Numerical solution of phase field models for two-phase flows

Phase-field models describe the motion of multiphase flows using smooth interfaces across which the composition changes continuously. The phase-field variable represents a measure of phase as it quantifies relative differences or fractions of the fluid s components. The Cahn-Hilliard equation was originally proposed to model spinodal decomposition and coarsening in binary alloys. To this day, it has become broad ranged in its applicability. This thesis focuses on solving the Cahn-Hilliard equation numerically. A review of the mathematical modelling is made in order to develop numerical methods. Different numerical simulations in two dimensions are implemented to study the numerical and physical properties. Two realistic physical examples are also numerically solved

Isogeometric analysis of Cahn-Hilliard phase field-based Binary-Fluid-Structure Interaction based on an ALE variational formulation

This thesis is concerned with the development of a computational model and simulation technique capable of capturing the complex physics behind the intriguing phenomena of Elasto-capillarity. Elastocapillarity refers to the ability of capillary forces or surface tensions to deform elastic solids through a complex interplay between the energy of the surfaces (interfaces) and the elastic strain energy in the solid bulk. The described configuration gives rise to a three-phase system featuring a fluid-fluid interface (for instance the interface of a liquid and an ambient fluid such as air) and two additional interfaces separating the elastic solid from the first and second fluids, respectively. This setup is encountered in the wetting of soft substrates which is an emerging young field of research with many potential applications in micro- and nanotechnology and biomechanics. By virtue of the fact that a lot of physical phenomena under the umbrella of the wetting of soft substrates (e.g. Stick-slip motion, Durotaxis, Shuttleworth effect, etc.) are not yet fully understood, numerical analysis and simulation tools may yield invaluable insights when it comes to understanding the underlying processes. The problem tackled in this work – dubbed Elasto-Capillary Fluid-Structure Interaction or Binary-Fluid-Structure Interaction (BFSI) – is of multiphysics nature and poses a tremendous and challenging complexity when it comes to its numerical treatment. The complexity is given by the individual difficulties of the involved Two-phase Flow and Fluid-Structure Interaction (FSI) subproblems and the additional complexity emerging from their complex interplay. The two-phase flow problems considered in this work are immiscible two-component incompressible flow problems which we address with a Cahn-Hilliard phase field-based two-phase flow model through the Navier-Stokes-Cahn-Hilliard (NSCH) equations. The phase field method – also known as the diffuse interface method – is based on models of fluid free energy and has a solid theoretical foundation in thermodynamics and statistical mechanics. It may therefore be perceived as a physically motivated extension of the level-set or volume-of-fluid methods. It differs from other Eulerian interface motion techniques by virtue of the fact that it does not feature a sharp, but a diffuse interface of finite width whose dynamics are governed by the joint minimization of a double well chemical energy and a gradientsquared surface energy – both being constituents of the fluid free energy. Particularly for two-phase flows, diffuse interface models have gained a lot of attention due to their ability to handle complex interface dynamics such moving contact lines on wetted surfaces, and droplet coalescence or segregation without any special procedures. Our computational model for the FSI subproblem is based on a hyperelastic material model for the solid. When modeling the coupled dynamics of FSI, one is confronted with the dilemma that the fluid model is naturally based on an Eulerian perspective while it is very natural to express the solid problem in Lagrangian formulation. The monolithic approach we take, uses a fully coupled Arbitrary Lagrangian– Eulerian (ALE) variational formulation of the FSI problem and applies Galerkin-based Isogeometric Analysis for the discretization of the partial differential equations involved. This approach solves the difficulty of a common variational description and facilitates a consistent Galerkin discretization of the FSI problem. Besides, the monolithic approach avoids any instability issues that are associated with partitioned FSI approaches when the fluid and solid densities approach each other. The BFSI computational model presented in this work is obtained through the combination of the above described phase field-based two-phase flow and the monolithic fluid-structure interaction models and yields a very robust and powerful method for the simulation of elasto-capillary fluid-structure interaction problems. In addition, we also show that it may also be used for the modeling of FSI with free surfaces, involving totally or partially submerged solids. Our BFSI model may be classified as “quasi monolithic” as we employ a two-step solution algorithm, where in the first step we solve the pure Cahn-Hilliard phase field problem and use its results in a second step in which the binary-fluid-flow, the solid deformation and the mesh regularization problems are solved monolithically

Approximation of Smectic-A liquid crystals

In this paper, we present energy-stable numerical schemes for a Smectic-A liquid crystal model. This model involve the hydrodynamic velocity-pressure macroscopic variables (u, p) and the microscopic order parameter of Smectic-A liquid crystals, where its molecules have a uniaxial orientational order and a positional order by layers of normal and unitary vector n. We start from the formulation given in [E’97] by using the so-called layer variable φ such that n = ∇φ and the level sets of φ describe the layer structure of the Smectic-A liquid crystal. Then, a strongly non-linear parabolic system is derived coupling velocity and pressure unknowns of the Navier-Stokes equations (u, p) with a fourth order parabolic equation for φ. We will give a reformulation as a mixed second order problem which let us to define some new energy-stable numerical schemes, by using second order finite differences in time and C 0 - finite elements in space. Finally, numerical simulations are presented for 2D-domains, showing the evolution of the system until it reachs an equilibrium configuration. Up to our knowledge, there is not any previous numerical analysis for this type of models.Ministerio de Economía y CompetitividadMinistry of Education, Youth and Sports of the Czech Republi

Phase-field modeling and effective simulation of non-isothermal reactive transport

We consider single-phase flow with solute transport where ions in the fluid can precipitate and form a mineral, and where the mineral can dissolve and release solute into the fluid. Such a setting includes an evolving interface between fluid and mineral. We approximate the evolving interface with a diffuse interface, which is modeled with an Allen-Cahn equation. We also include effects from temperature such that the reaction rate can depend on temperature, and allow heat conduction through fluid and mineral. As Allen-Cahn is generally not conservative due to curvature-driven motion, we include a reformulation that is conservative. This reformulation includes a non-local term which makes the use of standard Newton iterations for solving the resulting non-linear system of equations very slow. We instead apply L-scheme iterations, which can be proven to converge for any starting guess, although giving only linear convergence. The three coupled equations for diffuse interface, solute transport and heat transport are solved via an iterative coupling scheme. This allows the three equations to be solved more efficiently compared to a monolithic scheme, and only few iterations are needed for high accuracy. Through numerical experiments we highlight the usefulness and efficiency of the suggested numerical scheme and the applicability of the resulting model