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An explicit Eulerian method for multiphase flow with contact line dynamics and insoluble surfactant
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.ORCID iD: 0000-0001-7425-8029
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
2014 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 101, p. 50-63Article in journal (Refereed) Published
##### Abstract [en]

The flow behavior of many multiphase flow applications is greatly influenced by wetting properties and the presence of surfactants. We present a numerical method for two-phase flow with insoluble surfactants and contact line dynamics in two dimensions. The method is based on decomposing the interface between two fluids into segments, which are explicitly represented on a local Eulerian grid. It provides a natural framework for treating the surfactant concentration equation, which is solved locally on each segment. An accurate numerical method for the coupled interface/surfactant system is given. The system is coupled to the Navier-Stokes equations through the immersed boundary method, and we discuss the issue of force regularization in wetting problems, when the interface touches the boundary of the domain. We use the method to illustrate how the presence of surfactants influences the behavior of free and wetting drops.

##### Place, publisher, year, edition, pages
2014. Vol. 101, p. 50-63
##### Keywords [en]
Multiphase flow, Insoluble surfactant, Marangoni force, Moving contact line, Immersed boundary method
##### National Category
Fluid Mechanics and Acoustics
##### Identifiers
ISI: 000340851500005Scopus ID: 2-s2.0-84903152815OAI: oai:DiVA.org:kth-48763DiVA, id: diva2:458563
##### Funder
Swedish Research Council, 621-2007-6375
##### Note

QC 20140919. Updated from accepted to published.

Available from: 2011-11-23 Created: 2011-11-23 Last updated: 2017-12-08Bibliographically approved
##### In thesis
1. Spectral Accuracy in Fast Ewald Methods and Topics in Fluid Interface Simulation
Open this publication in new window or tab >>Spectral Accuracy in Fast Ewald Methods and Topics in Fluid Interface Simulation
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This work contains two separate but related parts: one on spectrally  accurate and fast Ewald methods for electrostatics and viscous flow,  and one on micro- and complex fluid interface problems.  In Part I we are concerned with fast and spectrally accurate methods  to compute sums of slowly decaying potentials over periodic  lattices. We consider two PDEs: Laplace (electrostatics, the Coulomb  potential) and Stokes (viscous flow, the Stokeslet''  potential). Moreover, we consider both full and planar periodicity,  the latter meaning that periodicity applies in two dimensions and  the third is free''. These are major simulation tasks in current  molecular dynamics simulations and in many areas of computational  fluid mechanics involving e.g. particle suspensions.   For each of the four combinations of PDE and periodic structure, we  give spectrally accurate and O(N log N) fast methods based on  Ewald's or Ewald-like decompositions of the underlying potential  sums. In the plane-periodic cases we derive the decompositions in a  manner that lets us develop fast methods. Associated error estimates  are developed as needed throughout. All four methods can be placed  in the P3M/PME (Particle Mesh Ewald) family. We argue that they  have certain novel and attractive features: first, they are spectral  accurate; secondly, they use the minimal amount of memory possible  within the PME family; third, each has a clear and reliable view of  numerical errors, such that parameters can be chosen  wisely. Analytical and numerical results are given to support these  propositions. We benchmark accuracy and performance versus an  established (S)PME method.  Part II deals with free boundary problems, specifically numerical  methods for multiphase flow. We give an interface tracking method  based on a domain-decomposition idea that lets us split the  interface into overlapping patches. Each patch is discretized on a  uniform grid, and accurate and efficient numerical methods are given  for the equations that govern interface transport. We demonstrate  that the method is accurate and how it's used in immersed boundary,  and interface, Navier-Stokes methods, as well as in a boundary  integral Stokes setting.  Finally, we consider a problem in complex fluidics where there is a  concentration of surfactants \emph{on} the interface and the  interface itself is in contact with a solid boundary (the contact  line problem). We argue that the domain-decomposition framework is  attractive for formulating and treating complex models  (e.g. involving PDEs on a dynamic interface) and proceed with  developing various aspects of such a method.

##### Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2011. p. xv, 104
##### Series
Trita-CSC-A, ISSN 1653-5723 ; 2011:19
##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:kth:diva-48805 (URN)978-91-7501-195-0 (ISBN)
##### Public defence
2011-12-16, Salongen, KTHB, Osquars backe 25, Stockholm, 10:00 (English)
##### Funder
Swedish e‐Science Research Center
##### Note
QC 20111125Available from: 2011-11-25 Created: 2011-11-23 Last updated: 2012-05-24Bibliographically approved
2. Computational methods for microfluidics
Open this publication in new window or tab >>Computational methods for microfluidics
2013 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this.

The first paper of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an explicit interface tracking method, wherein the interface between two fluids is decomposed into segments, which are represented locally on an Eulerian grid. The framework of this method provides a natural setting for solving the advection-diffusion equation governing the surfactant concentration on the interface. Open interfaces and moving contact lines are also incorporated into the method in a natural way, though we show that care must be taken when regularizing interface forces to the grid near the boundary of the computational domain.

In the second paper we present a boundary integral formulation for sedimenting particles in periodic Stokes flow, using the completed double layer boundary integral formulation. The long-range nature of the particle-particle interactions lead to the formulation containing sums which are not absolutely convergent if computed directly. This is solved by applying the method of Ewald summation, which in turn is computed in a fast manner by using the FFT-based spectral Ewald method. The complexity of the resulting method is O(N log N), as the system size is scaled up with the number of discretization points N. We apply the method to systems of sedimenting spheroids, which are discretized using the Nyström method and a basic quadrature rule.

The Ewald summation method used in the boundary integral method of the second paper requires a decomposition of the potential being summed. In the introductory chapters of the thesis we present an overview of the available methods for creating Ewald decompositions, and show how the methods and decompositions can be related to each other.

##### Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. p. viii, 39
##### Series
Trita-NA, ISSN 0348-2952 ; 2013:01
##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:kth:diva-116384 (URN)978-91-7501-625-2 (ISBN)
##### Presentation
2013-02-19, F3, Lindstedtsvägen 26, Kungliga Tekniska Högskolan, Stockholm, 10:00 (English)
##### Note

QC 20130124

Available from: 2013-01-24 Created: 2013-01-17 Last updated: 2013-01-24Bibliographically approved
3. Fast and accurate integral equation methods with applications in microfluidics
Open this publication in new window or tab >>Fast and accurate integral equation methods with applications in microfluidics
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this.

The first, and smaller, part of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an interface decomposition resulting in local, Eulerian grid representations. This provides a natural setting for solving the PDE governing the surfactant concentration on the interface.

The second, and larger, part of the thesis is concerned with a framework for simulating large systems of rigid particles in three-dimensional, periodic viscous flow using a boundary integral formulation. This framework can solve the underlying flow equations to high accuracy, due to the accurate nature of surface quadrature. It is also fast, due to the natural coupling between boundary integral methods and fast summation methods.

The development of the boundary integral framework spans several different fields of numerical analysis. For fast computations of large systems, a fast Ewald summation method known as Spectral Ewald is adapted to work with the Stokes double layer potential. For accurate numerical integration, a method known as Quadrature by Expansion is developed for this same potential, and also accelerated through a scheme based on geometrical symmetries. To better understand the errors accompanying this quadrature method, an error analysis based on contour integration and calculus of residues is carried out, resulting in highly accurate error estimates.

##### Abstract [sv]

Denna avhandling behandlar beräkningsmetoder för strömning på mikroskalan, även känt som mikrofluidik. Detta val av ämne motiveras av aktuell forskning inom biologisk fysik och miniatyrisering, där det ofta finns ett behov av att förstå komplexa flöden med strukturer på mikroskalan. Datorsimuleringar är ett viktigt verktyg för att öka den förståelsen.

Avhandlingens första, och mindre, del beskriver en numerisk metod för att simulera flerfasflöden med olösliga surfaktanter och rörliga kontaktlinjer. Metoden är baserad på en uppdelning av gränsskiktet, som tillåter det att representeras med lokala, Euleriska nät. Detta skapar naturliga förutsättningar för lösning av den PDE som styr surfaktantkoncentrationen på gränsskiktets yta.

Avhandlingens andra, och större, del beskriver ett ramverk för att med hjälp av en randintegralformulering simulera stora system av styva partiklar i tredimensionellt, periodiskt Stokesflöde. Detta ramverk kan lösa flödesekvationerna mycket noggrant, tack vare den inneboende höga noggrannheten hos metoder för numerisk integration på släta ytor. Metoden är också snabb, tack vare den naturliga kopplingen mellan randintegralmetoder och snabba summeringsmetoder.

Utvecklingen av ramverket för partikelsimuleringar täcker ett brett spektrum av ämnet numerisk analys. För snabba beräkningar på stora system används en snabb Ewaldsummeringsmetod vid namn spektral Ewald. Denna metod har anpassats för att fungera med den randintegralformulering för Stokesflöde som används. För noggrann numerisk integration används en metod kallad expansionskvadratur (eng. Quadrature by Expansion), som också har utvecklats för att passa samma Stokesformulering. Denna metod har även gjorts snabbare genom en nyutvecklad metod baserad på geometriska symmetrier. För att bättre förstå kvadraturmetodens inneboende fel har en analys baserad på konturintegraler och residykalkyl utförts, vilket har resulterat i väldigt noggranna felestimat.

##### Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. p. 51
##### Series
TRITA-MAT-A ; 2016:03
##### National Category
Computational Mathematics
##### Research subject
Applied and Computational Mathematics
##### Identifiers
urn:nbn:se:kth:diva-185758 (URN)978-91-7595-962-7 (ISBN)
##### Public defence
2016-06-02, F3, Lindstedtsvägen 26, Stockholm, 10:00 (English)
##### Funder
Swedish Research Council, 2011-3178Swedish Research Council, 2007-6375
##### Note

QC 20160427

Available from: 2016-04-27 Created: 2016-04-26 Last updated: 2016-04-27Bibliographically approved

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