Lattice Boltzmann simulations of liquid-gas and binary fluid systems

Hybrid lattice-Boltzmann-finite-difference approach for the simulation of micro-phase-change-material slurry in convective flow

In this paper, we present a hybrid numerical scheme that couples the lattice Boltzmann method (LBM) with the finite difference method (FDM) to model micro-phase-change-material (MPCM) suspensions in a minichannel. Within this framework, the LBM was employed to solve the continuity, momentum, and energy equations for the fluid domain, while a Lagrangian scheme replicates the motion of MPCM particles. The LBM is coupled with an FDM solver which operates under the lumped capacitance assumption to address the phase-change phenomena within the microparticles. This hybrid coupling eliminates the necessity for any specific treatment in handling phase transitions and tracking phase interfaces. The proposed method is first evaluated on classic particle cases, demonstrating its ability to achieve four-way coupling. Furthermore, the current model effectively adapted viscosity changes when integrating the microparticles, obviating the need for homogenous viscosity models. Subsequently, the potential of this approach is demonstrated by examining the influence of the near-wall thermal interaction of MPCM particles considering three scenarios based on particle density: light (ρ_{p}<ρ_{f}), neutrally buoyant (ρ_{p}≈ρ_{f}), and dense (ρ_{p}>ρ_{f}) microparticles. The hybrid approach further revealed insights into the impact of the volume fraction on the heat transfer coefficient as well as on the overall heat transfer coefficient and performance index from a Lagrangian perspective.

Phase-field lattice Boltzmann model with singular mobility for quasi-incompressible two-phase flows

In this paper, a lattice Boltzmann for quasi-incompressible two-phase flows is proposed based on the Cahn-Hilliard phase-field theory, which can be viewed as an improved model of a previous one [Yang and Guo, Phys. Rev. E 93, 043303 (2016)2470-004510.1103/PhysRevE.93.043303]. The model is composed of two LBE's, one for the Cahn-Hilliard equation (CHE) with a singular mobility, and the other for the quasi-incompressible Navier-Stokes equations (qINSE). Particularly, the LBE for the CHE uses an equilibrium distribution function containing a free parameter associated with the gradient of chemical potential, such that the variable (and even zero) mobility can be handled. In addition, the LBE for the qINSE uses an equilibrium distribution function containing another free parameter associated with the local shear rate, such that the large viscosity ratio problems can be handled. Several tests are first carried out to test the capability of the proposed LBE for the CHE in capturing phase interface, and the results demonstrate that the proposed model outperforms the original LBE model in terms of accuracy and stability. Furthermore, by coupling the hydrodynamic equations, the tests of double-stationary droplets and droplets falling problems indicate that the proposed model can reduce numerical dissipation and produce physically acceptable results at large time scales. The results of droplets falling and phase separation of binary fluid problems show that the present model can handle two-phase flows with large viscosity ratio up to the magnitude of 10^{4}.

A ternary mixture model with dynamic boundary conditions

The influence of short-range interactions between a multi-phase, multi-component mixture and a solid wall in confined geometries is crucial in life sciences and engineering. In this work, we extend the Cahn-Hilliard model with dynamic boundary conditions from a binary to a ternary mixture, employing the Onsager principle, which accounts for the cross-coupling between forces and fluxes in both the bulk and surface. Moreover, we have developed a linear, second-order and unconditionally energy-stable numerical scheme for solving the governing equations by utilizing the invariant energy quadratization method. This efficient solver allows us to explore the impacts of wall-mixture interactions and dynamic boundary conditions on phenomena like spontaneous phase separation, coarsening processes and the wettability of droplets on surfaces. We observe that wall-mixture interactions influence not only surface phenomena, such as droplet contact angles, but also patterns deep within the bulk. Additionally, the relaxation rates control the droplet spreading on surfaces. Furthermore, the cross-coupling relaxation rates in the bulk significantly affect coarsening patterns. Our work establishes a comprehensive framework for studying multi-component mixtures in confined geometries.

Growth regimes in three-dimensional phase separation of liquid-vapor systems

The liquid-vapor phase separation is investigated via lattice Boltzmann simulations in three dimensions. After expressing length and time scales in reduced physical units, we combined data from several large simulations (on 512^{3} nodes) with different values of viscosity, surface tension, and temperature, to obtain a single curve of rescaled length l[over ̂] as a function of rescaled time t[over ̂]. We find evidence of the existence of kinetic and inertial regimes with growth exponents α_{d}=1/2 and α_{i}=2/3 over several time decades, with a crossover from α_{d} to α_{i} at t[over ̂]≃1. This allows us to rule out the existence of a viscous regime with α_{v}=1 in three-dimensional liquid-vapor isothermal phase separation, differently from what happens in binary fluid mixtures. An in-depth analysis of the kinetics of the phase separation process, as well as a characterization of the morphology and the flow properties, are further presented in order to provide clues into the dynamics of the phase-separation process.

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.

Unrestricted component count in multiphase lattice Boltzmann: A fugacity-based approach

Studies of multiphase fluids utilizing the lattice Boltzmann method (LBM) are typically severely restricted by the number of components or chemical species being modeled. This restriction is particularly pronounced for multiphase systems exhibiting partial miscibility and significant interfacial mass exchange, which is a common occurrence in realistic multiphase systems. Modeling such systems becomes increasingly complex as the number of chemical species increases due to the increased role of molecular interactions and the types of thermodynamic behavior that become possible. The recently introduced fugacity-based LBM [Soomro et al., Phys. Rev. E 107, 015304 (2023)2470-004510.1103/PhysRevE.107.015304] has provided a thermodynamically consistent modeling platform for multicomponent, partially miscible LBM simulations. However, until now, this fugacity-based LB model had lacked a comprehensive demonstration of its ability to accurately reproduce thermodynamic behavior beyond binary mixtures and to remove any restrictions in a number of components for multiphase LBM. In this paper we closely explore these fugacity-based LBM capabilities by showcasing comprehensive, thermodynamically consistent simulations of multiphase mixtures of up to ten chemical components. The paper begins by validating the model against the Young-Laplace equation for a droplet composed of three components. The model is then applied to study mixtures with a range of component numbers from one to six, showing agreement with rigorous thermodynamic predictions and demonstrating linear scaling of computational time with the number of components. We further investigate ternary systems in detail by exploring a wide range of temperature, pressure, and overall composition conditions to produce various characteristic ternary diagrams. In addition, the model is shown to be unrestricted in the number of phases as demonstrated through simulations of a three-component three-phase equilibrium case. The paper concludes by demonstrating simulations of a ten-component, realistic hydrocarbon mixture, achieving excellent agreement with thermodynamics for both flat interface vapor-liquid equilibrium and curved interface spinodal decomposition cases. This study represents a significant expansion of the scope and capabilities of multiphase LBM simulations that encompass multiphase systems of keen interest in engineering.

Accurate and Wide-Voltage-Range Modeling of Electrowetting with a Lattice Boltzmann Approach

The lattice Boltzmann method (LBM) has been widely used in multi-phase fluid mechanics and is known to be more computationally efficient than the traditional method of numerically solving Navier-Stokes and Cahn-Hilliard equations. Electrowetting is an important component of interfacial sciences, in which the liquid-liquid and solid-liquid interfaces are tuned by electrostatics. Modeling electrowetting using the LBM can be categorized into surface and bulk methods. By modifying the surface tension scalar, the surface method easily reproduces the fundamental Young-Lippmann (YL) equation at low voltages but fails to capture contact angle saturation at high voltages. With fully coupled hydrodynamics and electrostatics in the form of spatially dependent matrices, the bulk method can successfully show contact angle saturation, but it is often unable to reproduce the YL equation due to its intrinsic inaccuracies. The inaccuracies are mainly due to the fact that while the hydrodynamics are all described by continuous physical quantities in the framework of diffusive interfaces, the interfacial electrostatics are governed by discontinuous electric fields caused by sheet charge density. In this paper, we show that accurately modeling electrowetting using the LBM is non-trivial. Additional modeling work, especially the treatment of interfacial electric fields, is needed to recover the fundamental YL equation at low voltages and predict contact angle saturation at high voltages, with a systematic model validation over key parameters and applications.

The relationship between structure and rheology in a three-dimensional sheared lamellar mesophase

The evolution of a lamellar mesophase from an initially disordered state under shear is examined using simulations of a mesoscale model based on a concentration field ψ that distinguishes the hydrophilic and hydrophobic components. The Landau-Ginzburg free-energy functional is augmented by a term that is minimised for sinusoidal modulations in the concentration field with wavelength λ = (2π/k), and the dynamical equations are the model H equations. The structure and rheology are determined by the relative magnitudes of the diffusion time for coarsening, (λ²/D) and the inverse of the strain rate ^-1, and the Ericksen number, which is the ratio of the shear stress and the layer stiffness. When the diffusion time is small compared with the inverse of the strain rate, there is a local formation of misaligned layers, which are deformed by the imposed flow. There is near-perfect ordering with isolated defects at low values of the Ericksen number, but the defects result in a significant increase in viscosity due to the high layer stiffness. At high values of the Ericksen number, the concentration field is deformed by the mean shear before layers form via diffusion. Cylindrical structures aligned along the flow direction form after about 8-10 strain units, and these evolve into layers with disorder through diffusion perpendicular to the flow. The layers are not perfectly ordered, even after hundreds of strain units, due to the creation and destruction of defects via shear. The excess viscosity is low because the layer stiffness is small compared with the applied shear at a high Ericksen number. This study provides guidance on how the material parameters and imposed flow can be tailored to achieve the desired rheological behaviour.

Coupled Lattice Boltzmann Modeling Framework for Pore-Scale Fluid Flow and Reactive Transport

In this paper, we propose a modeling framework for pore-scale fluid flow and reactive transport based on a coupled lattice Boltzmann model (LBM). We develop a modeling interface to integrate the LBM modeling code parallel lattice Boltzmann solver and the PHREEQC reaction solver using multiple flow and reaction cell mapping schemes. The major advantage of the proposed workflow is the high modeling flexibility obtained by coupling the geochemical model with the LBM fluid flow model. Consequently, the model is capable of executing one or more complex reactions within desired cells while preserving the high data communication efficiency between the two codes. Meanwhile, the developed mapping mechanism enables the flow, diffusion, and reactions in complex pore-scale geometries. We validate the coupled code in a series of benchmark numerical experiments, including 2D single-phase Poiseuille flow and diffusion, 2D reactive transport with calcite dissolution, as well as surface complexation reactions. The simulation results show good agreement with analytical solutions, experimental data, and multiple other simulation codes. In addition, we design an AI-based optimization workflow and implement it on the surface complexation model to enable increased capacity of the coupled modeling framework. Compared to the manual tuning results proposed in the literature, our workflow demonstrates fast and reliable model optimization results without incorporating pre-existing domain knowledge.

Improved partially saturated method for the lattice Boltzmann pseudopotential multicomponent flows

This paper extends the partially saturated method (PSM), used for curved or complex walls, to the lattice Boltzmann (LB) pseudopotential multicomponent model and adapts the wetting boundary condition to model the contact angle. The pseudopotential model is widely used for various complex flow simulations due to its simplicity. To simulate the wetting phenomenon within this model, the mesoscopic interaction force between the boundary fluid and solid nodes is used to mimic the microscopic adhesive force between the fluid and the solid wall, and the bounce-back (BB) method is normally adopted to achieve the no-slip boundary condition. In this paper, the pseudopotential interaction forces are computed with eighth-order isotropy since fourth-order isotropy leads to the condensation of the dissolved component on curved walls. Due to the staircase approximation of curved walls in the BB method, the contact angle is sensitive to the shape of corners on curved walls. Furthermore, the staircase approximation makes the movement of the wetting droplet on curved walls not smooth. To solve this problem, the curved boundary method may be used, but due to the interpolation or extrapolation process, most curved boundary conditions suffer from massive mass leakage when applied to the LB pseudopotential model. Through three test cases, it is found that the improved PSM scheme is mass conservative, that nearly identical static contact angles are observed on flat and curved walls under the same wetting condition, and that the movement of a wetting droplet on curved and inclined walls is smoother compared to the usual BB method. The present method is expected to be a promising tool for modeling flows in porous media and in microfluidic channels.

The crucial role of adhesion in the transmigration of active droplets through interstitial orifices

Active fluid droplets are a class of soft materials exhibiting autonomous motion sustained by an energy supply. Such systems have been shown to capture motility regimes typical of biological cells and are ideal candidates as building-block for the fabrication of soft biomimetic materials of interest in pharmacology, tissue engineering and lab on chip devices. While their behavior is well established in unconstrained environments, much less is known about their dynamics under strong confinement. Here, we numerically study the physics of a droplet of active polar fluid migrating within a microchannel hosting a constriction with adhesive properties, and report evidence of a striking variety of dynamic regimes and morphological features, whose properties crucially depend upon droplet speed and elasticity, degree of confinement within the constriction and adhesiveness to the pore. Our results suggest that non-uniform adhesion forces are instrumental in enabling the crossing through narrow orifices, in contrast to larger gaps where a careful balance between speed and elasticity is sufficient to guarantee the transition. These observations may be useful for improving the design of artificial micro-swimmers, of interest in material science and pharmaceutics, and potentially for cell sorting in microfluidic devices.

Equations of state in multiphase lattice Boltzmann method revisited

The single-component multiphase fluids can be described by a single equation of state (EOS), and various EOSs have been employed in the multiphase lattice Boltzmann (LB) method. In this work, we revisit five commonly used EOSs, including the van der Waals EOS, the Redlich-Kwong EOS, the Redlich-Kwong-Soave EOS, the Peng-Robinson EOS, and the Carnahan-Starling EOS. The recent multiphase LB model with self-tuning EOS is employed because of its thermodynamic consistency in a strict sense and clear physical picture at the microscopic level. First, the way to incorporate these multiphase EOSs is proposed. Two scaling factors are introduced to independently adjust the surface tension and interface thickness, and the lattice sound speed is EOS-dependent to ensure the numerical stability. Then, numerical tests are conducted to validate the incorporations of these EOSs and compare their numerical performances. The surface tension and interface thickness are set to the same values for different EOSs in the comparisons. The liquid and gas densities, surface tension, and interface thickness by the LB simulation agree well with the thermodynamic results. The maximum density ratios achieved with different EOSs are at the same level and could be very close to each other when the interface thickness is relatively small. The effects of multiphase EOS, density ratio, and dimensionless relaxation time on the spurious current are discussed in detail. It is interesting to find the van der Waals EOS shows the best numerical performance in reducing the spurious current.

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.

Free-Energy-Based Discrete Unified Gas Kinetic Scheme for van der Waals Fluid

The multiphase model based on free-energy theory has been experiencing long-term prosperity for its solid foundation and succinct implementation. To identify the main hindrance to developing a free-energy-based discrete unified gas-kinetic scheme (DUGKS), we introduced the classical lattice Boltzmann free-energy model into the DUGKS implemented with different flux reconstruction schemes. It is found that the force imbalance amplified by the reconstruction errors prevents the direct application of the free-energy model to the DUGKS. By coupling the well-balanced free-energy model with the DUGKS, the influences of the amplified force imbalance are entirely removed. Comparative results demonstrated a consistent performance of the well-balanced DUGKS despite the reconstruction schemes utilized. The capability of the DUGKS coupled with the well-balanced free-energy model was quantitatively validated by the coexisting density curves and Laplace's law. In the quiescent droplet test, the magnitude of spurious currents is reduced to a machine accuracy of 10-15. Aside from the excellent performance of the well-balanced DUGKS in predicting steady-state multiphase flows, the spinodal decomposition test and the droplet coalescence test revealed its stability problems in dealing with transient flows. Further improvements are required on this point.

Consistent and conservative phase-field-based lattice Boltzmann method for incompressible two-phase flows

In this work, we consider a general consistent and conservative phase-field model for the incompressible two-phase flows. In this model, not only the Cahn-Hilliard or Allen-Cahn equation can be adopted, but also the mass and the momentum fluxes in the Navier-Stokes equations are reformulated such that the consistency of reduction, consistency of mass and momentum transport, and the consistency of mass conservation are satisfied. We further develop a lattice Boltzmann (LB) method, and show that through the direct Taylor expansion, the present LB method can correctly recover the consistent and conservative phase-field model. Additionally, if the divergence of the extra momentum flux is seen as a force term, the extra force in the present LB method would include another term which has not been considered in the previous LB methods. To quantitatively evaluate the incompressibility and the consistency of the mass conservation, two statistical variables are introduced in the study of the deformation of a square droplet, and the results show that the present LB method is more accurate. The layered Poiseuille flow and a droplet spreading on an ideal wall are further investigated, and the numerical results are in good agreement with the analytical solutions. Finally, the problems of the Rayleigh-Taylor instability, a single rising bubble, and the dam break with the high Reynolds numbers and/or large density ratios are studied, and it is found that the present consistent and conservative LB method is robust for such complex two-phase flows.

Multiphase curved boundary condition in lattice Boltzmann method

The boundary treatment is fundamental for modeling fluid flows especially in the lattice Boltzmann method; the curved boundary conditions effectively improve the accuracy of single-phase simulations with complex-geometry boundaries. However, the conventional curved boundary conditions usually cause dramatic mass leakage or increase when they are directly used for multiphase flow simulations. We find that the principal reason for this is the absence of a nonideal effect in the curved boundary conditions, followed by a calculation error. In this paper, incorporating the nonideal effect into the linear interpolation scheme and compensating for the interpolating error, we propose a multiphase curved boundary condition to treat the wetting boundaries with complex geometries. A series of static and dynamic multiphase simulations with large density ratio verify that the present scheme is accurate and ensures mass conservation.

A lattice Boltzmann model for self-diffusiophoretic particles near and at liquid-liquid interfaces

We introduce a novel mesoscopic computational model based on a multiphase-multicomponent lattice Boltzmann method for the simulation of self-phoretic particles in the presence of liquid-liquid interfaces. Our model features fully resolved solvent hydrodynamics, and, thanks to its versatility, it can handle important aspects of the multiphysics of the problem, including particle wettability and differential solubility of the product in the two liquid phases. The method is extensively validated in simple numerical experiments, whose outcome is theoretically predictable, and then applied to the study of the behavior of active particles next to and trapped at interfaces. We show that their motion can be variously steered by tuning relevant control parameters, such as the phoretic mobilities, the contact angle, and the product solubility.

Lattice Boltzmann Modeling of Spontaneous Imbibition in Variable-Diameter Capillaries

Previous micro-scale studies of the effect of pore structure on spontaneous imbibition are mainly limited to invariable-diameter capillaries. However, in real oil and gas reservoir formations, the capillary diameters are changing and interconnected. Applying the lattice Boltzmann color gradient two-phase flow model and the parallel computation of CPUs, we simulated the spontaneous imbibition in variable-diameter capillaries. We explored the reasons for the nonwetting phase snap-off and systematically studied the critical conditions for the snap-off in spontaneous imbibition. The effects of pore-throat aspect ratio, throat diameter, and the pore-throat tortuosity of the capillary on spontaneous imbibition were studied. Through analyzing the simulated results, we found that the variation in the capillary diameter produces an additional resistance, which increases with the increase in the pore-throat ratio and the pore-throat tortuosity of a capillary. Under the action of this additional resistance, the snap-off phenomenon sometimes occurs in the spontaneous imbibition, which makes the recovery efficiency of the non-wetting phase extremely low. In addition, the main factors affecting this phenomenon are the pore-throat ratio and the pore-throat tortuosity, which is different from the conventional concept of tortuosity. When the snap-off does not occur, the spontaneous imbibition velocity increases when the throat diameter increases and the pore-throat aspect ratio is fixed, and when the period increases, i.e., the diameter changing rate decreases, the spontaneous imbibition velocity also increases. In addition, when the capillary throat diameter is fixed, a bigger pore diameter and a smaller period of sine function both inhibit the speed of spontaneous imbibition.

Comparative investigation of a lattice Boltzmann boundary treatment of multiphase mass transport with heterogeneous chemical reactions

Multiphase reactive transport in porous media is an important component of many natural and engineering processes. In the present study, boundary schemes for the continuum species transport-lattice Boltzmann (CST-LB) mass transport model and the multicomponent pseudopotential model are proposed to simulate heterogeneous chemical reactions in a multiphase system. For the CST-LB model, a lattice-interface-tracking scheme for the heterogeneous chemical reaction boundary is provided. Meanwhile, a local-average virtual density boundary scheme for the multicomponent pseudopotential model is formulated based on the work of Li et al. [Li, Yu, and Luo, Phys. Rev. E 100, 053313 (2019)10.1103/PhysRevE.100.053313]. With these boundary treatments, a numerical implementation is put forward that couples the multiphase fluid flow, interfacial species transport, heterogeneous chemical reactions, and porous matrix structural evolution. A series of comparison benchmark cases are investigated to evaluate the numerical performance for different pseudopotential wetting boundary treatments, and an application case of multiphase dissolution in porous media is conducted to validate the present models' ability to solve complex problems. By applying the present LB models with reasonable boundary treatments, multiphase reactive transport in various natural or engineering scenarios can be simulated accurately.

Lattice Boltzmann model for capillary interactions between particles at a liquid-vapor interface under gravity

A computational technique based on the lattice Boltzmann method (LBM) is developed to simulate the wettable particles adsorbed to a liquid-vapor interface under gravity. The proposed technique combines the improved smoothed-profile LBM for the treatment of moving solid particles in a fluid and the free-energy LBM for the description of a liquid-vapor system. Five benchmark two-dimensional problems are examined: (A) a stationary liquid drop in the vapor phase; a wettable particle adsorbed to a liquid-vapor interface in (B) the absence and (C) the presence of gravity; (D) two freely moving particles at a liquid-vapor interface in the presence of gravity (i.e., capillary flotation forces); and (E) two vertically constrained particles at a liquid-vapor interface (i.e., capillary immersion forces). The simulation results are in good quantitative agreement with theoretical estimations, demonstrating that the proposed technique can reproduce the capillary interactions between wettable particles at a liquid-vapor interface under gravity.

How Heterogeneous Pore Scale Distributions of Wettability Affect Infiltration into Porous Media

Wettability is an important parameter that significantly determines hydrology in porous media, and it especially controls the flow of water across the rhizosphere—the soil-plant interface. However, the influence of spatially heterogeneous distributions on the soil particles surfaces is scarcely known. Therefore, this study investigates the influence of spatially heterogeneous wettability distributions on infiltration into porous media. For this purpose, we utilize a two-phase flow model based on Lattice-Boltzmann to numerically simulate the infiltration in porous media with a simplified geometry and for various selected heterogeneous wettability coatings. Additionally, we simulated the rewetting of the dry rhizosphere of a sandy soil where dry hydrophobic mucilage depositions on the particle surface are represented via a locally increased contact angle. In particular, we can show that hydraulic dynamics and water repellency are determined by the specific location of wettability patterns within the pore space. When present at certain locations, tiny hydrophobic depositions can cause water repellency in an otherwise well-wettable soil. In this case, averaged, effective contact angle parameterizations such as the Cassie equation are unsuitable. At critical conditions, when the rhizosphere limits root water uptake, consideration of the specific microscale locations of exudate depositions may improve models of root water uptake.

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.

Droplet evaporation in finite-size systems: Theoretical analysis and mesoscopic modeling

The classical D^{2}-Law states that the square of the droplet diameter decreases linearly with time during its evaporation process, i.e., D^{2}(t)=D_{0}^{2}-Kt, where D_{0} is the droplet initial diameter and K is the evaporation constant. Though the law has been widely verified by experiments, considerable deviations are observed in many cases. In this work, a revised theoretical analysis of the single droplet evaporation in finite-size open systems is presented for both two-dimensional (2D) and 3D cases. Our analysis shows that the classical D^{2}-Law is only applicable for 3D large systems (L≫D_{0}, L is the system size), while significant deviations occur for small (L≤5D_{0}) and/or 2D systems. Theoretical solution for the temperature field is also derived. Moreover, we discuss in detail the proper numerical implementation of droplet evaporation in finite-size open systems by the mesoscopic lattice Boltzmann method (LBM). Taking into consideration shrinkage effects and an adaptive pressure boundary condition, droplet evaporation in finite-size 2D/3D systems with density ratio up to 328 within a wide parameter range (K=[0.003,0.18] in lattice units) is simulated, and remarkable agreement with the theoretical solution is achieved, in contrast to previous simulations. The present work provides insights into realistic droplet evaporation phenomena and their numerical modeling using diffuse-interface methods.

Coolant Wetting Simulation on Simplified Stator Coil Model by the Phase-Field Lattice Boltzmann Method

Stator coils of automobiles in operation generate heat and are cooled by coolant poured from above. The flow characteristic of the coolant depends on the coil structure, flow condition, solid–fluid interaction, and fluid property, which has not been clarified due to its complexities. Since straight coils are aligned and layered with an angle at the coolant-touchdown region, the coil structure is simplified to a horizontal square rod array referring to an actual coil size. To obtain the flow and wetting characteristics, two-phase fluid flow simulations are conducted by using the phase-field lattice Boltzmann method. First, the flow onto the single-layered rod array is discussed. The wetting area is affected both by the rod gap and the wettability, which is normalized by the gap and the averaged boundary layer thickness. Then, the flow onto the multi-layered rod arrays is investigated with different rod gaps. The top layer wetting becomes longitudinal due to the reduction of the flow advection by the second layer. The wetting area jumps up at the second layer and increases proportionally to the below layers. These become remarkable at the narrow rod gap case, and finally, the dimensionless wetting area is discussed at each layer.

Direct matching between the flow factor approach model and molecular dynamics simulation for nanochannel flows

Mathematically formulating nanochannel flows is challenging. Here, the values of the characteristic parameters were extracted from molecular dynamics simulation (MDS), and directly input to the closed-form explicit flow factor approach model (FFAM) for nanochannel flows. By this way, the physical nature of the simulated system in FFAM is the same with that in MDS. Two nano slit channel heights respectively with two different liquid-channel wall interactions were addressed. The flow velocity profiles across the channel height respectively calculated from MDS and FFAM were compared. By introducing the equivalent value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\Delta_{im} } \mathord{\left/ {\vphantom {{\Delta_{im} } D}} \right. \kern-\nulldelimiterspace} D}$$\end{document}Δim/D, FFAM fairly agrees with MDS for all the cases. The study values FFAM in simulating nanochannel flows.

Shaping the equation of state to improve numerical accuracy and stability of the pseudopotential lattice Boltzmann method

It has recently been shown that altering the shape of the metastable and unstable branches of an equation of state (EOS) can substantially improve the numerical accuracy of liquid and vapor densities in the pseudopotential lattice Boltzmann method [Peng et al., Phys. Rev. E 101, 063309 (2020)2470-004510.1103/PhysRevE.101.063309]. We found that this approach reduces stability of the method in nonequilibrium conditions and is unstable for bubbles at low reduced temperatures. Here we present an improved method for altering the shape of the metastable and unstable branches of the EOS which remains stable for both equilibrium and nonequilibrium situations and has no issues with bubbles. We also performed a detailed study of the stability of the methods for a droplet impact on a liquid film for reduced temperatures down to 0.35 with Reynolds number of 300. Our approach remained stable for a density ratio of up to 3.38×10^{4}.

Lattice Boltzmann and Jones matrix calculations for the determination of the director field structure in self-propelling nematic droplets

Nematic droplets immersed in aqueous surfactant solutions can show a self-propelled motion induced by a Marangoni flow in the droplet surface. In addition to the self-propulsion, the Marangoni flow induces within the droplet a convective flow which considerably influences the nematic director field of the droplet. We report numerical simulations aiming at the determination of the director field in the self-propelling droplet. The convective flow and the resulting structure of director field are described by a lattice Boltzmann model. The reliability of the obtained structures is proved by subsequent Jones matrix calculations which enable the direct comparison of experimental polarizing microscopy images of self-propelling droplets with calculated images based on the simulated structures.

Multiscale Modeling and Simulation of Polymer Blends in Injection Molding: A Review

Modeling and simulation of the morphology evolution of immiscible polymer blends during injection molding is crucial for predicting and tailoring the products’ performance. This paper reviews the state-of-the-art progress in the multiscale modeling and simulation of injection molding of polymer blends. Technological development of the injection molding simulation on a macroscale was surveyed in detail. The aspects of various models for morphology evolution on a mesoscale during injection molding were discussed. The current scale-bridging strategies between macroscopic mold-filling flow and mesoscopic morphology evolution, as well as the pros and cons of the solutions, were analyzed and compared. Finally, a comprehensive summary of the above models is presented, along with the outlook for future research in this field.

A unified lattice Boltzmann model and application to multiphase flows

In this work, we develop a unified lattice Boltzmann model (ULBM) framework that can seamlessly integrate the widely used lattice Boltzmann collision operators, including the Bhatnagar-Gross-Krook or single-relation-time, multiple-relaxation-time, central-moment or cascaded lattice Boltzmann method and multiple entropic operators (KBC). Such a framework clarifies the relations among the existing collision operators and greatly facilitates model comparison and development as well as coding. Importantly, any LB model or treatment constructed for a specific collision operator could be easily adopted by other operators. We demonstrate the flexibility and power of the ULBM framework through three multiphase flow problems: the rheology of an emulsion, splashing of a droplet on a liquid film and dynamics of pool boiling. Further exploration of ULBM for a wide variety of phenomena would be both realistic and beneficial, making the LBM more accessible to non-specialists. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Lattice Boltzmann simulation for phase separation with chemical reaction controlled by thermal diffusion

This study investigates phase separation behavior and pattern formation in a binary fluid with chemical reaction controlled by thermal diffusion. By incorporating the Arrhenius equation into the lattice Boltzmann method (LBM), the coupling effects of the pre-exponential factor K, viscosity [Formula: see text], and thermal diffusion D on phase separation were successfully evaluated. The effect of the competition between thermal diffusion and concentration on the phase separation morphology and dynamics of binary mixtures under a chemically reacting controlled by slow cooling is assessed based on the extended LBM. The calculations indicated that increases in viscosity and thermal diffusion can obtain interconnected structures (ISs) and lamellar structures (LSs) for cases with small K. However, concentric phase-separated structures (CSs) were observed in cases with large K. The increase in the degree and efficiency of phase separation were significantly greater in cases with decreased viscosity and increased thermal diffusion.

Hydrodynamics of immiscible binary fluids with viscosity contrast: a multiparticle collision dynamics approach

We present a multiparticle collision dynamics (MPC) implementation of layered immiscible fluids A and B of different shear viscosities separated by planar interfaces. The simulated flow profile for imposed steady shear motion and the time-dependent shear stress functions are in excellent agreement with our continuum hydrodynamics results for the composite fluid. The wave-vector dependent transverse velocity auto-correlation functions (TVAF) in the bulk-fluid regions of the layers decay exponentially, and agree with those of single-phase isotropic MPC fluids. In addition, we determine the hydrodynamic mobilities of an embedded colloidal sphere moving steadily parallel or transverse to a fluid-fluid interface, as functions of the distance from the interface. The obtained mobilities are in good agreement with hydrodynamic force multipoles calculations, for a no-slip sphere moving under creeping flow conditions near a clean, ideally flat interface. The proposed MPC fluid-layer model can be straightforwardly implemented, and it is computationally very efficient. Yet, owing to the spatial discretization inherent to the MPC method, the model can not reproduce all hydrodynamic features of an ideally flat interface between immiscible fluids.

Implementation of lattice Boltzmann free-surface and shallow water models and their two-way coupling

•A detailed, practical description of a 2D lattice Boltzmann (LB) free-surface model and its coupling with a 1D LB shallow water model is provided.•A Python code is provided, that implements the Gaussian droplet benchmark of the research article (Thorimbert et al., 2019) corresponding to this method article.•Particular attention is given to the details of the free-surface implementation which, in the literature, vary among authors. These ambiguities must be addressed in order to build a reproducible scheme, as well as the exact implementation and parameters of the coupling model proposed in the associated research article.

Alternative wetting boundary condition for the chemical-potential-based free-energy lattice Boltzmann model

The free-energy lattice Boltzmann (LB) method is a multiphase LB approach based on the thermodynamic theory. Compared with traditional free-energy LB models, which employ a nonideal thermodynamic pressure tensor, the chemical-potential-based free-energy LB model has attracted much attention in recent years as it avoids computing the thermodynamic pressure tensor and its divergence. In this paper, we propose an improved wetting boundary condition for the chemical-potential-based free-energy LB model. Different from the original wetting boundary condition in the literature, the improved wetting boundary condition utilizes a surface chemical potential that is compatible with the chemical potential of the fluid domain. Accordingly, the thermodynamic consistency of the chemical-potential-based free-energy LB model can be retained by the improved wetting boundary condition. Numerical simulations are performed for droplets resting on flat and cylindrical surfaces with different contact angles. The numerical results show that the improved wetting boundary condition yields more reasonable results and the maximum spurious velocities are found to be smaller by 2 ∼ 3 orders of magnitude than those produced by the original wetting boundary condition.

Lattice Boltzmann modeling of interfacial mass transfer in a multiphase system

In the present study, a numerical model based on the lattice Boltzmann method (LBM) is proposed to simulate multiphase mass transfer, referred to as the CST-LB model. This model introduced continuum species transfer (CST) formulation by an additional collision term to model the mass transfer across the multiphase interface. The boundary condition treatment of this model is also discussed. In order to verify the applicability, the CST-LB model is combined with the pseudopotential multiphase model to simulate a series of benchmark cases, including concentration jump near the interface, gas dissolution in a closed system, species transport during drainage in a capillary tube, and multiphase species transport in the porous media. This CST-LB model can also be coupled with other multiphase LBMs since the model depends on the phase fraction field, which is not explicitly limited to specified multiphase models.

Structure and isotropy of lattice pressure tensors for multirange potentials

We systematically analyze the tensorial structure of the lattice pressure tensors for a class of multiphase lattice Boltzmann models (LBM) with multirange interactions. Due to lattice discrete effects, we show that the built-in isotropy properties of the lattice interaction forces are not necessarily mirrored in the corresponding lattice pressure tensor. This finding opens a different perspective for constructing forcing schemes, achieving the desired isotropy in the lattice pressure tensors via a suitable choice of multirange potentials. As an immediate application, the obtained LBM forcing schemes are tested via numerical simulations of nonideal equilibrium interfaces and are shown to yield weaker and less spatially extended spurious currents with respect to forcing schemes obtained by forcing isotropy requirements only. From a general perspective, the proposed analysis yields an approach for implementing forcing symmetries, never explored so far in the framework of the Shan-Chen method for LBM. We argue this will be beneficial for future studies of nonideal interfaces.

Lattice Boltzmann Modeling of Drying of Porous Media Considering Contact Angle Hysteresis

Drying of porous media is governed by a combination of evaporation and movement of the liquid phase within the porous structure. Contact angle hysteresis induced by surface roughness is shown to influence multi-phase flows, such as contact line motion of droplet, phase distribution during drainage and coffee ring formed after droplet drying in constant contact radius mode. However, the influence of contact angle hysteresis on liquid drying in porous media is still an unanswered question. Lattice Boltzmann model (LBM) is an advanced numerical approach increasingly used to study phase change problems including drying. In this paper, based on a geometric formulation scheme to prescribe contact angle, we implement a contact angle hysteresis model within the framework of a two-phase pseudopotential LBM. The capability and accuracy of prescribing and automatically measuring contact angles over a large range are tested and validated by simulating droplets sitting on flat and curved surfaces. Afterward, the proposed contact angle hysteresis model is validated by modeling droplet drying on flat and curved surfaces. Then, drying of two connected capillary tubes is studied, considering the influence of different contact angle hysteresis ranges on drying dynamics. Finally, the model is applied to study drying of a dual-porosity porous medium, where phase distribution and drying rate are compared with and without contact angle hysteresis. The proposed model is shown to be capable of dealing with different contact angle hysteresis ranges accurately and of capturing the physical mechanisms during drying in different porous media including flat and curved geometries.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11242-021-01644-9.

Lattice Boltzmann Simulations of Multiphase Dielectric Fluids

The dynamic effect of an electric field on dielectric liquids is called liquid dielectrophoresis. It is widely used in several industrial and scientific applications, including inkjet printing, microfabrication, and optical devices. Numerical simulations of liquid-dielectrophoresis are necessary to understand the fundamental physics of the phenomenon, but also to explore situations that might be difficult or expensive to implement experimentally. However, such modeling is challenging, as one needs to solve the electrostatic and fluid dynamics equations simultaneously. Here, we formulate a new lattice-Boltzmann method capable of modeling the dynamics of immiscible dielectric fluids coupled with electric fields within a single framework, thus eliminating the need of using separate algorithms to solve the electrostatic and fluid dynamics equations. We validate the numerical method by comparing it with analytical solutions and previously reported experimental results. Beyond the benchmarking of the method, we study the spreading of a droplet using a dielectrowetting setup and quantify the mechanism driving the variation of the apparent contact angle of the droplet with the applied voltage. Our method provides a useful tool to study liquid-dielectrophoresis and can be used to model dielectric fluids in general, such as liquid-liquid and liquid-gas systems.

Hysteresis in spreading and retraction of liquid droplets on parallel fiber rails

Wetting and spreading of liquids on fibers occur in many natural and artificial processes. Unlike on a planar substrate, a droplet attached to one or more fibers can assume several different shapes depending on geometrical parameters such as liquid volume and fiber size and distance. This paper presents lattice Boltzmann simulations of the morphology of liquid droplets on two parallel cylindrical fibers. We investigate the final shapes resulting from spreading of an initially spherical droplet deposited on the fibers and from retraction of an initial liquid column deposited between the fibers. We observe three possible equilibrium configurations: barrel-shaped droplet, droplet bridges, and liquid columns. We determine the complete morphology diagram for varying inter-fiber spacings and liquid volumes and find a region of bistability that spans both the column regime and the droplet regime. We further present a simulation protocol that allows one to probe the hysteresis of transitions between different shapes. The results provide insights into energies and forces associated with shape transformations of droplets on fibers that can be used to develop fiber-based materials and microfluidic systems for manipulation of liquids at a small scale.

Consistent evaporation formulation for the phase-field lattice Boltzmann method

A consistent evaporation model is developed for the conservative Allen-Cahn-based phase-field lattice Boltzmann method that uses an appropriate source term to recover the advection-diffusion equation for the specific humidity. To evaluate the accuracy of the proposed scheme, simulations are conducted of a steady-state one-dimensional Stefan flow for a flat interface and a three-dimensional evaporating sessile droplet on a flat substrate for a curved interface. It is confirmed that the results for the evaporative mass flux of the Stefan flow agree well with those obtained from the analytical solution within a specific humidity range of 0.8 or less at the liquid-gas interface. For the sessile droplet case, the results for the dependence of the contact angle on the evaporative mass flux and its profile show good agreement with those obtained from the model of Hu and Larson [J. Phys. Chem. B 106, 1334 (2002)JPCBFK1520-610610.1021/jp0118322].

Optimal surface-tension isotropy in the Rothman-Keller color-gradient lattice Boltzmann method for multiphase flow

The Rothman-Keller color-gradient (CG) lattice Boltzmann method is a popular method to simulate two-phase flow because of its ability to deal with fluids with large viscosity contrasts and a wide range of interfacial tensions. Two fluids are labeled red and blue, and the gradient in the color difference is used to compute the effect of interfacial tension. It is well known that finite-difference errors in the color-gradient calculation lead to anisotropy of interfacial tension and errors such as spurious currents. Here, we investigate the accuracy of the CG calculation for interfaces between fluids with several radii of curvature and find that the standard CG calculations lead to significant inaccuracy. Specifically, we observe significant anisotropy of the color gradient of order 7% for high curvature of an interface such as when a pinchout occurs. We derive a second order accurate color gradient and find that the diagonal nearest neighbors can be weighted differently than in the usual color-gradient calculation such that anisotropy is minimized to a fraction of a percent. The optimal weights that minimize anisotropy for the smallest radius of curvature interface are found to be w=(0.298,0.284,0.275) for diagonal nearest neighbors for the cases of the interface smoothing parameter β=(0.5,0.7,0.99), somewhat higher than the w=0.25 value derived by Leclaire et al. [Leclaire, Reggio, and Trepanier, Computers and Fluids 48, 98 (2011)CPFLBI0045-793010.1016/j.compfluid.2011.04.001] based on obtaining isotropic errors to second order. We find that use of these optimal w values yields over a factor of 10 decrease in anisotropy and over a factor of 30 decrease in mean anisotropy relative to using the standard w=1 value. And we find a factor of about 2 decrease in the anisotropic error and up to factor 15 decrease in mean anisotropic error relative to the choice of w=0.25 for small radius of curvature interfaces. The improved CG calculations will allow the method to be more reliably applied to studies of phenomenology and pore scale processes such as viscous and capillary fingering, and droplet formation where surface-tension isotropy of narrow fingers and small droplets plays a crucial role in correctly capturing phenomenology. We present an example illustrating how different phenomena can be captured using the improved color-gradient method. Namely, we present simulations of a wetting fluid invading a fluid filled pipe where the viscosity ratio of fluids is unity in which droplets form at the transition to fingering using the improved CG calculations that are not captured using the standard CG calculations. We present an explanation of why this is so which relates to anisotropy of the surface tension, which inhibits the pinchouts needed to form droplets.

Modeling ternary fluids in contact with elastic membranes

We present a thermodynamically consistent model of a ternary fluid interacting with elastic membranes. Following a free-energy modeling approach for the fluid phases, we derive the governing equations for the dynamics of the ternary fluid flow and membranes. We also provide the numerical framework for simulating such fluid-structure interaction problems. It is based on the lattice Boltzmann method for the ternary fluid (Eulerian description) and a finite difference representation of the membrane (Lagrangian description). The ternary fluid and membrane solvers are coupled through the immersed boundary method. For validation purposes, we consider the relaxation dynamics of a two-dimensional elastic capsule placed at a fluid-fluid interface. The capsule shapes, resulting from the balance of surface tension and elastic forces, are compared with equilibrium numerical solutions obtained by surface evolver. Furthermore, the Galilean invariance of the proposed model is proven. The proposed approach is versatile, allowing for the simulation of a wide range of geometries. To demonstrate this, we address the problem of a capillary bridge formed between two deformable capsules.

Kinetic Simulations of Compressible Non-Ideal Fluids: From Supercritical Flows to Phase-Change and Exotic Behavior

We investigate a kinetic model for compressible non-ideal fluids. The model imposes the local thermodynamic pressure through a rescaling of the particle’s velocities, which accounts for both long- and short-range effects and hence full thermodynamic consistency. The model is fully Galilean invariant and treats mass, momentum, and energy as local conservation laws. The analysis and derivation of the hydrodynamic limit is followed by the assessment of accuracy and robustness through benchmark simulations ranging from the Joule–Thompson effect to a phase-change and high-speed flows. In particular, we show the direct simulation of the inversion line of a van der Waals gas followed by simulations of phase-change such as the one-dimensional evaporation of a saturated liquid, nucleate, and film boiling and eventually, we investigate the stability of a perturbed strong shock front in two different fluid mediums. In all of the cases, we find excellent agreement with the corresponding theoretical analysis and experimental correlations. We show that our model can operate in the entire phase diagram, including super- as well as sub-critical regimes and inherently captures phase-change phenomena.

Lattice Boltzmann Solver for Multiphase Flows: Application to High Weber and Reynolds Numbers

The lattice Boltzmann method, now widely used for a variety of applications, has also been extended to model multiphase flows through different formulations. While already applied to many different configurations in low Weber and Reynolds number regimes, applications to higher Weber/Reynolds numbers or larger density/viscosity ratios are still the topic of active research. In this study, through a combination of a decoupled phase-field formulation—the conservative Allen–Cahn equation—and a cumulant-based collision operator for a low-Mach pressure-based flow solver, we present an algorithm that can be used for higher Reynolds/Weber numbers. The algorithm was validated through a variety of test cases, starting with the Rayleigh–Taylor instability in both 2D and 3D, followed by the impact of a droplet on a liquid sheet. In all simulations, the solver correctly captured the flow dynamics andmatched reference results very well. As the final test case, the solver was used to model droplet splashing on a thin liquid sheet in 3D with a density ratio of 1000 and kinematic viscosity ratio of 15, matching the water/air system at We = 8000 and Re = 1000. Results showed that the solver correctly captured the fingering instabilities at the crown rim and their subsequent breakup, in agreement with experimental and numerical observations reported in the literature.

Dissolution process of a single bubble under pressure with a large-density-ratio multicomponent multiphase lattice Boltzmann model

A large-density-ratio and tunable-viscosity-ratio multicomponent multiphase pseudopotential lattice Boltzmann model is used to study the dissolution process of a bubble under pressure. The multi-relaxation-time collision operator, exact-difference-method external force scheme, and scaling coefficient k are applied to ensure the numerical stability of the model. The influence of k in the equation of state (EOS) and intermolecule interaction strength on the stationary bubble evolution process are discussed, and the effect of k on thermodynamic consistency is also analyzed. The results indicate that adjusting the scaling coefficient in the EOS changes the surface tension and interface thickness, and that the gas-liquid interface width w is proportional to 1/sqrt[k]. Considering the effect of k on the surface tension, interface thickness, and thermodynamic consistency, the scaling coefficient should be between 0.6 and 1. Furthermore, the dissolution process of a single bubble under pressure is studied using the developed model, and it is found that the dissolution mass and concentration of dissolved gas increase linearly with increases in the pressure difference, and that the concentration of dissolved gas is proportional to the gas pressure after the fluid system reaches equilibrium. These results are consistent with Henry's law.