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MA6L0 Advanced Topics in Fluids

Lecturer: Thomasina Ball

Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral Exam

Formal registration prerequisites: None

Assumed knowledge:

Familiarity with the mathematical description of fluid dynamics is necessary. It is possible for a strong student to do MA3D1 at the same time: contact the lecturer, if you intend to do it.

Experience with partial differential equations and methods of their solutions.

Useful background:

Some experience of modelling with PDEs. It is particularly important to do MA3J4, if you intend to take MA3D1 Fluid Dynamics at the same time as MA4L0.

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

Leads To:

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: Thin-film approximation, squeeze flows, surface tension, free-surface flows, gravity-driven flows
  • Complex fluids and non-Newtonian rheology: Constitutive laws for shear thinning, yield stress and viscoelastic fluids, squeeze flows and free surface flows for non-Newtonian rheologies
  • Hydrodynamic instability: The Rayleigh instability, Kelvin-Helmholtz 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 identify different non-Newtonian rheologies and describe their physical characteristics.
  • Be able to solve for the flow field for generalised non-Newtonian fluids in Poiseuille and Taylor-Couette flow, and for viscoelastic fluids in simplified shear and extensional flows.
  • 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