MAGIC102: Slow viscous flow

Course details

A specialist MAGIC course

Semester

Spring 2020
Monday, January 20th to Friday, March 27th

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
40

Timetable

Fridays
15:05 - 15:55 (UK)

Announcements

Time Change
Please note that the time slot for this module has changed from the original version of the Magic Timetable for this semester. It will now be lectured on Fridays from 3pm to 4pm, as is shown in the current timetable.

Background Material
In the Files section, you will find a set of notes on background material for this module.
The notes cover various bits of material that it will be necessary for those taking the module to be familiar with in advance. Hopefully most of those taking the module will already have seen most of he material here, but some of the content on tensors and the summation convention is more likely to be new. Some of the notation I will be using may be different from that which you have used before, and so I have also tried to make clear my notational conventions in the notes.
Please have a look through these notes before the first lecture, and have a deeper look at anything that you are not already familiar with.

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

Related courses

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

    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 20th April 2020 at 00:00 and is due in before Monday 4th May 2020 at 11:00.

The assessment for this course will be via a single take-home paper in Spring assessment period, with 2 weeks to complete and submit your answers online. There will be one question on each of the three topics, and all three questions should be attempted. Mark divisions for each part of each question will be shown on the paper. You will need the equivalent of 50% to pass.

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

Files

Only current consortium members and subscribers have access to these files.

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Lectures

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