Lattice-Boltzmann method combined with smoothed-profile method for particulate suspensions

Lattice Boltzmann equation for convection-diffusion flows with Neumann boundary condition

In this work, a lattice Boltzmann equation (LBE) is developed to convection-diffusion flows with Neumann boundary condition in complex geometry. The physical fluid domain together with the physical boundary is extended to a large domain similar to the fictitious domain method, and the original convection-diffusion equation (CDE) is reformulated for the large domain, where the Neumann boundary condition is naturally incorporated to CDE without separated explicit treatment. Based on this extended CDE for the large domain, the LBE solver is designed accordingly. Several classical simulations of pure thermal diffusion/natural convection between two concentrated cylinders, natural convection flow around a circular cylinder with constant heat flux in square cavity, and mixed convection flow in a square lid-driven cavity with a circular cylinder are carried out to validate the present method. Numerical results show that the predictions by present LBE agree well with theoretical or other results.

Conservative phase-field-based lattice Boltzmann equation for gas-liquid-solid flow

In this paper, a conservative phase-field based lattice Boltzmann equation (LBE) is developed to simulate gas-liquid-solid flows with large fluid density contrasts. In this model, the gas-liquid interface is captured by the conservative Allen-Cahn equation (CACE), where an additional source term is incorporated to realize the wettability of solid structure. Subsequently, a LBE is designed to solve this modified CACE (MCACE), while the two-phase flow field is resolved by using another classical incompressible LBE, and the fluid-solid interaction force is calculated by smoothed-profile method (SPM). Several classical simulations are conducted to demonstrate the capability of the present MCACE-LBE-SPM for simulating gas-liquid-solid flows, including a droplet spreading on a static wettable cylinder, a wettable cylinder floating on the gas-liquid interface without gravity, capillary interactions between two wettable cylinders under gravity, and multiple horizontal cylinders in gas-liquid channel flow. Numerical results indicate that the predictions by present MCACE-LBE-SPM are in good agreement with the theoretical or previous numerical results.

Microfluidic Shape Analysis of Non-spherical Graphite for Li-Ion Batteries via Viscoelastic Particle Focusing

The size and shape of graphite, which is a popular active anode material for lithium-ion batteries (LIBs), significantly affect the electrochemical performance of LIBs and the rheological properties of the electrode slurries used in battery manufacturing. However, the accurate characterization of its size and shape remains challenging. In this study, the edge plane of graphite in a cross-slot microchannel via viscoelastic particle focusing is characterized. It is reported that the graphite particles are aligned in a direction that shows the edge plane by a planar extensional flow field at the stagnation point of the cross-slot region. Accurate quantification of the edge size and shape for both spheroidized natural and ball-milled graphite is achieved when aligned in this manner. Ball-milled graphite has a smaller circularity and higher aspect ratio than natural graphite, indicating a more plate-like shape. The effects of these differences in graphite shape and size on the rheological properties of the electrode slurry, the structure of the coated electrodes, and electrochemical performance are investigated. This method can contribute to the quality control of graphite for the mass production of LIBs and enhance the electrochemical performance of LIBs.

Phase-field lattice Boltzmann equation for wettable particle fluid dynamics

In this paper a phase-field based lattice Boltzmann equation (LBE) is developed to simulate wettable particles fluid dynamics together with the smoothed-profile method (SPM). In this model the evolution of a fluid-fluid interface is captured by the conservative Allen-Cahn equation (CACE) LBE, and the flow field is solved by a classical incompressible LBE. The solid particle is represent by SPM, and the fluid-solid interaction force is calculated by direct force method. Some benchmark tests including a single wettable particle trapped at the fluid-fluid interface without gravity, capillary interactions between two wettable particles under gravity, and sinking of a horizontal cylinder through an air-water interface are carried out to validate present CACE LBE for fluid-fluid-solid flows. Raft sinking of multiple horizontal cylinders (up to five cylinders) through an air-water interface is further investigated with the present CACE LBE, and a nontrivial dynamics with an unusual nonmonotonic motion of the multiple cylinders is observed in the vertical plane. Numerical results show that the predictions by the present LBE are in good agreement with theoretical solutions and experimental data.

IBM-LBM-DEM Study of Two-Particle Sedimentation: Drafting-Kissing-Tumbling and Effects of Particle Reynolds Number and Initial Positions of Particles

Particle sedimentation is a fundamental process encountered in various industrial applications. In this study, we used immersed boundary lattice Boltzmann method and discrete element method (IBM-LBM-DEM) to investigate two-particle sedimentation. A lattice Boltzmann method was used to simulate fluid flow, a discrete element method was used to simulate particle dynamics, and an immersed boundary method was used to handle particle–fluid interactions. Via the IBM-LBM-DEM, the particles collision process in fluid or between rigid walls can be calculated to capture the information of particles and the flow field more efficiently and accurately. The numerical method was verified by simulating settling of a single three-dimensional particle. Then, the effects of Reynolds number (Re), initial distance, and initial angle of particles on two-particle sedimentation were characterized. A specific focus was to reproduce, analyze, and define the well-known phenomenon of drafting-kissing-tumbling (DKT) interaction between two particles. Further kinematic analysis to define DKT is meaningful for two-particle sedimentation studies at different particle locations. Whether a pair of particles has experienced DKT can be viewed from time plots of the distance between the particles (for kissing), the second-order derivative of distance to time (for drafting), and angular velocities of particles (for tumbling). Simulation results show that DKT’s signatures, including attraction, (near) contact, rotation, and in the end, separation, is only completely demonstrated when particles have nearly vertically aligned initial positions. Hence, not all initial positions of particles and Reynolds numbers lead to DKT and not all particle–particle hydrodynamic interactions are DKT. Whether particle–particle interaction is attractive or repulsive depends on the relative positions of particles and Re. Collision occurs when Re is high and the initial angle is small (<20°), almost independent of the initial distance.

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.

Smoothed profile method for direct numerical simulations of hydrodynamically interacting particles

A general method is presented for computing the motions of hydrodynamically interacting particles in various kinds of host fluids for arbitrary Reynolds numbers. The method follows the standard procedure for performing direct numerical simulations (DNS) of particulate systems, where the Navier-Stokes equation must be solved consistently with the motion of the rigid particles, which defines the temporal boundary conditions to be satisfied by the Navier-Stokes equation. The smoothed profile (SP) method provides an efficient numerical scheme for coupling the continuum fluid mechanics with the dispersed moving particles, which are allowed to have arbitrary shapes. In this method, the sharp boundaries between solid particles and the host fluid are replaced with a smeared out thin shell (interfacial) region, which can be accurately resolved on a fixed Cartesian grid utilizing a SP function with a finite thickness. The accuracy of the SP method is illustrated by comparison with known exact results. In the present paper, the high degree of versatility of the SP method is demonstrated by considering several types of active and passive particle suspensions.

Immersed boundary method for multiphase transport phenomena

Multiphase flows with momentum, heat, and mass transfer exist widely in a variety of industrial applications. With the rapid development of numerical algorithms and computer capacity, advanced numerical simulation has become a promising tool in investigating multiphase transport problems. Immersed boundary (IB) method has recently emerged as such a popular interface capturing method for efficient simulations of multiphase flows, and significant achievements have been obtained. In this review, we attempt to give an overview of recent progresses on IB method for multiphase transport phenomena. Firstly, the governing equations, the basic ideas, and different boundary conditions for the IB methods are introduced. This is followed by numerical strategies, from which the IB methods are classified into two types, namely the artificial boundary method and the authentic boundary method. Discussions on the implementation of various boundary conditions at the interphase surface with momentum, heat, and mass transfer for different IB methods are then presented, together with a summary. Then, the state-of-the-art applications of IB methods to multiphase flows, including the isothermal flows, the heat transfer flows, and the mass transfer problems are outlined, with particular emphasis on the latter two topics. Finally, the conclusions and future challenges are identified.

The first normal stress difference of non-Brownian hard-sphere suspensions in the oscillatory shear flow near the liquid and crystal coexistence region

We carry out a numerical study to investigate the dynamics of non-Brownian hard-sphere suspensions near the liquid and crystal coexistence region in small to large amplitude oscillatory shear flow. The first normal stress difference (N1) and related rheological functions are carefully analyzed, focusing on the strain stiffening phenomenon, which occurs in the large strain amplitude region. Under oscillatory shear, we observe several unique behaviors of N1. A negative nonzero mean value of N1 (N1,0) is observed for the applied strain amplitudes. The change of the sign, from negative to positive, at the maximum value of N1 (N1,max) is observed at a specific point, which is not consistent with the critical strain amplitude (γ0,c) at which the modulus begins to deviate from linear viscoelasticity. The behavior of N1 in the oscillatory shear flow is different from that of N1 in steady shear flow, that is, the characteristics of N1 in strain stiffening and shear thickening are quite distinguished from each other. We also perform structural analysis to confirm the relationship between the rheological properties and microstructure of the suspension. A strong correlation is observed between the global bond order parameter (Ψ6) and the distortions in both nonlinear shear and normal stresses. The most noticeable characteristic is captured through the maximum of the global bond order parameter (Ψ6,max). The strain amplitude at the slope change of Ψ6,max corresponds to the point where a unique behavior of N1 is observed, i.e. the change of the sign in N1,max, but a strong correlation is not captured at γ0,c. This demonstrates that the normal stress responds to particle ordering more sensitively than other rheological functions based on shear stress like dynamic moduli. As far as we are concerned, the behavior of N1 has rarely been fully explored and related with the strain stiffening of non-Brownian suspensions so far. Therefore, this study has significance as the first report to strictly analyze strain stiffening along with the first normal stress difference N1.

Lattice Boltzmann method for simulation of wettable particles at a fluid-fluid interface under gravity

A computational technique was developed to simulate wettable particles trapped at a fluid-fluid interface under gravity. The proposed technique combines the improved smoothed profile-lattice Boltzmann method (iSP-LBM) for the treatment of moving solid-fluid boundaries and the free-energy LBM for the description of isodensity immiscible two-phase flows. We considered five benchmark problems in two-dimensional systems, including a stationary drop, a wettable particle trapped at a fluid-fluid interface in the absence or presence of gravity, two freely moving particles at a fluid-fluid interface in the presence of gravity (i.e., capillary floatation forces), and two vertically constrained particles at a fluid-fluid interface (i.e., capillary immersion forces). The simulation results agreed well with theoretical estimations, demonstrating the efficacy of the proposed technique.

Effect of internal mass in the lattice Boltzmann simulation of moving solid bodies by the smoothed-profile method

A computational method for the simulation of particulate flows that can efficiently treat the particle-fluid boundary in systems containing many particles was developed based on the smoothed-profile lattice Boltzmann method (SPLBM). In our proposed method, which we call the improved SPLBM (iSPLBM), for an accurate and stable simulation of particulate flows, the hydrodynamic force on a moving solid particle is exactly formulated with consideration of the effect of internal fluid mass. To validate the accuracy and stability of iSPLBM, we conducted numerical simulations of several particulate flow systems and compared our results with those of other simulations and some experiments. In addition, we performed simulations on flotation of many lightweight particles with a wide range of particle size distribution, the results of which demonstrated the effectiveness of iSPLBM. Our proposed model is a promising method to accurately and stably simulate extensive particulate flows.

Investigation of particle inertial migration in high particle concentration suspension flow by multi-electrodes sensing and Eulerian-Lagrangian simulation in a square microchannel

The inertial migration of neutrally buoyant spherical particles in high particle concentration (αpi  > 3%) suspension flow in a square microchannel was investigated by means of the multi-electrodes sensing method which broke through the limitation of conventional optical measurement techniques in the high particle concentration suspensions due to interference from the large particle numbers. Based on the measured particle concentrations near the wall and at the corner of the square microchannel, particle cross-sectional migration ratios are calculated to quantitatively estimate the migration degree. As a result, particle migration to four stable equilibrium positions near the centre of each face of the square microchannel is found only in the cases of low initial particle concentration up to 5.0 v/v%, while the migration phenomenon becomes partial as the initial particle concentration achieves 10.0 v/v% and disappears in the cases of the initial particle concentration αpi  ≥ 15%. In order to clarify the influential mechanism of particle-particle interaction on particle migration, an Eulerian-Lagrangian numerical model was proposed by employing the Lennard-Jones potential as the inter-particle potential, while the inertial lift coefficient is calculated by a pre-processed semi-analytical simulation. Moreover, based on the experimental and simulation results, a dimensionless number named migration index was proposed to evaluate the influence of the initial particle concentration on the particle migration phenomenon. The migration index less than 0.1 is found to denote obvious particle inertial migration, while a larger migration index denotes the absence of it. This index is helpful for estimation of the maximum initial particle concentration for the design of inertial microfluidic devices.

Rheology and microstructure of non-Brownian suspensions in the liquid and crystal coexistence region: strain stiffening in large amplitude oscillatory shear

Concentrated hard-sphere suspensions in the liquid and crystal coexistence region show a unique nonlinear behavior under a large amplitude oscillatory shear flow, the so-called strain stiffening, in which the viscosity or modulus suddenly starts to increase near a critical strain amplitude. Even though this phenomenon has been widely reported in experiments, its key mechanism has never been investigated in a systematic way. To have a good understanding of this behavior, a numerical simulation was performed using the lattice Boltzmann method (LBM). Strain stiffening was clearly observed at large strain amplitudes, and the critical strain amplitude showed an angular frequency dependency. The distortion of the shear stress appeared near the critical strain amplitude, and the nonlinear behavior was quantified by the Fourier transformation (FT) and the stress decomposition methods. Above the critical strain amplitude, an increase in the global bond order parameter Ψ(6) was observed at the flow reversal. The maximum of Ψ(6) and the maximum shear stress occurred at the same strain. These results show how strongly the ordered structure of the particles is related to the stress distortion. The ordered particles maintained a bond number of "two" with alignment with the compressive axis, and they were distributed over a narrow range of angular distribution (110°-130°). In addition, the ordered structure was formed near the lowest shear rate region (the flow reversal). The characteristics of the ordered structure were remarkably different from those of the hydroclusters which are regarded as the origin of shear thickening. It is clear that strain stiffening and shear thickening originate from different mechanisms. Our results clearly demonstrate how the ordering of the particles induces strain stiffening in the liquid and crystal coexistence region.

Local shear stress and its correlation with local volume fraction in concentrated non-Brownian suspensions: lattice Boltzmann simulation

The local shear stress of non-Brownian suspensions was investigated using the lattice Boltzmann method coupled with the smoothed profile method. Previous studies have only focused on the bulk rheology of complex fluids because the local rheology of complex fluids was not accessible due to technical limitations. In this study, the local shear stress of two-dimensional solid particle suspensions in Couette flow was investigated with the method of planes to correlate non-Newtonian fluid behavior with the structural evolution of concentrated particle suspensions. Shear thickening was successfully captured for highly concentrated suspensions at high particle Reynolds number, and both the local rheology and local structure of the suspensions were analyzed. It was also found that the linear correlation between the local particle stress and local particle volume fraction was dramatically reduced during shear thickening. These results clearly show how the change in local structure of suspensions influences the local and bulk rheology of the suspensions.

Mesoscale models of dispersions stabilized by surfactants and colloids

In this paper we discuss and give an outlook on numerical models describing dispersions, stabilized by surfactants and colloidal particles. Examples of these dispersions are foams and emulsions. In particular, we focus on the potential of the diffuse interface models based on a free energy approach, which describe dispersions with the surface-active agent soluble in one of the bulk phases. The free energy approach renders thermodynamic consistent models with realistic sorption isotherms and adsorption kinetics. The free energy approach is attractive because of its ability to describe highly complex dispersions, such as emulsions stabilized by ionic surfactants, or surfactant mixtures and dispersions with surfactant micelles. We have classified existing numerical methods into classes, using either a Eulerian or a Lagrangian representation for fluid and for the surfactant/colloid. A Eulerian representation gives a more coarse-grained, mean field description of the surface-active agent, while a Lagrangian representation can deal with steric effects and larger complexity concerning geometry and (amphiphilic) wetting properties of colloids and surfactants. However, the similarity between the description of wetting properties of both Eulerian and Lagrangian models allows for the development of hybrid Eulerian/Lagrangian models having advantages of both representations.