamic variables (density, momentum, and pressure tensor) without storing the full set of discrete populations. This scheme shows significantly lower memory requirements and data access costs. A series of benchmark tests of relevance to soft matter, such as collisions of fluid droplets, is discussed to validate the method. The results can be of particular interest for high-performance simulations of soft matter systems on future exascale computers.

Color-gradient lattice Boltzmann model for immiscible fluids with density contrast

We present a color-gradient-based lattice Boltzmann model for immiscible fluids with a large density contrast. The model employs the velocity-based equilibrium distribution function, initially proposed for the phase-field-based model by Zu and He [Phys. Rev. E 87, 043301 (2013)1539-375510.1103/PhysRevE.87.043301], with a modification necessary to satisfy the kinematic condition at the interface. Different from the existing color-gradient models, the present model allows to specify interface mobility that is independent of the fluid density ratio. Further, we provide a unified framework, which uses the recursive representation of the lattice Boltzmann equation, to derive the governing equations of the system. The emergent color dynamics thus obtained, through an analysis of the segregation operator, is shown to obey the locally conservative Allen-Cahn equation. We use a series of benchmarks, which include a stationary drop, a layered Poiseuille flow, translation of a drop under a forced velocity field, the Rayleigh-Taylor instability, and the capillary intrusion test to demonstrate the model's ability in dealing with complex flow problems.

Rayleigh-Plateau Instability of a Particle-Laden Liquid Column: A Lattice Boltzmann Study

Colloidal particles known to be capable of stabilizing fluid-fluid interfaces have been widely applied in emulsion preparation, but their precise role and underlying influencing mechanism remain poorly understood. In this study, a perturbed liquid column with particles evenly distributed on its surface is investigated using a three-dimensional lattice Boltzmann method, which is built upon the color-gradient two-phase flow model but with a new capillary force model and a momentum exchange method for particle dynamics. The developed method is first validated by simulating the wetting behavior of a particle on a fluid interface and the classic Rayleigh-Plateau instability and is then used to explore the effects of particle concentration and contact angle on the capillary instability of the particle-laden liquid column. It is found that increasing the particle concentration can enhance the stability of the liquid column and thus delay the breakup, and the liquid column is most stable under slightly hydrophobic conditions, which corresponds to the lowest initial liquid-gas interfacial free energy. Due to different pressure gradients inside and outside the liquid column and the capillary force being directed away from the neck, hydrophobic particles tend to assemble in a less compact manner near the neck of the deformed liquid column, while hydrophilic particles prefer to gather far away from the neck. For hydrophobic particles, in addition to the influence of the initial liquid-gas interfacial free energy, the self-assembly of particles in a direction opposite to the liquid flow also contributes to opposing the rupture of the liquid column.

Achieving thermodynamic consistency in a class of free-energy multiphase lattice Boltzmann models

The free-energy lattice Boltzmann (LB) model is one of the major multiphase models in the LB community. The present study is focused on a class of free-energy LB models in which the divergence of thermodynamic pressure tensor or its equivalent form expressed by the chemical potential is incorporated into the LB equation via a forcing term. Although this class of free-energy LB models may be thermodynamically consistent at the continuum level, it suffers from thermodynamic inconsistency at the discrete lattice level owing to numerical errors [Guo et al., Phys. Rev. E 83, 036707 (2010)10.1103/PhysRevE.83.036707]. The numerical error term mainly includes two parts: one comes from the discrete gradient operator and the other can be identified in a high-order Chapman-Enskog analysis. In this paper, we propose an improved scheme to eliminate the thermodynamic inconsistency of the aforementioned class of free-energy LB models. The improved scheme is constructed by modifying the equation of state of the standard LB equation, through which the discretization of ∇(ρc_{s}^{2}) is no longer involved in the force calculation and then the numerical errors can be significantly reduced. Numerical simulations are subsequently performed to validate the proposed scheme. The numerical results show that the improved scheme is capable of eliminating the thermodynamic inconsistency and can significantly reduce the spurious currents in comparison with the standard forcing-based free-energy LB model.

Three-dimensional pseudopotential lattice Boltzmann model for multiphase flows at high density ratio

In this study, we extend the pseudopotential lattice Boltzmann model proposed by Huang and Wu [J. Comput. Phys. 327, 121 (2016)10.1016/j.jcp.2016.09.030] to a three-dimensional model for practical simulations of multiphase flows with high density ratio. In this model, an additional source term is introduced into the evolution function, and the performed high-order Chapman-Enskog analysis demonstrates that the Navier-Stokes equations with accurate pressure tensor are recovered. Also, an alternative geometric formulation is developed to obtain various contact angles and an iteration scheme is involved in the initialization to improve the stability of the model. Theoretical and numerical investigations both validate that the thermodynamic consistency and tuning surface tension independently of density ratio is achieved through varying the two free parameters in the source term. Numerical simulations of droplet wetting indicate that a large degree range of contact angles can be precisely realized with the implementation of the wetting boundary scheme. Further dynamic examinations of droplet impingement on a thin film and a dry surface also verify the stability and capability of the proposed pseudopotential lattice Boltzmann model.

Chromodynamic multirelaxation-time lattice Boltzmann scheme for fluids with density difference

We develop, after Dellar [Phys. Rev. E. 65, 036309 (2002)10.1103/PhysRevE.65.036309; J. Comput. Phys. 190, 351 (2003)10.1016/S0021-9991(03)00279-1], a multiple-relaxation-time (MRT), chromodynamic, multicomponent lattice Boltzmann equation (MCLBE) scheme for simulation of isothermal, immiscible fluid flow with a density contrast. It is based on Lishchuk's method [Brackbill, Kothe, and Zemach, J. Comp. Phys. 100, 335 (1992)10.1016/0021-9991(92)90240-Y; Lishchuk, Care, and Halliday, Phys. Rev. E. 67, 036701, (2003)10.1103/PhysRevE.76.036701] and the segregation of d'Ortona et al. [Phys. Rev. E. 51, 3718, (1995)10.1103/PhysRevE.51.3718]. We focus on fundamental model verifiability but do relate some of our data to that from previous approaches, due to Ba et al. [Phys. Rev. E 94, 023310 (2016)10.1103/PhysRevE.94.023310] and earlier Liu et al. [Phys. Rev. E 85, 046309 (2012)10.1103/PhysRevE.85.046309], who pioneered large density difference chromodynamic MCLBE and showed the practical benefits of an MRT collision model. Specifically, we test the extent to which chromodynamic MCLBE MRT schemes comply with the kinematic condition of mutual impenetrability and the continuous traction condition by developing analytical benchmarking flows. We conclude that our data, taken with those of Ba et al., verify the utility of MRT chromodynamic MCLBE.

Particle dynamics at fluid interfaces studied by the color gradient lattice Boltzmann method coupled with the smoothed profile method

We suggest a numerical method to describe particle dynamics at the fluid interface. We adopt a coupling strategy by combining the color gradient lattice Boltzmann method (CGLBM) and smoothed profile method (SPM). The proposed scheme correctly resolves the momentum transfer among the solid particles and fluid phases while effectively controlling the wetting condition. To validate the present algorithm (CGLBM-SPM), we perform several simulation tests like wetting a single solid particle and capillary interactions in two solid particles floating at the fluid interface. Simulation results show a good agreement with the analytical solutions available and look qualitatively reasonable. From these analyses, we conclude that the key features of the particle dynamics at the fluid interface are correctly resolved in our simulation method. In addition, we apply the present method for spinodal decomposition of a ternary mixture, which contains two-immiscible fluids with solid particles. By adding solid particles, fluid segregation is much suppressed than in the binary liquid mixture case. Furthermore, it has different morphology, such as with the jamming structure of the particles at the fluid interface, and captured images are similar to bicontinuous interfacially jammed emulsion gels in literature. From these results, we confirm the feasibility of the present method to describe soft matters; in particular, emulsion systems that contain solid particles at the interface.

Kinematics of chromodynamic multicomponent lattice Boltzmann simulation with a large density contrast

The utility of an enhanced chromodynamic, color gradient or phase-field multicomponent lattice Boltzmann (MCLB) equation for immiscible fluids with a density difference was demonstrated by Wen et al. [Phys. Rev. E 100, 023301 (2019)2470-004510.1103/PhysRevE.100.023301] and Ba et al. [Phys. Rev. E 94, 023310 (2016)2470-004510.1103/PhysRevE.94.023310], who advanced earlier work by Liu et al. [Phys. Rev. E 85, 046309 (2012)PLEEE81539-375510.1103/PhysRevE.85.046309] by removing certain error terms in the momentum equations. But while these models' collision scheme has been carefully enhanced by degrees, there is, currently, no quantitative consideration in the macroscopic dynamics of the segregation scheme which is common to all. Here, by analysis of the kinetic-scale segregation rule (previously neglected when considering the continuum behavior) we derive, bound, and test the emergent kinematics of the continuum fluids' interface for this class of MCLB, concurrently demonstrating the circular relationship with-and competition between-the models' dynamics and kinematics. The analytical and numerical results we present in Sec. V confirm that, at the kinetic scale, for a range of density contrast, color is a material invariant. That is, within numerical error, the emergent interface structure is isotropic (i.e., without orientation dependence) and Galilean-invariant (i.e., without dependence on direction of motion). Numerical data further suggest that reported restrictions on the achievable density contrast in rapid flow, using chromodynamic MCLB, originate in the effect on the model's kinematics of the terms deriving from our term F_{1i} in the evolution equation, which correct its dynamics for large density differences. Taken with Ba's applications and validations, this result significantly enhances the theoretical foundation of this MCLB variant, bringing it somewhat belatedly further into line with the schemes of Inamuro et al. [J. Comput. Phys. 198, 628 (2004)JCTPAH0021-999110.1016/j.jcp.2004.01.019] and the free-energy scheme [see, e.g., Phys. Rev. E. 76, 045702(R) (2007)10.1103/PhysRevE.76.045702, and references therein] which, in contradistinction to the present scheme and perhaps wisely, postulate appropriate kinematics a priori.