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## Boundary Integral Methods for Stokes Equations

A course intended for PhD and master-level students offered during Spring semester 2009, 7.5 hp.

The course started on Thursday, March 19, 2009. Please see the schedule of the course below.

### Description

Micro fluid dynamics or "micro-fluidics" is a rapidly growing research area within fluid mechanics, that deals with phenomena on the micro scale. At these small scales, the Reynolds numbers are typically low, and the Stokes equations are often applicable. This course focuses on numerical methods based on boundary integral equations for Stokes flow; in particular for multi-phase flows and flows with suspended particles of various shapes. The course includes topics ranging from relevant theory for integral equations to practical implementation issues, all ingredients that are needed in the design of a numerical method based on a boundary integral formulation.

### Course goals

After completing the course, you should be able to:
• Design a solution algorithm for a collocation scheme to solve the Stokes equations for a simple 3D-geometry, and motivate the choices made.
• Identify strengths and weaknesses about boundary integral methods. Argue for if a boundary integral method is advantageous to use for a specific problem, and how it compares to other solution methods, such as finite difference or finite element methods.
As smaller sub-goals, you should specifically be able to:
• Give examples of applications for which the Stokes equations are valid.
• Formulate the Stokes equations as a boundary integral equation (BIE).
• Master a few basic techniques to derive simple analytical solutions to the Stokes equations using singularity methods and slender body theory.
• Explain key concepts of the mathematical theory for integral equations (e.g. properties of integral equations of the first and second kind, practical consequences) and of theory specific to Stokes flow ( e.g. Lorentz reciprocal theorem).
• Explain what difficulties arise in the design of quadrature formulas for BIEs, and some techniques that can remedy these difficulties.
• Describe the need of so called "fast summation methods". Explain the underlying ideas of the Fast Multipole Method (FMM) and the particle-mesh Ewald method.

### List of topics:

Topics to be discussed include:
• Introduction to Stokes equations. Fundamental solutions of Stokes equations.
• Singularity methods for derivation of analytical solutions. Slender body approximations.
• Theory for integral equations, and specifically boundary integral equations. Starting with the Laplace equation, then moving on to Stokes equations. Single layer and double layer formulations.
• Numerical discretization of boundary integral equations. Collocation and Galerkin methods. Quadrature rules, including singularity treatments.
• Fast summation methods. FFT based methods, fast multipole method (FMM).
• Periodicity treatment. Theory and practical methods.

### Examination:

There will be two assignments, involving both theoretical questions and practical implementation, in the first part of the course. These assignments will be presented orally in class. The course ends with short individual oral exams.

For PhD students, there will also be a larger course project for which each student is to select his or her own subject. Students can also choose to work together on this project. The project work will continue after the oral exam. The class will meet again for oral project presentations in late August, and a written project report will be due shortly after.

The project part of the course is not mandatory for master students.

### Format of the course:

The course will run with one two hour lecture each week. The second hour of each lecture will be used to introduce a new topic, and the first hour of the lecture the following week will be used for a follow up discussion of that topic. In between, you will work with different discussion themes and two specific assignments.

### Schedule and lectures:

• Lecture 1, Thursday, March 19, 10-12 in D35
Formalia, outline of the course Introduction to Stokes equations (KG)
Fundamental solutions (AKT)
Chapter 1-2 in Kim&Karilla
• Lecture 2, Thursday, March 26, 10-12 in 4523
Follow up on lecture 1
Singularity methods (KG)
Chapter 3 in Kim&Karilla
• Lecture 3, Thursday, April 2, 16-18 in D42
Follow up on lecture 2
Theory for boundary integral equations (AKT)
• Lecture 4, Tuesday, April 14, 13-15 in 4523
Follow up on lecture 3
Theory for boundary integral equations, contd. (AKT)
• Lecture 5, Thursday, April 23, 13-15 in 4523
Follow up on lecture 4
Numerical discretization of boundary integral equations (KG)
Chapter 14 and 15 in Kim&Karilla
• Lecture 6, Tuesday, April 28, 13-15 in 4523
Follow up on lecture 5
Numerical discretization of boundary integral equations, contd. (KG)
Chapter 14 and 15 in Kim&Karilla
Chapter 18.5 (Singularity subtraction) in Kim&Karilla
• Lecture 7, Friday, May 8, 13-15 in 1537
Presentations and discussion on assignments
• Lecture 8, Tuesday, May 12, 13-15 in 4523
Follow up on lecture 6 and presentations during lecture 7.
Periodicity (AKT)
• Lecture 9, Tuesday, May 19, 13-15 in 4523
Follow up on lecture 8
Particle mesh methods & Fast multipole methods (AKT)
• Lecture 10, Friday, May 29, 13-15 in 4523
Presentations and discussion on Assignment 2.
Summary.
• Oral exams in week 23

### Handouts:

At each lecture, copies of the slides presented at that lecture will be given out. Theere will also be additional handouts, as listed below.

From Sean's applied math book (free resource on the web):
A short chapter on Dirac Delta functions.
A few sections on Green's functions in 1D .

### Assignments:

Assignment 1,part I. Part II of Assignment 1 will be handed out later. The due date for Assignment 1 is May 8, 2009.
NOTE: a new version 29/4-09. There was an error in the potential dipole.
Assignment 1,part II. The due date for Assignment 1, both part I and II, is May 8, 2009. NOTE: a new version 28/4-09. There was an error in the formula for the spherical coordinates in the earlier version.
Assignment 2. The due date for Assignment 2 is May 29, 2009.

### Literature:

We will use the book Microhydrodynamics - Principles and Selected Applications by S. Kim and S. J. Karilla (Dover Publications, 2005), together with selected pages from Boundary integral and singularity methods for linearized viscous flow, by C. Pozrikidis (Cambridge University Press, 1992). We will complement this with handouts regarding e.g. boundary integral equations for the Laplace equation, and will read several research papers on the topic (e.g. concerning the FMM method and periodicity treatment).

### Instructors:

The course will be taught by Anna-Karin Tornberg and Katarina Gustavsson.

### Contact information:

Anna-Karin Tornberg, email annak@nada.kth.se.
Office: Osquars backe 2, floor 5, room 4517.
Phone: 08-790 6266.
Katarina Gustavsson, email katarina@nada.kth.se.
Office: Osquars backe 2, floor 5, room 1517.
Phone: 08-790 6696.

Upp till Nadas hemsida.

Anna-Karin Tornberg