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Boundary integral methods for fast and accurate simulation of droplets in two-dimensional Stokes flow
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0003-2327-9633
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Accurate simulation of viscous fluid flows with deforming droplets creates a number of challenges. This thesis identifies these principal challenges and develops a numerical methodology to overcome them. Two-dimensional viscosity-dominated fluid flows are exclusively considered in this work. Such flows find many applications, for example, within the large and growing field of microfluidics; accurate numerical simulation is of paramount importance for understanding and exploiting them.

A boundary integral method is presented which enables the simulation of droplets and solids with a very high fidelity. The novelty of this method is in its ability to accurately handle close interactions of drops, and of drops and solid boundaries, including boundaries with sharp corners. The boundary integral method is coupled with a spectral method to solve a PDE for the time-dependent concentration of surfactants on each of the droplet interfaces. Surfactants are molecules that change the surface tension and are therefore highly influential in the types of flow problems which are considered herein.

A method’s usefulness is not dictated by accuracy alone. It is also necessary that the proposed method is computationally efficient. To this end, the spectral Ewald method has been adapted and applied. This yields solutions with computational cost O(N log N ), instead of O(N^2), for N source and target points.

Together, these innovations form a highly accurate, computationally efficient means of dealing with complex flow problems. A theoretical validation procedure has been developed which confirms the accuracy of the method.

Abstract [sv]

Att noggrant simulera viskösa flöden med deformerande droppar medför flera utmaningar. Denna avhandling identifierar de viktigaste utmaningarna och utvecklar numeriska metoder för att övervinna dem. Visköst dominerade tvådimensionella flöden studeras. Sådana flöden har många tillämpningar till exempel inom mikrofluidik och noggrann beräkning är av största vikt för att förstå och utnyttja dem.

En randintegralsmetod som möjliggör simulering av droppar och fasta ränder med en mycket hög noggrannhet presenteras. Metoden särskiljer sig från andra genom dess förmåga att hantera nära samspel mellan droppar och förekomst av hörn på de fasta ränderna. Randin- tegralsmetoden är kopplad till en spektral metod som möjliggör inkluderandet av surfaktanter i flödesproblemet. Surfaktanter är molekyler som förändrar ytspänningen och de är därför betydelsefulla för de typer av flödesproblem som beaktas här.

En metods användbarhet bestäms inte endast av dess noggrannhet. Det är också nödvändigt att den föreslagna metoden är effektiv. För detta ändamål har metoden spektral Ewald anpassats och tillämpats. Detta ger lösningar med beräkningskostnaden O(N log N ) istället för O(N^2), där N är antalet diskreta punkter i systemet.

Tillsammans utgör dessa innovationer ett mycket noggrant, be- räkningseffektivt sätt att hantera komplexa flödesproblem. Ett teoretiskt valideringsförfarande har utvecklats som bekräftar metodens noggrannhet.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2019. , p. 49
Series
TRITA-SCI-FOU ; 2019;50
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-264369ISBN: 978-91-7873-355-2 (print)OAI: oai:DiVA.org:kth-264369DiVA, id: diva2:1373113
Public defence
2019-12-18, F3, Lindstedtsvägen 26, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Göran Gustafsson Foundation for Research in Natural Sciences and MedicineAvailable from: 2019-11-27 Created: 2019-11-26 Last updated: 2019-11-27Bibliographically approved
List of papers
1. Simulation and validation of surfactant-laden drops in two-dimensional Stokes flow
Open this publication in new window or tab >>Simulation and validation of surfactant-laden drops in two-dimensional Stokes flow
2019 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 386, p. 218-247Article in journal (Refereed) Published
Abstract [en]

Performing highly accurate simulations of droplet systems is a challenging problem. This is primarily due to the interface dynamics which is complicated further by the addition of surfactants. This paper presents a boundary integral method for computing the evolution of surfactant-covered droplets in 2D Stokes flow. The method has spectral accuracy in space and the adaptive time-stepping scheme allows for control of the temporal errors. Previously available semi-analytical solutions (based on conformal-mapping techniques) are extended to include surfactants, and a set of algorithms is introduced to detail their evaluation. These semi-analytical solutions are used to validate and assess the accuracy of the boundary integral method, and it is demonstrated that the presented method maintains its high accuracy even when droplets are in close proximity. 

Place, publisher, year, edition, pages
Academic Press, 2019
Keywords
Insoluble surfactants, Stokes flow, Validation, Integral equations, Two-phase flow, Drop deformation, Special quadrature
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-251466 (URN)10.1016/j.jcp.2018.12.044 (DOI)000464675600011 ()2-s2.0-85062883114 (Scopus ID)
Note

QC 20190515

Available from: 2019-05-15 Created: 2019-05-15 Last updated: 2019-11-26Bibliographically approved
2. An integral equation method for closely interacting surfactant-covered droplets in wall-confined Stokes flow
Open this publication in new window or tab >>An integral equation method for closely interacting surfactant-covered droplets in wall-confined Stokes flow
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A highly accurate method for simulating surfactant-covered droplets in two-dimensional Stokes flow with solid boundaries is presented. The method handles both periodic channel flows of arbitrary shape and stationary solid constrictions. A boundary integral method together with a special quadrature scheme is applied to solve the Stokes equations to high accuracy, also for droplets in close interaction. The problem is considered in a periodic setting and Ewald decompositions for the Stokeslet and stresslet are derived to make the periodic sums convergent. Computations are sped up using the spectral Ewald method. The time evolution is handled with a fourth order, adaptive, implicit-explicit time-stepping scheme. The numerical method is tested through several convergence studies and other challenging examples and is shown to handle drops in close proximity both to other drops and solid objects to a high accuracy.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-264363 (URN)
Note

QC 20191126

Available from: 2019-11-26 Created: 2019-11-26 Last updated: 2019-11-26Bibliographically approved
3. An accurate integral equation method for Stokes flow with piecewise smooth boundaries
Open this publication in new window or tab >>An accurate integral equation method for Stokes flow with piecewise smooth boundaries
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Two-dimensional Stokes flow through a periodic channel is considered. The channel walls need only be Lipschitz continuous, in other words they are allowed to have corners. Boundary integral methods are an attractive numerical method to solve the Stokes equations, as the problem can be reformulated into a problem that must be solved only over the boundary of the domain. When the boundary is at least C1 smooth, the boundary integral kernel is a compact operator, and traditional Nyström methods can be used to obtain highly accurate solutions. In the case of Lipschitz continuous boundaries however, obtaining accurate solutions using the standard Nyström method can require high resolution. We adapt a technique known as recursively compressed inverse preconditioning to accurately solve the Stokes equations without requiring any more resolution than is needed to resolve the boundary. Combined with a periodic fast summation method we construct a method that is O(N log N ) where N is the number of quadrature points on the boundary. We demonstrate the robustness of this method by extending an existing boundary integral method for viscous drops to handle the movement of drops near corners.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-264366 (URN)
Note

QC 20191126

Available from: 2019-11-26 Created: 2019-11-26 Last updated: 2019-11-26Bibliographically approved
4. Spectrally accurate Ewald summation for the Yukawa potential in two dimensions
Open this publication in new window or tab >>Spectrally accurate Ewald summation for the Yukawa potential in two dimensions
(English)Manuscript (preprint) (Other academic)
Abstract [en]

An Ewald decomposition of the two-dimensional Yukawa potential and its derivative is presented for both the periodic and the free-space case. These modified Bessel functions of the second kind of zeroth and first degrees are used e.g. when solving the modified Helmholtz equation using a boundary integral method. The spectral Ewald method is used to compute arising sums at O(N log N ) cost for N source and target points. To facilitate parameter selection, truncation-error estimates are developed for both the real-space sum and the Fourier-space sum, and are shown to estimate the errors well.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-264367 (URN)
Note

QC 20191126

Available from: 2019-11-26 Created: 2019-11-26 Last updated: 2019-11-26Bibliographically approved
5. Adaptive time-stepping for surfactant-laden drops
Open this publication in new window or tab >>Adaptive time-stepping for surfactant-laden drops
2017 (English)In: Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11) / [ed] Chappell, D.J., 2017Conference paper, Published paper (Refereed)
Abstract [en]

An adaptive time-stepping scheme is presented aimed at computing the dynamics of surfactant-covered deforming droplets. This involves solving a coupled system, where one equation corresponds to the evolution of the drop interfaces and one to the surfactant concentration. The first is discretised in space using a boundary integral formulation which can be treated explicitly in time. The latter is a convection-diffusion equation solved with a spectral method and is advantageously solved with a semi-implicit method in time. The scheme is adaptive with respect to drop deformation as well as surfactant concentration and the adjustment of time-steps takes both errors into account. It is applied and demonstrated for simulation of the deformation of surfactant-covered droplets, but can easily be applied to any system of equations with similar structure. Tests are performed for both 2D and 3D formulations and the scheme is shown to meet set error tolerances in an efficient way.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-264368 (URN)978-0-9931112-9-7 (ISBN)978-1-912253-00-5 (ISBN)
Conference
UKBIM11, 10-11 July 2017,Nottingham Trent University
Note

QC 20191129

Available from: 2019-11-26 Created: 2019-11-26 Last updated: 2019-11-29Bibliographically approved

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