# MAGIC102: Slow viscous flow

## Course details

A specialist MAGIC course

### Semester

Spring 2024
Monday, January 29th to Friday, March 22nd; Monday, April 22nd to Friday, May 3rd

### Hours

Live lecture hours
10
Recorded lecture hours
0
40

### Timetable

Thursdays
12:05 - 12:55 (UK)

### Course forum

Visit the MAGIC102 forum

## Description

The Reynolds number gives the ratio of inertial to viscous effects in a fluid flow. When the Reynolds number is small, inertial effects are negligible and the Du/Dt term in the Navier-Stokes equations may be neglected. This simplifies the Navier-Stokes equations, making them linear and instantaneous. These simplifications make solving low-Reynolds-number flow problems much easier than high Reynolds number flows.

This module will consider the circumstances under which the Reynolds number will be small and examine the basic properties of low-Reynolds-number flows. We shall present a number of solution techniques, and show how they can be applied to a range of problems. In the course of this, students will meet various useful applied mathematics methods, including solution by potentials, boundary integral methods, and asymptotic approximations.

### Prerequisites

Essential
•  Vector Calculus (div, grad, curl, line,surface/volume integrals, divergence theorem)
•  Differential Equations (methods for first-order ordinary differential equations)
•  Basic Fluid Mechanics (introductory course in inviscid fluid mechanics)

Desirable / Complimentary
•  Further Fluid mechanics (introductory course in viscous fluid mechanics)
•  Tensors and the Einstein Summation Convention (some previous experience useful)
•  Non-dimensionalisation / scaling analysis
A summary of the material that it is most important to be familiar with can be found in the "Background Material" document in the "Files" section of this module on the Magic Website.

### Syllabus

• Introduction to low-Reynolds-number flow (3 lectures)
The Stokes equations and boundary conditions. Basic properties, uniqueness theorem, reciprocal theorem, minimum dissipation theorem. Oscillating Couette flow and Poiseuille flow.
• Fundamental solutions and representation by potentials (4 lectures)
Solution using potentials. Papkovich-Neuber potentials, flow past a rigid sphere. Boundary integrals and the multi-pole expansion.
• Slender-body theory (3 lectures)
Basic derivation. Applications to sedimenting slender objects and swimming micro-organisms.

## Lecturer

• ### Dr Robert Whittaker

University
University of East Anglia

## Bibliography

### Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

## Assessment

The assessment for this course will be released on Monday 13th May 2024 at 00:00 and is due in before Friday 24th May 2024 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hoursâ€™ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.