Decoupled, Linear, and Energy Stable Finite Element Method for the Cahn-Hilliard-Navier-Stokes-Darcy Phase Field Model

In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn—Hilliard—Navier—Stokes equations in the free flow region and Cahn—Hilliard—Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the interface conditions, the decoupled scheme needs to be constructed appropriately for the interface terms, and a modified discrete energy is introduced with an interface component. Furthermore, the scheme is linearized and energy stable. Hence, at each time step one need only solve a linear elliptic system for each of the two decoupled equations. Stability of the model and the proposed method is rigorously proved. Numerical experiments are presented to illustrate the features of the proposed numerical method and verify the theoretical conclusions. © 2018 Society for Industrial and Applied Mathematics

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

Topics in complex multiscale systems: theory and computations of noise-induced transitions and transport in heterogenous media

The present work seeks to address three different problems that have a multiscale nature, we apply different techniques from multiscale analysis to treat these problems. We introduce the field of multiscale analysis and motivate the need for techniques to bridge between scales, presenting the history of some common methods, and an overview of the current state of the field. The remainder of the work deals with the treatment of these problems, one motivated by reaction rate theory, and two from multiphase flow. These superficially have little relation with each other, but the approaches taken share similarities and the results are the same - an average picture of the microscopic description informs the macroscale. In Chapter 2 we address an asymmetric potential with a microscale, showing that the interaction between this microscale and the noise causes a first-order phase transition. This induces a metastable state which we observe and characterise: showing that the stability of this state depends on the strength of the tilt, and that the phase transition is inherently different to the symmetric case. In Chapter 3 we investigate the nucleation and coarsening process of a two-phase flow in a corrugated channel using a Cahn--Hilliard Navier--Stokes model. We show that several flow morphologies can be present depending on the channel geometry and the initial random condition. We rationalise this with a static energy model, predicting the preferential formation of one morphology over another and the existence of a first-order phase-transition from smooth slug flow to discontinuous motion when the channel is strongly corrugated. In Chapter 4 we address a model for interfacial flows in porous geometries, formulating an finite-element model for the equations. Within this framework we solve two equations in the microscale to obtain effective coefficients decoupling the two scales from each other. Finite-difference simulations of the macroscopic flow recover results from literature, supporting robustness of the method.Open Acces

Discontinuous Galerkin finite element method applied to the coupled Navier-Stokes/Cahn-Hilliard equations

International audienceTwo-phase flows driven by the interfacial dynamics is studied with a phase-field model to tract implicitly interfaces. The phase field obeys the Cahn-Hilliard equation. The fluid dynamics is described with the Stokes equations with an additional source term in the momentum equation taking into account the capillary forces. A discontinuous Galerkin finite element method is used to solve the coupled Stokes/Cahn-Hilliard equations. The Cahn-Hilliard equation is treated as a system of two coupled equations corresponding to the advection-diffusion equation for the phase field and a non-linear elliptic equation for the chemical potential. First, the variational formulation of the Cahn-Hilliard equation is presented. A numerical test is achieved showing the optimal-order in error bounds. Second, the variational formulation in discontinuous Galerkin finite element approach of the Stokes equations is recalled in which the same space of approximation is used for the velocity and the pressure with an adequate stabilization technique. Finally, numerical simulations describing the capillary rising in a tube is presented