Academic literature on the topic 'Navier-Stokes-Cahn-Hilliard model'

We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase-field variable, chemical potential, velocity and pressure in different discrete norms for the Cahn–Hilliard–Stokes phase-field model. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of our scheme.

In this paper we investigate the compressible Navier–Stokes–Cahn–Hilliard equations (the so-called NSCH model) derived by Lowengrub and Truskinovsky. This model describes the flow of a binary compressible mixture; the fluids are supposed to be macroscopically immiscible, but partial mixing is permitted leading to narrow transition layers. The internal structure and macroscopic dynamics of these layers are induced by a Cahn–Hilliard law that the mixing ratio satisfies. The PDE constitute a strongly coupled hyperbolic–parabolic system. We establish a local existence and uniqueness result for strong solutions.

In this paper, we propose a new generalization of the well-known Cahn–Hilliard two-phase model for the modeling of n-phase mixtures. The model is derived using the consistency principle: we require that our n-phase model exactly coincides with the classical two-phase model when only two phases are present in the system. We give conditions for the model to be well-posed. We also present numerical results (including simulations obtained when coupling the Cahn–Hilliard system with the Navier–Stokes so as to obtain a phase-field model for multiphase flows) to illustrate the capability of such modeling.

We derive a class of Navier–Stokes–Cahn–Hilliard systems that models two-phase flows with mass transfer coupled to the process of chemotaxis. These thermodynamically consistent models can be seen as the natural Navier–Stokes analogues of earlier Cahn–Hilliard–Darcy models proposed for modelling tumour growth, and are derived based on a volume-averaged velocity, which yields simpler expressions compared to models derived based on a mass-averaged velocity. Then, we perform mathematical analysis on a simplified model variant with zero excess of total mass and equal densities. We establish the existence of global weak solutions in two and three dimensions for prescribed mass transfer terms. Under additional assumptions, we prove the global strong well-posedness in two dimensions with variable fluid viscosity and mobilities, which also includes a continuous dependence on initial data and mass transfer terms for the chemical potential and the order parameter in strong norms.

We construct first- and second-order time discretization schemes for the Cahn–Hilliard–Navier–Stokes system based on the multiple scalar auxiliary variables (MSAV) approach for gradient systems and (rotational) pressure-correction for Navier–Stokes equations. These schemes are linear, fully decoupled, unconditionally energy stable, and only require solving a sequence of elliptic equations with constant coefficients at each time step. We carry out a rigorous error analysis for the first-order scheme, establishing optimal convergence rate for all relevant functions in different norms. We also provide numerical experiments to verify our theoretical results.

In this paper, we consider a stochastic Cahn–Hilliard Navier–Stokes system in a bounded domain of [Formula: see text] [Formula: see text]. The system models the evolution of an incompressible isothermal mixture of binary fluids under the influence of stochastic external forces. We prove the existence of a global weak martingale solution. The proof is based on the splitting-up method as well as some compactness method.

This paper reports a numerical study of coupling two-phase fluid flow in a free fluid region with two-phase Darcy flow in a homogeneous and anisotropic porous medium region. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations in the free fluid region and the two-phase Darcy law in the anisotropic porous medium region. A Robin-Robin domain decomposition method is used for the coupled Navier-Stokes and Darcy system with the generalized Beavers-Joseph-Saffman condition on the interface between the free flow and the porous media regions. Obtained results have shown the anisotropic properties effect on the velocity and pressure of the two-phase flow.

Les expériences de déplacement en milieu poreux sont la méthode habituellement utilisée pour étudier l'écoulement biphasique immiscible. Cependant, malgré les aspects de reproductibilité, un inconvénient majeur est que ces expériences de type "boîte noire" ne permettent pas d'observer et de capturer les phénomènes clés à l'échelle des pores, y compris les interactions interfaciales et les détails sur la mobilisation de l'huile piégée (par exemple, la taille et la distribution des ganglions résiduels). C'est pourquoi les dispositifs micromodèles microfluidiques sont désormais largement utilisés dans les expériences de récupération assistée d'huile (EOR) en laboratoire. Ils préservent les détails structurels de la roche tout en offrant des avantages tels que la facilité de nettoyage et la répétabilité. Le suivi visuel du déplacement des fluides est particulièrement important car il peut fournir plus de détails sur le comportement des phases mouillantes et non mouillantes dans les milieux poreux, aidant à élaborer des stratégies ciblées pour améliorer les taux de récupération du pétrole. Cette thèse explore la dynamique complexe des écoulements biphasiques immiscibles en combinant des modèles de milieux poreux microfluidiques, souvent appelés « réservoir-sur-puce », avec des simulations numériques.Dans nos expériences, nous avons utilisé des techniques morphologiques pour surveiller et enregistrer le comportement de déplacement dans un écoulement biphasique, en étudiant systématiquement les effets de différents nombres capillaires (Ca) et rapports de viscosité (M) sur les mécanismes d'écoulement et la mobilisation de l'huile résiduelle. Les résultats ont indiqué que pendant l'inondation par l'eau, le déplacement présentait des caractéristiques de doigté visqueux à des valeurs plus basses de Ca et M. En augmentant le débit pour améliorer Ca de dix fois, l'huile résiduelle montrait une invasion latérale et même arrière des chemins de flux sans changements significatifs dans la taille des grappes. Avec l'augmentation de M, la taille des grappes et la taille maximale des grappes ont diminué, conduisant à une distribution plus uniforme de l'huile résiduelle et à une Sor plus faible. Le mécanisme de mobilisation de l'huile résiduelle s'est manifesté par la rupture des ganglions, les nouveaux petits ganglions formés étant mobilisés sous des pressions plus élevées. La distribution des grappes d'huile résiduelle est conforme à la théorie de percolation, où l'exposant de mise à l'échelle τ est de 2,0. Tous les résultats expérimentaux pour Sor et les valeurs de Ca correspondantes se sont regroupés sur la courbe classique de désaturation capillaire (CDC).Les résultats expérimentaux ont servi de fondement pour développer un modèle numérique utilisant une approche de champ de phase. Ce modèle, basé sur le système d'équations de Cahn-Hilliard-Navier-Stokes, capture efficacement le comportement d'écoulement biphasique de fluides immiscibles dans des domaines confinés. Il intègre les équations de conservation de la masse et de la quantité de mouvement, enrichies par la dynamique de séparation de phase et les considérations d'énergie interfaciale. Les simulations numériques, exécutées sur la plateforme d'éléments finis en source ouverte Fenics, s'alignent qualitativement et quantitativement avec les observations expérimentales, confirmant la précision du modèle pour prédire les comportements fluidiques sous diverses conditions physiques, et avançant notre compréhension de la dynamique des fluides à l'échelle des pores. Les simulations se concentrent sur l'analyse de l'influence des propriétés des fluides et des conditions opérationnelles sur les mécanismes de déplacement à l'échelle des pores
The core-flood experiments are the usual method used to study the immiscible biphasic flow. However, beside reproducibility aspects, a significant drawback is that with these black box experiments, we cannot observe and capture key phenomena at the pore scale, including interfacial interactions and details about mobilization of the trapped oil (e.g. size and distribution of residual ganglia). This is why microfluidic micromodel devices are now extensively used in lab EOR experiments. They preserve the structural details of the rock while offering advantages such as easy cleaning and repeatability. Visual tracking of fluids displacement is particularly important as it can provide more details about the behavior of wetting and non-wetting phases in porous media, aiding in targeted strategies to enhance oil recovery rates. This thesis explores the intricate dynamics of immiscible two-phase flows combines microfluidic porous medium models, often referred to as “reservoir-on-a-chip”, with numerical simulations.In our experiments, we used morphological to monitor and record displacement behavior in biphasic flow, systematically studying the effects of different capillary numbers (Ca) and viscosity ratios (M) on the flow mechanisms and the mobilization of residual oil. The results indicated that during waterflooding, displacement exhibited characteristics of viscous fingering at lower Ca and M values. By increasing the flow rate to enhance Ca tenfold, the residual oil showing lateral and even backward invasion of flow paths without significant changes in cluster size. With increasing M, both the cluster size and the maximum cluster size decreased, leading to a more uniform distribution of residual oil and lower Sor. The mobilization mechanism of residual oil manifested as ganglia breakup, with newly formed smaller ganglia being mobilized under higher pressures. The distribution of residual oil clusters is consistent with percolation theory, where the scaling exponent τ is 2.0. All experimental results for Sor and corresponding Ca values collapsed onto the classical Capillary Desaturation Curve (CDC).The experimental findings served as a foundation for developing a numerical model using a phase-field approach. This model, based on the Cahn-Hilliard-Navier-Stokes system of equations, effectively captures the bi-phasic flow behavior of immiscible fluids within confined domains. It incorporates conservation of mass and momentum equations, enhanced by phase separation dynamics and interfacial energy considerations. The numerical simulations, executed on the open-source finite element platform Fenics, align qualitatively and quantitatively with experimental observations, affirming the accuracy of model in predicting fluid behaviors under varied physical conditions, advancing our understanding of pore-scale fluid dynamics. Simulations focus on dissecting the influence of fluid properties and operational conditions on the displacement mechanisms at the pore scale

This work presents a novel method to model immiscible incompressible fluids in a stable manner. Here, the immiscible behavior of the flow is described by the incompressible Navier-Stokes-Cahn-Hilliard model, which is based on a diffuse interface method. We introduce buoyancy effects in the model through the Boussinesq approximation in a consistent manner. A structure-preserving discretization is used to guarantee the linear stability of the discrete problem and to satisfy the incompressibility of the discrete solution at every point in space by construction. For the solution of the model, we developed the Portable Extensible Toolkit for Isogeometric Analysis with Multi-Field discretizations (PetIGA-MF), a high-performance framework that supports structure-preserving spaces. PetIGA-MF is built on top of PetIGA and the Portable Extensible Toolkit for Scientific Computation (PETSc), sharing all their user-friendly, performance, and flexibility features. Herein, we describe the implementation of our model in PetIGA-MF and the details of the numerical solution. With several numerical tests, we verify the convergence, scalability, and validity of our approach. We use highly-resolved numerical simulations to analyze the merging and rising of droplets. From these simulations, we detailed the energy exchanges in the system to evaluate quantitatively the quality of our simulations. The good agreement of our results when compared against theoretical descriptions of the merging, and the small errors found in the energy analysis, allow us to validate our approach. Additionally, we present the development of an unconditionally energy-stable generalized-alpha method for the Swift-Hohenberg model that offers control over the numerical dissipation. A pattern formation example demonstrates the energy-stability and convergence of our method.

We study physical systems composed of at least two immiscible fluids occu- pying different regions of space, the so-called phases. Flows of such multi-phase fluids are frequently met in industrial applications which rises the need for their numerical simulations. In particular, the research conducted herein is motivated by the need to model the float glass forming process. The systems of interest are in the present contribution mathematically described in the framework of the so-called diffuse interface models. The thesis consists of two parts. In the modelling part, we first derive standard diffuse interface models and their generalized variants based on the concept of multi-component continuous medium and its careful thermodynamic analysis. We provide a critical assessment of assumptions that lead to different models for a given system. Our newly formulated class of generalized models of Cahn-Hilliard-Navier-Stokes-Fourier (CHNSF) type is applicable in a non-isothermal setting. Each model belonging to that class describes a mixture of separable, heat conducting Newtonian fluids that are either compressible or incompressible. The models capture capillary and thermal effects in thin interfacial regions where the fluids actually mix. In the computational part, we focus on the development of an efficient and robust...

The least squares spectral element method (LS-SEM) offers many advantages in the implementation of the finite element model compared with the traditional weak Galerkin method. In this article, the LS-SEM is used to solve the Navier-Stokes (NS) and the Cahn-Hilliard (CH) equations. The NS equation is solved with both C0 and C1 basis functions and their performance is compared in terms of accuracy. A two-dimensional steady-state solver is verified with the case of Kovasznay flow and validated for the cavity flow, and a two-dimensional unsteady solver is verified by a transient manufactured solution case. The phenomenon of phase separation in binary system is described by the CH equation. Due to the fourth-order characteristics of the CH equation, only a high order continuity approximation is used by employing C1 basis function for both space and time domain. The obtained solutions are in accordance with previous results from the literature and show the fundamental characteristics of the NS and CH equations. The results in this study give the possibility of developing a solver for the coupled NS and CH equations.

In this paper, a discontinuous finite element method is presented for the fourth-order nonlinear Cahn-Hilliard equation that models multiphase flows together with the Navier-Stokes equations. A flux scheme suitable for the method is proposed and analyzed together with numerical results. The model is applied to simulate the droplet movement and numerical results are presented.

We consider the least-squares spectral element method to solve the phase field model for two immiscible, incompressible and density-matched fluids. The coupled Cahn-Hilliard and Navier-Stokes system is selected as the numerical model, which was introduced by Hohenberg et al. [1]. The least-squares spectral element scheme is combined with a time-space formulation where both time and space domains are discretized by the same finite element approach to cope with time dependent multidimensional problems in an efficient way. C1 Hermite basis functions are applied for approximating the coupled system. An element-by-element conjugated gradient method is used to facilitate parallelization of the solver. The convergence analysis is conducted to verify our solver, and two numerical experiments are addressed to show applicability of the solver in general situations. Energy dissipation with conserved phase field at equilibrium state is confirmed through the bubble coalescence case, and the influence of the interface mobility is studied with the two-phase lid-driven cavity flow example.

For interface-tracking simulation of two-phase flows in various micro-fluidics devices, the applicability of two versions of Navier-Stokes phase-field method (NS-PFM) was examined, combining NS equations for a continuous fluid with a diffuse-interface model based on the van der Waals-Cahn-Hilliard free-energy theory. Through the numerical simulations, the following major findings were obtained: (1) The first version of NS-PFM gives good predictions of interfacial shapes and motions in an incompressible, isothermal two-phase fluid with high density ratio on solid surface with heterogeneous wettability. (2) The second version successfully captures liquid-vapor motions with heat and mass transfer across interfaces in phase change of a non-ideal fluid around the critical point.

We simulate numerically a novel method for dispensing, mixing and ejecting of picolitre liquid samples in a single step. The system consists of a free liquid film, suspended in a frame and positioned in front of a droplet dispenser. On impact, a picolitre droplet merges with the film, but due to its momentum, passes through and forms a droplet that separates on the other side and continues its flight. Through this process the liquid in the droplet and that in the film is mixed in a controlled way. We model the flow using the Navier-Stokes together with the Cahn-Hilliard equations. This system allows us to simulate the motion of a free surface in the presence of surface tension during merging, mixing and ejection of droplets. The influence of dispensing conditions was studied and it was found that the residual velocity of droplets after passage through the thin liquid film matches the measured velocity from the experiment well.

The purpose of this study is to examine multi-physics computational fluid dynamics method, NS-PFM, which is a combination of Navier-Stokes (NS) equations with phase-field model (PFM) based on the free-energy theory, for interface-capturing/tracking simulation of two-phase flows. First, a new NS-PFM which we have proposed was applied to immiscible, incompressible, isothermal two-phase flow problems with a high density ratio equivalent to that of an air-water system. In this method, a Cahn-Hilliard equation was used for prediction of diffusive interface configuration. The numerical simulations demonstrated that (1) predicted collapse of two-dimensional liquid column in a gas under gravity agreed well with available data at aspect ratios of column = 1 and 2, and (2) coalescence of free-fall drops into a liquid film was successfully simulated in three dimensions. Second, we took heat transfer into account in another NS-PFM which solves a full set of NS equations and the van-der-Waals equation of state. Through a numerical simulation of a non-ideal fluid flow in the vicinity of the critical point, it was confirmed that the NS-PFM is applicable to thermal liquid-vapor flow problems with phase change.

For interface-tracking simulation of two-phase flows, we propose a new computational method, NS-PFM, combining Navier-Stokes (NS) equations with phase-field model (PFM). Based on the free energy theory, PFM describes an interface as a volumetric zone across which physical properties vary continuously. Surface tension is defined as an excessive free energy per unit area induced by density gradient. Consequently, PFM simplifies the interface-tracking procedure by use of a standard technique. The proposed NS-PFM was applied to several problems of incompressible, isothermal two-phase flow with the same density ratio as that of an air-water system. In this method, the Cahn-Hilliard (CH) equation was used for predicting interface configuration. It was confirmed through numerical simulations that (1) the flux driven by chemical potential gradient in the CH equation plays an important role in interfacial advection and reconstruction, (2) the NS-PFM gives good predictions for pressure increase inside a bubble caused by the surface tension, (3) coalescence of liquid film and single drop falling through a stagnant gas was well simulated, and (4) collapse of liquid column under gravity was predicted in good agreement with other available data. Then, another version of NS-PFM was proposed and applied to a direct simulation of bubble nucleation of a non-ideal fluid in the vicinity of the critical point, which demonstrated the capability of NS-PFM to capture liquid-vapor interface motion in boiling and condensation.

Abstract 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.