Lattice Boltzmann simulation of the trapping of a microdroplet in a well of surface energy

In this paper, a three-dimensional phase-field lattice Boltzmann method is used to simulate the dynamical behavior of a droplet, subject to an outer viscous flow, in a microchannel that contains a cylindrical hole etched into its top surface. The influence of the capillary number and the hole diameter (expressed as the ratio of hole diameter to channel height, b) is investigated. We demonstrate numerically that the surface energy gradient induced by the hole can create an anchoring force to resist the hydrodynamic drag from the outer flow, resulting in the droplet anchored to the hole when the capillary number is below a critical value. As b increases, the droplet can be anchored more easily. For b 2, the spherical cap of droplet reaches the top wall of the hole, making the hole depth into an additional important parameter. These observations are consistent with the previously reported experiments. However, the droplet does not fully fill the hole for b > 2, departing from the expectation of Dangla et al. [R. Dangla, S. Lee, C. N. Baroud, Trapping microfluidic drops in wells of surface energy, Phys. Rev. Lett. 107 (2011) 124501]. Also, it is observed in the anchored state that the rear of the droplet rests at a small distance away from the junction. Finally, the droplet undergoes a slow-down process only when itsrear passes through the hole, regardless of b

A lattice Boltzmann method for axisymmetric thermocapillary flows

In this work, we develop a two-phase lattice Boltzmann method (LBM) to simulate axisymmetric thermocapil- lary flows. This method simulates the immiscible axisymmetric two-phase flow by an improved color-gradient model, in which the single-phase collision, perturbation and recoloring operators are all presented with the axisymmetric effect taken into account in a simple and computational consistent manner. An additional lattice Boltzmann equation is introduced to describe the evolution of the axisymmetric temperature field, which is coupled to the hydrodynamic equations through an equation of state. This method is first validated by simulations of Rayleigh-B ́enard convection in a vertical cylinder and thermocapillary migration of a de- formable droplet at various Marangoni numbers. It is then used to simulate the thermocapillary migration of two spherical droplets in a constant applied temperature gradient along their line of centers, and the influence of the Marangoni number (Ca), initial distance between droplets (S0), and the radius ratio of the leading to trailing droplets (Λ) on the migration process is systematically studied. As Ma increases, the thermal wake behind the leading droplet strengthens, resulting in the transition of the droplet migration from coalescence to non-coalescence; and also, the final distance between droplets increases with Ma for the non-coalescence cases. The variation of S0 does not change the final state of the droplets although it has a direct impact on the migration process. In contrast, Λ can significantly influence the migration process of both droplets and their final state: at low Ma, decreasing Λ favors the coalescence of both droplets; at high Ma, the two droplets do not coalesce eventually but migrate with the same velocity for the small values of Λ, and decreasing Λ leads to a shorter equilibrium time and a faster migration velocity

A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects

In this paper, we develop a phase-field model for binary incompressible (quasi-incompressible) fluid with thermocapillary effects, which allows for the different properties (densities, viscosities and heat conductivities) of each component while maintaining thermodynamic consistency. The governing equations of the model including the Navier-Stokes equations with additional stress term, Cahn-Hilliard equations and energy balance equation are derived within a thermodynamic framework based on entropy generation, which guarantees thermodynamic consistency. A sharp-interface limit analysis is carried out to show that the interfacial conditions of the classical sharp-interface models can be recovered from our phase-field model. Moreover, some numerical examples including thermocapillary convections in a two-layer fluid system and thermocapillary migration of a drop are computed using a continuous finite element method. The results are compared to the corresponding analytical solutions and the existing numerical results as validations for our model

Walls and domain shape effects on the thermal marangoni migration of three-dimensional droplets

The thermocapillary motion of liquid droplets in fluid media depends on a variety of influential factors, including the not yet fully understood role played by the presence of the walls and other geometrical constraints. In order to address this specific question, in the present work we rely on a rigorous mathematical and numerical framework (including an adaptive mesh strategy), which are key to perform physically consistent and computationally reliable simulations of such a problem given the different space scales it involves. Our final aim is the proper discernment of the triadic relationship established among viscous phenomena, thermal effects and other specific behaviour due to the proximity of the droplet to a solid boundary. Different geometric configurations are considered (e.g., straight, converging and diverging channels, droplets located near a single or adjacent walls) and distinct regimes are examined (including both (Ma, Re)->0 and finite Ma flows). The results show that for straight channels the droplet generally undergoes a decrease in the migration velocity due to its proximity to the wall. Such a departure becomes larger as the Marangoni number is increased. In addition, a velocity component directed perpendicularly to the wall emerges. This effect tends to “pull” the droplet away from the solid boundary if adiabatic conditions are considered, whereas for thermally conducting sidewalls and relatively large values of the Marangoni number, the distortion of the temperature field in the region between the droplet and the wall results in a net force with a component directed towards the surface. For non-straight channels, the dynamics depend essentially on the balance between two counteracting factors, namely, the effective distribution of temperature established in the channel (for which we provide analytic solutions in the limit as Re->0) and the “blockage effect” due to the non-parallel configuration of the walls. The relative importance of these mechanisms is found to change according to the specific regime considered (creeping flow or Re=O(1))

A hybrid lattice Boltzmann and finite difference method for droplet dynamics with insoluble surfactants

Droplet dynamics in microfluidic applications is significantly influenced by surfactants. It remains a research challenge to model and simulate droplet behavior including deformation, breakup and coalescence, especially in the confined microfluidic environment. Here we propose a hybrid method to simulate interfacial flows with insoluble surfactants. The immiscible two-phase flow is solved by an improved lattice Boltzmann color-gradient model that incorporates a Marangoni stress resulting from non-uniform interfacial tension, while the convection-diffusion equation which describes the evolution of surfactant concentration in the entire fluid domain is solved by a finite difference method. The lattice Boltzmann and finite difference simulations are coupled through an equation of state, which describes how surfactant concentration influences interfacial tension. Our method is first validated for the surfactant-laden droplet deformation in a three- dimensional (3D) extensional flow and a 2D shear flow, and then applied to investigate the effect of surfactants on droplet dynamics in a 3D shear flow. Numerical results show that, at low capillary numbers, surfactants increase droplet deformation, due to reduced interfacial tension by the average surfactant concentration, and non-uniform effects from non-uniform capillary pressure and Marangoni stresses. The role of surfactants on critical capillary number (Cacr) of droplet breakup is investigated for various confinements (defined as the ratio of droplet diameter to wall separation) and Reynolds numbers. For clean droplets, Cacr first decreases and then increases with confinement, and the minimum value of Cacr is reached at the confinement of 0.5; for surfactant-laden droplets, Cacr exhibits the same variation in trend for the confinements lower than 0.7, but for higher confinements, Cacr is almost a constant. The presence of surfactants decreases Cacr for each confinement, and the decrease is also attributed to the reduction in average interfacial tension and non-uniform effects, which are found to prevent droplet breakup at low confinements but promote breakup at high confinements. In either clean or surfactant-laden cases, Cacr first keeps almost unchanged and then decreases with increasing Reynolds number, and a higher confinement or Reynolds number favors ternary breakup. Finally, we study the collision of two equal-sized droplets in a shear flow in both surfactant-free and surfactant-contaminated systems with the same effective capillary numbers. It is identified that the non-uniform effects in near-contact interfacial region immobilize the interfaces when two droplets are approaching each other and thus inhibit their coalescence

A lattice Boltzmann method for axisymmetric multicomponent flows with high viscosity ratio

A color-gradient lattice Boltzmann method (LBM) is proposed to simulate ax- isymmetric multicomponent flows. This method uses a collision operator that is a combination of three separate parts, namely single-component collision op- erator, perturbation operator, and recoloring operator. A source term is added into the single-component collision operator such that in each single-component region the axisymmetric continuity and momentum equations can be exactly re- covered. The interfacial tension effect is realized by the perturbation operator, in which an interfacial force of axisymmetric form is derived using the concept of continuum surface force. A recoloring operator proposed by Latva-Kokko and Rothman is extended to the axisymmetric case for phase segregation and maintenance of the interface. To enhance the method’s numerical stability for handling binary fluids with high viscosity ratio, a multiple-relaxation-time mod- el is used for the collision operator. Several numerical examples, including static droplet test, oscillation of a viscous droplet, and breakup of a liquid thread, are presented to test the capability and accuracy of the proposed color-gradient LB- M. It is found that the present method is able to accurately capture the phase interface and produce low spurious velocities. Also, the LBM results are all in good agreement with the analytical solutions and/or available experimental data for a very broad range of viscosity ratios

A Novel Multiple-Phase, Multiple-Component, Thermal Lattice Boltzmann Model

The lattice Boltzmann method (LBM) is gaining traction as a powerful approach to fluid flow simulation. In this work, developments toward the incorporation of more complex physical phenomena into the LBM are presented. As will be discussed, existing approaches are currently inadequate for thermal flows with wall interactions and multiple components. A novel methodology will be detailed, which enables the simulation of multiphase, multicomponent, thermal flows. The need for these simulation techniques is clear. As energy densities in electronic devices rapidly increase, improved two-phase microchannel heat exchanger designs are of great interest. Similarly, with the implementation of phase separation as a method of flow manipulation in microdevices, understanding the flow dynamics of multiple phases in microchannels is vital. However, experimental studies have thus far shown a great deal of variety in the flow patterns and instabilities that develop at the microscale level. Thus, numerical techniques capable of simulating such conditions are desirable. While traditional computational fluid dynamics (CFD) methods are based on macroscale equations, and molecular dynamics simulations seek to model the microscopic behavior of individual molecules, the LBM takes a mesoscopic approach. Based on the linearized kinetic lattice Boltzmann equation, particle interactions are directly implemented, while the movement of those particles is confined to a discrete lattice. This makes the LBM very useful in modeling interfacial dynamics and multiphase flows, while avoiding the enormous computational complexity of a direct MD simulation. The novel contributions of this work are: a) the combination of the Peng-Robinson equation of state with a recently developed linear approximation of the interparticle interaction gradient for the improvement of the multiphase, single-component, thermal (MPSC-T) LBM, b) the development of a thermally-dependent wall interaction model for dynamic contact angle simulation in the MPSC-T LBM, c) an analysis of the stability region of the interparticle interaction parameters in a multiphase, immiscible, multicomponent, isothermal (MPiMC-IT) model, and d) the development of a multiphase, immiscible, multicomponent, thermal (MPiMC-T) model using a density-weighted coupling of macroscopic properties

A brief review of the phase-field-based lattice Boltzmann method for multiphase flows

In this paper, we present a brief overview of the phase-field-based lattice Boltzmann method (LBM) that is a distinct and efficient numerical algorithm for multiphase flow problems. We first give an introduction to the mathematical theory of phase-field models for multiphase flows, and then present some recent progress on the LBM for the phase-field models which are composed of the classic Navier-Stokes equations and the Cahn-Hilliard or Allen-Cahn equation. Finally, some applications of the phase-field-based LBM are also discussed.Cited as: Wang, H., Yuan, X., Liang, H., Chai, Z., Shi, B. A brief review of the phase-field-based lattice Boltzmann method for multiphase flows. Capillarity, 2019, 2(3): 33-52, doi: 10.26804/capi.2019.03.0