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:
 MA3G1 Theory of Partial Differential Equations
 MA3H0 Numerical Analysis and PDEs
 MA4H0 Applied Dynamical Systems
 MA4J1 Continuum Mechanics
Content:
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: Thinfilm approximation, squeeze flows, surface tension, freesurface flows, gravitydriven flows
 Porous media flows: Darcy's law, Dupuit approximation, gravitydriven freeboundary flows, Richards equation for unsaturated flows
 Hydrodynamic instability: The Rayleigh instability, KelvinHelmholtz instability
Aims:

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 singlephase flow in porous media, and to use this to analyse gravity and pressuredriven 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