MMathPhys Advanced Fluid Dynamics : Complex and Non-Newtonian Fluids

Overall theme: fluids with an extra stress due to (small) embedded particles, polymer molecules etc, with connections to kinetic theory and MHD

Lecture notes (last updated 16th January 2023) and problem sheet

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 5th week (13th February 2024) in the Lindemann Lecture Theatre.

Homework is due by 11:59pm on Friday of 4th week (9th February 2024). Please hand in work via the course page on Canvas.
In case of problems, please email a scan to the TA and cc to

In HT24 Complex and Non-Newtonian Fluids will be lectured first


• 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.

Previous handwritten notes from online lectures in HT21:

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

See some experiments:

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.

Suggested books and chapters:

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)

book cover   É. Guazzelli & J. F. Morris (2011) A Physical Introduction to Suspension Dynamics (CUP, Google Books)
   Chapters 1, 2, 3.1, 3.2, 7.1 Read this book online in Oxford via Proquest or Cambridge Core

book cover   M. Renardy (2000) Mathematical Analysis of Viscoelastic Flows (SIAM, Google Books)

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

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

Further reading

book cover    S. Kim & S. J. Karrila (1991, 2005) Microhydrodynamics: Principles and Selected Applications (Dover, Google Books)

    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.

book cover    N. Phan-Thien (2015) Introduction to Suspension Rheology, chapter 1 of Rheology of Non-Spherical Particle Suspensions (Elsevier, Google Books)

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

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

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

book cover   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.