Module MATH3101: Fluid Mechanics III

Department: Mathematical Sciences

MATH3101: Fluid Mechanics III

Type	Level	Credits	Availability	Module Cap	Location
Open	3	20	Available in 2026/2027		Durham

Pre-requisites

Mathematical Methods II (MATH2811) OR Mathematical Methods in Physics (PHYS2611) OR Analysis in Many Variables II (MATH2031).

Corequisites

None.

Excluded Combinations

None.

Aims

To introduce the mathematical description of fluid flow and other continuous media.
To familiarise students with the successful applications of mathematics in this area of modelling.
To prepare students for future study of advanced topics.

Content

Kinematics: Eulerian and Lagrangian pictures, velocity field, streamlines and stream functions.
Compressibility, vorticity and circulation, integrals over material domains.
Dynamics of ideal fluids: derivation of the incompressible Euler equations, energy, vorticity dynamics, circulation and rotating frames.
Water waves: free-surface boundaries, linearisation, travelling and standing waves.
Compressible flow: barotropic fluids, sound waves, nonlinearity.
Hydrodynamic stability: normal mode analysis, Rayleigh-Taylor and Kelvin-Helmholtz instabilities.
Dynamics of viscous fluids: Newtonian fluids, derivation of the Navier-Stokes equations, exact solutions, dynamical similarity, boundary layers.

Learning Outcomes

Subject-specific Knowledge

Fundamental concepts in the kinematics of fluids.
Derivation of equations of motion for both ideal and viscous fluids, including incompressible and compressible flows.
Behaviour of ideal and viscous fluids, including conservation laws, dynamical similarity and boundary layers.
Water waves and free surface boundaries, and sound waves.
Basic concepts in hydrodynamic stability including normal mode analysis and common instabilities.

Subject-specific Skills

Be able to solve novel and/or complex problems in Fluid Mechanics.
Demonstrate an understanding of theoretical mathematics relevant to Fluid Mechanics and the ability to apply it appropriately.

Key Skills

Basic mathematical skills in the following areas: problem solving, modelling, computation.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

Lectures demonstrate what is required to be learned and the application of the theory to practical examples. 
Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding. 
Formatively and summatively assessed assignments provide practice in the application of logic and a high level of rigour as well as feedback for the students and the lecturer on the students’ progress. 
The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Activity	Number	Frequency	Duration	Total Hours	Attendance Monitored
Lectures	40	2 per week for 20 weeks	1 Hour	40
Problem Classes	10	4 in Michaelmas; 4 in Epiphany; 2 in Easter	1 Hour	10	Yes
Preparation and Reading				150
Total				200

Summative Assessment

Component	Component Weighting	Element	Length/Duration	Element Weighting	Resit Opportunity
Examination	70%	On Campus Written Examination	2 hours	100%
Summative Assignments	30%	Assignment		100%

Formative Assessment

Four assignments to be submitted.

Students who do not attend monitored activities shown under Teaching Methods and Learning Hours, or who fail to complete the summative or formative assessment(s) specified above, may be subject to the Academic Progress procedures defined in the University's General Regulation V, and may be required to leave the University.