Mesoscopic modeling of a two-phase flow in the presence of boundaries: The contact angle

Mesoscopic Model for Disjoining Pressure Effects in Nanoscale Thin Liquid Films and Evaporating Extended Meniscuses

Disjoining pressure effect is the key to describe contact line dynamics, micro/nanoscale liquid-vapor phase change heat transfer, and liquid transport in nanopores. In this paper, by combining a mesoscopic approach for nanoscale liquid-vapor interfacial transport and a mean-field approximation of the long-range solid-fluid molecular interaction, a mesoscopic model for the disjoining pressure effect in nanoscale thin liquid films is proposed. The capability of this model to delineate the disjoining pressure effect is validated. We demonstrate that the Hamaker constant determined from our model agrees very well with molecular dynamics (MD) simulation and that the transient evaporation/condensation mass flux predicted by this mesoscopic model is also consistent with the kinetic theory. Using this model, we investigate the characteristics of the evaporating extended meniscus in a nanochannel. The nonevaporating film region, the evaporating thin-film region, and the intrinsic meniscus region are successfully captured by our model. Our results suggest that the apparent contact angle and thickness of the nonevaporating liquid film are self-tuned according to the evaporation rate, and a higher evaporation rate results a in larger apparent contact angle and thinner nonevaporating liquid film. We also show that disjoining pressure plays a dominant role in the nonevaporating film region and suppresses the evaporation in this region, while capillary pressure dominates the intrinsic meniscus region. Strong evaporation takes place in the thin-film region, and both the disjoining pressure and capillary pressure contribute to the total pressure difference that delivers the liquid from the intrinsic meniscus region to the evaporating thin-film region, compensating for the liquid mass loss due to strong evaporation. Our work provides a new avenue for investigating thin liquid film spreading, liquid transport in nanopores, and microscopic liquid-vapor phase change heat/mass transfer mechanisms near the three-phase contact line region.

Pore-Scale Study on Shale Oil-CO2-Water Miscibility, Competitive Adsorption, and Multiphase Flow Behaviors

Due to the fracturing fluid imbibition and primary water, oil-water two-phase fluids generally exist in shale nanoporous media. The effects of water phase on shale oil recovery and geological carbon sequestration via CO2 huff-n-puff is non-negligible. Meanwhile, oil-CO2 miscibility after CO2 huff-n-puff also has an important effect on oil-water two-phase flow behaviors. In this work, by considering the oil-CO2 competitive adsorption behaviors and the effects of oil-CO2 miscibility on water wettability, an improved multicomponent and multiphase lattice Boltzmann method is proposed to study the effects of water phase on CO2 huff-n-puff. Additionally, the effects of oil-CO2 miscibility on oil-water flow behaviors and relative permeability are also discussed. The results show that due to Jamin's effect of water droplets in oil-wetting pores and the capillary resistance of bridge-like water phase in water-wetting pores, CO2 can hardly diffuse into the oil phase, causing a large amount of remaining oil. As water saturation increases, Jamin's effect and the capillary resistance become more pronounced, and the CO2 storage mass gradually decreases. Then, based on the results from molecular dynamics simulations, the influences of oil-CO2 miscibility on oil-water relative permeability in calcite nanoporous media are studied, and as the oil mass percentage in the oil-CO2 miscible system decreases, the oil/water relative permeability decreases/increases. The improved lattice Boltzmann model can be readily extended to quantitatively calculate geological CO2 storage mass considering water saturation and calculate the accurate oil-water relative permeability based on the real 3D digital core.

Improved partially saturated method for the lattice Boltzmann pseudopotential multicomponent flows

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

Equation-of-state-dependent surface free-energy density for wettability in lattice Boltzmann method

In thermodynamic theory, the liquid-vapor fluids can be described by a single multiphase equation of state and the surface wettability is usually characterized by the surface free-energy density. In this work, we propose an equation-of-state-dependent surface free-energy density for the wettability of the liquid-vapor fluids on a solid surface, which can lead to a simple closed-form analytical expression for the contact angle. Meanwhile, the thermodynamically derived equilibrium condition is equivalent to the geometric formulation of the contact angle. To numerically validate the present surface free-energy density, the mesoscopic multiphase lattice Boltzmann model with self-tuning equation of state, which is strictly consistent with thermodynamic theory, is employed, and the two-dimensional wetting condition treatment is extended to the three-dimensional situation with flat and curved surfaces. Two- and three-dimensional lattice Boltzmann simulations of static droplets on flat and curved surfaces are first performed, and the obtained contact angles agree well with the closed-form analytical expression. Then, the three-dimensional lattice Boltzmann simulation of a moving droplet on an inclined wall, which is vertically and sinusoidally oscillated, is carried out. The dynamic contact angles well satisfy the Cox-Voinov law. The droplet movement regimes are consistent with previous experiments and two-dimensional simulations. The dependence of the droplet overall velocity with respect to the dimensionless oscillation strength is also discussed in detail.

Wall wettability effect on process of collapse of single cavitation bubbles in near-wall region using pseudo-potential lattice Boltzmann method

This study investigates the effect of wall wettability on cavitation collapse based on a large-density-ratio lattice Boltzmann method (LBM) pseudo-potential model. The validity and superiority of the proposed model in simulation of cavitation under complex conditions are confirmed by comparing with theories, experiments, and numerical results by other models. Our simulations indicate that wall wettability has a significant influence on near-wall cavitation of an order no less than the effect of the initial bubble distance. A criterial initial distance exists in near-wall cavitation within which the micro-jet will direct toward the wall. This criterial distance is shown to be positively correlated with the contact angle by a cosine function. Within this distance, the lifetime of the bubble decreases by up to 50%, and the increase of the maximum micro-jet velocity and collapse pressure are up to 131% and 65%, respectively, when the contact angle increases from the hydrophilic 53° to the hydrophobic 113°. Without considering the shock-wave mechanism, the impact pressure transmitted to the hydrophilic wall is of the same order as the maximum collapse pressure while the impact velocity is an order smaller than the maximum micro-jet velocity. Wall wettability affects collapse through the Bjerknes force and the pressure around the bubble. Preliminary analysis also suggests that the relation between the pressure difference and the intensity of collapse exhibits more patterns than we have assumed, which fits a logistic curve well, and appears not changing with the contact angle or the initial bubble distance.

Multiphase curved boundary condition in lattice Boltzmann method

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

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

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

Characterization of condensation on nanostructured surfaces and associated thermal hydraulics using a thermal lattice Boltzmann method

The dynamics of the condensation process on nanostructured surfaces can be modulated substantially by tuning the surface architecture. Present study uses the mesoscopic framework of lattice Boltzmann method (LBM) to explore the role of surface morphology and cold spot temperature in determining the visual state of the condensate droplet, mode of nucleation, and associated rates of energy and mass interactions. A multiple relaxation time-(MRT)-based LBM solver, coupled with pseudopotential model, has been developed to simulate a rectangular domain of saturated vapor, housing a cold spot on the bottom rough surface. Superhydrophobicity has been achieved for certain combinations of surface parameters, with the intercolumn spacing being the most influential one. Gradual increase in the spacing modifies the nucleation mode from top through side to bottom, while the droplet changes from Cassie to Wenzel state. The Cassie drop in top nucleation mode exhibits the largest contact angle and least rate of surface heat transfer. Both types of Wenzel drops display large rate of condensation and two peaks in heat transfer, along with very short nucleation time in comparison with Cassie drops. Couple of phase diagrams have been developed combining all four scenarios of condensation predicted by the present model. One important novelty of the present study is the consideration of nonisothermal condition within LB structure. Enhancement in the degree of subcooling at the cold spot encourages greater condensation and Cassie-to-Wenzel transition.

Simulations of surface acoustic wave interactions on a sessile droplet using a three-dimensional multiphase lattice Boltzmann model

This study reports the development of a three-dimensional numerical model for acoustic interactions with a microscale sessile droplet under surface acoustic wave (SAW) excitation using the lattice Boltzmann method (LBM). We first validate the model before SAW interactions are added. The results demonstrate good agreement with the analytical results for thermodynamic consistency, Laplace law, static contact angle on a flat surface, and droplet oscillation. We then investigate SAW interactions on the droplet, with resonant frequencies ranging 61.7-250.1 MHz. According to our findings, an increase in wave amplitude elicits an increase in streaming velocity inside the droplet, causing internal mixing, and further increase in wave amplitude leads to pumping and jetting. The boundaries of wave amplitude at various resonant frequencies are predicted for mixing, pumping, and jetting modes. The modeling predictions on the roles of forces (SAW, interfacial tension, inertia, and viscosity) on the dynamics of mixing, pumping, and jetting of a droplet are in good agreement with observations and experimental data. The model is further applied to investigate the effects of SAW substrate surface wettability, viscosity ratio, and interfacial tension on SAW actuation onto the droplet. This work demonstrates the capability of the LBM in the investigation of acoustic wave interactions between SAW and a liquid medium.

Fluid-wall interactions in pseudopotential lattice Boltzmann models

Designing proper fluid-wall interaction forces to achieve proper wetting conditions is an important area of interest in pseudopotential lattice Boltzmann models. In this paper, we propose a modified fluid-wall interaction force that applies for pseudopotential models of both single-component fluids and partially miscible multicomponent fluids, such as hydrocarbon mixtures. A reliable correlation that predicts the resulting liquid contact angle on a flat solid surface is also proposed. This correlation works well over a wide variety of pseudopotential lattice Boltzmann models and thermodynamic conditions.

Assessing the influence of pore structure formation on heavy metal immobilization through image-based CFD

Porous media are widely adopted as immobilization sorbents in environmental engineering. The microscale difference in pore structure formation causes significant deflection in a vast landscape. Computational fluid dynamics (CFD) offers a comparative approach to evaluate the individual influence from pore structure formation with strictly controlled surface and volume properties. This paper presents a comprehensive comparison between the performance of cylindrical media and spherical-media in heavy metal immobilization. Digital testing was performed to measure the surface area, specific surface area, density and porosity. Image-based input technique was developed to reconstruct the cylindrical media. It was found that although the surface area, specific surface area and porosity were the same, the spherical media still had an accelerated immobilization rate. Results further showed that the spherical media in floatation arrangement had an immobilization rate of 16% higher than the cylindrical media with the same surface properties. Non-floatation arrangement of the spherical media caused a reduction in immobilization capacity up to 32.8% lower than the cylindrical media. The cylindrical media demonstrated an advantage of being structurally stable under high porosity, the latter of which resulted in an increased immobilization capacity compared with the spherical-media. The results suggest that the cylindrical bio-microstructure is desirable for heavy metal immobilization in a non-flotational environment. The computational approach provides a digital solution to evaluate the immobilization in 3D architected media. The proposed testing methods are feasible for both experimentally obtained images and structures from algorithm-generation.

Algorithmic augmentation in the pseudopotential-based lattice Boltzmann method for simulating the pool boiling phenomenon with high-density ratio

The pseudopotential-based lattice Boltzmann method (LBM), despite enormous potential in facilitating natural development and migration of interfaces during multiphase simulation, remains restricted to low-density ratios, owing to inherent thermodynamic inconsistency. The present paper focuses on augmenting the basic algorithm by enhancing the isotropy of the discrete equation and thermodynamic consistency of the overall formulation, to expedite simulation of pool boiling at higher-density ratios. Accordingly, modification is suggested in the discrete form of the updated interparticle interaction term, by expanding the discretization to the eighth order. The proposed amendment is successful in substantially reducing the spurious velocity in the vicinity of a static droplet, while allowing stable simulation at a much higher-density ratio under identical conditions, which is a noteworthy improvement over existing Single Relaxation Time (SRT)-LBM algorithms. Various pool boiling scenarios have been explored for a reduced temperature of 0.75, which itself is significantly lower than reported in comparable literature, in both rectangular and cylindrical domains, and also with micro- and distributed heaters. All three regimes of pool boiling have aptly been captured with both plain and structured heaters, allowing the development of the boiling curve. The predicted value of critical heat flux for the plain heater agrees with Zuber correlation within 10%, illustrating both quantitative and qualitative capability of the proposed algorithm.

Impact of the prequench state of binary fluid mixtures on surface-directed spinodal decomposition

Using lattice Boltzmann simulations we investigate the impact of the amplitude of concentration fluctuations in binary fluid mixtures prior to demixing when in contact with a surface that is preferentially wet by one of the components. We find a bicontinuous structure near the surface for an initial, prequench state of the mixture close to the critical point where the amplitude of concentration fluctuations is large. In contrast, if the initial state of the mixture is not near the critical point and concentration fluctuations are relatively weak, then the morphology is not bicontinuous but remains layered until the very late stages of coarsening. In both cases, it is the morphology of a depletion layer rich in the nonpreferred component that dictates the growth exponent of the thickness of the fluid layer that is in direct contact with the substrate. In the early stages of demixing, we find a growth exponent consistent with a value of 1/4 for a prequench state away from the critical point, which is different from the usual diffusive scaling exponent of 1/3 that we recover for a prequench state close to the critical point. We attribute this to the structure of a depletion layer that is penetrated by tubes of the preferred fluid, connecting the wetting layer to the bulk fluid even in the early stages if the initial state is characterized by concentration fluctuations that are large in amplitude. Furthermore, we find that in the late stages of demixing the flow through these tubes results in significant in-plane concentration variations near the substrate, leading to dropletlike structures with a concentration lower than the average concentration in the wetting layer. This causes a deceleration in the growth of the wetting layer in the very late stages of the demixing. Irrespective of the prequench state of the mixture, the late stages of the demixing process produce the same growth law for the layer thickness, with a scaling exponent of unity usually associated with the impact of hydrodynamic flow fields.

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

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

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.

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 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.

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.

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.

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.

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.

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.

Implementation of contact angles in pseudopotential lattice Boltzmann simulations with curved boundaries

The pseudopotential multiphase lattice Boltzmann (LB) model is a very popular model in the LB community for simulating multiphase flows. When the multiphase modeling involves a solid boundary, a numerical scheme is required to simulate the contact angle at the solid boundary. In this work, we aim at investigating the implementation of contact angles in the pseudopotential LB simulations with curved boundaries. In the pseudopotential LB model, the contact angle is usually realized by employing a solid-fluid interaction or specifying a constant virtual wall density. However, it is shown that the solid-fluid interaction scheme yields very large spurious currents in the simulations involving curved boundaries, while the virtual-density scheme produces an unphysical thick mass-transfer layer near the solid boundary although it gives much smaller spurious currents. We also extend the geometric-formulation scheme in the phase-field method to the pseudopotential LB model. Nevertheless, in comparison with the solid-fluid interaction scheme and the virtual-density scheme, the geometric-formulation scheme is relatively difficult to implement for curved boundaries and cannot be directly applied to three-dimensional space. By analyzing the features of these three schemes, we propose an improved virtual-density scheme to implement contact angles in the pseudopotential LB simulations with curved boundaries, which does not suffer from a thick mass-transfer layer near the solid boundary and retains the advantages of the original virtual-density scheme, i.e., simplicity, easiness for implementation, and low spurious currents.

Lattice Boltzmann method for contact-line motion of binary fluids with high density ratio

Within the phase-field framework, we present an accurate and robust lattice Boltzmann (LB) method for simulating contact-line motion of immiscible binary fluids on the solid substrate. The most striking advantage of this method lies in that it enables us to handle two-phase flows with mass conservation and a high density contrast of 1000, which is often unavailable in the existing multiphase LB models. To simulate binary fluid flows, the present method utilizes two LB evolution equations, which are respectively used to solve the conservative Allen-Cahn equation for interface capturing, and the incompressible Navier-Stokes equations for hydrodynamic properties. Besides, to account for the substrate wettability, two popular contact angle models including the cubic surface-energy model and the geometrical one are incorporated into the present method, and their performances are numerically evaluated over a wide range of contact angles. The contact-angle hysteresis effect, which is inherent to a rough or chemically inhomogeneous substrate, is also introduced in the present LB approach through the strategy proposed by Ding and Spelt [J. Fluid Mech. 599, 341 (2008)10.1017/S0022112008000190]. The present method is first validated by simulating droplet spreading and capillary intrusion on the ideal or smooth pipes. It is found that the cubic surface-energy and geometrical wetting schemes both offer considerable accuracy for predicting a static contact angle within its middle region, while the former is more stable at extremely small contact angles. Besides, it is shown that the geometrical wetting scheme enables us to obtain better accuracy for predicting dynamic contact points in capillary pipe. Then we use the present LB method to simulate the droplet shearing processes on a nonideal substrate with contact angle hysteresis. The geometrical wetting model is found to be capable of reproducing four typical motion modes of contact line, while the surface-energy wetting scheme fails to predict the hysteresis behaviors in some cases. At last, a complex contact-line dynamic problem of three-dimensional microscale droplet impact on a wettable solid is simulated, and it is found that the numerical results for droplet shapes agree well with the experimental data.

Droplet Misalignment Limit for Inkjet Printing into Cavities on Textured Surfaces

The control of droplets deposited onto textured surfaces is of great importance for both engineering and medical applications. This research investigates the dynamics of a single droplet deposited into a confined space and its final equilibrium morphology, with emphasis given to droplet deposition under print head misalignment, the effect of nonuniform wettability, and deposition of droplets with varying sizes. A multiphase pseudopotential lattice Boltzmann methodology is used to simulate the process of deposition. The print quality is characterized in terms of a parameter referred to as the wetted fraction, which describes the proportion of the cavity that is wetted by the droplet. Our results show how single and multiple axis misalignment affect the final equilibrium morphology, and it was found for comparable configurations that multiaxis misalignment resulted in a higher wetted fraction. Investigations into wettabilities of the substrate and cavity wall revealed how larger ratios of the contact angles between the two enhance the ability for the droplet to self-align within the cavity. Additionally, a range of uniform wettabilities between the substrate and cavity were found, which mitigate against misalignment. Investigations into varying droplet sizes relative to the cavity revealed how misalignment can be compensated for with larger droplets, and limits for filling a cavity with a single drop are defined. Finally, we explore the deposition with misalignment into closely positioned cavities where it is found that the spacing between cavities is a key factor in determining the maximum permissible misalignment.

Interaction pressure tensor on high-order lattice Boltzmann models for nonideal fluids

In this work we address the application of pseudopotentials directly on high-order lattice Boltzmann models. We derive a general expression for the pressure tensor on high-order lattices considering all nonideal interactions, including intra- and intermolecular interactions, following the discrete lattice theory introduced by X. Shan [Phys. Rev. E 77, 066702 (2008)PLEEE81539-375510.1103/PhysRevE.77.066702]. From the derived expression, a generalized continuum approximation, truncated at fourth-order isotropy, is obtained that is readily applicable to high-order lattices. With this, we demonstrate that high-order lattice models with pseudopotentials can satisfy thermodynamic consistency. The derived generalized expression and continuum approximation are validated for the case of a flat interface and compared against the standard definition available from the literature. The generalized expression is also shown to accurately reproduce the Laplace experiment for a variety of high-order lattice structures. This work sets the preliminary steps towards the application of high-order lattice models for simulating nonideal fluid mixtures.

A Comparative Study of Multiphase Lattice Boltzmann Methods for Bubble-Dendrite Interaction during Solidification of Alloys

This paper presents a comparative study between the pseudopotential Shan-Chen model and the phase field multiphase lattice Boltzmann method for simulating bubble dynamics during dendritic solidification of binary alloys. The Shan-Chen method is an efficient lattice Boltzmann multiphase method despite having some limitations, including the generation of large spurious currents. The phase field model solves the Cahn-Hilliard equation in addition to the Navier-Stokes equation to track the interface between phases. The phase field method is more accurate than the Shan-Chen model for simulation of fluids with a high-density ratio since it generates an acceptable small spurious current, though at the expense of higher computational costs. For the simulations in this article, the multiphase lattice Boltzmann model was coupled with the cellular automata and finite difference methods to solve temperature and concentration fields. The simulated results were presented and compared regarding the ability of each model to simulate phenomena at a microscale resolution, such as Marangoni convection, the magnitude of spurious current, and the computational costs. It is shown that although Shan-Chen methods can replicate some qualitative features of bubble-dendrite interaction, the generated spurious current is unacceptably large, particularly for practical values of the density ratio between fluid and gas phases. This occurs even after implementation of several enhancements to the original Shan-Chen method. This serious limitation makes the Shan-Chen models unsuitable to simulate fluid flow phenomena, such as Marangoni convection, because the large spurious currents mask completely the physical flow.

Fluid flow simulations meet high-speed video: Computer vision comparison of droplet dynamics

HYPOTHESIS

While multiphase flows, particularly droplet dynamics, are ordinary in nature as well as in industrial processes, their mathematical and computational modelling continue to pose challenging research tasks - patent approaches for tackling them are yet to be found. The lack of analytical flow field solutions for non-trivial droplet dynamics hinders validation of computer simulations and, hence, their application in research problems. High-speed videos and computer vision algorithms can provide a viable approach to validate simulations directly against experiments.

EXPERIMENTS

Droplets of water (or glycerol-water mixtures) impacting on both hydrophobic and superhydrophobic surfaces were imaged with a high-speed camera. The corresponding configurations were simulated using a lattice-Boltzmann multiphase scheme. Video frames from experiments and simulations were compared, by means of computer vision, over entire droplet impact events.

FINDINGS

The proposed experimental validation procedure provides a detailed, dynamic one-on-one comparison of a droplet impact. The procedure relies on high-speed video recording of the experiments, computer vision, and on a software package for the analyzation routines. The procedure is able to quantitatively validate computer simulations against experiments and it is widely applicable to multiphase flow systems in general.

Chemical-potential-based lattice Boltzmann method for nonideal fluids

Chemical potential, as an important thermodynamic quantity, has been popularly used in thermodynamic modeling for complex systems, especially for those involving the phase transitions and chemical reactions. Here we present a chemical-potential-based multiphase lattice Boltzmann model, in which the nonideal force is directly evaluated by a chemical potential. The numerical computation is more efficient than the pressure-tensor-based model [Wen et al. Europhys. Lett. 112, 44002 (2015)10.1209/0295-5075/112/44002] because the calculations of the pressure tensor and its divergence are avoided. We have derived several chemical potentials of the popular equations of state from the free-energy density function. The theoretical analyses and numerical results support that the present model satisfies thermodynamics and Galilean invariance. An effective chemical-potential boundary condition is also implemented to investigate the wettability of a solid surface, and the contact angle can be linearly tuned by the surface chemical potential.

Analysis of force treatment in the pseudopotential lattice Boltzmann equation method

In this paper, different force treatments are analyzed in detail for a pseudopotential lattice Boltzmann equation (LBE), and the contribution of third-order error terms to pressure tensor with a force scheme is analyzed by a higher-order Chapman-Enskog expansion technique. From the theoretical analysis, the performance of the original force treatment of Shan-Chen (SC), Ladd, Guo et al., and the exact difference method (EDM) are ɛ_{Ladd}<ɛ_{Guo}<ɛ_{EDM}≤ɛ_{SC} with the relaxation time τ≥1, while ɛ_{Ladd}<ɛ_{Guo}<ɛ_{SC}<ɛ_{EDM} with τ<1; here ɛ is a parameter related to the mechanical stability and the subscripts are the corresponding force scheme. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to show that the predictions of modified force treatment of the pseudopotential LBE are all in good agreement with the analytical solution and other predictions.

Pseudopotential lattice Boltzmann equation method for two-phase flow: A higher-order Chapmann-Enskog expansion

In this paper, a higher order Chapmann-Enskog expansion technique is applied to pseudopotential lattice Boltzmann equation (LBE), and the contribution of third order error terms to pressure tensor is analyzed in detail. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to validate the LBE, and the results show that the predictions of the present LBE agree well with the analytical solution and other predictions.

Cascaded lattice Boltzmann method with improved forcing scheme for large-density-ratio multiphase flow at high Reynolds and Weber numbers

A recently developed forcing scheme has allowed the pseudopotential multiphase lattice Boltzmann method to correctly reproduce coexistence curves, while expanding its range to lower surface tensions and arbitrarily high density ratios [Lycett-Brown and Luo, Phys. Rev. E 91, 023305 (2015)PLEEE81539-375510.1103/PhysRevE.91.023305]. Here, a third-order Chapman-Enskog analysis is used to extend this result from the single-relaxation-time collision operator, to a multiple-relaxation-time cascaded collision operator, whose additional relaxation rates allow a significant increase in stability. Numerical results confirm that the proposed scheme enables almost independent control of density ratio, surface tension, interface width, viscosity, and the additional relaxation rates of the cascaded collision operator. This allows simulation of large density ratio flows at simultaneously high Reynolds and Weber numbers, which is demonstrated through binary collisions of water droplets in air (with density ratio up to 1000, Reynolds number 6200 and Weber number 440). This model represents a significant improvement in multiphase flow simulation by the pseudopotential lattice Boltzmann method in which real-world parameters are finally achievable.

A hydro-kinetic scheme for the dynamics of hydrogen bonds in water-like fluids

A hydro-kinetic scheme for water-like fluids, based on a lattice version of the Boltzmann equation, is presented and applied to the popular TIP3P model of liquid water. By proceeding in much larger steps than molecular dynamics, the scheme proves to be very effective in attaining global minima of classical pair potentials with directional and radial interactions, as confirmed by further simulations using the three-dimensional Ben-Naim water potential. The scheme is shown to reproduce the propensity of water to form nearly four hydrogen bonds per molecule, as well as their statistical distribution in the presence of thermal fluctuations, at a linear cost of computational time with the system size.

The importance of chemical potential in the determination of water slip in nanochannels

We investigate the slip properties of water confined in graphite-like nanochannels by non-equilibrium molecular dynamics simulations, with the aim of identifying and analyze separately the influence of different physical quantities on the slip length. In a system under confinement but connected to a reservoir of fluid, the chemical potential is the natural control parameter: we show that two nanochannels characterized by the same macroscopic contact angle--but a different microscopic surface potential--do not exhibit the same slip length unless the chemical potential of water in the two channels is matched. Some methodological issues related to the preparation of samples for the comparative analysis in confined geometries are also discussed.

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

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

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

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

Contact angles in the pseudopotential lattice Boltzmann modeling of wetting

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

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

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

Tuning drop motion by chemical patterning of surfaces

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