Lecture notes (last updated 16

Lectures for HT24 are 10–11 on Mondays and 12–1 on Tuesdays in the Lindemann Lecture Theatre

The class will be 4–6pm on Tuesday of 5

Homework is due by 11:59pm on Friday of 4

In case of problems, please email a scan to the TA and cc to mmathphys@maths.ox.ac.uk

In HT24 Complex and Non-Newtonian Fluids will be lectured

• Low Reynolds number hydrodynamics. The Stokes flow regime, general mathematical results, flow past a sphere. Stresses due to suspended rigid particles. Calculation of the Einstein viscosity for a dilute suspension.

• Stresses due to Hookean bead-spring dumb-bells. Derivation of the upper convected Maxwell model for a viscoelastic fluid. Properties of such fluids.

• Suspensions of orientable particles, Jeffery's equation, very brief introduction to active suspensions and liquid crystals.

Lecture 1 Stokes flow regime – minimum dissipation, uniqueness and reciprocal theorems

Lecture 2 Stokes flow around a sphere in translation, rotation, and strain

Lecture 3 Stokes drag, Fáxen relations, Einstein viscosity of a dilute suspension of spheres

Lecture 4 Suspensions of bead-spring pairs and the upper convected Maxwell viscoelastic model

Lecture 5 Resistance matrix formulation for arbitrary and axisymmetric bodies. Jeffery's equation for axisymmetric torque-free bodies

Lecture 6 Suspensions of spheroidal particles, anisotropic viscosity, connection to Braginskii MHD

Rheological Behaviour of Fluids (MIT
TechTV, YouTube,
Film Notes)

Low Reynolds Number Flow (
MIT
TechTV, YouTube,
Film Notes)

Find the complete set of films by the
US National Committee for Fluid Mechanics
Films here.

This course will assume the incompressible Navier–Stokes
equations as a starting point. You can find a traditional
continuum mechanics-style derivation of the Navier–Stokes
equations in chapter 1 of the lecture
notes for the Maths Part B course Viscous
Flow. The compressible Navier–Stokes equations were derived
from the Boltzmann equation in MMathPhys/MTP Kinetic Theory.

Link to ORLO
reading list with scans of some material not otherwise available
online (single sign-on required)

Chapters 1, 2, 3.1, 3.2, 7.1 Read this book online in Oxford via Proquest or Cambridge Core

M. Renardy (2000)

Chapters 1–3. Read this book online in Oxford.

These chapters of these 2 books match the course contents quite closely.

S. Kim & S. J. Karrila (1991, 2005)

Chapter 2, section 2.5, especially example 2.1 on effective stresses in suspensions of rigid particles, section 3.5 on Faxen relations, sections 5.5 & 5.6 on dilute suspensions of spheroids.

N. Phan-Thien (2015)

Concise introduction to suspensions of spheres and spheroids. Read this book online in Oxford.

R. B. Bird, R. C. Armstrong & O. Hassager

R. B. Bird, C. F. Curtiss, R. C. Armstrong
& O. Hassager** **(1987)** **
Dynamics of Polymeric Liquids, 2^{nd} edition, volume 2,
Kinetic Theory (substantial changes from 1^{st }edition,
1977)

Chapter 13 in volume 2 on bead-spring pairs (called “elastic dumbbell models”) or chapter 10 in volume 2 of the 1^{st} edition.

P. Oswald (2009)*Rheophysics: * *The Deformation and Flow of Matter*
(CUP,
Google
Books)

Sections 1.5, 3.7, 7.5.1, appendices 7.A and 7.C.

A. Morozov & S. E. Spagnolie (2015)*Introduction to
Complex Fluids,* chapter 1 of *Complex Fluids in
Biological Systems* (Springer)

J. Happel & H. Brenner (1965,1983)*Low Reynolds Number
Hydrodynamics *(Springer,
Google
Books)

**Some relevant papers:**

D. Saintillan & M. J. Shelley (2013) *Active suspensions
and their nonlinear models*, Comptes
Rendus Physique **14** 497–517

G. K. Batchelor (1970)*The stress system in a suspension of
force-free particles*, J.
Fluid Mech. **41** 545–570

Original paper on how to define the average stress in a suspension without knowing the stress inside rigid particles

E. J. Hinch & L. G. Leal (1972)*The effect of Brownian
motion on the rheological properties of a suspension of
non-spherical particles*, J.
Fluid Mech. **52 **683–712

Fokker–Planck equation for suspensions of passive rods

P. J. Dellar (2014)*Lattice Boltzmann formulation for linear
viscoelastic fluids using an abstract second stress*, SIAM
J. Sci. Comput. **36** A2507–A2532 (reprint)

Compares the evolution of the pressure tensor in kinetic theory with the upper and lower convected Maxwell stress evolution equations

D. F. James (2009)* Boger Fluids* Annu.
Rev. Fluid Mech. **41** 129–142

Describes a class of real fluids that are viscoelastic but not shear-thinning, so their behaviour is reasonably well described by the upper convected Maxwell and Oldroyd-B models.

Chapter 13 in volume 2 on bead-spring pairs (called “elastic dumbbell models”) or chapter 10 in volume 2 of the 1

P. Oswald (2009)

Sections 1.5, 3.7, 7.5.1, appendices 7.A and 7.C.

A. Morozov & S. E. Spagnolie (2015)

J. Happel & H. Brenner (1965,1983)

G. K. Batchelor (1970)

Original paper on how to define the average stress in a suspension without knowing the stress inside rigid particles

E. J. Hinch & L. G. Leal (1972)

Fokker–Planck equation for suspensions of passive rods

P. J. Dellar (2014)

Compares the evolution of the pressure tensor in kinetic theory with the upper and lower convected Maxwell stress evolution equations

D. F. James (2009)

Describes a class of real fluids that are viscoelastic but not shear-thinning, so their behaviour is reasonably well described by the upper convected Maxwell and Oldroyd-B models.