Lattice Boltzmann model for simulating flows with multiple phases and components

Multipseudopotential interaction: a solution for thermodynamic inconsistency in pseudopotential lattice Boltzmann models

Pseudopotential lattice Boltzmann (LB) models have been recognized as efficient numerical tools to simulate complex fluid systems, including those at thermodynamic equilibrium states and with phase transitions. However, when the equation of state (EOS) of real fluids is implemented, the existing pseudopotential LB models suffer from thermodynamic inconsistency. This study presents a multipseudopotential interparticle interaction (MPI) scheme, which is fully consistent with thermodynamics and applicable to engineering applications. In this framework, multiple pseudopotentials are employed to represent dominant interaction potentials at different extents of the mean free path of particles. By simulating van der Waals and Carnahan-Starling fluids, it is demonstrated that the MPI scheme can correctly simulate the physical nature of two-phase systems on the lattice including the continuum predictions of liquid-vapor coexistence states and the sound speeds in liquid and vapor phases. It is also shown that the lattice interactions of the MPI scheme represent underlying molecular interactions as they vary in a broad range from strong short-distance repulsions to weak long-distance attractions during phase transitions. Consequently, the MPI is proved to be a reliable LB scheme as it avoids generating unphysical potentials in implementing the EOSs of real fluids and limiting the spurious velocities at the interface of two-phase systems. Additionally, a straightforward procedure is suggested and discussed to preset the MPI system with the two-phase properties of a selected fluid.

Improved forcing scheme in pseudopotential lattice Boltzmann methods for multiphase flow at arbitrarily high density ratios

The pseudopotential lattice Boltzmann method has been widely used to simulate many multiphase flow applications. However, there still exist problems with reproducing realistic values of density ratio and surface tension. In this study, a higher-order analysis of a general forcing term is derived. A forcing scheme is then constructed for the pseudopotential method that is able to accurately reproduce the full range of coexistence curves. As a result, multiphase flow of arbitrarily high density ratios independent of the surface tension can be simulated. Furthermore, the interface width can be tuned to allow for grid refinement and systematic error reduction. Numerical results confirm that the proposed scheme enables independent control of density ratio, surface tension, and interface width simultaneously.

Fluctuating multicomponent lattice Boltzmann model

Current implementations of fluctuating lattice Boltzmann equations (FLBEs) describe single component fluids. In this paper, a model based on the continuum kinetic Boltzmann equation for describing multicomponent fluids is extended to incorporate the effects of thermal fluctuations. The thus obtained fluctuating Boltzmann equation is first linearized to apply the theory of linear fluctuations, and expressions for the noise covariances are determined by invoking the fluctuation-dissipation theorem directly at the kinetic level. Crucial for our analysis is the projection of the Boltzmann equation onto the orthonormal Hermite basis. By integrating in space and time the fluctuating Boltzmann equation with a discrete number of velocities, the FLBE is obtained for both ideal and nonideal multicomponent fluids. Numerical simulations are specialized to the case where mean-field interactions are introduced on the lattice, indicating a proper thermalization of the system.

Lattice Boltzmann simulations of supercritical CO2-water drainage displacement in porous media: CO2 saturation and displacement mechanism

CO2 geosequestration in deep aquifers requires the displacement of water (wetting phase) from the porous media by supercritical CO2 (nonwetting phase). However, the interfacial instabilities, such as viscous and capillary fingerings, develop during the drainage displacement. Moreover, the burstlike Haines jump often occurs under conditions of low capillary number. To study these interfacial instabilities, we performed lattice Boltzmann simulations of CO2-water drainage displacement in a 3D synthetic granular rock model at a fixed viscosity ratio and at various capillary numbers. The capillary numbers are varied by changing injection pressure, which induces changes in flow velocity. It was observed that the viscous fingering was dominant at high injection pressures, whereas the crossover of viscous and capillary fingerings was observed, accompanied by Haines jumps, at low injection pressures. The Haines jumps flowing forward caused a significant drop of CO2 saturation, whereas Haines jumps flowing backward caused an increase of CO2 saturation (per injection depth). We demonstrated that the pore-scale Haines jumps remarkably influenced the flow path and therefore equilibrium CO2 saturation in crossover domain, which is in turn related to the storage efficiency in the field-scale geosequestration. The results can improve our understandings of the storage efficiency by the effects of pore-scale displacement phenomena.

Interface-capturing lattice Boltzmann equation model for two-phase flows

In this work, an interface-capturing lattice Boltzmann equation (LBE) model is proposed for two-phase flows. In the model, a Lax-Wendroff propagation scheme and a properly chosen equilibrium distribution function are employed. The Lax-Wendroff scheme is used to provide an adjustable Courant-Friedrichs-Lewy (CFL) number, and the equilibrium distribution is presented to remove the dependence of the relaxation time on the CFL number. As a result, the interface can be captured accurately by decreasing the CFL number. A theoretical expression is derived for the chemical potential gradient by solving the LBE directly for a two-phase system with a flat interface. The result shows that the gradient of the chemical potential is proportional to the square of the CFL number, which explains why the proposed model is able to capture the interface naturally with a small CFL number, and why large interface error exists in the standard LBE model. Numerical tests, including a one-dimensional flat interface problem, a two-dimensional circular droplet problem, and a three-dimensional spherical droplet problem, demonstrate that the proposed LBE model performs well and can capture a sharp interface with a suitable CFL number.

Dynamic stability characteristics of fluid flow in CO2 miscible displacements in porous media

The dynamic stability characteristics of fluid flow in miscible displacement processes were investigated by using a magnetic resonance imaging apparatus and simulated by a lattice-Boltzmann method at elevated pressures.

Extraction of macroscopic and microscopic adjoint concepts using a lattice Boltzmann method and discrete adjoint approach

In the present research, we tried to improve the performance of the lattice Boltzmann (LB) -based adjoint approach by utilizing the mesoscopic inherent of the LB method. In this regard, two macroscopic discrete adjoint (MADA) and microscopic discrete adjoint (MIDA) approaches are used to answer the following two challenging questions. Is it possible to extend the concept of the macroscopic and microscopic variables of the flow field to the corresponding adjoint ones? Further, similar to the conservative laws in the LB method, is it possible to find the comparable conservation equations in the adjoint approach? If so, then a definite framework, similar to that used in the flow solution by the LB method, can be employed in the flow sensitivity analysis by the MIDA approach. This achievement can decrease the implementation cost and coding efforts of the MIDA method in complicated sensitivity analysis problems. First, the MADA and MIDA equations are extracted based on the LB method using the duality viewpoint. Meanwhile, using an elementary case, inverse design of a two-dimensional unsteady Poiseuille flow in a periodic channel with constant body forces, the procedure of analytical evaluation of the adjoint variables is described. The numerical results show that similar correlations between the distribution functions can be seen between the corresponding adjoint ones. Besides, the results are promising, emphasizing the flow field adjoint variables can be evaluated via the adjoint distribution functions. Finally, the adjoint conservative laws are introduced.

Phase-field-based lattice Boltzmann model for axisymmetric multiphase flows

In this paper, a phase-field-based lattice Boltzmann (LB) model is proposed for axisymmetric multiphase flows. Modified equilibrium distribution functions and some source terms are properly added into the evolution equations such that multiphase flows in the axisymmetric coordinate system can be described. Different from previous axisymmetric LB multiphase models, the added source terms that arise from the axisymmetric effect contain no additional gradients, and therefore the present model is much simpler. Furthermore, through the Chapmann-Enskog analysis, the axisymmetric Chan-Hilliard equation and Navier-Stokes equations can be exactly derived from the present model. The model is also able to deal with flows with density contrast. A variety of numerical experiments, including planar and curve interfaces, an elongation field, a static droplet, a droplet oscillation, breakup of a liquid thread, and dripping of a liquid droplet under gravity, have been conducted to test the proposed model. It is found that the present model can capture accurate interface and the numerical results of multiphase flows also agree well with the analytical solutions and/or available experimental data.

Numerical simulation of condensation on structured surfaces

Condensation of liquid droplets on solid surfaces happens widely in nature and industrial processes. This phase-change phenomenon has great effect on the performance of some microfluidic devices. On the basis of micro- and nanotechnology, superhydrophobic structured surfaces can be well-fabricated. In this work, the nucleating and growth of droplets on different structured surfaces are investigated numerically. The dynamic behavior of droplets during the condensation is simulated by the multiphase lattice Boltzmann method (LBM), which has the ability to incorporate the microscopic interactions, including fluid-fluid interaction and fluid-surface interaction. The results by the LBM show that, besides the chemical properties of surfaces, the topography of structures on solid surfaces influences the condensation process. For superhydrophobic surfaces, the spacing and height of microridges have significant influence on the nucleation sites. This mechanism provides an effective way for prevention of wetting on surfaces in engineering applications. Moreover, it suggests a way to prevent ice formation on surfaces caused by the condensation of subcooled water. For hydrophilic surfaces, however, microstructures may be submerged by the liquid films adhering to the surfaces. In this case, microstructures will fail to control the condensation process. Our research provides an optimized way for designing surfaces for condensation in engineering systems.

Contact angles in the pseudopotential lattice Boltzmann modeling of wetting

In this paper we investigate the implementation of contact angles in the pseudopotential lattice Boltzmann modeling of wetting at a large density ratio ρ_{L}/ρ_{V}=500. The pseudopotential lattice Boltzmann model [X. Shan and H. Chen, Phys. Rev. E 49, 2941 (1994)10.1103/PhysRevE.49.2941] is a popular mesoscopic model for simulating multiphase flows and interfacial dynamics. In this model the contact angle is usually realized by a fluid-solid interaction. Two widely used fluid-solid interactions, the density-based interaction and the pseudopotential-based interaction, as well as a modified pseudopotential-based interaction formulated in the present paper are numerically investigated and compared in terms of the achievable contact angles, the maximum and the minimum densities, and the spurious currents. It is found that the pseudopotential-based interaction works well for simulating small static (liquid) contact angles θ<90^{∘}, however, it is unable to reproduce static contact angles close to 180^{∘}. Meanwhile, it is found that the proposed modified pseudopotential-based interaction performs better in light of the maximum and the minimum densities and is overall more suitable for simulating large contact angles θ>90^{∘} as compared with the two other types of fluid-solid interactions. Furthermore, the spurious currents are found to be enlarged when the fluid-solid interaction force is introduced. Increasing the kinematic viscosity ratio between the vapor and liquid phases is shown to be capable of reducing the spurious currents caused by the fluid-solid interactions.

Consistent lattice Boltzmann equations for phase transitions

Unlike conventional computational fluid dynamics methods, the lattice Boltzmann method (LBM) describes the dynamic behavior of fluids in a mesoscopic scale based on discrete forms of kinetic equations. In this scale, complex macroscopic phenomena like the formation and collapse of interfaces can be naturally described as related to source terms incorporated into the kinetic equations. In this context, a novel athermal lattice Boltzmann scheme for the simulation of phase transition is proposed. The continuous kinetic model obtained from the Liouville equation using the mean-field interaction force approach is shown to be consistent with diffuse interface model using the Helmholtz free energy. Density profiles, interface thickness, and surface tension are analytically derived for a plane liquid-vapor interface. A discrete form of the kinetic equation is then obtained by applying the quadrature method based on prescribed abscissas together with a third-order scheme for the discretization of the streaming or advection term in the Boltzmann equation. Spatial derivatives in the source terms are approximated with high-order schemes. The numerical validation of the method is performed by measuring the speed of sound as well as by retrieving the coexistence curve and the interface density profiles. The appearance of spurious currents near the interface is investigated. The simulations are performed with the equations of state of Van der Waals, Redlich-Kwong, Redlich-Kwong-Soave, Peng-Robinson, and Carnahan-Starling.

Assembling ellipsoidal particles at fluid interfaces using switchable dipolar capillary interactions

How to dynamically tune an assembly of anisotropic colloidal particles adsorbed at fluid-fluid interfaces using dipolar capillary interactions is demonstrated. A previously discovered first-order phase transition is exploited and it is shown how to spontaneously turn off these dipolar capillary interactions by exceeding a critical field strength, providing unprecedented control of the bottom-up fabrication of soft materials.

Deformation and breakup of a liquid droplet past a solid circular cylinder: a lattice Boltzmann study

In this paper, we present a numerical study on the deformation and breakup behavior of liquid droplet past a solid circular cylinder by using an improved interparticle-potential lattice Boltzmann method. The effects of the eccentric ratio β, viscosity ratio λ between the droplet and the surrounding fluid, surface wettability, and Bond number (Bo) on the dynamic behavior of the liquid droplet are considered. The parameter β represents the degree that the solid cylinder deviates from the center line, and Bo is the ratio between the inertial force and capillary force. Numerical results show that there are two typical patterns, i.e., breakup and no breakup, which are greatly influenced by the aforementioned parameters. When β increases to a critical value βc, the droplet can pass the circular cylinder without a breakup, otherwise, the breakup phenomenon occurs. The critical eccentric ratio βc increases significantly with increasing Bo for case with λ>1, while for the case with λ<1, the viscosity effects on the βc is not obvious when Bo is large. For the breakup case, the amount of deposited liquid on the tip of the circular cylinder is almost unaffected by β. In addition, the results also show that the viscosity ratio and wettability affect the deformation and breakup process of the droplet. For case with λ<1, the viscosity ratio plays a minor role in the thickness variations of the deposited liquid, which decreases to a nonzero constant eventually; while for λ>1, the increase of the viscosity ratio significantly accelerates the decrease of the deposited liquid, and finally no fluid deposits on the cylinder. In term of the wettability, there occurs continuous gas phase trapped by the wetting droplet, but this does not happen for nonwetting droplet. Besides, for λ<1, the time required to pass the cylinder (tp) decreases monotonically with decreasing contact angle, while a nonmonotonic decrease appears for λ>1. It is also found that tp decreases monotonically with increasing Bo and is less sensitive to λ at a large Bo.

Domain and droplet sizes in emulsions stabilized by colloidal particles

Particle-stabilized emulsions are commonly used in various industrial applications. These emulsions can present in different forms, such as Pickering emulsions or bijels, which can be distinguished by their different topologies and rheology. We numerically investigate the effect of the volume fraction and the uniform wettability of the stabilizing spherical particles in mixtures of two fluids. For this, we use the well-established three-dimensional lattice Boltzmann method, extended to allow for the added colloidal particles with non-neutral wetting properties. We obtain data on the domain sizes in the emulsions by using both structure functions and the Hoshen-Kopelman (HK) algorithm, and we demonstrate that both methods have their own (dis)advantages. We confirm an inverse dependence between the concentration of particles and the average radius of the stabilized droplets. Furthermore, we demonstrate the effect of particles detaching from interfaces on the emulsion properties and domain-size measurements.

Depinning of drops on inclined smooth and topographic surfaces: experimental and lattice Boltzmann model study

In this study, the dynamics of initially stationary liquid drops on smooth and topographic inclined silicon surfaces was investigated experimentally and by lattice Boltzmann simulations. The transient contact angles and the critical angle of inclination were measured systematically for different liquids, drop sizes, and surfaces having different wettability and surface roughness. In general, the critical angle of inclination is larger for hydrophilic than for hydrophobic surfaces, irrespective of the liquids, and increases with increasing contact angle hysteresis and decreasing drop sizes. A two-phase liquid-vapor lattice Boltzmann model based on the Shan and Chen approach was developed for two dimensions which incorporates the wetting and topographic characteristics of the surface. The simulation results matched the experimentally found features quantitatively and allowed one to explore the roll-off behavior even in cases that can hardly be accessed experimentally.

Interface deformations affect the orientation transition of magnetic ellipsoidal particles adsorbed at fluid-fluid interfaces

Manufacturing new soft materials with specific optical, mechanical and magnetic properties is a significant challenge. Assembling and manipulating colloidal particles at fluid interfaces is a promising way to make such materials. We use lattice-Boltzmann simulations to investigate the response of magnetic ellipsoidal particles adsorbed at liquid-liquid interfaces to external magnetic fields. We provide further evidence for the first-order orientation phase transition predicted by Bresme and Faraudo [Journal of Physics: Condensed Matter, 2007, 19, 375110]. We show that capillary interface deformations around the ellipsoidal particle significantly affect the tilt-angle of the particle for a given dipole-field strength, altering the properties of the orientation transition. We propose scaling laws governing this transition, and suggest how to use these deformations to facilitate particle assembly at fluid-fluid interfaces.

Deformation and breakup of viscoelastic droplets in confined shear flow

The deformation and breakup of Newtonian and viscoelastic droplets are studied in confined shear flow. Our numerical approach is based on a combination of lattice-Boltzmann models and finite difference schemes, the former used to model two immiscible fluids with variable viscosity ratio and the latter used to model the polymer dynamics. The kinetics of the polymers is introduced using constitutive equations for viscoelastic fluids with finitely extensible nonlinear elastic dumbbells with Peterlin's closure. We quantify the droplet response by changing the polymer relaxation time τ(P), the maximum extensibility L of the polymers, and the degree of confinement, i.e., the ratio of the droplet diameter to wall separation. In unconfined shear flow, the effects of droplet viscoelasticity on the critical capillary number Ca(cr) for breakup are moderate in all cases studied. However, in confined conditions a different behavior is observed: The critical capillary number of a viscoelastic droplet increases or decreases, depending on the maximum elongation of the polymers, the latter affecting the extensional viscosity of the polymeric solution. Force balance is monitored in the numerical simulations to validate the physical picture.

Beyond Cassie equation: local structure of heterogeneous surfaces determines the contact angles of microdroplets

The application of Cassie equation to microscopic droplets is recently under intense debate because the microdroplet dimension is often of the same order of magnitude as the characteristic size of substrate heterogeneities, and the mechanism to describe the contact angle of microdroplets is not clear. By representing real surfaces statistically as an ensemble of patterned surfaces with randomly or regularly distributed heterogeneities (patches), lattice Boltzmann simulations here show that the contact angle of microdroplets has a wide distribution, either continuous or discrete, depending on the patch size. The origin of multiple contact angles observed is ascribed to the contact line pinning effect induced by substrate heterogeneities. We demonstrate that the local feature of substrate structure near the contact line determines the range of contact angles that can be stabilized, while the certain contact angle observed is closely related to the contact line width.

Timescales of emulsion formation caused by anisotropic particles

Particle stabilized emulsions have received much interest in the recent past, but our understanding of the dynamics of emulsion formation is still limited. For simple spherical particles, the time dependent growth of fluid domains is dominated by the formation of droplets, particle adsorption and coalescence of droplets (Ostwald ripening), which eventually can be almost fully blocked due to the presence of the particles. Ellipsoidal particles are known to be more efficient stabilizers of fluid interfaces than spherical particles and their anisotropic shape and the related additional rotational degrees of freedom have an impact on the dynamics of emulsion formation. In this paper, we investigate this point by means of simple model systems consisting of a single ellipsoidal particle or a particle ensemble at a flat interface as well as a particle ensemble at a spherical interface. By applying combined multicomponent lattice Boltzmann and molecular dynamics simulations we demonstrate that the anisotropic shape of ellipsoidal particles causes two additional timescales to be of relevance in the dynamics of emulsion formation: a relatively short timescale can be attributed to the adsorption of single particles and the involved rotation of particles towards the interface. As soon as the interface is jammed, however, capillary interactions between the particles cause a local reordering on very long timescales leading to a continuous change in the interface configuration and increase of the interfacial area. This effect can be utilized to counteract the thermodynamic instability of particle stabilized emulsions and thus offers the possibility to produce emulsions with exceptional stability.

Enhanced permeation of a hydrophobic fluid through particles with hydrophobic and hydrophilic patterned surfaces

The wetting properties of solid surfaces are significant in oil/gas and liquid displacement processes. It is difficult for hydrophobic fluids to permeate channels filled with hydrophilic particles and an aqueous phase, and this is thought to be the primary cause of low yields in low permeability reservoir operations. Using three-dimensional lattice Boltzmann simulations, we show that particles with hydrophobic and hydrophilic patterned surfaces can greatly improve hydrophobic fluid permeation. Specifically, a hydrophobic fluid can easily access micro-channels in the hydrophobic regions, which extend rapidly even to the hydrophilic regions and accelerate hydrophobic fluid escape. This work enriches understanding of multiphase flow in porous media at the pore scale and fracture conductivity and is expected to have great significance in the exploitation of low permeability reservoirs and shale gas.

Direct evidence of plastic events and dynamic heterogeneities in soft-glasses

By using fluid-kinetic simulations of confined and concentrated emulsion droplets, we investigate the nature of space non-homogeneity in soft-glassy dynamics and provide quantitative measurements of the statistical features of plastic events in the proximity of the yield-stress threshold. Above the yield stress, our results show the existence of a finite stress correlation scale, which can be mapped directly onto the cooperativity scale, recently introduced in the literature to capture non-local effects in the soft-glassy dynamics. In this regime, the emergence of a separate boundary (wall) rheology with higher fluidity than the bulk is highlighted in terms of near-wall spontaneous segregation of plastic events. Near the yield stress, where the cooperativity scale cannot be estimated with sufficient accuracy, the system shows a clear increase of the stress correlation scale, whereas plastic events exhibit intermittent clustering in time, with no preferential spatial location. A quantitative measurement of the space-time correlation associated with the motion of the interface of the droplets is key to spot the elastic rigidity of the system.

Acoustic equations of state for simple lattice Boltzmann velocity sets

The lattice Boltzmann (LB) method typically uses an isothermal equation of state. This is not sufficient to simulate a number of acoustic phenomena where the equation of state cannot be approximated as linear and constant. However, it is possible to implement variable equations of state by altering the LB equilibrium distribution. For simple velocity sets with velocity components ξ(iα)∈(-1,0,1) for all i, these equilibria necessarily cause error terms in the momentum equation. These error terms are shown to be either correctable or negligible at the cost of further weakening the compressibility. For the D1Q3 velocity set, such an equilibrium distribution is found and shown to be unique. Its sound propagation properties are found for both forced and free waves, with some generality beyond D1Q3. Finally, this equilibrium distribution is applied to a nonlinear acoustics simulation where both mechanisms of nonlinearity are simulated with good results. This represents an improvement on previous such simulations and proves that the compressibility of the method is still sufficiently strong even for nonlinear acoustics.

Feedback control of inertial microfluidics using axial control forces

Inertial microfluidics is a promising tool for many lab-on-a-chip applications. Particles in channel flows with Reynolds numbers above one undergo cross-streamline migration to a discrete set of equilibrium positions in square and rectangular channel cross sections. This effect has been used extensively for particle sorting and the analysis of particle properties. Using the lattice Boltzmann method, we determined the equilibrium positions in square and rectangular cross sections and classify their types of stability for different Reynolds numbers, particle sizes, and channel aspect ratios. Our findings thereby help to design microfluidic channels for particle sorting. Furthermore, we demonstrated how an axial control force, which slows down the particles and shifts the stable equilibrium position towards the channel center. Ultimately, the particles then stay on the centerline for forces exceeding the threshold value. This effect is sensitive to the particle size and channel Reynolds number and therefore suggests an efficient method for particle separation. In combination with a hysteretic feedback scheme, we can even increase the particle throughput.

Pattern formation in liquid-vapor systems under periodic potential and shear

In this paper the phase behavior and pattern formation in a sheared nonideal fluid under a periodic potential is studied. An isothermal two-dimensional formulation of a lattice Boltzmann scheme for a liquid-vapor system with the van der Waals equation of state is presented and validated. Shear is applied by moving walls and the periodic potential varies along the flow direction. A region of the parameter space, where in the absence of flow a striped phase with oscillating density is stable, will be considered. At low shear rates the periodic patterns are preserved and slightly distorted by the flow. At high shear rates the striped phase loses its stability and traveling waves on the interface between the liquid and vapor regions are observed. These waves spread over the whole system with wavelength only depending on the length of the system. Velocity field patterns, characterized by a single vortex, will also be shown.

Single-bubble dynamics in pool boiling of one-component fluids

We numerically investigate the pool boiling of one-component fluids with a focus on the effects of surface wettability on the single-bubble dynamics. We employed the dynamic van der Waals theory [Phys. Rev. E 75, 036304 (2007)], a diffuse-interface model for liquid-vapor flows involving liquid-vapor transition in nonuniform temperature fields. We first perform simulations for bubbles on homogeneous surfaces. We find that an increase in either the contact angle or the surface superheating can enhance the bubble spreading over the heating surface and increase the bubble departure diameter as well and therefore facilitate the transition into film boiling. We then examine the dynamics of bubbles on patterned surfaces, which incorporate the advantages of both hydrophobic and hydrophilic surfaces. The central hydrophobic region increases the thermodynamic probability of bubble nucleation while the surrounding hydrophilic region hinders the continuous bubble spreading by pinning the contact line at the hydrophobic-hydrophilic intersection. This leads to a small bubble departure diameter and therefore prevents the transition from nucleate boiling into film boiling. With the bubble nucleation probability increased and the bubble departure facilitated, the efficiency of heat transfer on such patterned surfaces is highly enhanced, as observed experimentally [Int. J. Heat Mass Transfer 57, 733 (2013)]. In addition, the stick-slip motion of contact line on patterned surfaces is demonstrated in one-component fluids, with the effect weakened by surface superheating.

Thermal multicomponent lattice Boltzmann model for catalytic reactive flows

Catalytic reactions are of great interest in many applications related to power generation, fuel reforming and pollutant abatement, as well as in various biochemical processes. A recently proposed lattice Boltzmann model for thermal binary-mixture gas flows [J. Kang, N. I. Prasianakis, and J. Mantzaras, Phys. Rev. E. 87, 053304 (2013)] is revisited and extended for the simulation of multispecies flows with catalytic reactions. The resulting model can handle flows with large temperature and concentration gradients. The developed model is presented in detail and validated against a finite volume Navier-Stokes solver in the case of channel-flow methane catalytic combustion. The surface chemistry is treated with a one-step global reaction for the catalytic total oxidation of methane on platinum. In order to take into account thermal effects, the catalytic boundary condition of S. Arcidiacono, J. Mantzaras, and I. V. Karlin [Phys. Rev. E 78, 046711 (2008)] is adapted to account for temperature variations. Speed of sound simulations further demonstrate the physical integrity and unique features of the model.

Lattice Boltzmann scheme for electrolytes by an extended Maxwell-Stefan approach

This paper presents an extended multicomponent lattice Boltzmann model for the simulation of electrolytes. It is derived by means of a finite discrete velocity model and its discretization. The model recovers momentum and mass transport according to the incompressible Navier-Stokes equation and Maxwell-Stefan formulation, respectively. It includes external driving forces (e.g., electric field) on diffusive and viscous scales, concentration-dependent Maxwell-Stefan diffusivities, and thermodynamic factors. The latter take into account nonideal diffusion behavior, which is essential as electrolytes involve charged species and therefore nonideal long and short-range interactions among the molecules of the species. Furthermore, we couple our scheme to a finite element method to include electrostatic interactions on the macroscopic level. Numerical experiments show the validity of the presented model.

Effect of the forcing term in the pseudopotential lattice Boltzmann modeling of thermal flows

The pseudopotential lattice Boltzmann (LB) model is a popular model in the LB community for simulating multiphase flows. Recently, several thermal LB models, which are based on the pseudopotential LB model and constructed within the framework of the double-distribution-function LB method, were proposed to simulate thermal multiphase flows [G. Házi and A. Márkus, Phys. Rev. E 77, 026305 (2008); L. Biferale, P. Perlekar, M. Sbragaglia, and F. Toschi, Phys. Rev. Lett. 108, 104502 (2012); S. Gong and P. Cheng, Int. J. Heat Mass Transfer 55, 4923 (2012); M. R. Kamali et al., Phys. Rev. E 88, 033302 (2013)]. The objective of the present paper is to show that the effect of the forcing term on the temperature equation must be eliminated in the pseudopotential LB modeling of thermal flows. First, the effect of the forcing term on the temperature equation is shown via the Chapman-Enskog analysis. For comparison, alternative treatments that are free from the forcing-term effect are provided. Subsequently, numerical investigations are performed for two benchmark tests. The numerical results clearly show that the existence of the forcing-term effect will lead to significant numerical errors in the pseudopotential LB modeling of thermal flows.

Lattice-Boltzmann method for the simulation of multiphase mass transfer and reaction of dilute species

Despite the popularity of the lattice-Boltzmann method (LBM) in simulating multiphase flows, a general approach for modeling dilute species in multiphase systems is still missing. In this report we propose to modify the collision operator of the solute by introducing a modified redistribution scheme. This operator is based on local fluid variables and keeps the parallelism inherent to LBM. After deriving macroscopic transport equations, an analytical equation of state of the solute is exhibited and the method is proven constituting a unified framework to simulate arbitrary solute distribution between phases, including single-phase soluble compounds, amphiphilic species with a partition coefficient, and surface-adsorbed compounds.

Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows

In this paper, a phase-field-based multiple-relaxation-time lattice Boltzmann (LB) model is proposed for incompressible multiphase flow systems. In this model, one distribution function is used to solve the Chan-Hilliard equation and the other is adopted to solve the Navier-Stokes equations. Unlike previous phase-field-based LB models, a proper source term is incorporated in the interfacial evolution equation such that the Chan-Hilliard equation can be derived exactly and also a pressure distribution is designed to recover the correct hydrodynamic equations. Furthermore, the pressure and velocity fields can be calculated explicitly. A series of numerical tests, including Zalesak's disk rotation, a single vortex, a deformation field, and a static droplet, have been performed to test the accuracy and stability of the present model. The results show that, compared with the previous models, the present model is more stable and achieves an overall improvement in the accuracy of the capturing interface. In addition, compared to the single-relaxation-time LB model, the present model can effectively reduce the spurious velocity and fluctuation of the kinetic energy. Finally, as an application, the Rayleigh-Taylor instability at high Reynolds numbers is investigated.

High-accuracy approximation of high-rank derivatives: isotropic finite differences based on lattice-Boltzmann stencils

We propose isotropic finite differences for high-accuracy approximation of high-rank derivatives. These finite differences are based on direct application of lattice-Boltzmann stencils. The presented finite-difference expressions are valid in any dimension, particularly in two and three dimensions, and any lattice-Boltzmann stencil isotropic enough can be utilized. A theoretical basis for the proposed utilization of lattice-Boltzmann stencils in the approximation of high-rank derivatives is established. In particular, the isotropy and accuracy properties of the proposed approximations are derived directly from this basis. Furthermore, in this formal development, we extend the theory of Hermite polynomial tensors in the case of discrete spaces and present expressions for the discrete inner products between monomials and Hermite polynomial tensors. In addition, we prove an equivalency between two approaches for constructing lattice-Boltzmann stencils. For the numerical verification of the presented finite differences, we introduce 5th-, 6th-, and 8th-order two-dimensional lattice-Boltzmann stencils.

Tuning drop motion by chemical patterning of surfaces

We report the results of extensive experimental studies of the sliding of water drops on chemically heterogeneous surfaces formed by square and triangular hydrophobic domains printed on glass surfaces and arranged in various symmetric patterns. Overall, the critical Bond number, that is, the critical dimensionless force needed to depin the drop, is found to be strongly affected by the shape and the spatial arrangement of the domains. Soon after the droplet begins to move, stick-slip motion is observed on all surfaces, although it is less pronounced than that on striped surfaces. On the triangular patterns, anisotropic behavior is found with drops sliding down faster when the tips of the glass hydrophilic triangles are pointing in the down-plane direction. Away from the critical Bond number, the dynamic regime depends mainly on the static contact angle and weakly on the actual surface pattern. Lattice Boltzmann numerical simulations are performed to validate the experimental results and test the importance of the viscous ratio between the droplet phase and the outer phase.

SIMULATION OF STENOSIS GROWTH IN THE CAROTID ARTERY BY LATTICE BOLTZMANN METHOD

Atherosclerosis, as the leading cause of mortality, is usually regarded as a systemic disease and several well-identified risk factors have been implicated in its pathogenesis. Low or highly oscillatory wall shear stress has mainly been linked to the development of atherosclerosis. Conditions under which human blood can be considered Newtonian for the purpose of arterial flow modeling are investigated with emphasis on near wall shear stresses. The Lattice Boltzmann method is implemented in parallel for both Newtonian and non-Newtonian models of blood and then examined in the context of steady and oscillatory flows. As the lattice method permits to adjust the morphology of the computational domain during the solving process, the artery walls are reshaped in a recursive manner by the progressive accumulation of deposits according to the conventional OSI criterion. Regions subjected to partial obstructions identified qualitatively well with those susceptible to atherosclerosis in the in vivo sample, thereby approving this criterion by verifying its accumulative effect. The present work demonstrates the suitability of LB method for studying flows across geometries that transform due to atherosclerotic progression and permits to explain the trend of deposit distribution across time.

Investigation of the kinetic model equations

Currently the Boltzmann equation and its model equations are widely used in numerical predictions for dilute gas flows. The nonlinear integro-differential Boltzmann equation is the fundamental equation in the kinetic theory of dilute monatomic gases. By replacing the nonlinear fivefold collision integral term by a nonlinear relaxation term, its model equations such as the famous Bhatnagar-Gross-Krook (BGK) equation are mathematically simple. Since the computational cost of solving model equations is much less than that of solving the full Boltzmann equation, the model equations are widely used in predicting rarefied flows, multiphase flows, chemical flows, and turbulent flows although their predictions are only qualitatively right for highly nonequilibrium flows in transitional regime. In this paper the differences between the Boltzmann equation and its model equations are investigated aiming at giving guidelines for the further development of kinetic models. By comparing the Boltzmann equation and its model equations using test cases with different nonequilibrium types, two factors (the information held by nonequilibrium moments and the different relaxation rates of high- and low-speed molecules) are found useful for adjusting the behaviors of modeled collision terms in kinetic regime. The usefulness of these two factors are confirmed by a generalized model collision term derived from a mathematical relation between the Boltzmann equation and BGK equation that is also derived in this paper. After the analysis of the difference between the Boltzmann equation and the BGK equation, an attempt at approximating the collision term is proposed.

Free-energy-based lattice Boltzmann model for the simulation of multiphase flows with density contrast

A free-energy-based phase-field lattice Boltzmann method is proposed in this work to simulate multiphase flows with density contrast. The present method is to improve the Zheng-Shu-Chew (ZSC) model [Zheng, Shu, and Chew, J. Comput. Phys. 218, 353 (2006)] for correct consideration of density contrast in the momentum equation. The original ZSC model uses the particle distribution function in the lattice Boltzmann equation (LBE) for the mean density and momentum, which cannot properly consider the effect of local density variation in the momentum equation. To correctly consider it, the particle distribution function in the LBE must be for the local density and momentum. However, when the LBE of such distribution function is solved, it will encounter a severe numerical instability. To overcome this difficulty, a transformation, which is similar to the one used in the Lee-Lin (LL) model [Lee and Lin, J. Comput. Phys. 206, 16 (2005)] is introduced in this work to change the particle distribution function for the local density and momentum into that for the mean density and momentum. As a result, the present model still uses the particle distribution function for the mean density and momentum, and in the meantime, considers the effect of local density variation in the LBE as a forcing term. Numerical examples demonstrate that both the present model and the LL model can correctly simulate multiphase flows with density contrast, and the present model has an obvious improvement over the ZSC model in terms of solution accuracy. In terms of computational time, the present model is less efficient than the ZSC model, but is much more efficient than the LL model.

Lattice Boltzmann simulations of multiple-droplet interaction dynamics

A lattice Boltzmann (LB) formulation, which is consistent with the phase-field model for two-phase incompressible fluid, is proposed to model the interface dynamics of droplet impingement. The interparticle force is derived by comparing the macroscopic transport equations recovered from LB equations with the governing equations of the continuous phase-field model. The inconsistency between the existing LB implementations and the phase-field model in calculating the relaxation time at the phase interface is identified and an approximation is proposed to ensure the consistency with the phase-field model. It is also shown that the commonly used equilibrium velocity boundary for the binary fluid LB scheme does not conserve momentum at the wall boundary and a modified scheme is developed to ensure the momentum conservation at the boundary. In addition, a geometric formulation of the wetting boundary condition is proposed to replace the popular surface energy formulation and results show that the geometric approach enforces the prescribed contact angle better than the surface energy formulation in both static and dynamic wetting. The proposed LB formulation is applied to simulating droplet impingement dynamics in three dimensions and results are compared to those obtained with the continuous phase-field model, the LB simulations reported in the literature, and experimental data from the literature. The results show that the proposed LB simulation approach yields not only a significant speed improvement over the phase-field model in simulating droplet impingement dynamics on a submillimeter length scale, but also better accuracy than both the phase-field model and the previously reported LB techniques when compared to experimental data. Upon validation, the proposed LB modeling methodology is applied to the study of multiple-droplet impingement and interactions in three dimensions, which demonstrates its powerful capability of simulating extremely complex interface phenomena.

Spinodal decomposition in homogeneous and isotropic turbulence

We study the competition between domain coarsening in a symmetric binary mixture below critical temperature and turbulent fluctuations. We find that the coarsening process is arrested in the presence of turbulence. The physics of the process shares remarkable similarities with the behavior of diluted turbulent emulsions and the arrest length scale can be estimated with an argument similar to the one proposed by Kolmogorov and Hinze for the maximal stability diameter of droplets in turbulence. Although, in the absence of flow, the microscopic diffusion constant is negative, turbulence does effectively arrest the inverse cascade of concentration fluctuations by making the low wavelength diffusion constant positive for scales above the Hinze length.