MATHEMATICAL FLUID MECHANICS WITH ADVANCED TOPICS - 2026/7

Module Overview

This module introduces the ideas of viscous fluid flows which build on the ideas some students would have seen in MAT2050: Inviscid Fluid Dynamics. By the end of the module, students should be familiar with the Navier-Stokes equations, and should be able to solve these equations in various simplified situations as well as in a variety of geometries. A selection of advanced topics will also be covered, enabling students to appreciate some of the finer points of viscous fluid mechanics.

Module provider

Mathematics & Physics

Module Leader

NOBILI Camilla (Maths & Phys)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 7

Module cap (Maximum number of students): N/A

Module Availability

Semester 2

Prerequisites / Co-requisites

N/A

Module content

Indicative content includes:

• Introduction to fluid mechanics 

  Definition of a fluid and examples of fluid flows in nature and engineering.  

• Kinematics of fluids 

  Eulerian and Lagrangian descriptions of motion, material derivative, Jacobian of the flow map and Euler's identity. Reynolds transport theorem and conservation of mass

• Conservation laws 

  Cauchy's stress theorem, properties of the stress tensor, conservation of momentum and derivation of Cauchy's momentum equation.

• Newtonian fluids 

  Constitutive laws for viscous fluids, rate-of-strain tensor and the Newtonian stress tensor. 

• The incompressible Navier-Stokes equations  

  Derivation of the Navier-Stokes equations from conservation of mass and momentum. Boundary conditions (no-slip and free-surface conditions).

• Vorticity and energy 

  Definition of vorticity, vorticity transport equation and interpretation of viscous diffusion of vorticity. Dissipation of energy in viscous flows. Conserved and dissipated quantities in 2d/3d viscous flows.

• Exact solutions of the Navier-Stokes equations 

  Unidirectional flows including Couette flow, Poiseuille flow and related canonical solutions. 

• Dimensionless analysis and similarity 
  Non-dimensionalisation of the Navier-Stokes equations and the role of the Reynolds number.  

  Together with an appropriate selection from Advanced Topics such as: 

• High Reynolds number flows

  Boundary-layer theory and asymptotic approximations. Derivation of Prandtl's boundary-layer equations and similarity solutions such as the Blasius boundary layer. 

• Boundary layer phenomena  

  Variable external flows, breakdown of boundary-layer approximations and flow separation. 

• Low Reynolds number flows 

  Stokes flows past obstacles including cylinders and spheres. 

• Lubrication theory 

  Thin-film flows and applications to bearings and squeeze-film flows. 

• Instabilities and pattern formation   

  Hele-Shaw flows and the Saffman-Taylor instability (viscous fingering). 

• Applications of asymptotic and PDE methods 

  Mathematical analysis of viscous flow phenomena arising in engineering and natural systems. 

Assessment pattern

Alternative Assessment

N/A

Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate: 

• Understanding of the methods required to solve complex fluid flow problems. 
• Subject knowledge through the recall of definitions as well as explaining why certain simplifications to the velocity field can be made and under what conditions certain approximations breakdown, also through a physical understanding of fluid problems. 
• Analytic ability through the solution of unseen and seen similar problems in the test and exam. 

Thus, the summative assessment for this module consists of: 

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 to 5. 
• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1 to 8. 

Formative assessment  
 
There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.   

Feedback 
 
Students receive individual written feedback on the formative unassessed coursework and the in-semester test. The feedback is timed so that feedback from the first unassessed coursework assists students with preparation for the in-semester test. The feedback from both unassessed courseworks and the in-semester test assists students with preparation for the end-of-semester examination. This written feedback is complemented by verbal and written feedback given in tutorials. Students also receive verbal and written feedback in office hours. 

Module aims

• Introduce students to viscous fluids in various simple geometries, and with various boundary conditions.
• Enable students to solve the Navier-Stokes equations in simple situations.
• Illustrate how fluid mechanics is connected to various problems in the real world, such as in engineering, and how the techniques learnt in this course can be applied to these problems.

Learning outcomes

Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A thorough account of exact solutions to the Navier-Stokes equations in a variety of geometries and under various simplifications.  
• Experience (through demonstration) of the methods and techniques used to solve problems in fluid mechanics. 

The learning and teaching methods include: 

• Three one-hour lectures per week for eleven weeks. Students will receive typeset notes to complement the lectures: these contain incomplete sections which will be filled in during lectures. The lectures provide a structured learning environment with opportunities for students to ask questions and to practice methods taught.  A suitable number of additional lectures will be provided to cover the advanced topic material. 
• Students will reinforce lectures by tackling a wide range of problems. 
• There are two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.  
• Lectures may be recorded. Lecture recordings are intended to give students the opportunity to review parts of the session that they might not have understood fully and should not be seen as an alternative to attendance at lectures.  
• Students should also use the books listed as background reading on the subject. 

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Reading list

https://readinglists.surrey.ac.uk
Upon accessing the reading list, please search for the module using the module code: MATM076

Other information

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2026/7 academic year.