MAGIC102: Slow viscous flow

Course details

Semester

Spring 2019
Monday, January 21st to Friday, March 29th

Hours

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

Timetable

Tuesdays
10:05 - 10:55

Announcements

Week 10 Lecture
  • There were a few a typos in the slides and lecture notes for this week's lecture. I'm sorry about that. I've now uploaded corrected versions of both to the Magic website.
Week 8 Lecture
  • I have just made a few updates to the lecture notes and slides following today's lecture, as I spotted in the middle of the lecture that there was a slight inconsistency in the dimensions of some of the Slender Body Theory expressions. If you assume L=O(1) it doesn't make any difference, but I thought it better to be more precise in the notes. I've also added a bit more to the notes about why the centreline far-field approximation is valid even for points on the surface of the slender body.
Week 6 Lecture
  • I didn't have time to do this before the lecture today, but I've now added plots of the streamlines for flow past a rigid sphere to both the lecture notes and the slides.
Week 3 Lecture
  • I did the Minimum Dissipation Theorem example using slightly different notation for the different regions than what was in the lecture notes. I've now updated the notes to match the lecture and uploaded a new version to the course page.
  • Also, I'm sorry that I missed the typed question from York about setting p=0. I've added a few more comments in the typeset notes that may help to explain things. If your questions aren't answered there, then please email me or post in the forum.
Week 2 Lecture
  • I'm afraid there was a technical issue with the Visimeet servers this week, which meant that a number of Magic lectures (including mine) did not get recorded. As far as I know, there's nothing we can do now to get the videos back.

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

  • RW

    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

Description

The assessment for this course will be via a single take-home paper in April with 2 weeks to complete and submit online. There will be one question on each of the three topics, and all three questions should be attempted.

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Files

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Lectures

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