Domain growth in ternary fluids: A level set approach

Spontaneous phase separation of ternary fluid mixtures

We computationally study the spontaneous phase separation of ternary fluid mixtures using the lattice Boltzmann method both when all the surface tensions are equal and when they have different values. To rationalise the phase diagram of possible phase separation mechanisms, previous theoretical works typically rely on analysing the sign of the eigenvalues resulting from a simple linear stability analysis, but we find this does not explain the observed simulation results. Here, we classify the possible separation pathways into four basic mechanisms, and develop a phenomenological model that captures the composition regimes where each mechanism is prevalent. We further highlight that the dominant mechanism in ternary phase separation involves enrichment and instability of the minor component at the fluid-fluid interface, which is absent in the case of binary fluid mixtures.

Reduction-consistent phase-field lattice Boltzmann equation for N immiscible incompressible fluids

In this paper, we develop a reduction-consistent conservative phase-field method for interface-capturing among N (N≥ 2) immiscible fluids, which is governed by conservative Allen-Cahn equation (CACE); here the reduction-consistent property is that if only M (1≤M≤N-1) immiscible fluids are present in a N-phase system, the governing equations for N immiscible fluids must reduce to the corresponding M immiscible fluids system. Then we propose a reduction-consistent lattice Boltzmann equation (LBE) method for solving N immiscible incompressible fluids with high density and viscosity contrasts. Some numerical simulations are carried out to validate the present LBE such as stationary droplets, spreading of a liquid lens, and spinodal decomposition together with the reduction-consistent property, and the numerical results predicted by present LBE are in good agreement with the analytical solutions/other numerical results, which also demonstrate the reduction-consistent property by present LBE.

Multiphase flows of N immiscible incompressible fluids: Conservative Allen-Cahn equation and lattice Boltzmann equation method

In this paper, we develop a conservative phase-field method for interface-capturing among N (N≥2) immiscible fluids, the evolution of the fluid-fluid interface is captured by conservative Allen-Cahn equation (CACE), and the interface force of N immiscible fluids is incorporated to Navier-Stokes equation (NSE) by chemical potential form. Accordingly, we propose a lattice Boltzmann equation (LBE) method for solving N (N≥2) immiscible incompressible NSE and CACE at high density and viscosity contrasts. Numerical simulations including stationary droplets, Rayleigh-Taylor instability, spreading of liquid lenses, and spinodal decompositions are carried out to show the accuracy and capability of present LBE, and the results show that the predictions by use of the present LBE agree well with the analytical solutions and/or other numerical results.

Phase-field-theory-based lattice Boltzmann equation method for N immiscible incompressible fluids

From the phase field theory, we develop a lattice Boltzmann equation (LBE) method for N (N≥2) immiscible incompressible fluids, and the Cahn-Hilliard equation, which could capture the interfaces between different phases, is also solved by LBE for an N-phase system. In this model, the interface force of N immiscible incompressible fluids is incorporated by chemical potential form, and the fluid-fluid surface tensions could be directly calculated and independently tuned. Numerical simulations including two stationary droplets, spreading of a liquid lens with and without gravity and two immiscible liquid lenses, and phase separation are conducted to validate the present LBE, and numerical results show that the predictions by LBE agree well with the analytical solutions and other numerical results.

Numerical investigation of bubbles coalescence in a shear flow with diffuse-interface model

In this article, we apply the diffuse-interface model [developed by Shah and Yuan (2011) [21]] for collision and coalescence of two bubbles in a linear shear flow. The governing equations consist of a system of coupled nonlinear partial differential equations for conservation of mass, momentum and phase transport. In the two-phase flow, the diffuse-interface model relaxes certain numerical difficulties for tracking the moving interface. An artificial compressibility based numerical scheme is implemented to study the effects of surface tension on bubbles coalescence and separation. We found the critical value of the surface tension coefficient and observed that lowering the surface tension coefficient from the critical value prevent bubbles to coalesce.

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.

A level set method for determining critical curvatures for drainage and imbibition

An accurate description of the mechanics of pore level displacement of immiscible fluids could significantly improve the predictions from pore network models of capillary pressure-saturation curves, interfacial areas and relative permeability in real porous media. If we assume quasi-static displacement, at constant pressure and surface tension, pore scale interfaces are modeled as constant mean curvature surfaces, which are not easy to calculate. Moreover, the extremely irregular geometry of natural porous media makes it difficult to evaluate surface curvature values and corresponding geometric configurations of two fluids. Finally, accounting for the topological changes of the interface, such as splitting or merging, is nontrivial. We apply the level set method for tracking and propagating interfaces in order to robustly handle topological changes and to obtain geometrically correct interfaces. We describe a simple but robust model for determining critical curvatures for throat drainage and pore imbibition. The model is set up for quasi-static displacements but it nevertheless captures both reversible and irreversible behavior (Haines jump, pore body imbibition). The pore scale grain boundary conditions are extracted from model porous media and from imaged geometries in real rocks. The method gives quantitative agreement with measurements and with other theories and computational approaches.

Encapsulated drop breakup in shear flow

We investigate the deformation and breakup in shear flow of an encapsulated drop in which both the core and shell are Newtonian fluids. The equations of motion are solved numerically using a level set method to track interface motion. We consider the case of a drop stretched to a given length in constant shear and then allowed to relax. A range of morphologies is produced, and novel kinematics occur, due to the interaction of the core and outer interfaces. A phase diagram is presented to describe the morphologies produced over a range of capillary numbers and core interfacial tensions.

Dynamics of a drop at a fluid interface under shear

We analyze the dynamics of a two-dimensional drop lying on a fluid interface, sometimes called a liquid lens, subjected to simple shear flow. The three fluids, the drop and the two external fluids, meet at a triple point (or a triple line in three dimensions). A requirement for steady drop shapes is that the triple points are stationary. This leads to a flow topology different than that of a freely suspended drop. Results are substantiated with numerical results using a level set method for interface evolution and treatment of triple points. Possible implications for new drop instabilities are also discussed.

Scale invariance in coarsening of binary and ternary fluids

Phase separation in binary and ternary fluids is studied using a two-dimensional lattice gas automata. The lengths given by the the first zero crossing point of the correlation function and the total interface length is shown to exhibit power law dependence on time. In binary mixtures, our data clearly indicate the existence of a regime having more than one length scale, where the coarsening process proceeds through the rupture and reassociation of domains. In ternary fluids; in the case of symmetric mixtures there exists a regime with a single length scale having dynamic exponent 1/2, while in asymmetric mixtures our data establish the break down of scale invariance.

Hydrodynamic coarsening of binary fluids

By suitable interpretation of results from the linear analysis of interface dynamics, it is found that the hydrodynamic growth of the size L of domains that follow spinodal decomposition in fluid mixtures scales with time as L approximately t(alpha), with alpha = 4/7 in the inertial regime. The previously proposed exponent alpha = 2/3 is shown to indicate only the scaling of the oscillatory frequency omega(-2/3) approximately L of the largest structures of the system. The viscous dissipation in the system occurs within a layer of thickness L(d) that also follows a power law of the form L(d) approximately L3/4 in the inertial regime. In the viscous regime the growth is linear in time L approximately t and the dissipative region remains constant L(d) approximately L0.