Jun-Jie Huang

Physics-Informed Neural Networks (PINNs) provide a promising framework for solving partial differential equations (PDEs). By incorporating temporal causality, Causal PINN improves training stability in time-dependent problems. However, applying Causal PINN to higher-order nonlinear PDEs, such as the Cahn–Hilliard equation (CHE), presents notable challenges due to the inefficient utilization of temporal information. This inefficiency often results in numerical instabilities and physically inconsistent solutions. This study systematically analyzes the limitations of Causal PINN in solving the one-dimensional CHE. To resolve these issues, we propose a novel framework called APM (Adaptive Progressive Marching)-PINN that enhances temporal representation and improves model robustness. APM-PINN mainly integrates a progressive temporal marching strategy, a causality-based adaptive sampling algorithm, and a residual-based adaptive loss weighting mechanism (effective with the chemical potential reformulation). Comparative experiments on two one-dimensional CHE test cases show that APM-PINN achieves relative errors consistently near 10⁻³ or even 10⁻⁴. It also preserves mass conservation and energy dissipation better. The promising results highlight APM-PINN’s potential for the accurate, stable modeling of complex high-order dynamic systems.

Coalescence-induced droplet jumping (CIDJ) has significant applications in heat transfer, anti-frosting, and corrosion prevention. The thin-walled-lattice (TWL) microstructure design enhances CIDJ by continuously sieving condensing droplets on solid surfaces. However, the simultaneous appearance of many droplets has the risk of film formation, which can degrade surface performance. This study addresses this challenge by proposing symmetric four-TWL units to improve CIDJ efficiency. Using three-dimensional conservative phase-field lattice Boltzmann simulations with large density ratios, we demonstrate that these units achieve superior CIDJ performance compared to existing designs, with droplet jumping velocities reaching up to 10.15 m/s under certain conditions. The height of the TWL structure, the radius of the superhydrophilic spots (Rspot), and the wall contact angle (θw) were found to affect the jumping performance of the droplets significantly. Our results indicate that the strict requirements for θw (160 ± 10 deg) commonly seen in current CIDJ research could be relaxed to θw>120 deg on the proposed surface. A phase diagram for droplet jumping with the radius of the spot Rspot and the contact angle θw as independent variables was generated to provide essential guidance for the design of similar lattice structures. Finally, after reviewing the numerical results, we proposed two innovative surface designs, which could provide new design paradigms for future CIDJ research and applications.

Compound droplets have received increasing attention due to their applications in many several areas, including medicine and materials. Previous works mostly focused on compound droplets on planar surfaces and, as such, the effects of curved walls have not been studied thoroughly. In this paper, the influence of the properties of curved solid wall (including the shape, curvature, and contact angle) on the wetting behavior of compound droplets is explored. The axisymmetric lattice Boltzmann method, based on the conservative phase field formulation for ternary fluids, was used to numerically study the wetting and spreading of a compound droplet of the Janus type on various curved solid walls at large density ratios, focusing on whether the separation of compound droplets occurs. Several types of wall geometries were considered, including a planar wall, a concave wall with constant curvature, and a convex wall with fixed or variable curvature (specifically, a prolate or oblate spheroid). The effects of surface wettability, interfacial angles, and the density ratio (of droplet to ambient fluid) on the wetting process were also explored. In general, it was found that, under otherwise identical conditions, droplet separation tends to happen more likely on more hydrophilic walls, under larger interfacial angles (measured inside the droplet), and at larger density ratios. On convex walls, a larger radius of curvature of the surface near the droplet was found to be helpful to split the Janus droplet. On concave walls, as the radius of curvature increases from a small value, the possibility to observe droplet separation first increases and then decreases. Several phase diagrams on whether droplet separation occurs during the spreading process were produced for different kinds of walls to illustrate the influences of various factors.

The wetting and spreading of droplets on solid walls are commonly seen in nature. The study of such a phenomenon can deepen our understanding of solid-liquid interaction and promote the development of relevant cutting-edge technological applications. In this work, the lattice Boltzmann method based on phase field theory is used to investigate the wetting and spreading of a compound droplet on a wedge. This method combines the finite-difference solution of the Cahn-Hilliard equations for ternary fluids to capture the interface dynamics and the lattice Boltzmann method for the hydrodynamics of the flow. Symmetric compound droplets with equal interfacial tensions on a wedge are considered first. Through theoretical analysis and numerical simulation, it is found that the wetted area on the wedge increases with the decrease of the contact angle of the wedge surface and the wedge apex angle. Depending on these two factors, the droplet may or may not split on the wedge. We also find that the droplet near the critical state predicted not to split by static equilibrium analysis could split during the spreading along the wall of the wedge under certain density and viscosity ratios. Based on the simulation results, a phase diagram of the droplet splitting state is generated with the density ratio and viscosity ratio as the coordinates. As the density ratio and kinematic viscosity ratio increase, the inertia effect becomes more prominent in the wetting and spreading process and the droplet is more likely to split. By comparing the phase diagrams in different initial conditions, it is found that under the same conditions, the compound droplet with an equilibrium initial state is less likely to split than that with an unequilibrium initial state, which is possibly because the initial total energy of the former is relatively small. Our study also shows that the kinematic viscosity ratio between the left half and the right half droplet may affect the results of droplet splitting. The increase of such a viscosity difference is conducive to the splitting of the compound droplet. Besides, asymmetric compound droplets with unequal interfacial tensions are also simulated, and it is found that the greater the wrapping degree between the left half and right half, the more difficult it is to separate the compound droplet.

A simplified method is proposed to implement a wetting boundary condition on curved surfaces within the conservative phase-field lattice-Boltzmann (LB) simulation framework. It combines the idea of Huang et al. [“An alternative method to implement contact angle boundary condition and its application in hybrid lattice-Boltzmann finite-difference simulations of two-phase flows with immersed surfaces,” Eur. Phys. J. E 41, 17 (2018)] to find the order parameter on the other side of the wall with the conservative Allen–Cahn equation (CACE) for interface evolution solved by the LB equations. It inherits the advantage of the original method using the Cahn–Hilliard equation to avoid complicated interpolations under different geometries. By using the CACE, the boundary condition for the chemical potential is circumvented (making it more simplified), and the overshooting of the order parameter is also greatly suppressed, enabling it to simulate two-phase flows with solid objects of various shapes and wettabilities at large density and viscosity ratios. Several two-dimensional, axisymmetric, and three-dimensional problems, including some previously studied by experiments, were simulated and the capability of the proposed method is demonstrated through its good agreement with theoretical predictions and/or experimental observations.

In this paper, a hybrid lattice-Boltzmann finite-difference method is developed for the simulation of ternary fluids near immersed solid objects of general shapes. The flow equations are solved by the lattice-Boltzmann method and the coupled Cahn–Hilliard equations for interface evolutions are solved by the finite-difference method. A special implementation of the wetting boundary condition on a surface of general shapes immersed inside the domain was extended for ternary fluids within the phase-field framework with no need to use complicated interpolations. Several two and three dimensional problems with three immiscible fluids were studied by using the proposed method and the results agree well with analytical predictions and/or previous numerical and experimental studies. In particular, the inclusion of properly chosen free energy to handle total spreading enabled us to numerically reproduce the encapsulation of a small droplet by another bigger one of different component on a round fiber. The proposed method is expected to be useful to investigate a variety of multiphase problems involving ternary fluids and surfaces with different configurations, including the challenging total spreading regime.

The lattice Boltzmann method (LBM) has gained increasing popularity in incompressible viscous flow simulations, but it uses many distribution functions (far more than the flow variables) and is often memory demanding. This disadvantage was overcome by a recent approach that solves the more actual macroscopic equations obtained through Taylor series expansion analysis of the lattice Boltzmann equations [Lu et al., J. Comput. Phys. 415, 109546 (2020)]. The key is to keep some small additional terms (SATs) to stabilize the numerical solution of the weakly compressible Navier-Stokes equations. However, there are many SATs that complicate the implementation of their method. Based on some analyses and numerous tests, we ultimately pinpoint two essential ingredients for stable simulations: (1) suitable density (pressure) diffusion added to the continuity equation and (2) proper numerical dissipation related to the velocity divergence added to the momentum equations. Then we propose a simplified method that is not only easier to implement but noticeably faster than the original method and the LBM. It contains much simpler SATs that only involve the density (pressure) derivatives, and it requires no intermediate steps or variables. As well, it is extended for thermal flows with small temperature variations and for two-phase flows with uniform density and viscosity. Several test cases, including some two-phase problems under two-dimensional, axisymmetric, and three-dimensional geometries, are presented to demonstrate its capability. This work may help pave the way for the simplest simulation of incompressible viscous flows on collocated grids based on the artificial compressibility methodology.

The general pressure equation (GPE) is a new method proposed recently by Toutant (J. Comput. Phys., 374:822-842 (2018)) for incompressible flow simulation. It circumvents the Poisson equation for the pressure and performs better than the classical artificial compressibility method. Here it is generalized for two-phase incompressible viscous flows with variable density and viscosity. First, the pressure evolution equation is modified to account for the density variation. Second, customized discretizations are proposed to deal with the viscous stress terms with variable viscosity. Besides, additional terms related to the bulk viscosity are included to stabilize the simulation. The interface evolution and surface tension effects are handled by a phase-field model coupled with the GPE-based flow equations. The pressure and momentum equations are discretized on a stagger grid using the second order centered scheme and marched in time using the third order total variation diminishing Runge-Kutta scheme. Several unsteady two-phase problems in two dimensional, axisymmetric and three dimensional geometries at intermediate density and viscosity ratios were simulated and the results agreed well with those obtained by other incompressible solvers and/or the lattice-Boltzmann method (LBM) simulations. Similar to the LBM, the proposed GPE-based method is fully explicit and easy to be parallelized. Although slower than the LBM, it requires much less memory than the LBM. Thus, it can be a good alternative to simulate two-phase flows with limited memory resource.

Existing energy-based models on coalescence-induced droplet jumping on superhydrophobic surfaces often assume that the viscously dissipated energy is proportional to the Ohnesorge (Oh) number (linear form). Based on the numerical results by two different methods, we found that the energy dissipation for droplet coalescence follows a power law form. What is more, the energy dissipation at the time of droplet jumping was found to be larger than that of droplet coalescence. Besides, existing energy-based models assume a fixed portion of translational kinetic energy (KE) within the droplet's total KE, whereas it actually varies with the Oh number from our simulations. Incorporating a special form of energy dissipation and an Oh-dependent portion of translational KE, we obtained an improved model that can predict the jumping velocity of the coalesced droplet of tens of nanometer- to micrometer-size on nonwetting surfaces much more accurately than the existing energy-based models. Finally, we extended the model for superhydrophobic surfaces with finite adhesion and demonstrated its capability through extensive comparisons with relevant experimental and numerical results.

Coalescence-induced droplet jumping from a round hydrophobic fiber was studied by phase-field-based hybrid lattice-Boltzmann finite-difference simulations in which the interface dynamics is handled by the finite-difference solution of the Cahn–Hilliard equation and the hydrodynamics is handled by the lattice-Boltzmann method. It was found that at a small Ohnesorge number of O(0.01), several different outcomes, including normal jumping at a positive velocity, jumping with a negative velocity, jumping after wrapping the fiber, and oscillating on the fiber, may occur after droplet coalescence, depending on the wettability of the fiber and the droplet-to-fiber radius ratio. In accordance with previous reports, normal droplet jumping from the fiber happens only when the radius ratio exceeds some critical value for a given contact angle. The critical radius ratio decreases as the fiber becomes more hydrophobic. For a given contact angle and radius ratio, it was observed that there exists an Ohnesorge number at which the jumping speed achieves a maximum value.