The University of Southampton
Courses

## Module Overview

Modelling fluid flow requires us first to extend vector calculus to include volumes that change with time. This will allow us to rephrase Newtonâ€™s second law of motion, that the force is equal to the time derivative of the linear momentum, in a way that can be applied to materials that flow and do not have a constant shape, i.e. to fluids. The final resulting equations are called the Navier-Stokes equations and are at the foundation of all fluid studies, from the microscopic motion of a bacterium to the hypersonic flow around a missile. In this module we will just touch on the simplest of the cases model by them: exact solutions of steady flows, water in a sloping channel, or of time dependent flows, driven by pulsating pressure (like blood flow). We will conclude by studying one of the most intriguing aspects of fluid dynamics, namely surface tension, the phenomenon responsible for the round shape of rain drops or soap bubbles. We will study its physical origin and how to model it in the context of the Navier-Stokes equations; we will finish by considering some fluid configurations where surface tension plays a dominant role (e.g. the capillary effect and soap bubbles).

### Aims and Objectives

#### Module Aims

The aim of the module is to provide the students with the mathematical tools and the physical understanding required to model, solve and understand simple viscous flows, such as water flowing in a pipe under a steady pressure, or blood being pumped in an artery by the periodic contractions of the heart.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Have knowledge and understanding of vector calculus of moving bodies
• Derive the continuum conservation laws and explain their physical meaning
• Derive the 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
• Define and model surface tension and use it study fluid configuration where its effect are particularly significant.

### Syllabus

1. Course overview 2. Vector and tensor calculus of moving bodies. 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 Time independent flows 4.2 Pulsatile flows. 5. Surface tension 5.1 Definition of the contact angle and derivation of the Young relation. 5.2 Derivation of the boundary conditions on the Navier-Stokes equations in the presence of surface tension. 5.3 Analysis of fluid configurations where surface tension plays a dominant role, e.g. capillary action and soap bubbles.

### Learning and Teaching

#### Teaching and learning methods

Lectures, worksheets and private study

TypeHours
Independent Study90
Teaching60
Total study time150

Acheson DJ (1990). Elementary Fluid Dynamics.

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

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

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

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

Fung YC (1997). Biomechanics: Circulation.

Matthews PC (1998). Vector Calculus.

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%

#### Repeat Information

Repeat type: Internal & External

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

### 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.