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MA4L0 Advanced Topics in Fluids

Lecturer: Ellen Luckins

Term(s): Term 1

Status for Mathematics students: List C

Commitment: 30 lectures

Assessment: 100% Examination

Formal registration prerequisites: None

Assumed knowledge:

Familiarity with the mathematical description of fluid dynamics is necessary.

Experience with partial differential equations and methods of their solutions.

Useful background:

Some experience of modelling with PDEs.

Synergies: The following modules would go well with Advanced Topics in Fluids:


Fluid dynamics forms a core subject with applications in a number of disciplines including, engineering, nanotechnology, biology, medicine and geosciences. Principles of fluid dynamics serves as an anchor to describe natural phenomena by providing a common language and set of tools for describing, analyzing and understanding observations and experiments in such a diverse array of disciplines. Continuing on from MA3D1: Fluid dynamics, in this module we will study selected advanced topics in fluid dynamics that provides a core understanding of fluid dynamical phenomena.

The module will cover the following topics:

  • Stokes flow: Properties of Stokes Flows, spherical harmonic solutions, flow around a translating rigid sphere
  • Lubrication theory: Thin-film approximation, squeeze flows, surface tension, free-surface flows, gravity-driven flows
  • Porous media flows: Darcy's law, Dupuit approximation, gravity-driven free-boundary flows, Richards equation for unsaturated flows
  • Hydrodynamic instability: The Rayleigh instability, Kelvin-Helmholtz instability


  • Students will be able to apply the governing principles of fluid dynamics to specific phenomena, possibly involving some systematic simplification methods.

  • They will be introduced to some advanced techniques for analyzing fluid flow.

  • They will be able to relate observations in nature to the aforementioned analysis techniques.

Learning outcomes:

  • Be able to state and prove properties of Stokes flow and use these properties to formulate solutions in spherical geometries.
  • Be able to describe the approximations of lubrication theory and derive the governing equations.
  • Be able to apply lubrication theory to squeeze flow and free surface flows and solve the associated partial differential equations for their flow characteristics.
  • Be able to state and derive Darcy's law for single-phase flow in porous media, and to use this to analyse gravity- and pressure-driven flows.
  • Be able to describe the effect of capillary forces on unsaturated flows in porous media, and to use the Richards equation to solve for such flow fields.
  • Be able to describe the method of linear stability analysis.
  • Be able to use linear stability analysis to classify the stability of layered inviscid and viscous flows.

Further reading:

D.J. Acheson, Elementary Fluid Dynamics, Oxford University Press.

G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press

H. Ockendon, J. R. Ockendon, Viscous Flow, Cambridge University Press

A. Morosov, S. E. Spagnolie, Complex Fluids in Biological Systems, Chapter 1: Introduction to Complex Fluids, Springer

Additional Resources