# MAGIC102: Slow viscous flow

## Course details

A specialist MAGIC course

### Semester

Spring 2021
Monday, January 25th to Friday, March 19th; Monday, April 26th to Friday, May 7th

### Hours

Live lecture hours
10
Recorded lecture hours
0
40

### Timetable

Wednesdays
14:05 - 14:55 (UK)

### Announcements

Apologies for another change of time, but I've had to move the final 10th lecture of this module to avoid a clash with some other teaching I have to do at UEA. It will now be on the same day, but now start an hour earlier at 1pm. The 9th lecture will take place at the usual time. So the last two lectures will now be as follows:
• Lecture 9: Wednesday 28th April, 2pm-3pm
• Lecture 10: Wednesday 5th May, 1pm-2pm
The Magic timetable has been updated to reflect this.

## 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

### 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.

• Elementary Fluid Dynamics (Acheson, D. J., )
• Viscous Flow (Ockendon & Ockendon, )
• An Introduction to Fluid Dynamics (Batchelor, G. K., )
• Low Reynolds Number Hydrodynamics (Happel & Brenner, )
• Boundary Integral and Singularity Methods for Linearised Viscous Flow (Pozrikidis, C., )
• Microhydrodynamics (Kim & Karrila, )

## Assessment

The assessment for this course will be released on Monday 10th May 2021 at 00:00 and is due in before Monday 24th May 2021 at 11:00.

The assessment for this course will be via a single take-home paper in the Semester 2 Magic Assessment Period. There will be one question on each of the three topics, and all three questions should be attempted.

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