A finite difference formulation inspired by the pseudopotential lattice Boltzmann method

The coalescence of liquid lenses is relevant in various applications, including inkjet printing and fog harvesting. However, the dynamics of liquid-lens coalescence have been relatively underexplored, particularly in the case of liquid lenses with larger contact angles. The coalescence of low-viscosity liquid lenses is numerically investigated by means of the pseudopotential multi-component lattice Boltzmann method over a wide range of contact angles. In two-dimensional simulations, numerical results on the growth of the bridge height are in quantitative agreement with experimental measurements for small contact angles. In addition, a comparison of the simulation results with a theoretical approach based on the thin-sheet equations for liquid lenses shows that these equations accurately capture the bridge-growth dynamics up to moderate contact angles. For the three-dimensional case, the growth of the bridge radius is independent of the equilibrium contact angle of the liquid lenses at the initial stage of growth. The dependency between the growth of the bridge height and the bridge radius exhibits a non-linear to linear transition. Document Type: Original article Cited as: Xie, Q., Harting, J. Simulations of inertial liquid-lens coalescence with the pseudopotential lattice Boltzmann method. Capillarity, 2026, 18(2): 41-49. https://doi.org/10.46690/capi.2026.02.01

Boiling heat transfer plays a crucial role in various engineering applications, requiring accurate numerical modeling to capture phase-change dynamics. This study employs the pseudopotential lattice Boltzmann method (LBM) to simulate boiling heat transfer at different reduced temperatures, aiming to provide deeper insights into bubble dynamics and heat transfer mechanisms. The LBM framework incorporates a multi-relaxation-time approach and the Peng–Robinson equation of state to enhance numerical stability and thermodynamic consistency. Simulations were performed to analyze bubble nucleation, growth, and detachment across varying reduced temperatures, considering the influence of surface wettability, surface tension and gravitational acceleration. The results indicate a strong dependence of bubble behavior on the reduced temperature, affecting both heat flux and boiling regimes. The numerical findings show reasonable agreement with theoretical predictions and experimental trends, validating the effectiveness of the LBM approach for phase-change simulations. Additionally, this study highlights the role of contact angle variation in modifying boiling characteristics, emphasizing the necessity of accurate surface interaction modeling. The outcomes of this work contribute to advancing computational methodologies for boiling heat transfer, supporting improved thermal management in industrial applications.

Investigating snap-off behavior during spontaneous imbibition in 3D pore-throat model by pseudopotential lattice Boltzmann method

In polymer electrolyte membrane fuel cells, the gas diffusion layer (GDL) is composed of porous media and serves a critical role as a mass transport layer, facilitating reactant gas diffusion, removal of water generated in the catalyst layer, and electron transport. Artificial spacings known as perforations can be introduced to improve water management within this mass transport system. However, the impact of these perforations on the effective electrical conductivity has not been adequately studied. This study employs numerical methods to investigate water management and effective electrical conductivity in the presence of perforations, aiming to provide indicators for optimal design. The pseudopotential lattice Boltzmann method is utilized, which is particularly advantageous for modeling two-phase flow and electron transport in complex geometries. Using this numerical approach, we analyze water penetration in GDL structures and effective electrical conductivity based on electric potential fields focusing on geometric parameters such as the perforation size. Our results demonstrate a relationship between water management efficiency and effective electrical conductivity, suggesting the existence of an optimal perforation diameter. Moreover, when there is a water-induced penetration pattern due to the perforated structure, both the effective electrical conductivity and water management are enhanced at a lower porosity of the GDL structure.

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

Pore-scale modelling of water sorption in nanopore systems of shale

Lattice boltzmann investigation of droplet interactions with non-uniform chemically patterned surfaces

Development and assessment of the interface lattice Boltzmann flux solvers for multiphase flows

In this work, a numerical model for isothermal liquid-vapor phase change (evaporation) of the two-component air-water system is proposed based on the pseudopotential lattice Boltzmann method. Through the Chapman-Enskog multiscale analysis, we show that the model can correctly recover the macroscopic governing equations of the multicomponent multiphase system with a built-in binary diffusion mechanism. The model is verified based on the two-component Stefan problem where the measured binary diffusivity is consistent with theoretical analysis. The model is then applied to convective drying of a dual-porosity porous medium at the pore scale. The simulation captures a classical transition in the drying process of porous media, from the constant rate period (CRP, first phase) showing significant capillary pumping from large to small pores, to the falling rate period (FRP, second phase) with the liquid front receding in small pores. It is found that, in the CRP, the evaporation rate increases with the inflow Reynolds number (Re), while in the FRP, the evaporation curves almost collapse at different Res. The underlying mechanism is elucidated by introducing an effective Péclet number (Pe). It is shown that convection is dominant in the CRP and diffusion in the FRP, as evidenced by Pe > 1 and Pe < 1, respectively. We also find a log-law dependence of the average evaporation rate on the inflow Re in the CRP regime. The present work provides new insights into the drying physics of porous media and its direct modeling at the pore scale.

Fluid-structure interaction using lattice Boltzmann method: Moving boundary treatment and discussion of compressible effect

This work shows that, by adequately rewriting the one-dimensional convection-diffusion equation in a diffusional form, all the usual numerical procedures, that is, the finite element, finite volume and finite difference procedures, lead to the same and optimal steady-state stencil for numerical solutions. Additionally, a lumped capacitance (mass) matrix first-order time integration technique is presented. The resulting scheme, expressed with a transient, special lumped capacitance matrix (PMGV or prevailing main grid value method), was used to solve the problem of onedimensional transient saturated soil solute transport. A benchmark problem with analytical solution, representing a moving concentration front in a saturated soil was solved by both the diffusional method and by the Two-Step Taylor-Galerkin (TSTG) finite element method. The diffusional (PMGV) method proved itself to be an excellent option for solving advection dominated problems such as soil solute transport problems. The simulation results showed that the explicit PMGV scheme leads to the same accuracy as the TSTG scheme. The TSTG scheme shows numerical dispersion while the diffusional method showed false diffusion. However, the stability region for the PMGV is broader. The PMGV scheme is more effective than the TSTG in problems involving either low Pe numbers or high Pe numbers, if mesh refinement is made.

Revisiting the second-order convergence of the lattice Boltzmann method with reaction-type source terms

This article addresses the abilities and limitations of the Lattice Boltzmann (LB) method in solving advection-dominated mass transport problems. Several schemes of the LB method, including D2Q4, D2Q5, and D2Q9, were assessed in the simulation of two-dimensional advection-dispersion equations. The concept of Single Relaxation Time (SRT) and Multiple Relaxation Time (MRT) in addition to linear and quadratic Equilibrium Distribution Functions (EDF) were taken into account. The results of LB models were compared to the well-known Finite Difference (FD) solutions, including Explicit Finite Difference (EFD) and Crank-Nicolson (CN) methods. All LB models are more accurate than the aforementioned FD schemes. The results also indicate the high potency of D2Q5 SRT and D2Q9 SRT in describing advection-controlled mass transfer problems. The numerical artificial oscillations are observed when the Grid Peclet Number (GPN) is greater than 10, 25, 20, 25, and 10 regarding D2Q4 SRT, D2Q5 SRT, D2Q5 MRT, D2Q9 SRT and D2Q9 MRT, respectively, while the corresponding GPN values obtained for the EFD and CN methods were 2 and 5, respectively. Finally, a coupled system of groundwater and solute transport equations were solved satisfactorily using several LB models. Considering computational time, all LB models are much faster than CN method.

The curvature and temperature dependency of the liquid-vapor surface tension has a significant influence on the accurate prediction of the nanobubble/nanodrop nucleation process. In this work, a mesoscopic approach combining the pseudo-potential multiphase lattice Boltzmann method (LBM), the principle of dynamic similarity, and the van der Waals theory of corresponding states is adopted to quantitatively investigate the curvature and temperature dependency of the surface tension and Tolman length for real fluids (water and R134a). By Tolman length, we mean the distance from the surface of tension to the equimolar surface, which measures the extent by which the surface tension of a nanodrop/nanobubble deviates from the corresponding flat interface limit. We show that the Tolman lengths for flat liquid-vapor interfaces (δF) increase with the increase of temperature and are proportional to (1−Tr)−1.044. Equations for predicting surface tensions of water and R134a with effects of temperature and curvature radius taken into consideration are proposed. We demonstrate that the surface tensions increase while the Tolman lengths (δB) decrease with the increase of curvature for nanobubbles. For nanodroplets, however, the surface tensions decrease while the Tolman lengths (δD) increase with the increase of curvature. Effects of the equation of state for real fluids, which determines the interparticle interaction force in the pseudo-potential LBM, are also discussed. This mesoscopic approach can quantify the curvature dependency of liquid-vapor surface tensions for various real fluids in a wide temperature range with low computation costs, providing a new avenue for the accurate prediction of nucleation processes in micro-/nanoscale phase change heat transfer with applications to boiling, evaporation, and condensation.

Investigation of CO2 and CH4 capture from syngas in adsorption process and breakthrough modeling by modified Lattice-Boltzmann mesoscale method