Improved locality of the phase-field lattice-Boltzmann model for immiscible fluids at high density ratios

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.

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

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

Simplified wetting boundary scheme in phase-field lattice Boltzmann model for wetting phenomena on curved boundaries

In this work, a simplified wetting boundary scheme in the phase-field lattice Boltzmann model is developed for wetting phenomena on curved boundaries. The proposed scheme combines the advantages of the fluid-solid interaction scheme and geometric scheme-easy to implement (no need to interpolate the values of parameters exactly on solid boundaries and find proper characteristic vectors), the value of contact angle can be directly prescribed, and no unphysical spurious mass layer-and avoids mass leakage. Different from previous works, the values of the order parameter gradient on fluid boundary nodes are directly determined according to the geometric formulation rather than indirectly regulated through the order parameters on ghost solid nodes (i.e., ghost contact-line region). For this purpose, two numerical approaches to evaluate the order parameter gradient on fluid boundary nodes are utilized, one with the prevalent isotropic central scheme and the other with a local gradient scheme that utilizes the distribution functions. The simplified wetting boundary schemes with both numerical approaches are validated and compared through several numerical simulations. The results demonstrate that the proposed model has good ability and satisfactory accuracy to simulate wetting phenomena on curved boundaries under large density ratios.

Fluid interfaces laden by force dipoles: towards active matter-driven microfluidic flows

In recent years, nonlinear microfluidics in combination with lab-on-a-chip devices has opened a new avenue for chemical and biomedical applications such as droplet formation and cell sorting. In this article, we integrate ideas from active matter into a microfluidic setting, where two fluid layers with identical densities but different viscosities flow through a microfluidic channel. Most importantly, the fluid interface is laden with active particles that act with dipolar forces on the adjacent fluids and thereby generate flows. We perform lattice-Boltzmann simulations and combine them with phase field dynamics of the interface and an advection-diffusion equation for the density of active particles. We show that only contractile force dipoles can destabilize the flat fluid interface. It develops a viscous finger from which droplets break up. For interfaces with non-zero surface tension, a critical value of activity equal to the surface tension is necessary to trigger the instability. Since activity depends on the density of force dipoles, the interface can develop steady deformation. Lastly, we demonstrate how to control droplet formation using switchable activity.

Interaction between a rising bubble and a stationary droplet immersed in a liquid pool using a ternary conservative phase-field lattice Boltzmann method

When a stationary bubble and a stationary droplet immersed in a liquid pool are brought into contact, they form a bubble-droplet aggregate. Its equilibrium morphology and stability largely depend on the combination of different components' surface tensions, known as the "spreading factor." In this study, we look at the interaction between a rising bubble and a stationary droplet to better understand the dynamics of coalescence and rising and morphological changes for the bubble-droplet aggregate. A systematic study is conducted on the interaction processes with various bubble sizes and spreading factors in two dimensions. The current simulation framework consists of the ternary conservative phase-field lattice Boltzmann method (LBM) for interface tracking and the velocity-pressure LBM for hydrodynamics, which is validated by benchmark cases such as the liquid lens and parasitic currents around a static droplet with several popular surface tension formulations. We further test our LBM for the morphology changes of two droplets initially in contact with various spreading factors and depict the final morphologies in a phase diagram. The separated, partially engulfed, and completely engulfed morphologies can be replicated by systematically altering the sign of the spreading factors. The rising bubble and stationary droplet interaction are simulated based on the final morphologies obtained under stationary conditions by imposing an imaginary buoyancy force on the rising bubble. The results indicate that the bubble-droplet aggregate with double emulsion morphology can minimize the distortion of the bubble-droplet aggregate and achieve a greater terminal velocity than the aggregate with partially engulfed morphology.

Lattice Boltzmann method for interface capturing

Accurately solving phase interface plays a great role in modeling an immiscible multiphase flow system. In this paper, we propose an accurate interface-capturing lattice Boltzmann method from the perspective of the modified Allen-Cahn equation (ACE). The modified ACE is built based on the commonly used conservative formulation via the relation between the signed-distance function and the order parameter also maintaining the mass-conserved characteristic. A suitable forcing term is carefully incorporated into the lattice Boltzmann equation for correctly recovering the target equation. We then test the proposed method by simulating some typical interface-tracking problems of Zalesaks disk rotation, single vortex, deformation field and demonstrate that the present model can be more numerically accurate than the existing lattice Boltzmann models for the conservative ACE, especially at a small interface-thickness scale.

Quantifying the Permeability Enhancement from Blast-Induced Microfractures in Porphyry Rocks Using a Cumulant Lattice Boltzmann Method

The permeability of rocks is important in a range of geoscientific applications, including $$\hbox {CO}_{2}$$ CO 2 sequestration, geothermal energy extraction, and in situ mineral recovery. This work presents an investigation of the change in permeability in porphyry rock samples due to blast-induced fracturing. Two samples were analysed before and after exposure to stress waves induced by the detonation of an explosive charge. Micro-computed tomography was used to image the interior of the samples at a pixel resolution of $$10.3\,\mu m$$ 10.3 μ m . The images were segmented into void, matrix, and grain to help quantify the differences in the rock samples. Following this, they were binarised as void or solid and the cumulant lattice Boltzmann method (LBM) was applied to simulate the flow of fluid through the connected void space. A correction required with the use of inlet and outlet reservoirs in computational permeability assessment was also proposed. Interrogation of the steady-state flow field allowed the pre- and post-loading permeability to be extracted. Conclusions were then drawn as to the effectiveness of blasting for enhancing fluid accessibility via the generation of microfractures in the rock matrix within the vicinity of a detonated charge. This paper makes contributions in three fundamental areas relating to the numerical assessment of permeability and the enhancement of fluid accessibility in low-porosity rocks. Firstly, a correction factor was proposed to account for the reservoirs commonly imposed on digitised rock samples when investigating sample permeability through numerical methods. Secondly, it validates the benefits of the LBM in handling complex geometries that would be intractable with conventional computational fluid dynamics methods that require body-fitted meshing. This is done with a novel implementation of the cumulant LBM in the open-source TCLB code. Finally, the improvement in fluid accessibility in low-permeability rock samples was shown through the assessment of multiple regions within two blasted samples. It was found that the blast-induced loading can generate extended microfractures that results in multiple orders of magnitude of permeability enhancement if the target rock possesses existing weaknesses and/or mineralisation.

Analysis and reconstruction of the multiphase lattice Boltzmann flux solver for multiphase flows with large density ratios

The multiphase lattice Boltzmann flux solver (MLBFS) has been proposed to tackle complex geometries with nonuniform meshes. It also has been proven to have good numerical stability for multiphase flows with large density ratios. However, the reason for the good numerical stability of MLBFS at large density ratios has not been well established. The present paper reveals the relation between MLBFS and the macroscopic weakly compressible multiphase model by recovering the macroscopic equations of MLBFS (MEs-MLBFS) with actual numerical dissipation terms. By directly solving MEs-MLBFS, the reconstructed MLBFS (RMLBFS) that involves only macroscopic variables in the computational processes is proposed. The analysis of RMLBFS indicates that by combining the predictor step, the corrector step of MLBFS introduces some numerical dissipation terms which contribute to the good numerical stability of MLBFS. By retaining these numerical dissipation terms, RMLBFS can maintain the numerical stability of MLBFS even at large density ratios.

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.

Modified radius-weighted lattice Boltzmann model to address singularities in axisymmetric multiphase flows

The radius-weighted lattice Boltzmann model has achieved great success in the simulation of axisymmetric flows. However, severe spurious currents near the axis are observed when this model is extended to simulate axisymmetric multiphase flows. In this study, to determine the origin of this singularity, we conducted a truncation error analysis based on high-order Taylor series expansion and identified the leading error terms through dimensionless analysis. By neglecting the error terms in proportion to the radius, we obtained the final forms of the singular terms in the axisymmetric lattice Boltzmann model. We proposed a modified model by including an additional correction term, to remove the singularity at the third order. We validated the proposed model using numerical tests for flat and spherical interfaces. Results showed that the present modified model reduced the spurious currents near the axis by two orders of magnitude compared with the original model. This modified model also has been successfully applied to predict bubble dynamics in an air-water system. Our numerical results are in excellent agreement with available experimental observations in terms of bubble shapes and terminal velocities.

A Droplet-Manipulation Method Based on the Magnetic Particle-Stabilized Emulsion and Its Direct Numerical Simulation

Droplet manipulation has found broad applications in various engineering fields, such as microfluidic systems. This work reports a droplet-manipulation method based on particle-stabilized emulsions, where the magnetic particles adsorbed to the droplet surface serve as the actuator. The movement and the release of the droplet can be controlled by applying an external magnetic field. A lattice Boltzmann model for a three-phase system containing liquids and solid particles is adopted, which could provide a full coupling between fluids and particles. The effectiveness of the present droplet-manipulation method is validated through experiments and numerical simulations. Furthermore, the numerical simulation can provide insight into the interactions between the magnetic particles and the droplet during the droplet-driven process. To drive the droplet successfully, the magnetic particle needs to adhere to its surface and act as an "engine" to provide the driving force. As it is a surface-tension-dominant problem, the capillary effect can be considered as an "energy transfer station". The magnetic driving force on the particle is transmitted primarily to the droplet through interfacial capillary forces at the three-phase contact line, which assists the droplet in overcoming the viscous resistance and moving forward. A dimensionless number is proposed as a predictor of droplet transport and particle detachment.

Phase-field lattice Boltzmann model for two-phase flows with large density ratio

In this work, a lattice Boltzmann (LB) model based on the phase-field method is proposed for simulating large density ratio two-phase flows. An improved multiple-relaxation-time (MRT) LB equation is first developed to solve the conserved Allen-Cahn (AC) equation. By utilizing a nondiagonal relaxation matrix and modifying the equilibrium distribution function and discrete source term, the conserved AC equation can be correctly recovered by the proposed MRT LB equation with no deviation term. Therefore, the calculations of the temporal derivative term in the previous LB models are successfully avoided. Numerical tests demonstrate that satisfactory accuracy can be achieved by the present model to solve the conserved AC equation. What is more, the discrete force term of the MRT LB equation for the incompressible Navier-Stokes equations is also simplified and modified in the present work. An alternative scheme to calculate the gradient terms of the order parameter involved in the discrete force term through the nonequilibrium part of the distribution function is also developed. To validate the ability of the present LB model for simulating large density ratio two-phase flows, series of benchmarks, including two-phase Poiseuille flow, droplet impacting on thin liquid film, and planar Taylor bubble are simulated. It is found that the results predicted by the present LB model agree well with the analytical, numerical, and experimental results.

Instability of a liquid sheet with viscosity contrast in inertial microfluidics

Flows at moderate Reynolds numbers in inertial microfluidics enable high throughput and inertial focusing of particles and cells with relevance in biomedical applications. In the present work, we consider a viscosity-stratified three-layer flow in the inertial regime. We investigate the interfacial instability of a liquid sheet surrounded by a density-matched but more viscous fluid in a channel flow. We use linear stability analysis based on the Orr-Sommerfeld equation and direct numerical simulations with the lattice Boltzmann method (LBM) to perform an extensive parameter study. Our aim is to contribute to a controlled droplet production in inertial microfluidics. In the first part, on the linear stability analysis we show that the growth rate of the fastest growing mode [Formula: see text] increases with the Reynolds number [Formula: see text] and that its wavelength [Formula: see text] is always smaller than the channel width w for sufficiently small interfacial tension [Formula: see text]. For thin sheets we find the scaling relation [Formula: see text], where m is viscosity ratio and [Formula: see text] the sheet thickness. In contrast, for thicker sheets [Formula: see text] decreases with increasing [Formula: see text] or m due to the nearby channel walls. Examining the eigenvalue spectra, we identify Yih modes at the interface. In the second part on the LBM simulations, the thin liquid sheet develops two distinct dynamic states: waves traveling along the interface and breakup into droplets with bullet shape. For smaller flow rates and larger sheet thicknesses, we also observe ligament formation and the sheet eventually evolves irregularly. Our work gives some indication how droplet formation can be controlled with a suitable parameter set [Formula: see text].

Modified phase-field-based lattice Boltzmann model for incompressible multiphase flows

Based on the phase-field theory, a multiple-relaxation-time (MRT) lattice Boltzmann model is proposed for the immiscible multiphase fluids. In this model, the local Allen-Chan equation is chosen as the target equation to capture the phase interface. Unlike previous MRT schemes, an off-diagonal relaxation matrix is adopted in the present model so that the target phase-field equation can be recovered exactly without any artificial terms. To check the necessity of removing those artificial terms, comparative studies were carried out among different MRT schemes with or without correction. Results show that the artificial terms can be neglected at low March number but will cause unphysical diffusion or interface undulation instability for the relatively large March number cases. The present modified model shows superiority in reducing numerical errors by adjusting the free parameters. As the interface transport coupled to the fluid flow, a pressure-evolution lattice Boltzmann equation is adopted for hydrodynamic properties. Several benchmark cases for multiphase flow were conducted to test the validity of the present model, including the static drop test, Rayleigh-Taylor instability, and single rising bubble test. For the rising bubble simulation at high density ratios, bubble dynamics obtained by the present modified MRT lattice Boltzmann model agree well with those obtained by the FEM-based level set and FEM-based phase-field models.

Improved dynamic imaging of multiphase flow by constrained tomographic reconstruction

Dynamic tomography has become an important technique to study fluid flow processes in porous media. The use of laboratory X-ray tomography instruments is, however, limited by their low X-ray brilliance. The prolonged exposure times, in turn, greatly limit temporal resolution. We have developed a tomographic reconstruction algorithm that maintains high image quality, despite reducing the exposure time and the number of projections significantly. Our approach, based on the Simultaneous Iterative Reconstruction Technique, mitigates the problem of few and noisy exposures by utilising a high-quality scan of the system before the dynamic process is started. We use the high-quality scan to initialise the first time step of the dynamic reconstruction. We further constrain regions of the dynamic reconstruction with a segmentation of the static system. We test the performance of the algorithm by reconstructing the dynamics of fluid separation in a multiphase system. The algorithm is compared quantitatively and qualitatively with several other reconstruction algorithms and we show that it can maintain high image quality using only a fraction of the normally required number of projections and with a substantially larger noise level. By robustly allowing fewer projections and shorter exposure, our algorithm enables the study of faster flow processes using laboratory tomography instrumentation but it can also be used to improve the reconstruction quality of dynamic synchrotron experiments.

Numerical Study on Bubble Rising in Complex Channels Saturated with Liquid Using a Phase-Field Lattice-Boltzmann Method

Packed bed reactors have been widely applied in industrial production, such as for catalytic hydrogenation. Numerical simulations are essential for the design and scale-up of packed beds, especially direct numerical simulation (DNS) methods, such as the lattice-Boltzmann method (LBM), which are the focus of future researches. However, the large density difference between gas and liquid in packed beds often leads to numerical instability near phase interface when using LBM. In this paper, a lattice-Boltzmann (LB) model based on diffuse-interface phase-field is employed to simulate bubble rising in complex channels saturated with liquid, while the numerical problems caused by large liquid-to-gas density ratio are solved. Among them, the channel boundaries are constructed with regularly arranged circles and semicircles, and the bubbles pass through the channels accompanied by deformation, breakup, and coalescence behaviors. The phase-field LB model is found to exhibit good numerical stability and accuracy in handing the problem of the bubbles rising through the high-density liquid. The effects of channel structures, gas-liquid physical properties, and operating conditions on bubble deformation, motion velocity, and drag coefficient are simulated in detail. Moreover, different flow patterns are distinguished according to bubble behavior and are found to be associated with channel structure parameters, gravity Reynolds number (ReGr), and Eötvös number (Eo).

Study of phase-field lattice Boltzmann models based on the conservative Allen-Cahn equation

Conservative phase-field (CPF) equations based on the Allen-Cahn model for interface tracking in multiphase flows have become more popular in recent years, especially in the lattice-Boltzmann (LB) community. This is largely due to their simplicity and improved efficiency and accuracy over their Cahn-Hilliard-based counterparts. Additionally, the improved locality of the resulting LB equation (LBE) for the CPF models makes them more ideal candidates for LB simulation of multiphase flows on nonuniform grids, particularly within an adaptive-mesh refinement framework and massively parallel implementation. In this regard, some modifications-intended as improvements-have been made to the original CPF-LBE proposed by Geier et al. [Phys. Rev. E 91, 063309 (2015)PLEEE81539-375510.1103/PhysRevE.91.063309] which require further examination. The goal of the present study is to conduct a comparative investigation into the differences between the original CPF model proposed by Geier et al. [Phys. Rev. E 91, 063309 (2015)PLEEE81539-375510.1103/PhysRevE.91.063309] and the so-called improvements proposed by Ren et al. [Phys. Rev. E 94, 023311 (2016)2470-004510.1103/PhysRevE.94.023311] and Wang et al. [Phys. Rev. E 94, 033304 (2016)2470-004510.1103/PhysRevE.94.033304]. Using the Chapman-Enskog analysis, we provide a detailed derivation of the governing equations in each model and then examine the efficacy of the above-mentioned models for some benchmark problems. Several test cases have been designed to study different configurations ranging from basic yet informative flows to more complex flow fields, and the results are compared with finite-difference simulations. Furthermore, as a development of the previously proposed CPF-LBE model, axisymmetric formulations for the proposed model by Geier et al. [Phys. Rev. E 91, 063309 (2015)PLEEE81539-375510.1103/PhysRevE.91.063309] are derived and presented. Finally, two benchmark problems are designed to compare the proposed axisymmetric model with the analytical solution and previous work. We find that the accuracy of the model for interface tracking is roughly similar for different models at high viscosity ratios, high density ratios, and relatively high Reynolds numbers, while the original CFP-LBE without the additional time-dependent terms outperforms the so-called improved models in terms of efficiency, particularly on distributed parallel machines.

Simulation of high-viscosity-ratio multicomponent fluid flow using a pseudopotential model based on the nonorthogonal central-moments lattice Boltzmann method

In this research, the development of a pseudopotential multicomponent model with the capability of simulating high-viscosity-ratio flows is discussed and examined. The proposed method is developed based on the non-orthogonal central moments model in the lattice Boltzmann method, and the exact difference model (EDM) is used to apply the intercomponent interaction force. In contrast to the original Shan-Chen model, in which the applying force has the viscosity-dependent error term, the error term of this model does not depend on the viscosity. A GPU parallel cuda code has been developed and is employed to study the proposed method. Different cases are considered to evaluate the ability of the model, including the Laplace test, a static droplet, and a two-component concurrent channel flow. Also, wetting and nonwetting relative permeabilities for flows with dynamic viscosity ratios between 0.0002 and 5000 are predicted. Numerical results are compared with those of available analytical solutions. Very good agreement between these results are observed. The model has the capability of simulating multicomponent flows with very low kinematic viscosities of the order of 10^{-5} and dynamic viscosity ratios of up to an order of 10^{4}, which is a much wider range compared with that of existing pseudopotential models. Furthermore, the results showed that the parallel processing on GPU significantly accelerated computations. The present parallel performance evaluation shows that the cuda parallel can achieve about 41 times improvement than the CPU serial implementation. The aforementioned enhancement increases the flexibility of the multicomponent lattice Boltzmann method and its applicability to a broader spectrum of engineering applications.

Numerical study of droplet impact on a flexible substrate

Droplets interacting with deformable moving boundaries is ubiquitous. The flexible boundaries may dramatically affect the hydrodynamic behavior of droplets. A numerical method for simulating droplet impact on flexible substrates is developed. The effect of flexibility is investigated. To reduce the contact time and increase the remaining upward momentum in the flexible cases, the Weber number should be larger than a critical value. Moreover, the ratio of the natural frequency of the plate to that of the droplet F_{r} should approximately equal to the reciprocal of the contact time of droplets impact on the rigid surfaces (t_{ctr}) at the same We, e.g., F_{r}≈1/t_{ctr}. Only under this circumstance would the kinetic energy convert into the surface energy of the droplet and the elastic energy of the plate simultaneously, and vice versa. Moreover, based on a double spring model, we proposed scaling laws for the maximal deflection of the plate and spreading diameter of the drop. Finally, the droplet impact under different wettability is qualitatively studied. We found that the flexibility may contribute to the droplet bouncing at a smaller contact angle.

Lattice Boltzmann model for ternary fluids with solid particles

On the basis of phase-field theory, we develop a lattice Boltzmann model for ternary fluids containing solid. We develop a modified bounce-back method to describe the interactions between the solid and N-phase (N≥2) fluids. We derive a wetting boundary condition for three-phase flows from the point of mass conservation and propose a scheme for implementing the wetting condition on curved boundaries. We develop a diffuse interface method to compute the capillary force acting on the moving solid objects at the ternary fluids-sold contact lines. In addition, this model can deal with problems involving high density and viscosity contrasts. The proposed method is examined through several test cases. We test the modified bounce-back scheme, wetting boundary condition, and capillary force model in three different cases, and the numerical results agree well with the analytical solutions. Finally, we apply the model to two three-dimensional problems to assess its numerical accuracy and stability.

Reduction-consistent phase-field lattice Boltzmann equation for N immiscible incompressible fluids

In this paper, we develop a reduction-consistent conservative phase-field method for interface-capturing among N (N≥ 2) immiscible fluids, which is governed by conservative Allen-Cahn equation (CACE); here the reduction-consistent property is that if only M (1≤M≤N-1) immiscible fluids are present in a N-phase system, the governing equations for N immiscible fluids must reduce to the corresponding M immiscible fluids system. Then we propose a reduction-consistent lattice Boltzmann equation (LBE) method for solving N immiscible incompressible fluids with high density and viscosity contrasts. Some numerical simulations are carried out to validate the present LBE such as stationary droplets, spreading of a liquid lens, and spinodal decomposition together with the reduction-consistent property, and the numerical results predicted by present LBE are in good agreement with the analytical solutions/other numerical results, which also demonstrate the reduction-consistent property by present LBE.

Numerical simulation of dissolved air flotation using a lattice Boltzmann method

In this paper, the behavior of a bubble and droplet rising in a system, namely, a dissolved air flotation system, is investigated under different conditions. A lattice Boltzmann model which is based on the Cahn-Hilliard equations for ternary flows is implemented. This model can handle high density and viscosity ratios, remove parasitic currents, and capture partial and total spreading conditions. Two classical problems, such as spreading of a liquid lens and the Rayleigh-Taylor instability are used to determine the accuracy of the model. As a practical application, three-component flow in a tank is studied and the dynamics of bubble and droplet under different conditions is investigated. We then concentrate on the dimensionless average velocity and locations of bubble and droplet at different density ratios, viscosity ratios, and diameter ratios. Also, total spreading and partial spreading conditions are compared. The numerical results are justifiable physically and show the ability of this model to simulate three-component flows.

Interface tracking characteristics of color-gradient lattice Boltzmann model for immiscible fluids

We study the interface tracking characteristics of a color-gradient-based lattice Boltzmann model for immiscible flows. Investigation of the local density change in one of the fluid phases, via a Taylor series expansion of the recursive lattice Boltzmann equation, leads to the evolution equation of the order parameter that differentiates the fluids. It turns out that this interface evolution follows a conservative Allen-Cahn equation with a mobility which is independent of the fluid viscosities and surface tension. The mobility of the interface, which solely depends upon lattice speed of sound, can have a crucial effect on the physical dynamics of the interface. Further, we find that, when the equivalent lattice weights inside the segregation operator are modified, the resulting differential operators have a discretization error that is anisotropic to the leading order. As a consequence, the discretization errors in the segregation operator, which ensures a finite interface width, can act as a source of the spurious currents. These findings are supported with the help of numerical simulations.

Interactions of Oil Drops Induced by the Lateral Capillary Force and Surface Tension Gradients

The interacting forces on the objects at the air-liquid interface are important for researching self-assembly of objects. Due to the interacting forces among objects, the objects self-assemble at the air-liquid interface and organize into ordered structures. In general, for the interacting of oil drops, they can attract coalescence or repel dispersion. However, according to our former research, we find that interactions of oil drops can be motion like interactions of gas atoms. The oil drops cannot coalesce to form a big oil drop or separate to form ordered structures. They attract at a long distance but repel when they almost contact. In this study, according to our simulation, we demonstrate that the atomic-like motion of oil drops is mainly caused by the surface tension gradient and the lateral capillary force. Based on the results, our research will facilitate the fundamental understanding about interactions among oil drops. Apart from giving detailed demonstration about the interacting forces among oil drops, our research may also put forward research about self-assembly, oil emulsification, and microfluidics.

Phase-field-based lattice Boltzmann model for liquid-gas-solid flow

Based on phase-field theory, we develop a lattice Boltzmann (LB) model for liquid-gas-solid flow from multiphase and particle dynamics algorithms. A modified bounce-back method is developed for the velocity-based LB approach. A curved boundary treatment with second-order accuracy based on velocity interpolation is developed. We propose a predictor-corrector scheme algorithm for specifying the three-phase contact angle on curved boundaries within the framework of structured Cartesian grids. In order to make the algorithm more stable, we combine the implicit particle velocity update scheme and the Galilean invariant momentum exchange method. The proposed method is validated through several single- and multicomponent fluid test cases. It was found the surface tension force associated with the interface acting on the solid structures can be captured. We simulate the sinking of a circular cylinder due to gravity, the numerical results agree well with the experimental data. Finally, we apply the method to the self-assembly process of multiple floating cylinders on water surface.

Numerical investigation of vibration-induced droplet shedding on smooth surfaces with large contact angles

In this work, numerical simulations are performed to study the droplet response to the vertical vibration of the substrate, under various frequencies and amplitudes using the multiphase lattice Boltzmann method. First, the numerical results are validated against published experimental data. The effects of droplet size, surface wettability, amplitude, and frequency of the vibrating substrate on droplet detachment are studied. For high contact angles, regardless of the droplet size, when the vibration frequency matches the droplet resonance frequency the droplet is easily removed from the surface. For lower contact angles, the resonance frequency is higher and the detachment amplitude increases significantly. It was also found that viscous forces do not affect the resonance frequency, but have a noticeable impact on the detachment amplitude. The findings of this study can be useful in applications where droplet shedding is crucial, e.g., condensation heat transfer.

Coalescence-induced droplet detachment on low-adhesion surfaces: A three-phase system study

Coalescing water droplets on superhydrophobic surfaces can detach from the surface without the aid of any external forces. This self-propelled droplet detachment mechanism is useful in many applications, such as phase change heat transfer enhancement, self-cleaning surfaces, and anti-icing and antidew coatings. In this article, the coalescence-induced droplet jumping in a three-phase system is numerically investigated. The gaps between the surface structures are filled with a liquid that is immiscible with water, e.g., lubricant. A mass-conserving lattice Boltzmann method is implemented to study the effects of several parameters, such as interfacial tensions, droplet size, and surface wettability on the jumping process. The numerical results show that for relatively high values of lubricant-water interfacial tensions and large surface-water contact angles (>150^{∘}) the water droplets are capable of detaching. The critical droplet size for jumping is also highly dependent on the lubricant-water interfacial properties. The results of this study provide insights into the fluid-fluid and fluid-solid interactions and shed light on the underlying mechanisms involved in the droplet coalescence process on such surfaces.

Numerical investigation of vibration-induced droplet shedding on microstructured superhydrophobic surfaces

The vibration-induced droplet shedding mechanism on microstructured superhydrophobic surfaces was simulated using the lattice Boltzmann method. The numerical simulations of natural droplet oscillations for various surface structures show that the natural frequency of the droplet is strongly dependent on surface morphology. The results show good agreement with basic theoretical values. Furthermore, simulations of the motion of the droplet subjected to vertical surface vibration demonstrate that droplets in the Cassie wetting state are easily removed from the surface, whereas for Wenzel state droplets, pinch-off occurs and only partial removal is possible. Microstructure spacing was found to be a key factor in the shedding process. On a surface with small microstructure spacing, the increased surface adhesion leads to a decrease of droplet departure velocity. In contrast, for large roughness spacing, the droplet is impaled on the microstructures, which causes the departure velocity to decrease. Reperforming the simulations under different vibration intensities reveals that as the vibration amplitude is increased, the optimum frequency for droplet removal decreases. The findings of this study shed light on the underlying mechanisms involved in forced vibrations of droplets and can be helpful in engineering applications in which droplet shedding processes are critical.

Phase-field method based on discrete unified gas-kinetic scheme for large-density-ratio two-phase flows

In this paper, a phase-field method under the framework of discrete unified gas-kinetic scheme (DUGKS) for incompressible multiphase fluid flows is proposed. Two kinetic models are constructed to solve the conservative Allen-Cahn equation that accounts for the interface behavior and the incompressible hydrodynamic equations that govern the flow field, respectively. With a truncated equilibrium distribution function as well as a temporal derivative added to the source term, the macroscopic governing equations can be exactly recovered from the kinetic models through the Chapman-Enskog analysis. Calculation of source terms involving high-order derivatives existed in the quasi-incompressible model is simplified. A series of benchmark cases including four interface-capturing tests and four binary flow tests are carried out. Results compared to that of the lattice Boltzmann method (LBM) have been obtained. A convergence rate of second order can be guaranteed in the test of interface diagonal translation. The capability of the present method to track the interface that undergoes a severe deformation has been verified. Stationary bubble and spinodal decomposition problems, both with a density ratio as high as 1000, are conducted and reliable solutions have been provided. The layered Poiseuille flow with a large viscosity ratio is simulated and numerical results agree well with the analytical solutions. Variation of positions of the bubble front and spike tip during the evolution of Rayleigh-Taylor instability has been predicted precisely. However, the detailed depiction of complicated interface patterns appearing during the evolution process is failed, which is mainly caused by the relatively large numerical dissipation of DUGKS compared to that of LBM. A high-order DUGKS is needed to overcome this problem.

Hybrid Allen-Cahn-based lattice Boltzmann model for incompressible two-phase flows: The reduction of numerical dispersion

In this work a phase field-lattice Boltzmann model for incompressible two-phase flows is presented. In this model, the interface tracking equation is a linear combination of the local and nonlocal Allen-Cahn equations. We also propose a multiple-relaxation-time lattice Boltzmann model for solving the hybrid Allen-Cahn equation. The second-order convergence rate of the present model in space is validated by simulating the diagonal translation of circular interface. Three other numerical tests, including static bubble immersed in another fluid, bubble rising under gravity, and droplet splashing on a thin liquid film, are simulated to verify the performance of the present model in reducing the numerical dispersion. The numerical results indicate that the order parameter fluctuation can be reduced by one order of magnitude in bulk region.

Lattice Boltzmann model with self-tuning equation of state for multiphase flows

A lattice Boltzmann (LB) model for multiphase flows is developed that complies with the thermodynamic foundations of kinetic theory. By directly devising the collision term for the LB equation at the discrete level, a self-tuning equation of state is achieved, which can be interpreted as the incorporation of short-range molecular interaction. A pairwise interaction force is introduced to mimic the long-range molecular interaction, which is responsible for interfacial dynamics. The derived pressure tensor is naturally consistent with thermodynamic theory, and surface tension and interface thickness can be independently prescribed. Numerical tests, including static and dynamic cases, are carried out to validate the present model and good results are obtained. As a further application, head-on collision of equal-sized droplets is simulated and the elusive "bouncing" regime is successfully reproduced.