# 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:

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
• 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

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 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.