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

Development of a central-moment phase-field lattice Boltzmann model for thermocapillary flows: Droplet capture and computational performance

This study develops a computationally efficient phase-field lattice Boltzmann model with the capability to simulate thermocapillary flows. The model was implemented into the open-source simulation framework, waLBerla, and extended to conduct the collision stage using central moments. The multiphase model was coupled with both a passive-scalar thermal LB, and a RK solution to the energy equation in order to resolve temperature-dependent surface tension phenomena. Various lattice stencils (D3Q7, D3Q15, D3Q19, D3Q27) were tested for the passive-scalar LB and both the second- and fourth-order RK methods were investigated. There was no significant difference observed in the accuracy of the LB or RK schemes. The passive scalar D3Q7 LB discretisation tended to provide computational benefits, while the second order RK scheme is superior in memory usage. This paper makes contributions relating to the modelling of thermocapillary flows and to understanding the behaviour of droplet capture with thermal sources analogous to thermal tweezers. Four primary contributions to the literature are identified. First, a new 3D thermocapillary, central-moment phase-field LB model is presented and implemented in the open-source software, waLBerla. Second, the accuracy and computational performance of various techniques to resolve the energy equation for multiphase, incompressible fluids is investigated. Third, the dynamic droplet transport behaviour in the presence of thermal sources is studied and insight is provided on the potential ability to manipulate droplets based on local domain heating. Finally, a concise analysis of the computational performance together with near-perfect scaling results on NVIDIA and AMD GPU-clusters is shown. This research enables the detailed study of droplet manipulation and control in thermocapillary devices

Motion of an air bubble under the action of thermocapillary and buoyancy forces

A novel way to handle surface tension gradient driven flows is developed in the volume-of-fluid (VoF) framework. Using an open source Navier-Stokes solver, {\it Basilisk}, and the present formulation, we investigate thermocapillary migration of drops/bubbles in a surrounding medium. Several validation exercises have been performed, which demonstrate that the present solver is a robust one to investigate interfacial flows with variable surface tension. It is well known that it is a challenging task to numerically model the tangential and normal surface forces arising due to interfacial tension. We have shown that the present method does not require the artificial smearing of surface tension about the interface, and thus predicts the theoretical value of the terminal velocity of bubble/drop migrating due to an imposed temperature gradient very well. It is also demonstrated that the present solver provides accurate results for problems exhibiting the gravity and thermocapillary forces simultaneously, and useful for systems with high viscosity and density ratios.Comment: 30 pages, 16 figures, submitted to Computers and Fluid

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))

Generalized equilibria for color-gradient lattice Boltzmann model based on higher-order Hermite polynomials: A simplified implementation with central moments

We propose generalized equilibria of a three-dimensional color-gradient lattice Boltzmann model for two-component two-phase flows using higher-order Hermite polynomials. Although the resulting equilibrium distribution function, which includes a sixth-order term on the velocity, is computationally cumbersome, its equilibrium central moments (CMs) are velocity-independent and have a simplified form. Numerical experiments show that our approach, as in Wen et al. [{Phys. Rev. E \textbf{100}, 023301 (2019)}] who consider terms up to third order, improves the Galilean invariance compared to that of the conventional approach. Dynamic problems can be solved with high accuracy at a density ratio of 10; however, the accuracy is still limited to a density ratio of 1000. For lower density ratios, the generalized equilibria benefit from the CM-based multiple-relaxation-time model, especially at very high Reynolds numbers, significantly improving the numerical stability.Comment: 22 pages, 8 figure

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