Lattice Boltzmann model for simulating flows with multiple phases and components

Validation of pore network modeling for determination of two-phase transport in fibrous porous media

Pore network modeling (PNM) has been widely investigated in the study of multiphase transport in porous media due to its high computational efficiency. The advantage of PNM is achieved in part at the cost of using simplified geometrical elements. Therefore, the validation of pore network modeling needs further verification. A Shan-Chen (SC) multiphase lattice Boltzmann model (LBM) was used to simulate the multiphase flow and provided as the benchmark. PNM using different definitions of throat radius was performed and compared. The results showed that the capillary pressure and saturation curves agreed well when throat radius was calculated using the area-equivalent radius. The discrepancy of predicted phase occupations from different methods was compared in slice images and the reason can be attributed to the capillary pressure gradients demonstrated in LBM. Finally, the relative permeability was also predicted using PNM and provided acceptable predictions when compared with the results using single-phase LBM.

Pore-Scale Transport and Two-Phase Fluid Structures in Fibrous Porous Layers: Application to Fuel Cells and Beyond

We present pore-scale simulations of two-phase flows in a reconstructed fibrous porous layer. The three-dimensional microstructure of the material, a fuel cell gas diffusion layer, is acquired via X-ray computed tomography and used as input for lattice Boltzmann simulations. We perform a quantitative analysis of the multiphase pore-scale dynamics, and we identify the dominant fluid structures governing mass transport. The results show the existence of three different regimes of transport: a fast inertial dynamics at short times, characterised by a compact uniform front, a viscous-capillary regime at intermediate times, where liquid is transported along a gradually increasing number of preferential flow paths of the size of one–two pores, and a third regime at longer times, where liquid, after having reached the outlet, is exclusively flowing along such flow paths and the two-phase fluid structures are stabilised. We observe that the fibrous layer presents significant variations in its microscopic morphology, which have an important effect on the pore invasion dynamics, and counteract the stabilising viscous force. Liquid transport is indeed affected by the presence of microstructure-induced capillary pressures acting adversely to the flow, leading to capillary fingering transport mechanism and unstable front displacement, even in the absence of hydrophobic treatments of the porous material. We propose a macroscopic model based on an effective contact angle that mimics the effects of the such a dynamic capillary pressure. Finally, we underline the significance of the results for the optimal design of face masks in an effort to mitigate the current COVID-19 pandemic.

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

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

Inhibiting Ostwald Ripening by Scaffolding Droplets

Nanoemulsions as colloidal dispersions of deformable nanodroplets promise wide range of applications in pharmaceuticals, cosmetics, and agriculture. The main limitation that reduces their industrial applications is stability, with Ostwald ripening acting as the main destabilization mechanism. Different from the conventional methods by functionalizing nanoemulsions with adequate ripening inhibitors, here we propose an alternative strategy to stabilize nanoemulsions by inhibiting Ostwald ripening. We report via Lattice Boltzmann method (LBM) and theoretical analysis that the evolution of droplets can be manipulated with the help of solid substrates, either along or against the direction of Ostwald ripening. It turns out that through pinning contact line of sessile droplets, heterogeneous substrates or solid nanoparticles can behave as a scaffold to suppress Ostwald ripening, to regulate droplet morphology and to enhance droplet stability. The identical curvature and unexpected stability of scaffolding droplets are then interpreted with free energy analysis. In addition, by simulating substrates with various heterogeneities and solid particles of different shapes, we demonstrate that it is a common phenomenon that scaffolding droplets can evolve beyond Ostwald ripening.

Force approach for the pseudopotential lattice Boltzmann method

One attractive feature of the original pseudopotential method consists on its simplicity of adding a force dependent on a nearest-neighbor potential function. In order to improve the method, regarding thermodynamic consistency and control of surface tension, different approaches were developed in the literature, such as multirange interactions potential and modified forcing schemes. In this work, a strategy to combine these enhancements with an appropriate interaction force field using only nearest-neighbor interactions is devised, starting from the desired pressure tensor. The final step of our procedure is implementing this external force by using the classical Guo forcing scheme. Numerical tests regarding static and dynamic flow conditions were performed. Static tests showed that current procedure is suitable to control the surface tension and phase densities. Based on thermodynamic principles, it is devised a solution for phase densities in a droplet, which states explicitly dependence on the surface tension and interface curvature. A comparison with numerical results suggest a physical inconsistency in the pseudopotential method. This fact is not commonly discussed in the literature, since most of studies are limited to the Maxwell equal area rule. However, this inconsistency is shown to be dependent on the equation of state (EOS), and its effects can be mitigated by an appropriate choice of Carnahan-Starling EOS parameters. Also, a droplet oscillation test was performed, and the most divergent solution under certain flow conditions deviated 7.5% from the expected analytical result. At the end, a droplet impact test against a solid wall was performed to verify the method stability, and it was possible to reach stable simulation results with density ratio of almost 2400 and Reynolds number of Re=373. The observed results corroborate that the proposed method is able to replicate the desired macroscopic multiphase behavior.

Benchmark cases for a multi-component Lattice-Boltzmann method in hydrostatic conditions

Hydrostatic properties of partially saturated granular materials at the pore scale are evaluated by the lattice Boltzmann method (LBM) using Palabos implementation of the multi-component multiphase Shan-Chen model. Benchmark cases are presented to quantify the discretization errors and the sensitivity to geometrical and physical properties. This work offers practical guidelines to design LBM simulations of multiphase problems in porous media. Namely, a solid walls retraction procedure is proposed to reduce discretization errors significantly, leading to quadratic convergence. On this basis the equilibrium shapes of pendular bridges simulated numerically are in good agreement with the Young-Laplace equation. Likewise, entry capillary pressure and meniscus profiles in tubes of various cross-sectional shapes are in agreement with analytical predictions. The main points of this article are summarized as:•Benchmark cases for a multi-component Lattice-Boltzmann method are illustrated to be a guideline to calibrate the method in hydrostatic conditions.•A wall retraction procedure is introduced to minimize discretization errors.

Lattice Boltzmann Method Applied to Nuclear Reactors—A Systematic Literature Review

Nuclear engineering requires computationally efficient methods to simulate different components and systems of plants. The Lattice Boltzmann Method (LBM), a numerical method with a mesoscopic approach to Computational Fluid Dynamic (CFD) derived from the Boltzmann equation and the Maxwell–Boltzmann distribution, can be an adequate option. The purpose of this paper is to present a review of the recent applications of the Lattice Boltzmann Method in nuclear engineering research. A systematic literature review using three databases (Web of Science, Scopus, and ScienceDirect) was done, and the items found were categorized by the main research topics into computational fluid dynamics and neutronic applications. The features of the problem addressed, the characteristics of the numerical method, and some relevant conclusions of each study are resumed and presented. A total of 45 items (25 for computational fluid dynamics applications and 20 for neutronics) was found on a wide range of nuclear engineering problems, including thermal flow, turbulence mixing of coolant, sedimentation of impurities, neutron transport, criticality problem, and other relevant issues. The LBM results in being a flexible numerical method capable of integrating multiphysics and hybrid schemes, and is efficient for the inner parallelization of the algorithm that brings a widely applicable tool in nuclear engineering problems. Interest in the LBM applications in this field has been increasing and evolving from early stages to a mature form, as this review shows.

Study on the Collapse Process of Cavitation Bubbles Near the Concave Wall by Lattice Boltzmann Method Pseudo-Potential Model

In this paper, the lattice Boltzmann pseudo-potential model coupled the Carnahan–Starling (C-S) equation of state and Li’s force scheme are used to study the collapse process of cavitation bubbles near the concave wall. It mainly includes the collapse process of the single and double cavitation bubbles in the near-wall region. Studies have shown that the collapse velocity of a single cavitation bubble becomes slower as the additional pressure reduces, and the velocity of the micro-jet also decreases accordingly. Moreover, the second collapse of the cavitation bubble cannot be found if the additional pressure reduces further. When the cavitation bubble is located in different angles with vertical direction, its collapse direction is always perpendicular to the wall. If the double cavitation bubbles are arranged vertically, the collapse process of the upper bubble will be quicker, as the relative distance increases. When the relative distance between the bubbles is large enough, no second collapse can be found for the upper bubble. On the other hand, when two cavitation bubbles are in the horizontal arrangement, the suppression effect between cavitation bubbles decreases as the relative distance between the bubbles increases and the collapse position of cavitation bubbles moves from the lower part to the upper part.

Multipseudopotential interaction models for thermal lattice Boltzmann method simulations

In this work, in the first instance, the multipseudopotential interaction (MPI) model's capabilities are extended for hydrodynamic simulations. This is achieved by combining MPI with the multiple-relaxation-time collision operator and with surface tension modification methods. A method of approaching thermodynamic consistency is also proposed, which consists of splitting the ɛ_{j} term into separate terms. One of these terms is used in the calculation of the interparticle force, and the second one is used in the forcing scheme. Secondly, MPI is combined with thermal models in order to simulate droplet evaporation and bubble nucleation in pool boiling. Thermal coupling is implemented using a double distribution function thermal model and a hybrid thermal model. It is found that MPI thermal models obey the D^{2}-law closely for droplet evaporation. MPI is also found to correctly simulate bubble nucleation and departure from the heating element during nucleate pool boiling. It can be suggested that MPI thermal models are comparatively better suited to thermal simulations at low reduced temperatures than single pseudopotential interaction models, although such cases remain very challenging. Droplet evaporation simulations are carried out at a reduced temperature (T_{r}) of 0.6 by setting the parameters in the Peng-Robinson equation of state to a=1/6272 and b=1/168.

Mesoscopic method to study water flow in nanochannels with different wettability

Molecular dynamics (MD) simulations is currently the most popular and credible tool to model water flow in nanoscale where the conventional continuum equations break down due to the dominance of fluid-surface interactions. However, current MD simulations are computationally challenging for the water flow in complex tube geometries or a network of nanopores, e.g., membrane, shale matrix, and aquaporins. We present a novel mesoscopic lattice Boltzmann method (LBM) for capturing fluctuated density distribution and a nonparabolic velocity profile of water flow through nanochannels. We incorporated molecular interactions between water and the solid inner wall into LBM formulations. Details of the molecular interactions were translated into true and apparent slippage, which were both correlated to the surface wettability, e.g., contact angle. Our proposed LBM was tested against 47 published cases of water flow through infinite-length nanochannels made of different materials and dimensions-flow rates as high as seven orders of magnitude when compared with predictions of the classical no-slip Hagen-Poiseuille (HP) flow. Using the developed LBM model, we also studied water flow through finite-length nanochannels with tube entrance and exit effects. Results were found to be in good agreement with 44 published finite-length cases in the literature. The proposed LBM model is nearly as accurate as MD simulations for a nanochannel, while being computationally efficient enough to allow implications for much larger and more complex geometrical nanostructures.

Simulation of Boiling Heat Transfer at Different Reduced Temperatures with an Improved Pseudopotential Lattice Boltzmann Method

The pseudopotential Lattice Boltzmann Method has attracted much attention in the recent years for the simulation of boiling heat transfer. Many studies have been published recently for the simulation of the bubble cycle (nucleation, growth and departure from a heated surface). This paper puts forward two-dimensional simulations of bubble nucleation, growth and departure using an improved pseudopotential Lattice Boltzmann Model from the literature at different reduced temperatures, Tr=0.76 and Tr=0.86. Two different models using the Bhatnagar–Gross–Krook (BGK) and the Multiple-Relaxation-Time (MRT) collision operators with appropriate forcing schemes are used. The results for pool boiling show that the bubbles exhibit axial symmetry during growth and departure. Numerical results of departure diameter and release period for pool boiling are compared against empirical correlations from the literature by varying the gravitational acceleration. Reasonable agreement is observed. Nucleate boiling trends with heat flux are also captured by the simulations. Numerical results of flow boiling simulations are compared by varying the Reynolds number for both reduced temperatures with the MRT model. It was found that the departure diamenter and release period decreases with the increase of the Reynolds number. These results are a direct effect of the drag force. Proper conclusions are commented at the end of the paper.

Lattice Boltzmann Modeling of Condensation Heat Transfer on Downward-Facing Surfaces with Different Wettabilities

The model of vapor condensation heat transfer on downward-facing surfaces with different wettabilities is built by a two-dimensional (2D) lattice Boltzmann method. Dynamic evolution of condensate microdroplets on different wettability surfaces is simulated and the influence on heat transfer performance is analyzed. Moreover, the mechanism of a heterogeneous wettability surface enhancing condensation heat transfer is explored by investigating the condensate behaviors in the process of condensation. The numerical results indicate that as the contact angle of the homogeneous wettability surface increases, the initial nucleation time of the condensate is prolonged, while the departure time of the condensate is reduced significantly. The temperature adjacent to the gas-liquid interface, especially in the three-phase contact line region, is much higher than elsewhere due to the release of latent heat during condensation. Coalescence and detachment behaviors of condensate droplets cause the average heat flux to fluctuate locally with time. For the hybrid wettability surface, if the proportion of hydrophobic regions is small, the condensation heat transfer performance will be deteriorated. However, increasing the hydrophobic-hydrophilic ratio has a positive effect on enhancing heat transfer. It is found that a critical hydrophobic-hydrophilic ratio exists to optimize the heat transfer performance. For the gradient wettability surface, directional migration induced by capillary force facilitates the removal of condensate droplets, thereby enhancing the condensation heat transfer. Furthermore, a larger wetting gradient benefits to further improve the heat transfer performance. The results are valuable for optimally designing the heat transfer enhancement of vapor condensation on functionalized surfaces with heterogeneous wettability.

Modeling and scale-bridging using machine learning: nanoconfinement effects in porous media

Fine-scale models that represent first-principles physics are challenging to represent at larger scales of interest in many application areas. In nanoporous media such as tight-shale formations, where the typical pore size is less than 50 nm, confinement effects play a significant role in how fluids behave. At these scales, fluids are under confinement, affecting key properties such as density, viscosity, adsorption, etc. Pore-scale Lattice Boltzmann Methods (LBM) can simulate flow in complex pore structures relevant to predicting hydrocarbon production, but must be corrected to account for confinement effects. Molecular dynamics (MD) can model confinement effects but is computationally expensive in comparison. The hurdle to bridging MD with LBM is the computational expense of MD simulations needed to perform this correction. Here, we build a Machine Learning (ML) surrogate model that captures adsorption effects across a wide range of parameter space and bridges the MD and LBM scales using a relatively small number of MD calculations. The model computes upscaled adsorption parameters across varying density, temperature, and pore width. The ML model is 7 orders of magnitude faster than brute force MD. This workflow is agnostic to the physical system and could be generalized to further scale-bridging applications.

Chemical-potential multiphase lattice Boltzmann method with superlarge density ratios

The liquid-gas density ratio is a key property of multiphase flow methods to model real fluid systems. Here, a chemical-potential multiphase lattice Boltzmann method is constructed to realize extremely large density ratios. The simulations show that the method reaches very low temperatures, at which the liquid-gas density ratio is more than 10^{14}, while the thermodynamic consistency is still preserved. Decoupling the mesh space from the momentum space through a proportional coefficient, a smaller mesh step provides denser lattice nodes to exactly describe the transition region and the resulting dimensional transformation has no loss of accuracy. A compact finite-difference method is applied to calculate the discrete derivatives in the mesh space with high-order accuracy. These enhance the computational accuracy of the nonideal force and suppress the spurious currents to a very low level, even if the density ratio is up to tens of thousands. The simulation of drop splashing verifies that the present model is Galilean invariant for the dynamic flow field. An upper limit of the chemical potential is used to reduce the influence of nonphysical factors and improve the stability.

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

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

Attainment of rigorous thermodynamic consistency and surface tension in single-component pseudopotential lattice Boltzmann models via a customized equation of state

The lack of thermodynamic consistency is a well-recognized problem in the single-component pseudopotential lattice Boltzmann models which prevents them from replicating accurate liquid and vapor phase densities; i.e., current models remain unable to exactly match coexisting density values predicted by the associated thermodynamic model. Most of the previous efforts had attempted to solve this problem by introducing tuning parameters, whose determination required empirical trial and error until acceptable thermodynamic consistency was achieved. In this study, we show that the problem can be alternatively solved by properly designing customized equations of state (EOSs) that replace any cubic EOS of choice during the computation of effective mass used in Shan-Chen forces. A two-parameter cubic-shaped customized EOS is introduced. Contrary to previous efforts, customization parameters in the new EOS are nonempirical and are rather derived from solving the integral mechanical stability equation, which neglects the need for any type of tuning for the attainment of rigorous thermodynamic consistency. The proposed approach reduces the errors of the coexisting densities and saturated pressure in the simulation to a maximum of 0.01% within the liquid-vapor density ratio range from O(1) to O(10^{4}), which had not been achieved in any of the previous tuning-based efforts. A straightforward way for achieving the desired surface tension via the customized EOS is also provided.

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.

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.

Effects of hysteresis window on contact angle hysteresis behaviour at large Bond number

Contact angle hysteresis, defined as the difference between advancing and receding contact angles, is an important phenomenon in multiphase flow on a wetting surface. In this study, a modified pseudo-potential lattice Boltzmann (LB) multiphase model with tunable surface tension is proposed, which is further coupled with the geometrical formulation contact angle scheme to investigate the motion of droplets invoking the contact angle hysteresis. We focus on the dynamic behaviour of droplets driven by a body force at the Bond number ranging from 1 to 6, which is defined as the ratio of the body force to the capillary force. The droplet morphology change is examined by varying (i) the Bond number and (ii) the hysteresis window. Results show the droplet morphology evolution can be classified into different stages, including stretch, relaxation, and equilibrium. The droplet oscillation phenomenon at large Bond numbers at the equilibrium stage is observed for the first time. In addition, it is found that such oscillation can lead to the breakup and/or coalescence of droplets when the surface waves spread on the top of the droplet. Furthermore, there is slight oscillation of the normalized length, width and height at the equilibrium stage for the neutral hysteresis window while more dramatic oscillation will appear for the hydrophobic hysteresis window.

Numerical simulations of self-diffusiophoretic colloids at fluid interfaces

The dynamics of active colloids is very sensitive to the presence of boundaries and interfaces which therefore can be used to control their motion. Here we analyze the dynamics of active colloids adsorbed at a fluid-fluid interface. By using a mesoscopic numerical approach which relies on an approximated numerical solution of the Navier-Stokes equation, we show that when adsorbed at a fluid interface, an active colloid experiences a net torque even in the absence of a viscosity contrast between the two adjacent fluids. In particular, we study the dependence of this torque on the contact angle of the colloid with the fluid-fluid interface and on its surface properties. We rationalize our results via an approximate approach which accounts for the appearance of a local friction coefficient. By providing insight into the dynamics of active colloids adsorbed at fluid interfaces, our results are relevant for two-dimensional self assembly and emulsion stabilization by means of active colloids.

Central-Moments-Based Lattice Boltzmann for Associating Fluids: A New Integrated Approach

Dynamic and thermodynamic behaviors of associating fluids play a crucial role in various science and engineering disciplines. Cubic plus association equation of state (CPA EOS) is implemented in a central-moments-based lattice Boltzmann method (LBM) in order to mimic the thermodynamic behavior of associating fluids. The pseudopotential approach is selected to model the multiphase thermodynamic characteristics such as reduced density of associating fluids. The priority of central-moments-based approach over multiple-relaxation-time collision operator is highlighted by performing double shear layers. The integration of central-moments-based LBM and CPA EOS is useful to simulate the dynamic and thermodynamic characteristics of associating fluids at high flow rate conditions, which is extended to high-density ratio scenarios by increasing the anisotropy order of gradient operator. In order to increase the stability of the model, a higher anisotropy order of the gradient operator is implemented; about 34 present reduction in spurious velocities is noticed in some cases. The type of gradient operator considerably affects the model thermodynamic consistency. Finally, the model is validated by observing a straight line in the Laplace law test. Prediction of thermodynamic behaviors of associating fluids is of significance in various applications including biological processes as well as fluid flow in porous media.

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 Investigation of the Effect of Viscoelasticity on Drop Retraction and the Evaluation of Interfacial Tension between Polymer Melts

The purpose of this paper is twofold. The first is to numerically investigate and reveal the effect of polymer viscoelasticity on the retraction of a deformed drop using the lattice Boltzmann (LB) method and polymer kinetic theory. More importantly, the second is to propose a novel method to evaluate the interfacial tension between polymer melts based on the numerical study. Compared with the conventional deformed drop retraction method (DDRM), the present method is designed to greatly reduce the impact of polymer viscoelasticity on measuring interfacial tension. To verify, the interfacial tension between molten PP and POE is evaluated using the proposed method and obviously closer result to the true value is shown.

Effect of Wetting Transition during Multiphase Displacement in Porous Media

The effects of wettability on multiphase displacement in porous media have been studied extensively in the past, and the contact angle is identified as an important factor influencing the displacement patterns. At the same time, it has been found that the effective contact angle can vary drastically in a time-dependent manner on rough surfaces due to the Cassie-Wenzel wetting transition. In this study, we develop a theoretical model at the pore scale describing the apparent contact angle on a rough interface as a function of time. The theory is then incorporated into the lattice Boltzmann method for simulation of multiphase displacement in disordered porous media. A dimensionless time ratio, Dy, describing the relative speed of the wetting transition and pore invasion is defined. We show that the displacement patterns can be significantly influenced by Dy, where more trapped defending ganglia are observed at large Dy values, leading to lower displacement efficiency. We investigate the mobilization of trapped ganglia through identifying different mobilization dynamics during displacement, including translation, coalescence, and fragmentation. Agreement is observed between the mobilization statistics and the total pressure gradient across a wide range of Dy values. Understanding the effect of the wetting transition during multiphase displacement in porous media is of importance for applications such as carbon geosequestration and oil recovery, especially for porous media where solid surface roughness cannot be neglected.

Application of high-order lattice Boltzmann pseudopotential models

Higher-order lattice Boltzmann (LB) pseudopotential models have great potential for solving complex fluid dynamics in various areas of modern science. The discreteness of the lattice discretization makes these models an attractive choice due to their flexibility, capacity to capture hydrodynamic details, and inherent adaptability to parallel computations. Despite those advantages, the discreteness makes high-order LB models difficult to apply due to the larger lattice structure, for which basic fundamental properties, namely diffusion coefficient and contact angle, remain unknown. This work addresses this by providing general continuum solutions for those two basic properties and demonstrating these solutions to compare favorably against known theory. Various high-order LB models are shown to reproduce the sinusoidal decay of a binary miscible mixture accurately and consistently. Furthermore, these models are shown to reproduce neutral, hydrophobic, and hydrophilic contact angles. Discrete differences are shown to exist, which are captured at the discrete level and confirmed through droplet shape analysis. This work provides practical tools that allow for high-order LB pseudopotential models to be used to simulate multicomponent flows.

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.

Lattice Boltzmann model for weakly compressible flows

We present an energy conserving lattice Boltzmann model based on a crystallographic lattice for simulation of weakly compressible flows. The theoretical requirements and the methodology to construct such a model are discussed. We demonstrate that the model recovers the isentropic sound speed in addition to the effects of viscous heating and heat flux dynamics. Several test cases for acoustics and thermal and thermoacoustic flows are simulated to show the accuracy of the proposed model.

Discrete fluidization of dense monodisperse emulsions in neutral wetting microchannels

The rheology of pressure-driven flows of two-dimensional dense monodisperse emulsions in neutral wetting microchannels is investigated by means of mesoscopic lattice Boltzmann simulations, capable of handling large collections of droplets, in the order of several hundreds. The simulations reveal that the fluidization of the emulsion proceeds through a sequence of discrete steps, characterized by yielding events whereby layers of droplets start rolling over each other, thus leading to sudden drops of the relative effective viscosity. It is shown that such discrete fluidization is robust against loss of confinement, namely it persists also in the regime of small ratios of the droplet diameter over the microchannel width. We also develop a simple phenomenological model which predicts a linear relation between the relative effective viscosity of the emulsion and the product of the confinement parameter (global size of the device over droplet radius) and the viscosity ratio between the disperse and continuous phases. The model shows excellent agreement with the numerical simulations. The present work offers new insights to enable the design of microfluidic scaffolds for tissue engineering applications and paves the way to detailed rheological studies of soft-glassy materials in complex geometries.

A Numerical Study of Micro-Droplet Spreading Behaviors on Wettability-Confined Tracks Using a Three-Dimensional Phase-Field Lattice Boltzmann Model

Wettability-confined tracks have been extensively used in open-surface microfluidic devices for their high capacity of transporting droplet pumplessly. Inspired by the experimental work of Sen et al. [ Langmuir 2018 , 34 , 1899 - 1907 ], in the present study, a three-dimensional phase-field lattice Boltzmann model is developed and used to investigate the spreading behaviors of microdroplet on a series of wettability-confined tracks. The experimental findings are successfully reproduced through our simulation, where three distinct stages of droplet spreading on the horizontal wettability-confined diverging track are fairly exhibited, that is, the initial stage with droplet front spreading quickly, the intermediate stage with both droplet front and bulge moving forward at a constant speed, and the final stage with droplet front decelerating gradually. Moreover, a parametric study of track divergence angle is further performed, and the influential mechanism of track divergence angle on droplet spreading is further revealed. It is demonstrated that track divergence is responsible for the Laplace pressure gradient and capillary force inside the droplet, which drives the droplet bulge to move forward on the diverging track. With an increase in divergence angle, the capillary force increases linearly, which increases the droplet spreading speed at the initial and intermediate stages, while the peak capillary force comes earlier, and consequently the final decelerating stage comes earlier. On the basis of the parametric study and droplet volume conservation rule, a power law relation between track divergence angle and droplet spreading is proposed, which helps to identify the start of final decelerating stage. Finally, the droplet spreading over various inclined tracks is explored, which can be achieved only when the capillary force at the beginning is larger than the droplet gravity component along the inclined track surface.

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.

Multiphase flows of N immiscible incompressible fluids: Conservative Allen-Cahn equation and lattice Boltzmann equation method

In this paper, we develop a conservative phase-field method for interface-capturing among N (N≥2) immiscible fluids, the evolution of the fluid-fluid interface is captured by conservative Allen-Cahn equation (CACE), and the interface force of N immiscible fluids is incorporated to Navier-Stokes equation (NSE) by chemical potential form. Accordingly, we propose a lattice Boltzmann equation (LBE) method for solving N (N≥2) immiscible incompressible NSE and CACE at high density and viscosity contrasts. Numerical simulations including stationary droplets, Rayleigh-Taylor instability, spreading of liquid lenses, and spinodal decompositions are carried out to show the accuracy and capability of present LBE, and the results show that the predictions by use of the present LBE agree well with the analytical solutions and/or other numerical results.

Thermodynamic consistency of a pseudopotential lattice Boltzmann fluid with interface curvature

Thermodynamic consistency of pseudopotential lattice Boltzmann models is a major topic that needs comprehensive evaluations. When interface is flat, pseudopotential models can give density-pressure isotherms in excellent agreement with those from equation of state. When interface is curved, thermodynamic equilibriums are affected by interface curvature, and consistency of pseudopotential models has not been systematically evaluated. In this study, we show that the effect of Laplace pressure on phase equilibrium is quantitatively consistent with Kelvin equation at high reduced temperatures (≥0.7). At low temperatures, inconsistency that can be attributed to the effect of orientation of the interface was noted, and it can be improved by tuning of the pseudopotential. By relating interfacial tension of a simulated fluid to that of a real fluid, the lattice spacing of pseudopotential model is found to be on the order of several molecular diameters, the typical range of intermolecular interactions. Interfacial thickness at different temperatures in pseudopotential model compared well with experiments and molecular dynamics simulations, which confirms that the calculated length scale is reasonable. Evaluation of a free energy lattice Boltzmann model indicate that it is consistent with Kelvin equation at high temperatures. The free energy model, however, is not as accurate as the tested pseudopotential model, and discrepancies may come from the relative inaccuracies in the predictions of vapor densities and the thinner interfaces.

Multicomponent flow on curved surfaces: A vielbein lattice Boltzmann approach

We develop and implement a finite difference lattice Boltzmann scheme to study multicomponent flows on curved surfaces, coupling the continuity and Navier-Stokes equations with the Cahn-Hilliard equation to track the evolution of the binary fluid interfaces. The standard lattice Boltzmann method relies on regular Cartesian grids, which makes it generally unsuitable to study flow problems on curved surfaces. To alleviate this limitation, we use a vielbein formalism to write the Boltzmann equation on an arbitrary geometry, and solve the evolution of the fluid distribution functions using a finite difference method. Focusing on the torus geometry as an example of a curved surface, we demonstrate drift motions of fluid droplets and stripes embedded on the surface of the torus. Interestingly, they migrate in opposite directions: fluid droplets to the outer side while fluid stripes to the inner side of the torus. For the latter we demonstrate that the global minimum configuration is unique for small stripe widths, but it becomes bistable for large stripe widths. Our simulations are also in agreement with analytical predictions for the Laplace pressure of the fluid stripes, and their damped oscillatory motion as they approach equilibrium configurations, capturing the corresponding decay timescale and oscillation frequency. Finally, we simulate the coarsening dynamics of phase separating binary fluids in the hydrodynamics and diffusive regimes for tori of various shapes, and compare the results against those for a flat two-dimensional surface. Our finite difference lattice Boltzmann scheme can be extended to other surfaces and coupled to other dynamical equations, opening up a vast range of applications involving complex flows on curved geometries.

Hybridized method of pseudopotential lattice Boltzmann and cubic-plus-association equation of state assesses thermodynamic characteristics of associating fluids

It is crucial to properly describe the associating fluids in terms of phase equilibrium behaviors, which are needed for design, operation, and optimization of various chemical and energy processes. Pseudopotential lattice Boltzmann method (LBM) appears to be a reliable and efficient approach to study thermodynamic behaviors and phase transition of complex fluid systems. However, when cubic equations of state (EOSs) are incorporated into single-component multiphase LBM, simulation results are not well matched with experimental data. This study presents the utilization of cubic-plus-association (CPA) EOS in the LBM structure to obtain more accurate modeling results for associating fluids. An approach based on the global search optimization algorithm is introduced to find the optimal association parameters of CPA EOS for water and primary alcohols in the lattice units. The thermodynamic consistency is verified by the Maxwell construction and is also improved by the forcing scheme of [Q. Li, K. H. Luo, and X. J. Li, Phys. Rev. E 86, 016709 (2012)10.1103/PhysRevE.86.016709]. The spurious velocity is reduced with increasing isotropy in the gradient operator. Furthermore, an extended version of CPA EOS is introduced, which increases the system stability at low reduced temperatures. There is a very good match between the LBM results and experimental data, confirming the reliability of the model developed in the present study. The introduced approach has potential to be employed for simulating transport phenomena and interfacial characteristics of associating fluids in porous systems.

Density gradient calculation in a class of multiphase lattice Boltzmann models

The multiphase lattice Boltzmann (LB) models based on pairwise interactions show great potential for simulating multiphase flows due to the conceptual and computational simplicity. Although the dynamics of multiphase flows are reproduced by the pairwise interaction force, the gradient of density (or effective density, i.e., pseudopotential) is implicitly involved in these models via the specialized forcing scheme or the consistent scheme for ɛ^{3}-order term. This work focuses on the calculation of density gradient in this class of multiphase LB models. Theoretical analyses are first carried out to reveal the involvement and calculation of density gradient. On the basis of a low Mach number approximation, an improved scheme is then proposed to calculate the density gradient for the recent LB model with self-tuning equation of state. Analytical and numerical calculations show that the improved scheme is more accurate and can help to reduce the numerical error when the reduced temperature is relatively low.

Lattice Boltzmann Simulation on Droplet Flow through 3D Metal Foam

The hydrodynamics of droplets passing through metal foam is investigated using the lattice Boltzmann method (LBM). The accurate 3D porous structure for the simulation is generated by X-ray micro-computed tomography. The simulated results are in good agreement with the experimental ones using high-speed video. The simulated results show that for droplets passing metal foam, there is a critical capillary number, Cac (around 0.061), above which the droplet continues to deform until it breaks up. The simulated results show that the capillary number, droplet size, pores diameter, and thickness of metal foam have the significant effect of droplets deforming and breaking up when the droplets pass through the metal foam. To avoid the calescence of two droplets at the inlet zone of the metal foam, the distance between droplets should be larger than three times the diameter of the droplet.

Hydrodynamic correlations of viscoelastic fluids by multiparticle collision dynamics simulations

The emergent fluctuating hydrodynamics of a viscoelastic fluid modeled by the multiparticle collision dynamics (MPC) approach is studied. The fluid is composed of flexible, Gaussian phantom polymers that interact by local momentum-conserving stochastic MPCs. For comparison, the analytical solution of the linearized Navier-Stokes equation is calculated, where viscoelasticity is taken into account by a time-dependent shear relaxation modulus. The fluid properties are characterized by the transverse velocity autocorrelation function in Fourier space as well as in real space. Various polymer lengths are considered-from dumbbells to (near-)continuous polymers. Viscoelasticity affects the fluid properties and leads to strong correlations, which overall decay exponentially in Fourier space. In real space, the center-of-mass velocity autocorrelation function of individual polymers exhibits a long-time tail, independent of the polymer length, which decays as t^-3/2, similar to a Newtonian fluid, in the asymptotic limit t → ∞. Moreover, for long polymers, an additional power-law decay appears at time scales shorter than the longest polymer relaxation time with the same time dependence, but negative correlations, and the polymer length dependence L^-1/2. Good agreement is found between the analytical and simulation results.