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A wall treatment for confined Stokes flow
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.ORCID-id: 0000-0002-8998-985X
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
(Engelska)Artikel i tidskrift (Övrigt vetenskapligt) Submitted
##### Abstract [en]

The study of bodies immersed in Stokes flow is crucial in various microfluidic applications. Recasting the governing equations in a boundary integral formulation reduces the three-dimensional problem to two-dimensional integral equations to be discretized over the surface of the submerged objects. The present work focuses on the development and validation of a wall treatment where the wall is discretized in the same fashion as the immersed bodies. For this purpose, a set of high-order quadrature rules for the numerical integration of integrals containing the singular Green’s function-the so-called Stokeslet - has been developed. By coupling the wall discretization to the discretization of immersed objects, we exploit the structure of the block matrix corresponding to the wall discretization in order to substantially reduce the memory usage. For validation, the classical problem of a sedimenting sphere in the vicinity of solid walls is studied.

##### Nationell ämneskategori
Datavetenskap (datalogi)
##### Identifikatorer
OAI: oai:DiVA.org:kth-29762DiVA, id: diva2:397505
##### Ingår i avhandling
1. Quadrature rules for boundary integral methods applied to Stokes flow
Öppna denna publikation i ny flik eller fönster >>Quadrature rules for boundary integral methods applied to Stokes flow
2011 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
##### Abstract [en]

Fluid phenomena dominated by viscous effects can, in many cases, be modeled by the Stokes equations. The boundary integral form of the Stokes equations reduces the number of degrees of freedom in a numerical discretization by reformulating the threedimensional problem to two-dimensional integral equations to be discretized over the boundaries of the domain. Hence for the study of objects immersed in a fluid, such as drops or elastic or solid particles, integral equations are to be discretized over the surfaces of these objects only. As outer boundaries or confinements are added these must also be included in the formulation. This work is focused on the development and validation of such a wall treatment. An inherent difficulty in the numerical treatment of boundary integrals for Stokes flow is the integration of the singular fundamental solution of the Stokes equations – the so called Stokeslet. To alleviate this problem we developed a set of high-order quadrature rules for the numerical integration of the Stokeslet over a flat surface. Such a quadrature rule was first designed for singularities of the type 1/|x|. To assess the convergence properties of this quadrature rule a theoretical analysis has been performed. The slightly more complicated singularity of the Stokeslet required certain modifications of the integration rule developed for 1/|x|. To validate the quadrature rule developed for the Stokeslet against a physical model we use it in a classical problem in fluid dynamics, the sedimentation of a sphere onto a flat plate. This involves a direct discretization of the plane wall and at the same time of the immersed sphere. Without any special treatment the algebraic system given by the discrete problem is quite memory consuming since matrix blocks are full. By exploring the structure of the block matrices that build up the system we have found that the wall discretization leads to a matrix which is generated by only three of its columns. This information together with certain preconditioning considerations allowed us to use the Schur complement method thus leading to a less memory expensive solution to the algebraic system. As a final step it is shown that the numerical simulations match the analytical solution, within the limitations of the model. This wall treatment can be easily extended to the problem of two parallel walls, and it is also shown that the simulation is in good agreement with some known results for the two parallel walls problem.

##### Ort, förlag, år, upplaga, sidor
Stockholm: KTH Royal Institute of Technology, 2011. s. vii, 30
##### Serie
Trita-CSC-A, ISSN 1653-5723 ; 2011:01
##### Nationell ämneskategori
Datavetenskap (datalogi)
##### Identifikatorer
urn:nbn:se:kth:diva-29763 (URN)978-91-7415-862-5 (ISBN)
##### Presentation
2011-02-15, Sal E32, KTH, Lindstedtsvägen 3, Stockholm, 13:00 (Engelska)
##### Forskningsfinansiär
Swedish e‐Science Research Center
##### Anmärkning
2. Boundary integral methods for Stokes flow: Quadrature techniques and fast Ewald methods
Öppna denna publikation i ny flik eller fönster >>Boundary integral methods for Stokes flow: Quadrature techniques and fast Ewald methods
2012 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
##### Abstract [en]

Fluid phenomena dominated by viscous effects can, in many cases, be modeled by the Stokes equations. The boundary integral form of the Stokes equations reduces the number of degrees of freedom in a numerical discretization by reformulating the three-dimensional problem to two-dimensional integral equations to be discretized over the boundaries of the domain.

Hence for the study of objects immersed in a fluid, such as drops or elastic/solid particles, integral equations are to be discretized over the surfaces of these objects only. As outer boundaries or confinements are added these must also be included in the formulation.

An inherent difficulty in the numerical treatment of boundary integrals for Stokes flow is the integration of the singular fundamental solution of the Stokes equations, e.g. the so called Stokeslet. To alleviate this problem we developed a set of high-order quadrature rules for the numerical integration of the Stokeslet over a flat surface. Such a quadrature rule was first designed for singularities of the type $1/|\mathbf{x}|$. To assess the convergence properties of this quadrature rule a theoretical analysis has been performed. The slightly more complicated singularity of the Stokeslet required certain modifications of the integration rule developed for $1/|\mathbf{x}|$. An extension of this type of quadrature rule to a cylindrical surface is also developed. These quadrature rules are tested also on physical problems that have an analytic solution in the literature.

Another difficulty associated with boundary integral problems is introduced by periodic boundary conditions. For a set of particles in a periodic domain periodicity is imposed by requiring that the motion of each particle has an added contribution from all periodic images of all particles all the way up to infinity. This leads to an infinite sum which is not absolutely convergent, and an additional physical constraint which removes the divergence needs to be imposed. The sum is decomposed into two fast converging sums, one that handles the short range interactions in real space and the other that sums up the long range interactions in Fourier space. Such decompositions are already available in the literature for kernels that are commonly used in boundary integral formulations. Here a decomposition in faster decaying sums than the ones present in the literature is derived for the periodic kernel of the stress tensor.

However the computational complexity of the sums, regardless of the decomposition they stem from, is $\mathcal{O}(N^{2})$. This complexity can be lowered using a fast summation method as we introduced here for simulating a sedimenting fiber suspension. The fast summation method was initially designed for point particles, which could be used for fibers discretized numerically almost without any changes. However, when two fibers are very close to each other, analytical integration is used to eliminate numerical inaccuracies due to the nearly singular behavior of the kernel and the real space part in the fast summation method was modified to allow for this analytical treatment. The method we have developed for sedimenting fiber suspensions allows for simulations in large periodic domains and we have performed a set of such simulations at a larger scale (larger domain/more fibers) than previously feasible.

##### Ort, förlag, år, upplaga, sidor
Stockholm: KTH Royal Institute of Technology, 2012. s. ix, 59
##### Serie
Trita-NA, ISSN 0348-2952 ; 2012:14
##### Nyckelord
boundary integral, Stokes flow, quadrature rule, Ewald decomposition
##### Nationell ämneskategori
Beräkningsmatematik
##### Identifikatorer
urn:nbn:se:kth:diva-105540 (URN)978-91-7501-578-1 (ISBN)
##### Disputation
2012-12-14, D2, Lindstedtsvägen, 5, Stockholm, 10:15 (Engelska)
##### Forskningsfinansiär
Swedish e‐Science Research Center

QC 20121122

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#### Personposter BETA

Gustavsson, Katarina

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##### Av författaren/redaktören
Marin, OanaGustavsson, KatarinaTornberg, Anna-Karin
##### Av organisationen
Numerisk analys, NANumerisk Analys och Datalogi, NADA
##### I ämnet
Datavetenskap (datalogi)

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