Skip to main navigationSkip to main content
The University of Southampton

MATH3072 Advanced Fluid Dynamics

Module Overview

First, a derivation of the governing Navier-Stokes equations will be carried out in general, during which the appropriate constitutive law for a Newtonian fluid will be introduced in terms of the stress tensor. Having derived the governing Navier-Stokes equations for a viscous fluid, the relevant boundary conditions will then be discussed, before moving on to consider the equations in non-dimensional form. At this point the Reynolds number, which characterises the relative importance of inertia versus viscosity in a given flow, will be introduced. In general, the Navier-Stokes equations are difficult to solve, however, in a certain simple situations exact solutions do exist and we shall examine a number of these solutions in a variety of biological, engineering and physical examples. One of the pre-requisites for: MATH6149

Aims and Objectives

Module Aims

To examine a number of everyday fluid flows that arise in a biological, physical and engineering context.

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

  • Manipulate Cartesian tensors and know how to use tensors to represent physical quantities
  • Have knowledge and understanding of vector calculus of moving bodies
  • Derive the continuum conservation laws and explain their physical meaning
  • Recognize how constitutive relations are used to model different types of material
  • Demonstrate the derivation of the governing Navier-Stokes equations
  • Derive a number of exact solutions to the Navier-Stokes equations and understand under what conditions they are a faithful representation of physical flows
  • Non-dimensionalise the Navier-Stokes equations, and understand the relevance of the Reynolds number
  • Take the lubrication, inviscid, and slow flow limits of the Navier-Stokes equations
  • Recognise a number of biological, physical and engineering examples in which such flows are relevant


1. Course overview 2. Background 2.1 What is biological fluid dynamics? 2.2 Vector and tensor calculus 3. Descriptions of Fluids 3.1 Eulerian and Lagrangian coordinates 3.2 Conservation equations 3.3 Constitutive laws 3.4 The Navier-Stokes equations 3.5 Boundary conditions 4. Exact Solutions to the Navier-Stokes Equations 4.1 Poiseuille and Couette flow 4.2 Drainage of tear films 4.3 Blood flow 4.4 Steady blood flow 4.5 Pulsatile blood flow 5. Beyond exact solutions to the Navier-Stokes equations 5.1 Nondimensionalisation One or more of the following: 5.2 Large Reynolds number flows 5.3 Small Reynolds number flows (Stokes flow) 5.4 The lubrication Approximation 5.5 Stability of exact solutions of the Navier-Stokes equations

Learning and Teaching

Teaching and learning methods

Lectures, worksheets and private study

Independent Study90
Total study time150

Resources & Reading list

Batchelor GK (1967). An Introduction to Fluid Dynamics. 

Acheson DJ (1990). Elementary Fluid Dynamics. 

Ockendon H and Ockendon JR (1995). Viscous Flow. 

Childress S (1981). Mechanics of swimming and flying. 

Matthews PC (1998). Vector Calculus. 

Fung YC (1997). Biomechanics: Circulation. 

Mazumdar J (1989). An Introduction to Mathematical Physiology and Biology. 

Howison S (2005). Practical Applied Mathematics, Modeling, Analysis, Approximation. 



MethodPercentage contribution
Coursework assignment(s) 10%
Exam  (120 minutes) 80%
Test 10%


MethodPercentage contribution
Exam 100%

Repeat Information

Repeat type: Internal & External

Linked modules

Pre-requisites: (MATH1057 AND MATH2038 AND MATH2044 AND MATH2045) OR (MATH1006 AND MATH1007 AND MATH2015) OR (MATH1008 AND MATH1009 AND MATH2015)


Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at

Share this module Share this on Facebook Share this on Twitter Share this on Weibo

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.