Phase-Field Models for Multi-Component Fluid Flows

We review our recent results on fingering, bamboo waves, and drop breakup in low Reynolds number flows composed of two viscous liquids under shear.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

The Cahn-Hilliard phase field method for two-phase flow has gained particular attention due to its unique features including its flexibility for complex morphological and topological changes, the intrinsic property of conserving mass, and the natural approach to account for the surface tension. The essential idea of the method is to use a phase field function to describe the two-phase system, while a thin smooth transition layer (interfacial area) connects the two immiscible fluids, where the value of phase field function varies continuously. The application of phase field method to two-phase flows has become more widespread recently, but to the best of our knowledge, very little progress has been made for the method being applied to the two-phase flows with phase change. This includes evaporation, condensation and boiling, which plays an important role in enhanced heat transfer in power electronics, energy, and aerospace engineering. In previous work, in order to face the challenge of large density contrast and high Reynolds number of practical engineering problems, we developed a stabilized phase field method that can handle two-phase flow with density ratio over 1000, at high Reynolds number over 10,000, and applied the method to simulate slug initiation in a long circular pipe. In this paper, inspired by the boiling model widely used in the level-set method, we propose a new boiling model that assumes that boiling takes place in the whole interfacial layer. The method is then used to solve the non-solenoidal Navier-Stokes equations. The boiling model is validated by simulating a vapor bubble growing in super-heated liquid. For this case, the growth rate of the bubble has an analytical solution, and it is used as a benchmark case in volume of fluid (VOF) and level-set methods extensively. For three different refrigerants, namely water, R134a and HFE7100, our phase field method with the boiling model can obtain accurate simulation results. Moreover, the method and model are applied to predict the three-dimensional boiling heat transfer in a rectangular micro-channel that contains a water vapor bubble with various inlet super-heat conditions. We found that the predicted bubble shape is very similar to that visualized in existing experiment. From our simulation of boiling flow using the phase field method, We have found that the required mesh resolution for the phase field method is comparable with that of VOF and level-set methods.

PurposeThe purpose of this study is to perform a detailed review on the numerical modeling of multiphase and multicomponent flows in microfluidic system using phase-field method. The phase-field method is of emerging importance in numerical computation of transport phenomena involving multiple phases and/or components. This method is not only used to model interfacial phenomena typical to multiphase flows encountered in engineering and nature but also turns out to be a promising tool in modeling the dynamics of complex fluid-fluid interfaces encountered in physiological systems such as dynamics of vesicles and red blood cells). Intrinsically, a priori unknown topological evolution of interfaces offers to be the most concerning challenge toward accurate modeling of moving boundary problems. However, the numerical difficulties can be tackled simultaneously with numerical convenience and thermodynamic rigor in the paradigm of the phase field method.Design/methodology/approachThe phase-field method replaces the macroscopically sharp interfaces separating the fluids by a diffuse transition layer where the interfacial forces are smoothly distributed. As against the moving mesh methods (Lagrangian) for the explicit tracking of interfaces, the phase-field method implicitly captures the same through the evolution of a phase-field function (Eulerian). In contrast to the deployment of an artificially smoothing function for the interface as used in the volume of a fluid or level set method, however, the phase-field method uses mixing free energy for describing the interface. This needs the consideration of an additional equation for an order parameter. The dynamic evolution of the system (equation for order parameter) can be described by Allen–Cahn or Cahn–Hilliard formulation, which couples with the Navier–Stokes equation with the aid of a forcing function that depends on the chemical potential and the gradient of the order parameter.FindingsIn this review, first, the authors discuss the broad motivation and the fundamental theoretical foundation associated with phase-field modeling from the perspective of computational microfluidics. They subsequently pinpoint the outstanding numerical challenges, including estimations of the model-free parameters. They outline some numerical examples, including electrohydrodynamic flows, to demonstrate the efficacy of the method. Finally, they pinpoint various emerging issues and futuristic perspectives connecting the phase-field method and computational microfluidics.Originality/valueThis paper gives unique perspectives to future directions of research on this topic.

Understanding pinchoff in a liquid-liquid jet is one of the fundamental problems in the physics of fluid. Pinchoff has a wide variety of applications such as in ink-jet printers. We have numerically investigated the breakup of a forced liquid jet into drops in immiscible liquid-liquid systems with a phase-field model. In the phase-field model, the classical sharp interface between the two immiscible fluids is represented by a transition region of small but finite width. Across this width the composition of one of the two fluids changes continuously. The phase-field method can deal with topological transitions such as breakup and reconnection smoothly without ad hoc “cut and connect” or smoothing procedures. We compared the numerical results on the pinchoff of liquidliquid jets with surface tension finding good agreement with experimental data. In particular, we investigated axial velocity and vorticity structures around the jet neck before and after pinchoff.

The phase field method (PFM) can be used as an approach to dislocation dynamics simulations alternative to the line DD method discussed in Chapter 10. The degrees of freedom in PFM are continuous smooth fields occupying the entire simulation volume, and dislocations are identified with locations where the field values change rapidly. As we will see later, as an approach to dislocation dynamics simulations PFM holds several advantages. First, it is easier to implement into a computer code than a line DD model. In particular, the complex procedures for making topological changes (Section 10.4) are no longer necessary. Second, the implementation of PFM can take advantage of well-developed and efficient numerical methods for solving partial differential equations (PDEs). Another important merit of PFM is its applicability in a wide range of seemingly different situations. For example, it is possible to simulate the interaction and co-evolution of several types of material microstructures, such as dislocations and alloying impurities, within a unified model. PFM has become popular among physicists and materials scientists over the last 20 years, but as a numerical method it is not new. After all, it is all about solving PDEs on a grid. Numerical integration of PDEs is a vast and mature area of computational mathematics. A number of efficient methods have already been developed, such as the finite difference method [121], the finite element method [122], and spectral methods [123], all of which have been used in PFM simulations. The relatively new aspects of PFM are associated with the method’s formulation and applications, which are partly driven by the growing interest in understanding material microstructures. In Section 11.1, we begin with the general aspects of PFM demonstrated by two simple applications of the method not related to dislocations. Section 11.2 describes the elements required to adapt PFM to dislocation simulations. There we will briefly venture into the field of micromechanics and consider the concept of eigenstrain. The elastic energy of an arbitrary eigenstrain field is derived in Section 11.3. Section 11.4 discusses an example in which the PFM equations for dislocations are solved using the fast Fourier transform method.

Numerical simulation of two-phase flow in porous media using a wavelet based phase-field method

Phase field method has been widely utilized to study multiphase flow problems, but has seldom been applied to the study of imbibition. Previous methods used to simulate imbibition, such as moving mesh method, need to specify capillary pressure as a boundary condition a priori, whereas phase field method can calculate capillary pressure automatically for various geometries. Therefore, phase field method would be a versatile tool for the study of imbibition in various geometries. In this paper, phase field method is employed to solve dynamical imbibition problem in various geometries, including straight tube, conical tube and structures in which the topology changes. The variation of the imbibition height with respect to time from phase field simulation is verified with theoretical predictions from Lucas-Washburn law in a straight capillary tube with three gravitational scenarios. In addition, the capillary pressure and velocity field are found to be consistent with Laplace-Young equation and Hagen-Poiseuille equation in various geometries. The applicability and accuracy of the phase field method for the study of imbibition in structures with changing topology are also discussed. Cited as : Xiao, J., Luo, Y., Niu, M., Wang, Q., Wu, J., Liu, X., Xu, J. Study of imbibition in various geometries using phase field method. Capillarity, 2019, 2(4): 57-65, doi: 10.26804/capi.2019.04.01

The simulation of multiphase and multicomponent fluid flows often requires large coupled systems of nonlinear partial differential equations. These equations are convection‐dominated, with important local diffusive effects. An operator splitting technique is used to address these different phenomena. Adaptive local grid refinement methods are then presented to resolve the moving internal boundary layers which arise and often govern the mass transfer between the phases. Owing to the large size of many application, efficiency in computation is critical. Concepts of domain decomposition for dynamic adaptive grid refinement are presented. Techniques for dividing the problems into pieces for parallel computation on separate processors are also discussed.

Multiphase and multi-component fluid flows are highly intricate, exhibiting dynamic behaviors that challenge conventional simulation methods, especially across a wide range of operating conditions. Adding fluid-solid interactions further escalates this complexity, requiring simulations that are both physically accurate and computationally more tractable. Existing methods often fall short in aspects of either accuracy, efficiency, or stability, and sometimes may also consume excessive memory, rendering them unsuitable for many practical applications. In this paper, we introduce a novel, more practical solution combining a collision-enhanced lattice Boltzmann method with a finite-volume approach. This hybrid solver is designed to effectively simulate multiphase, multi-component flows via the phase-field modeling with a novel wetting boundary treatment in the finite-volume framework, while significantly reducing memory consumption. It enables simulations of multiple fluid phases under varied conditions, capturing rich interfacial behaviors. Additionally, it incorporates an immersed-boundary method to efficiently handle complex fluid-solid interactions. Validation tests confirm that our method produces physically consistent results and demonstrates accuracy across multiple evaluations. We provide comprehensive simulation results, along with comparisons of memory consumption and efficiency, highlighting the practical utility and superiority of our solver over conventional methods.

A generalized continuous surface tension force formulation for phase-field models for multi-component immiscible fluid flows

An immersed boundary-lattice Boltzmann method for single- and multi-component fluid flows

Mass transfer characteristics of a single bubble in immiscible fluids

The lattice Boltzmann method is a promising approach for modeling single and multicomponent fluid flow in complex geometries like porous materials. Here, we review some of our previous work and discuss some recent developments concerning fluid flow in multiple pore size materials. After presenting some simple test cases to validate the model, results from large scale simulations of single and multi-component fluid flow through digitized Fontainebleau sandstone, generated by X-Ray microtomography, are given. Reasonably good agreement was found when compared to experimentally determined values of permeability for similar rocks. Finally, modification of the lattice Boltzmann equations, to describe flow in microporous materials, is described. The potential for modeling flows in other microstructures of interest to concrete technology will be discussed.

A novel probability approach is utilized to solve the coupled nonlinear partial differential equations describing multiphase and multicomponent fluid flow in a porous medium. The approach generalizes diffusion-limited aggregation (DLA) to finite two-fluid mobility ratios and to more complete descriptions of physical systems. The limits of validity of the usual DLA method are determined. Two-dimensional saturation and concentration profiles are determined as a function of the mobility ratio. Solutions are obtained for the full range of flow regimes (stable to highly unstable).