- Department: Mathematics
- Credit value: 20 credits
- Credit level: M
- Academic year of delivery: 2024-25
- See module specification for other years: 2023-24
Fluid Dynamics aims to describe the movement of liquids and gases. It formulates arguably one of the most successful applied mathematical theories to reveal the mechanisms for flow over many orders of magnitude from swimming microorganisms to large scale ocean circulation and beyond. This course will explore the fundamentals of fluid dynamics and apply tools from vector calculus and complex analysis to help to understand problems relating to flying, swimming, sinking and gliding, of both inanimate objects and biological organisms.
Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
You can’t take both H and M level versions.
Students should have passed appropriate courses covering the solution of ordinary and partial differential equations, vector calculus and complex analysis. Students should also have some experience of classical dynamics or introductory concepts in continuum dynamics.
Occurrence | Teaching period |
---|---|
A | Semester 2 2024-25 |
Fluid Dynamics aims to describe the movement of liquids and gases. It formulates arguably one of the most successful applied mathematical theories to reveal the mechanisms for flow over many orders of magnitude from swimming microorganisms to large scale ocean circulation and beyond. This course will explore the fundamentals of fluid dynamics and apply tools from vector calculus and complex analysis to help to understand problems relating to flying, swimming, sinking and gliding, of both inanimate objects and biological organisms.
By the end of the module, students will be able to:
Analyse characteristics of a particular flow
Formulate the governing equations and boundary conditions
Solve these equations analytically in simple cases
Analyse and interpret different flow regimes
Explain and apply theories underlying fluid flow dynamics
Carry out exact and approximate calculations for a range of fluid flow problem
Critically assess theory and literature on elements of the module content (including additional advanced M-level content).
Formulate equations for novel systems, determine analytical solutions and discuss their interpretation and limitations.
Fundamental concepts of fluid mechanics: continuous medium approximation and its applicability; the Lagrangian and Eulerian frameworks for a continuous medium. The transport theorems. Conservation of mass and momentum.
Newtonian fluids. The Navier-Stokes equations. The boundary conditions of Fluid Dynamics.
Hydrostatics. Elementary flows: uniform and shear flows; Poiseuille flow; spherically symmetric and circular flows; sources and sinks.
Motion of a body in an inviscid fluid. Flow past a sphere moving in an infinite fluid. Cavitation. D’Alembert’s paradox.
The Reynolds number. Low and high Reynolds number flows. Drag force on a body in a fluid. The drag coefficient. Notions of laminar and turbulent flows.
Classical aerofoil theory: two-dimensional inviscid flows. The complex variables formalism. The method of images. Milne-Thompson circle theorem. Circulation and lift. Blasius theorem. Kutta-Joukowski lift theorem. Conformal mapping. Joukowski transformation: the wing profile.
Very viscous (Stokes) flows: equations and boundary conditions; reversibility; the Stokes drag for a sphere and its generalisations to bodies of complex shape.
Swimming at low Reynolds numbers. Flow reversibility and the scallop theorem. Taylor’s wavy sheet - dimensional analysis. Resistive force theory for swimming micro-organisms: kinematics of flagellated swimmers; the approximation; thrust; mechanical efficiency.
Applications of elementary flows.
Boundary layer analysis.
Taylor’s wavy sheet.
The Stokes paradox: very viscous flow around a cylinder.
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
None
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
D J Acheson, Elementary Fluid Dynamics, Oxford University Press.
L M Milne-Thompson, Theoretical Hydrodynamics, Dover.
L D Landau and E M Lifshitz, Fluid Mechanics, Butterworth-Heinemann.
G K Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press.
S. Childress, Mechanics of swimming and flying, Cambridge Studies in Mathematical Biology (2), C.U.P., 1981.