Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows

Theoretical and numerical study on the well-balanced regularized lattice Boltzmann model for two-phase flow

In the multiphase flow simulations based on the lattice Boltzmann equation (LBE), the spurious velocity near the interface and the inconsistent density properties are frequently observed. In this paper, a well-balanced regularized lattice Boltzmann (WB-RLB) model with Hermite expansion up to third order is developed for two-phase flows. To this end, the equilibrium distribution function and the modified force term proposed by Guo [Phys. Fluids 33, 031709 (2021)1070-663110.1063/5.0041446] are directly introduced into the regularization of the transformed distribution functions when considering the LBE with trapezoidal integral. First, to give a detailed comparison of the well-balanced lattice Boltzmann equation (WB-LBE), WB-RLB, and second-order mixed difference scheme (SOMDS) proposed by Lee and Fischer [Phys. Rev. E 74, 046709 (2006)1539-375510.1103/PhysRevE.74.046709], the theoretical analyses on the force balance of LBE with two different gradient operators, isotropic central scheme (ICS) and SOMDS, as well as the numerical simulations of the stationary droplet are carried out. The force analysis shows that SOMDS can achieve a higher accuracy than ICS for the force balance, which has been validated in the simulations of stationary droplet cases. For the stationary droplet cases, all three models (WB-LBE, WB-RLB, and SOMDS) can capture the physical equilibrium state even at a large density ratio of 1000. Also, the numerical investigations of the WB-RLB model with third-order expansion (WB-RLB3) demonstrate that adjusting the relaxation parameters of the third-order moment can further improve the accuracy and stability of the WB-RLB model. Then, both the droplet coalescence and the phase separation cases are investigated with considering the effect of different interface thickness, which demonstrates that the performance of the WB-RLB for the two-phase dynamic problems is still quite well, and it exhibits better numerical stability when compared with the WB-LBE. In addition, the contact angle problem is investigated by the present WB-RLB model; the numerical results show that the predicted values of the contact angles agree well with the analytical solutions, but the well-balance property is not validated, especially near the three-phase junction. Overall, the present WB-RLB model exhibits excellent numerical accuracy and stability for both static and dynamic interface problems.

Phase-field-based lattice Boltzmann model for simulating thermocapillary flows

This paper proposes a simple and accurate lattice Boltzmann model for simulating thermocapillary flows, which can deal with the contrast between thermodynamic parameters. In this model, two lattice Boltzmann equations are utilized to solve the conservative Allen-Cahn equation and the incompressible Navier-Stokes equations, while another lattice Boltzmann equation is used for solving the temperature field, where the collision term is delicately designed such that the influence of the contrast between thermodynamic parameters is incorporated. In contrast to the previous lattice Boltzmann models for thermocapillary flows, the most distinct feature of the current model is that the forcing term used in the present thermal lattice Boltzmann equation is not needed to calculate space derivatives of the heat capacitance or the order parameter, making the scheme much more straightforward and able to retain the main merits of the lattice Boltzmann method. The developed model is first validated by considering the thermocapillary flows in a heated microchannel with two superimposed planar fluids. It is then used to simulate the thermocapillary migration of a two-dimensional deformable droplet, and its accuracy is consistent with the theoretical prediction when the Marangoni number approaches zero. Finally, we numerically study the motion of two recalcitrant bubbles in a two-dimensional channel where the relationship between surface tension and temperature is assumed to be a parabolic function. It is observed that due to the competition between the inertia and thermal effects, the bubbles can move against the liquid's bulk motion and towards areas with low surface tension.

Numerical Simulation on Insoluble Surfactant Mass Transfer on Deformable Bubble Interface in a Couette Flow by Phase-Field Lattice Boltzmann Method-Finite-Difference Method Hybrid Approach

Elaborate management on bubble shape and transportations depends on the balance between multiple physical behaviors for two-phase flow with Marangoni stress and the interface mass transfer. In this paper, a new model combining PFLBM (phase-field lattice Boltzmann method) and FDM ( finite-difference method) coupling with the ghost-cell method was built. The PFLBM-FDM was validated for the high accuracy, less computational cost, and low mass loss compared to other methods. Based on the PFLBM-FDM, a surfactant-laden bubble deformed and transported in a laminar Couette flow was investigated. The deformation ratio and transportation velocity were explored with different density ratios, surface tensions, shear velocities, and diffusion coefficients. The numerical results showed that the equilibrium state of the bubble deformation was decided only by the dimensionless numbers when the Sh number was higher than 100. Moreover, the transportation velocity of the bubble can be controlled by the balance between the Marangoni stress and shear velocity. When the Sh is lower than 100, the Marangoni stress from the surfactant is not a long-range force, which only works at the early flow. Otherwise, the Marangoni stress will be a long-range force that provides a persistent force to accelerate the bubble by ∼10%. Increasing ReH will further intensify the effect. Based on all the data, a correlation of the bubble deformation including with the densities of two fluids was obtained and the error range is less 5%.

Contact Angle Measurement on Curved Wetting Surfaces in Multiphase Lattice Boltzmann Method

Contact angle is an essential physical quantity that characterizes the wettability of a substrate. Although it is widely used in the studies of surface wetting, capillary phenomena, and moving contact lines, the contact angle measurements in simulations and experiments are still complicated and time-consuming. In this paper, we present an efficient scheme for the measurement of contact angle on curved wetting surfaces in lattice Boltzmann simulations. The measuring results are in excellent agreement with the theoretical predictions without considering the gravity effect. A series of simulations with various drop sizes and surface curvatures confirm that the present scheme is grid-independent. Then, the scheme is verified in gravitational environments by simulating the deformations of sessile and pendent droplets on the curved wetting surface. The numerical results are highly consistent with experimental observations and support the theoretical analysis that the microscopic contact angle is independent of gravity. Furthermore, the method utilizes only the microscopic geometry of the contact angle and does not depend on the droplet profile; therefore, it can be applied to nonaxisymmetric shapes or moving contact lines. The scheme is applied to capture the dynamic contact angle hysteresis on homogeneous or chemically heterogeneous curved surfaces. Importantly, the accurate contact angle measurement enables the dynamic mechanical analysis of moving contact lines. The present measurement is simple and efficient and can be extended to implementations in various multiphase lattice Boltzmann models.

Phase-field numerical study on the dynamic process of thermocapillary patterning

It is well known that surface tension is dependent on temperature, and thus a nonuniform temperature may cause thermocapillary flow which is referred to as the Marangoni effect. For a thin liquid-air film confined between a flat hot plate and a topographical cold template, it undergoes deformation due to thermocapillary flow. This phenomenon is termed as thermocapillary patterning, and has been used to fabricate micro- and nanostructure in polymer films. In most cases, the obtained structure conforms to the template; i.e., it can be considered as a replication technique. In this paper, we developed a two-phase flow numerical model based on the phase field to study the dynamic process of thermocapillary patterning. As a remeshing-free method, the phase field enables the incorporation of thermal field and multiphase flow with free surface deformation. The numerical model was employed to study the dynamic process of thermocapillary patterning. Meanwhile, the effects of some parameters, e.g., temperature, geometry parameters, and contact angle, were also investigated.

Improved phase-field-based lattice Boltzmann method for thermocapillary flow

In this paper, we present an improved phase-field-based lattice Boltzmann (LB) method for thermocapillary flows with large density, viscosity, and thermal conductivity ratios. The present method uses three LB models to solve the conservative Allen-Cahn equation, the incompressible Navier-Stokes equations, and the temperature equation. To overcome the difficulty caused by the convection term in solving the convection-diffusion equation for the temperature field, we first rewrite the temperature equation as a diffuse equation where the convection term is regarded as the source term and then construct an improved LB model for the diffusion equation. The macroscopic governing equations can be recovered correctly from the present LB method; moreover, the present LB method is much simpler and more efficient. In order to test the accuracy of this LB method, several numerical examples are considered, including the planar thermal Poiseuille flow of two immiscible fluids, the two-phase thermocapillary flow in a nonuniformly heated channel, and the thermocapillary Marangoni flow of a deformable bubble. It is found that the numerical results obtained from the present LB method are consistent with the theoretical prediction and available numerical data, which indicates that the present LB method is an effective approach for the thermocapillary flows.

Tricoupled hybrid lattice Boltzmann model for nonisothermal drying of colloidal suspensions in micropore structures

A tricoupled hybrid lattice Boltzmann model (LBM) is developed to simulate colloidal liquid evaporation and colloidal particle deposition during the nonisothermal drying of colloidal suspensions in micropore structures. An entropic multiple-relaxation-time multirange pseudopotential two-phase LBM for isothermal interfacial flow is first coupled to an extended temperature equation for simulating nonisothermal liquid drying. Then the coupled model is further coupled with a modified convection diffusion equation to consider the nonisothermal drying of colloidal suspensions. Two drying examples are considered. First, drying of colloidal suspensions in a two-pillar micropore structure is simulated in two dimensions (2D), and the final configuration of colloidal particles is compared with the experimental one. Good agreement is observed. Second, at the temperature of 343.15 K (70^{∘}C), drying of colloidal suspensions in a complex spiral-shaped micropore structure containing 220 pillars is simulated (also in 2D). The drying pattern follows the designed spiral shape due to capillary pumping, i.e., transport of the liquid from larger pores to smaller ones by capillary pressure difference. Since the colloidal particles are passively carried with liquid, they accumulate at the small menisci as drying proceeds. As liquid evaporates at the small menisci, colloidal particles are deposited, eventually forming solid structures between the pillars (primarily), and at the base of the pillars (secondarily). As a result, the particle deposition conforms to the spiral route. Qualitatively, the simulated liquid and particle configurations agree well with the experimental ones during the entire drying process. Quantitatively, the model demonstrates that the evaporation rate and the particle accumulation rate slowly decrease during drying, similar to what is seen in the experimental results, which is due to the reduction of the liquid-vapor interfacial area. In conclusion, the hybrid model shows the capability and accuracy for simulating nonisothermal drying of colloidal suspensions in a complex micropore structure both qualitatively and quantitatively, as it includes all the required physics and captures all the complex features observed experimentally. Such a tricoupled LBM has a high potential to become an efficient numerical tool for further investigation of real and complex engineering problems incorporating drying of colloidal suspensions in porous media.

Hybrid Allen-Cahn-based lattice Boltzmann model for incompressible two-phase flows: The reduction of numerical dispersion

In this work a phase field-lattice Boltzmann model for incompressible two-phase flows is presented. In this model, the interface tracking equation is a linear combination of the local and nonlocal Allen-Cahn equations. We also propose a multiple-relaxation-time lattice Boltzmann model for solving the hybrid Allen-Cahn equation. The second-order convergence rate of the present model in space is validated by simulating the diagonal translation of circular interface. Three other numerical tests, including static bubble immersed in another fluid, bubble rising under gravity, and droplet splashing on a thin liquid film, are simulated to verify the performance of the present model in reducing the numerical dispersion. The numerical results indicate that the order parameter fluctuation can be reduced by one order of magnitude in bulk region.

Prediction of immiscible two-phase flow properties in a two-dimensional Berea sandstone using the pore-scale lattice Boltzmann simulation

Immiscible two-phase flow in porous media is commonly encountered in industrial processes and environmental issues, such as enhanced oil recovery and the migration of fluids in an unsaturated zone. To deepen the current understanding of its underlying mechanism, this work focuses on the factors that influence the relative permeability and specific interfacial length of a two-phase flow in porous media, i.e., fluid saturation, viscosity ratio and contact angle. The lattice Boltzmann color-gradient model is adopted for pore-scale investigations, and the main findings are obtained as follows. Firstly, the relative permeability of each fluid increases as its saturation increases. The specific interfacial length first increases and then decreases as the saturation of the wetting fluid increases, and reaches a maximum when the permeabilities of both fluids are equal. Secondly, as the viscosity ratio of wetting to non-wetting fluids increases, the relative permeability of the wetting fluid will increase while that of the non-wetting fluid will decrease. The specific interfacial length will increase with increasing the viscosity difference between fluids. Finally, as the contact angle (measured from the wetting fluid) increases, the relative permeability of the wetting fluid overall increases while that of the non-wetting fluid decreases. Increasing contact angle always leads to a decrease in the specific interfacial length.

Thermohydrodynamics of an evaporating droplet studied using a multiphase lattice Boltzmann method

In this paper, the thermohydrodynamics of an evaporating droplet is investigated by using a single-component pseudopotential lattice Boltzmann model. The phase change is applied to the model by adding source terms to the thermal lattice Boltzmann equation in such a way that the macroscopic energy equation of multiphase flows is recovered. In order to gain an exhaustive understanding of the complex hydrodynamics during evaporation, a single droplet is selected as a case study. At first, some tests for a stationary (non-)evaporating droplet are carried out to validate the method. Then the model is used to study the thermohydrodynamics of a falling evaporating droplet. The results show that the model is capable of reproducing the flow dynamics and transport phenomena of a stationary evaporating droplet quite well. Of course, a moving droplet evaporates faster than a stationary one due to the convective transport. Our study shows that our single-component model for simulating a moving evaporating droplet is limited to low Reynolds numbers.

Lattice Boltzmann simulations of droplet formation in confined channels with thermocapillary flows

Based on mesoscale lattice Boltzmann simulations with the "Shan-Chen" model, we explore the influence of thermocapillarity on the breakup properties of fluid threads in a microfluidic T-junction, where a dispersed phase is injected perpendicularly into a main channel containing a continuous phase, and the latter induces periodic breakup of droplets due to the cross-flowing. Temperature effects are investigated by switching on-off both positive-negative temperature gradients along the main channel direction, thus promoting a different thread dynamics with anticipated-delayed breakup. Numerical simulations are performed at changing the flow rates of both the continuous and dispersed phases, as well as the relative importance of viscous forces, surface tension forces, and thermocapillary stresses. The range of parameters is broad enough to characterize the effects of thermocapillarity on different mechanisms of breakup in the confined T-junction, including the so-called "squeezing" and "dripping" regimes, previously identified in the literature. Some simple scaling arguments are proposed to rationalize the observed behavior, and to provide quantitative guidelines on how to predict the droplet size after breakup.

Experimental Droplet Study of Inverted Marangoni Effect of a Binary Liquid Mixture on a Nonuniform Heated Substrate

We present an experimental study on the inversion of the Marangoni effect of a binary mixture droplet under a horizontal temperature gradient. In particular, we studied the dynamics and the evaporation behavior under these conditions. We show that a binary mixture (97% water-3% butanol) droplet has a tendency to migrate to warmer areas, as opposed to spreading in pure fluids. During the evaporation process, we distinguish three stages of evaporation that are correlated to the dynamics of the droplet.

Lattice Boltzmann modeling of three-phase incompressible flows

In this paper, based on multicomponent phase-field theory we intend to develop an efficient lattice Boltzmann (LB) model for simulating three-phase incompressible flows. In this model, two LB equations are used to capture the interfaces among three different fluids, and another LB equation is adopted to solve the flow field, where a new distribution function for the forcing term is delicately designed. Different from previous multiphase LB models, the interfacial force is not used in the computation of fluid velocity, which is more reasonable from the perspective of the multiscale analysis. As a result, the computation of fluid velocity can be much simpler. Through the Chapman-Enskog analysis, it is shown that the present model can recover exactly the physical formulations for the three-phase system. Numerical simulations of extensive examples including two circular interfaces, ternary spinodal decomposition, spreading of a liquid lens, and Kelvin-Helmholtz instability are conducted to test the model. It is found that the present model can capture accurate interfaces among three different fluids, which is attributed to its algebraical and dynamical consistency properties with the two-component model. Furthermore, the numerical results of three-phase flows agree well with the theoretical results or some available data, which demonstrates that the present LB model is a reliable and efficient method for simulating three-phase flow problems.

Flow-parametric regulation of shear-driven phase separation in two and three dimensions

The Cahn-Hilliard equation with an externally prescribed chaotic shear flow is studied in two and three dimensions. The main goal is to compare and contrast the phase separation in two and three dimensions, using high-resolution numerical simulation as the basis for the study. The model flow is parametrized by its amplitudes (thereby admitting the possibility of anisotropy), length scales, and multiple time scales, and the outcome of the phase separation is investigated as a function of these parameters as well as the dimensionality. In this way, a parameter regime is identified wherein the phase separation and the associated coarsening phenomenon are not only arrested but in fact the concentration variance decays, thereby opening up the possibility of describing the dynamics of the concentration field using the theories of advection diffusion. This parameter regime corresponds to long flow correlation times, large flow amplitudes and small diffusivities. The onset of this hyperdiffusive regime is interpreted by introducing Batchelor length scales. A key result is that in the hyperdiffusive regime, the distribution of concentration (in particular, the frequency of extreme values of concentration) depends strongly on the dimensionality. Anisotropic scenarios are also investigated: for scenarios wherein the variance saturates (corresponding to coarsening arrest), the direction in which the domains align depends on the flow correlation time. Thus, for correlation times comparable to the inverse of the mean shear rate, the domains align in the direction of maximum flow amplitude, while for short correlation times, the domains initially align in the opposite direction. However, at very late times (after the passage of thousands of correlation times), the fate of the domains is the same regardless of correlation time, namely alignment in the direction of maximum flow amplitude. A theoretical model to explain these features is proposed. These features and the theoretical model carry over to the three-dimensional case, albeit that an extra degree of freedom pertains, such that the dynamics of the domain alignment in three dimensions warrant a more detailed consideration, also presented herein.

Importance of wave-number dependence of Biot numbers in one-sided models of evaporative Marangoni instability: Horizontal layer and spherical droplet

A one-sided model of the thermal Marangoni instability owing to evaporation into an inert gas is developed. Two configurations are studied in parallel: a horizontal liquid layer and a spherical droplet. With the dynamic gas properties being admittedly negligible, one-sided approaches typically hinge upon quantifying heat and mass transfer through the gas phase by means of transfer coefficients (like in the Newton's cooling law), which in dimensionless terms eventually corresponds to using Biot numbers. Quite a typical arrangement encountered in the literature is a constant Biot number, the same for perturbations of different wavelengths and maybe even the same as for the reference state. In the present work, we underscore the relevance of accounting for its wave-number dependence, which is especially the case in the evaporative context with relatively large values of the resulting effective Biot number. We illustrate the effect in the framework of the Marangoni instability thresholds. As a concrete example, we consider HFE-7100 (a standard refrigerant) for the liquid and air for the inert gas.

Multirelaxation-time lattice Boltzmann model for droplet heating and evaporation under forced convection

We investigate the evaporation of a droplet surrounded by superheated vapor with relative motion between phases. The evaporating droplet is a challenging process, as one must take into account the transport of mass, momentum, and heat. Here a lattice Boltzmann method is employed where phase change is controlled by a nonideal equation of state. First, numerical simulations are compared to the D(2) law for a vaporizing static droplet and good agreement is observed. Results are then presented for a droplet in a Lagrangian frame under a superheated vapor flow. Evaporation is described in terms of the temperature difference between liquid-vapor and the inertial forces. The internal liquid circulation driven by surface-shear stresses due to convection enhances the evaporation rate. Numerical simulations demonstrate that for higher Reynolds numbers, the dynamics of vaporization flux can be significantly affected, which may cause an oscillatory behavior on the droplet evaporation. The droplet-wake interaction and local mass flux are discussed in detail.

Thermocapillary-actuated contact-line motion of immiscible binary fluids over substrates with patterned wettability in narrow confinement

We investigate thermocapillary-driven contact-line dynamics of two immiscible fluids in a narrow fluidic confinement comprising substrates with patterned wettability variations. Our study, based on phase field formalism, demonstrates that the velocity of the contact line is a strong function of the combined consequences of the applied thermal gradient and the substrate wetting characteristics. Finally, we evaluate different energy transfer rates and show that the dissipation due to fluid slip over the solid surface plays a dominating role in transferring energy into the contact-line motion. Our analysis, in effect, provides an elegant way of controlling the capillary filling rate in a narrow fluidic confinement by tailoring the applied temperature gradient and the substrate wettability in tandem.

Nonequilibrium scheme for computing the flux of the convection-diffusion equation in the framework of the lattice Boltzmann method

In this paper, we propose a local nonequilibrium scheme for computing the flux of the convection-diffusion equation with a source term in the framework of the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). Both the Chapman-Enskog analysis and the numerical results show that, at the diffusive scaling, the present nonequilibrium scheme has a second-order convergence rate in space. A comparison between the nonequilibrium scheme and the conventional second-order central-difference scheme indicates that, although both schemes have a second-order convergence rate in space, the present nonequilibrium scheme is more accurate than the central-difference scheme. In addition, the flux computation rendered by the present scheme also preserves the parallel computation feature of the LBM, making the scheme more efficient than conventional finite-difference schemes in the study of large-scale problems. Finally, a comparison between the single-relaxation-time model and the MRT model is also conducted, and the results show that the MRT model is more accurate than the single-relaxation-time model, both in solving the convection-diffusion equation and in computing the flux.

Effect of the forcing term in the pseudopotential lattice Boltzmann modeling of thermal flows

The pseudopotential lattice Boltzmann (LB) model is a popular model in the LB community for simulating multiphase flows. Recently, several thermal LB models, which are based on the pseudopotential LB model and constructed within the framework of the double-distribution-function LB method, were proposed to simulate thermal multiphase flows [G. Házi and A. Márkus, Phys. Rev. E 77, 026305 (2008); L. Biferale, P. Perlekar, M. Sbragaglia, and F. Toschi, Phys. Rev. Lett. 108, 104502 (2012); S. Gong and P. Cheng, Int. J. Heat Mass Transfer 55, 4923 (2012); M. R. Kamali et al., Phys. Rev. E 88, 033302 (2013)]. The objective of the present paper is to show that the effect of the forcing term on the temperature equation must be eliminated in the pseudopotential LB modeling of thermal flows. First, the effect of the forcing term on the temperature equation is shown via the Chapman-Enskog analysis. For comparison, alternative treatments that are free from the forcing-term effect are provided. Subsequently, numerical investigations are performed for two benchmark tests. The numerical results clearly show that the existence of the forcing-term effect will lead to significant numerical errors in the pseudopotential LB modeling of thermal flows.