The University of Southampton

MATH3072 Biological 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.

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:

  • Take the lubrication, inviscid, and slow flow limits of the Navier-Stokes equations
  • Demonstrate the derivation of the governing Navier-Stokes equations
  • Manipulate Cartesian tensors and know how to use tensors to represent physical quantities
  • Explain the physical meaning of the continuum conservation laws
  • Derive a number of exact solutions to the Navier-Stokes equations
  • Recognize how constitutive relations are used to model different types of material
  • Non-dimensionalise the Navier-Stokes equations, and understand the relevance of the Reynolds number
  • Recognise a number of biological examples in which such flows are relevant
  • Non-dimensionalise the Navier-Stokes equations, and understand the relevance of the Reynolds number;


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

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

Fung YC (1997). Biomechanics: Circulation. 

Matthews PC (1998). Vector Calculus. 

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

Howison S (2005). Practical Applied MathemaYcs, Modeling, Analysis, ApproximaYon. 

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

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

Acheson DJ (1990). Elementary Fluid Dynamics. 



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


MethodPercentage contribution
Exam 100%

Linked modules


To study this module, you will need to have studied the following module(s):

MATH2008Introduction to Applied Mathematics
MATH2045Vector Calculus and Complex Variable
MATH2044Applications of Vector Calculus


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

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