Transport Processes at Fluidic Interfaces

Table of Contents

1. Frontmatter

2. Numerical Methods for Sharp Interface Models

   1. Frontmatter

   2. Chapter 1. ALE-FEM for Two-Phase and Free Surface Flows with Surfactants

      Sashikumaar Ganesan, Andreas Hahn, Kristin Simon, Lutz Tobiska

      Abstract

      We study two-phase and free surface flows with soluble and insoluble surfactants. A numerical analysis of the contained convection-diffusion equations is carried out. The surface equation is stabilized by Local Projection Stabilization. The benefit of Local Projection Stabilization on surfaces is shown by a numerical example. An advanced finite element method that allow for a robust and accurate numerical simulation is presented. The arbitrary Langrangian-Eulerian framework is utilized to capture the moving surface. This allows the usage of a fitted finite element mesh. A decoupling strategy is used to divide the origin problem into subproblems easier to solve. Different time discretizations are considered and the problem of spurious velocities for the spatial discretization is discussed. Numerical examples in 2d and 3d illustrate the potential of the proposed algorithm. The comparison to mathematical predicted values validates the obtained results.

   3. Chapter 2. High Order Unfitted Finite Element Methods for Interface Problems and PDEs on Surfaces

      Christoph Lehrenfeld, Arnold Reusken

      Abstract

      In this contribution we treat a special class of recently developed higher order unfitted finite element methods for the discretization of mass and surfactant transport equations. To achieve higher order accuracy for such PDEs on stationary geometries we combine standard techniques for numerical integration and a discretization with a special mesh transformation. This results in a new class of isoparametric unfitted finite element methods. For the treatment of such PDEs on evolving geometries we apply space-time variational formulations. These unfitted finite element techniques result in robust and accurate discretization methods for mass and surfactant transport problems in realistic two-phase flow simulations based on a sharp interface formulation. We present these finite element discretization methods, give theoretical error bounds for classes of model problems and present results of numerical simulations both for such model problems and for challenging two-phase flow applications.

   4. Chapter 3. An Extended Discontinuous Galerkin Framework for Multiphase Flows

      Thomas Utz, Christina Kallendorf, Florian Kummer, Björn Müller, Martin Oberlack

      Abstract

      We present a framework for handling cut cells in a high order discontinuous Galerkin (DG) context. To describe the boundary between fluid phases, we use a level-set formulation. When the interface cuts a computational cell, we discretize the resulting sub-cells with the same DG method as used on standard cells. This requires a suitable quadrature procedure. Within this framework, we present a solver for the two-phase Navier-Stokes equation, a reinitialization procedure for the level-set and a solver for transport-processes on the surface.

   5. Chapter 4. Building Blocks for a Strictly Conservative Generalized Finite Volume Projection Method for Zero Mach Number Two-Phase Flows

      Matthias Waidmann, Stephan Gerber, Michael Oevermann, Rupert Klein

      Abstract

      Building blocks for a generalized fully conservative finite volume projection method for numerical simulation of immiscible zero Mach number two-phase flows on Cartesian grids are presented, focusing on the crucial issues of interface propagation, fluid phase conservation and discretization of the singular contribution due to surface tension, each in a discretely conservative fashion. Additionally, a solution approach for solving Poisson-type equations for two-phase flows at arbitrary ratio of coefficients is sketched. Further, (intermediate) results applying these building blocks are presented and open issues and future developments are proposed.

   6. Chapter 5. Time Discretization for Capillary Flow: Beyond Backward Euler

      Stephan Weller, Eberhard Bänsch

      Abstract

      Development and analysis of numerical methods for two-phase flow has mainly focussed on spatial aspects of the discretization so far. In turn, most of the methods available are of first order in time at most and only conditionally stable. For many applications, however, these shortcomings may constitute the computational bottleneck, since both disadvantages dictate very small time step sizes in order to arrive at a decent approximation of the underlying problem. In this article we therefore focus on the time discretization of capillary free surface flows with the following features: higher order in time convergence, unconditional stability and low dissipativity. Applying the method of lines we use stiff integrators from the class of Rosenbrock and W-methods. As an alternative, a space-time Galerkin method is presented. These methods are compared computationally for examples of one- and two-phase capillary flows.

   7. Chapter 6. Upwind Schemes for Scalar Advection-Dominated Problems in the Discrete Exterior Calculus

      Michael Griebel, Christian Rieger, Alexander Schier

      Abstract

      We present the discrete exterior calculus (DEC) to solve discrete partial differential equations on discrete objects such as cell complexes. To cope with advection-dominated problems, we introduce a novel stabilization technique to the DEC. To this end, we use the fact that the DEC coincides in special situations with known discretization schemes such as finite volumes or finite differences. Thus, we can carry over well-established upwind stabilization methods introduced for these classical schemes to the DEC. This leads in particular to a stable discretization of the Lie-derivative. We present the numerical features of this new discretization technique and study its numerical properties for simple model problems and for advection-diffusion processes on simple surfaces.

   8. Chapter 7. Discrete Exterior Calculus (DEC) for the Surface Navier-Stokes Equation

      Ingo Nitschke, Sebastian Reuther, Axel Voigt

      Abstract

      We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on staggered grids in flat space for which we demonstrate second order convergence. We compare computational results with a vorticity-stream function approach for surfaces with genus \(g(\mathcal{S}) = 0\) and demonstrate the interplay between topology, geometry and flow properties. Our discretization also allows to handle harmonic vector fields, which we demonstrate on a torus.

3. Analysis and Simulation of Diffusive Interface Models

   1. Frontmatter

   2. Chapter 8. Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities

      Helmut Abels, Harald Garcke, Günther Grün, Stefan Metzger

      Abstract

      Diffuse interface models have become an important analytical and numerical method to model two-phase flows. In this contribution we review the subject and discuss in detail a thermodynamically consistent model with a divergence free velocity field for two-phase flows with different densities. The model is derived using basic thermodynamical principles, its sharp interface limits are stated, existence results are given, different numerical approaches are discussed and computations showing features of the model are presented.

   3. Chapter 9. Sharp Interface Limits for Diffuse Interface Models for Two-Phase Flows of Viscous Incompressible Fluids

      Helmut Abels, YuNing Liu, Andreas Schöttl

      Abstract

      We consider the mathematical relation between diffuse interface and sharp interface models for the flow of two viscous, incompressible Newtonian fluids like oil and water. In diffuse interface models a partial mixing of the macroscopically immiscible fluids on a small length scale ɛ > 0 and diffusion of the mass particles are taken into account. These models are capable to describe such two-phase flows beyond the occurrence of topological singularities of the interface due to collision or droplet formation. Both for theoretical and numerical purposes a deeper understanding of the limit ɛ → 0 in dependence of the scaling of the mobility coefficient m _ɛ is of interest. Here the mobility is the inverse of the Peclet number and controls the strength of the diffusion. We discuss several rigorous mathematical results on convergence and non-convergence of solutions of diffuse interface to sharp interface models in dependence of the scaling of the mobility.

   4. Chapter 10. Two-Phase Flow with Surfactants: Diffuse Interface Models and Their Analysis

      Helmut Abels, Harald Garcke, Kei Fong Lam, Josef Weber

      Abstract

      New diffuse interface and sharp interface models for soluble and insoluble surfactants fulfilling energy inequalities are introduced. We discuss their relation with the help of asymptotic analysis and present an existence result for a particular diffuse interface model.

   5. Chapter 11. Phase Field Models for Two-Phase Flow with Surfactants and Biomembranes

      Sebastian Aland

      Abstract

      We give an overview on recent developments of phase field models for two-phase flows with surfactants and lipid bilayer membranes. Starting from the two-phase flow model of a clean fluid-fluid interface we discuss the time discretization and boundary conditions for dynamic and static contact angles. Using the adsorption models of Henry and Langmuir, soluble surfactants are included in the diffuse interface formulation. To consider lipid bilayer membranes the model is extended by membrane bending stiffness and membrane inextensibility. We present phase field models to include these elastic effects, with a particular focus on the inextensibility constraint for which we discuss different phase field variants from the literature and present numerical tests.

   6. Chapter 12. Micro-Macro-Models for Two-Phase Flow of Dilute Polymeric Solutions: Macroscopic Limit, Analysis, and Numerics

      Günther Grün, Stefan Metzger

      Abstract

      We derive a diffuse-interface model for two-phase flow of incompressible fluids with dissolved noninteracting polymers. Describing the polymers as bead chains governed by general elastic spring potentials, including in particular Hookean and finitely extensible, nonlinear elastic (FENE) potentials, it couples a Fokker-Planck type equation describing distribution and orientation of the polymer chains with Cahn–Hilliard and Navier–Stokes type equations describing the balance of mass and momentum. Allowing for different solubility properties which are modelled by Henry type energy functionals, the presented model covers the case of one Newtonian fluid and one non-Newtonian fluid as well as the case of two non-Newtonian fluids. In the case of Hookean spring potentials, we derive a macroscopic diffuse-interface model for two-phase flow of Oldroyd-B-type liquids.

      In the case of dumbbell models, we show existence of solutions and present numerical simulations in two space dimensions on oscillating polymeric droplets.

   7. Chapter 13. Fully Adaptive and Integrated Numerical Methods for the Simulation and Control of Variable Density Multiphase Flows Governed by Diffuse Interface Models

      Michael Hintermüller, Michael Hinze, Christian Kahle, Tobias Keil

      Abstract

      The present work is concerned with the simulation and optimal control of two-phase flows. We provide stable time discretization schemes for the simulation based on both, smooth and non-smooth free energy densities, which we combine with a practical, reliable and efficient adaptive mesh refinement concept for the spatial variables. Furthermore, we consider optimal control problems for two-phase flows and, among other things, derive first order optimality conditions. In the presence of smooth free energies we encounter classical Karush-Kuhn-Tucker (KKT) conditions, while in the case of non-smooth free energies we can prove C(larke)-stationarity. Moreover, we propose a dual weighted residual concept for spatial mesh adaptivity which is based on the newly derived stationarity conditions. We also address future research directions, including closed-loop control concepts and model order reduction techniques for simulation and control of variable density multiphase flows.

   8. Chapter 14. Dynamical Stability of Diffuse Phase Boundaries in Compressible Fluids

      Heinrich Freistühler, Matthias Kotschote

      Abstract

      Aiming at an understanding of the nonlinear stability of moving fluidic phase boundaries, this project has provided (a) solution theories for three basic systems of nonlinear partial differential equations that model two-phase fluid flow as well as (b) results on the existence of corresponding traveling waves and the spectrum of the operators that result from linearizing the PDEs about these waves.

      The three models are the Navier-Stokes-Korteweg (NSK), the Navier-Stokes-Allen-Cahn (NSAC), and the Navier-Stokes-Cahn-Hilliard (NSCH) equations. For compressible NSAC and NSCH, the theories of strong solutions obtained seem to be the first ones in the literature. For NSK, a new theory of strong solutions has been developed, which in particular provides an alternative to the ‘quasi-incompressible’ approach that Abels et al. pursue for NSCH in the case of two separately incompressible phases of different density.

      While for NSK the existence of traveling waves representing phase boundaries was known before, corresponding results have been newly obtained for NSAC and NSCH. For all three contexts, the project has established the spectral stability of these traveling waves. Viscous shock waves providing useful heuristic inspiration, fluidic interfaces corresponding to phase boundaries turn out to have their own flavour.

4. Experimental and Numerical Investigation of Interfacial Phenomena

   1. Frontmatter

   2. Chapter 15. Experimental and Computational Analysis of Fluid Interfaces Influenced by Soluble Surfactant

      Chiara Pesci, Holger Marschall, Talmira Kairaliyeva, Vamseekrishna Ulaganathan, Reinhard Miller, Dieter Bothe

      Abstract

      The present contribution is the result of a collaboration between the Max Planck Institute of Colloids and Interfaces and the Technical University of Darmstadt (MMA group). The main objective is to give a quantitative description of fluid interfaces influenced by surfactants, comparing experimental and computational results. Surfactants are amphiphilic molecules subject to ad- and desorption processes at fluid interfaces. In fact, they accumulate at the interface, modifying the respective interfacial properties. Since these interfaces are moving, continuously deforming and expanding, the local time-dependent interfacial coverage is the most relevant quantity. The description of such processes poses severe challenges both to the experimental and to the simulation sides. Two prototypical problems are considered for comparison between experiments and simulations: the formation of droplets under the influence of surfactants and rising bubbles in aqueous solutions contaminated by surfactants. Direct Numerical Simulations (DNS) provide valuable insights into local quantities such as local surfactant distribution and surface tension, but at high computational costs and restricted to short time frames. On the other hand, experiments can give global quantities necessary for the validation of the numerical procedures and can afford longer time frames. The two methodologies thus yield complementary results which help to understand such complex interfacial phenomena.

   3. Chapter 16. Complex Patterns and Elementary Structures of Solutal Marangoni Convection: Experimental and Numerical Studies

      Kerstin Eckert, Thomas Köllner, Karin Schwarzenberger, Thomas Boeck

      Abstract

      The transfer of a solute between two liquid layers is susceptible to convective instabilities of the time-dependent diffusive concentration profile that may be caused by the Marangoni effect or buoyancy. Marangoni instabilities depend on the change of interfacial tension and Rayleigh instabilities on the change of liquid densities with solute concentration. Such flows develop increasingly complex cellular or wavy patterns with very fine structures in the concentration field due to the low solute diffusivity. They are important in several applications such as extraction or coating processes. A detailed understanding of the patterns is lacking although a general phenomenological classification has been developed based on previous experiments. We use both highly resolved numerical simulations and controlled experiments to examine two exemplary systems. In the first case, a stationary Marangoni instability is counteracted by a stable density stratification producing a hierarchical cellular pattern. In the second case, Rayleigh instability is opposed by the Marangoni effect causing solutal plumes and eruptive events with short-lived Marangoni cells on the interface. A good qualitative and acceptable quantitative agreement between the experimental visualizations and measurements and the corresponding numerical results is achieved in simulations with a planar interface, and a simple linear model for the interface properties, i.e. no highly specific properties of the interface are required for the complex patterns. Simulation results are also used to characterize the mechanisms involved in the pattern formation.

   4. Chapter 17. Transport at Interfaces in Lipid Membranes and Enantiomer Separation

      Oleg Boyarkin, Stefan Burger, Thomas Franke, Thomas Fraunholz, Ronald H. W. Hoppe, Simon Kirschler, Kevin Lindner, Malte A. Peter, Florian Strobl, Achim Wixforth

      Abstract

      We study the dynamics and formation of differently ordered lateral phases of interfacial lipid layers for two types of lipid systems, a vesicle-supported bilayer and a Langmuir–Blodgett monolayer, both in experiment and by simulation. Similarly, we investigate the dynamics of objects embedded in a simpler interface given by an air–water surface and demonstrate the surface-acoustic-wave-actuated separation of enantiomers (chiral objects) on the surface of the carrier fluid. It turns out that the dynamics and the separation of the phases do not only depend on parameters such as temperature, mobilities and line tension but also on the mechanics of the lipid layers subjected to exterior forces as, for instance, compression, extensional and shear forces in film-balance experiments. Since the mechanical behavior of lipid layers is viscoelastic, we use a modeling approach based on the incompressible Navier–Stokes equations with a viscoelastic stress term and a capillary term, a convective Jeffrey (Oldroyd) equation of viscoelasticity, and the Cahn–Hilliard equation with a transport term. The numerical simulations are based on C ⁰-interior-penalty discontinuous-Galerkin methods for the Cahn–Hilliard equation. Model-validation results and the verification of the simulation results by experimental data are presented. The feasibility of enantiomer separation by surface-acoustic-wave-generated vorticity patterns is shown both experimentally and through numerical simulations. This technique is cost-effective and provides an extremely high time resolution of the dynamics of the separation process compared to more traditional approaches. The experimental setup is an enhanced Langmuir–Blodgett film balance with a surface-acoustic-wave-generated vorticity pattern of the fluid, where model enantiomers (custom-made photoresist particles) float on the surface of the carrier fluid. For the simulations, we propose a finite element immersed boundary method (FEIBM) for deformable enantiomers and a fictitious-domain approach based on a distributed Lagrangian multiplier finite element immersed boundary method (DLM-FEIBM) for rigid chiral objects, both of which lead to simulation results consistent with experiments.

   5. Chapter 18. Structure Formation in Thin Liquid-Liquid Films

      Sebastian Jachalski, Dirk Peschka, Stefan Bommer, Ralf Seemann, Barbara Wagner

      Abstract

      We revisit the problem of a liquid polymer that dewets from another liquid polymer substrate with the focus on the direct comparison of results from mathematical modeling, rigorous analysis, numerical simulation and experimental investigations of rupture, dewetting dynamics and equilibrium patterns of a thin liquid-liquid system. The experimental system uses as a model system a thin polystyrene (PS)/polymethylmethacrylate (PMMA) bilayer of a few hundred nm. The polymer systems allow for in situ observation of the dewetting process by atomic force microscopy (AFM) and for a precise ex situ imaging of the liquid-liquid interface. In the present study, the molecular chain length of the used polymers is chosen such that the polymers can be considered as Newtonian liquids. However, by increasing the chain length, the rheological properties of the polymers can be also tuned to a viscoelastic flow behavior. The experimental results are compared with the predictions based on the thin film models. The system parameters like contact angle and surface tensions are determined from the experiments and used for a quantitative comparison. We obtain excellent agreement for transient drop shapes on their way towards equilibrium, as well as dewetting rim profiles and dewetting dynamics.

5. Taylor Bubbles: Experiments, Simulation and Validation

   1. Frontmatter

   2. Chapter 19. Taylor Bubbles in Small Channels: A Proper Guiding Measure for Validation of Numerical Methods for Interface Resolving Simulations

      Martin Wörner

      Abstract

      Taylor bubbles moving in a vertical pipe are elongated, bullet-shaped bubbles that almost fill the channel cross-section and are separated from the wall by a thin liquid film. Taylor bubbles and Taylor flow, which consists of a sequence of Taylor bubbles separated by liquid slugs, are of interest for various technical applications. This article introduces some characteristic features of Taylor bubbles and laminar Taylor flow in small channels to facilitate the understanding of the subsequent chapters in this book. Furthermore, the specific advantages of Taylor flow as guiding measure for the DFG Priority Programme SPP 1506 “Transport Processes at Fluidic Interfaces” are highlighted.

   3. Chapter 20. X-Ray Microtomography of Taylor Bubbles with Mass Transfer and Surfactants in Capillary Two-Phase Flow

      Stephan Boden, Mohammadreza Haghnegahdar, Uwe Hampel

      Abstract

      Development and application of techniques to measure local properties of dynamic flows is in the focus of the work of the Institute of Fluid Dynamics at HZDR and of the AREVA Endowed Chair of Imaging Techniques in Energy and Process Engineering at TU Dresden. In this paper we report on the application of enhanced X-ray microradiography and microtomography techniques to measure Taylor bubble shapes in micro- and milli-channels. Further, experiments to investigate the mass transport and the influence of surfactants were conducted. The resulting flow structural data will foster meso- and microscalic numerical flow model development for small channel multiphase flow. Data and material of the presented study can be freely downloaded from the website of SPP 1506 (http://www.dfg-spp1506.de/taylor-bubble).

   4. Chapter 21. Experimental Investigation and Modelling of Local Mass Transfer Rates in Pure and Contaminated Taylor Flows

      Sven Kastens, Christoph Meyer, Marko Hoffmann, Michael Schlüter

      Abstract

      In many industrial applications of chemical and bio-chemical engineering, new insights into mass transfer processes across fluidic interfaces are of high interest. Mass transfer processes across gas-liquid interfaces have been investigated for decades to understand the coupling of hydrodynamics and mass transport processes and to describe and correlate them for various gas-liquid flow apparatus and process parameters. The investigation of the linked transport processes and the understanding of their interaction is fundamental for the optimization of multiphase reactors and for the validation of numerical simulations, which are pointing at problems of higher complexity during the last years. One challenge for the investigation of gas-liquid flows is the highly stochastic behaviour of gas bubbles rising in liquids under turbulent flow conditions. For the investigation of local mass transfer processes at fluidic interfaces and the validation of numerical simulations, more well-defined and reproducible conditions are necessary. A suitable setup to study mass transfer at fluidic interfaces under well-defined and reproducible conditions is the gas-liquid flow through a small, straight capillary, called “Taylor bubble” for single bubbles and “Taylor flow” for bubbles in a chain. Taylor flows and Taylor bubbles have ideal properties for detailed investigation on the influence of hydrodynamics and mass transfer at clean and contaminated interfaces, where the shape oscillations are suppressed and the Taylor bubbles are self-centering within vertical channels. Therefore, in this work the local hydrodynamics and mass transfer processes in Taylor flows and at Taylor bubbles have been investigated with laser measurement techniques, to obtain a deeper insight into mass transfer processes at fluidic interfaces. Furthermore, experimental data for the guiding measure “Taylor flow” has been provided. The guiding measure has been established within the SPP 1506 to generate a reliable data basis for the validation of numerical simulations.

   5. Chapter 22. Comparative Simulations of Taylor Flow with Surfactants Based on Sharp- and Diffuse-Interface Methods

      Sebastian Aland, Andreas Hahn, Christian Kahle, Robert Nürnberg

      Abstract

      We present a quantitative comparison of simulations based on diffuse- and sharp-interface models for two-phase flows with soluble surfactants. The test scenario involves a single Taylor bubble in a counter-current flow. The bubble assumes a stationary position as liquid inflow and gravity effects cancel each other out, which makes the scenario amenable to high resolution experimental imaging. We compare the accuracy and efficiency of the different numerical models and four different implementations in total.

   6. Chapter 23. Direct Numerical Simulations of Taylor Bubbles in a Square Mini-Channel: Detailed Shape and Flow Analysis with Experimental Validation

      Holger Marschall, Carlos Falconi, Christoph Lehrenfeld, Rufat Abiev, Martin Wörner, Arnold Reusken, Dieter Bothe

      Abstract

      The Priority Program SPP 1506 “Transport Processes at Fluidic Interfaces” by the German Research Foundation DFG has established a benchmark problem for validation of two-phase flow solvers by means of specifically designed experiments for Taylor Bubble Flow. This chapter is devoted to results from Direct Numerical Simulations (DNS) of a single rising Taylor bubble and of Taylor flow in a square channel, where both bubble shape and flow pattern around the bubble have been thoroughly analyzed. Comparisons have been accomplished to highly resolved experiments providing detailed and local benchmark data for validation. An interesting three-dimensional backflow effect of technological relevance has been revealed. A criterion to estimate its occurrence is deducted from thorough analysis of local simulation data.