The University of Southampton
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# 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:

• Demonstrate the derivation of the governing Navier-Stokes equations
• Take the lubrication, inviscid, and slow flow limits of the Navier-Stokes equations
• Manipulate Cartesian tensors and know how to use tensors to represent physical quantities
• Recognize how constitutive relations are used to model different types of material
• Derive a number of exact solutions to the Navier-Stokes equations
• Explain the physical meaning of the continuum conservation laws
• 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;

### Syllabus

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

TypeHours
Independent Study90
Teaching60
Total study time150

Matthews PC (1998). Vector Calculus.

Fung YC (1997). Biomechanics: Circulation.

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

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.

Acheson DJ (1990). Elementary Fluid Dynamics.

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

### Assessment

#### Summative

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

#### Referral

MethodPercentage contribution
Exam 100%

#### Pre-requisites

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

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

### Costs

#### 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 www.calendar.soton.ac.uk.