MA0235: Elementary Fluid Dynamics

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA0235
External Subject Code G121
Number of Credits 10
Level L5
Language of Delivery English
Module Leader Dr Christopher Davies
Semester Spring Semester
Academic Year 2014/5

Outline Description of Module

The behaviour of fluid flows is important in a very wide variety of systems.  In weather and climate change studies it is necessary to predict and understand the general motion of both the air and the ocean. In medicine, it is important to know how blood flows in the arteries and the heart, and how air flows in the lungs. For instance, mathematically based simulations of the motion of blood in the heart have recently become sophisticated enough to guide surgeons when they take interventive action to treat various heart problems. In order to design aircraft it is necessary to know how wings can create a lifting force and how so-called viscous skin-friction can increase drag forces. If some means could be found to reduce skin-friction drag forces on aircraft by even just a few per cent, then this would translate into billions of pounds of savings in fuel costs for the airline industry every year.

The fundamental Euler and Navier-Stokes equations of fluid dynamics have been known for about two hundred and fifty years and a hundred and fifty years, respectively. Yet there remain many open and interesting questions about their solutions. This is despite the fact that, using a suitably compact notation, the equations are so short that they can each be written down in two lines.  The behaviour of turbulent flows, for example, can be described by solutions of these equations. Turbulent flows are ubiquitous in the natural world, as well as in engineered systems.  But no systematic means of obtaining turbulent solutions is known. Thus turbulence is an area of work that continues to attract the attention of many thousands of researchers, both in industry and over a range of academic departments within universities.

This module aims to provide students with a first look at the equations that govern the motion of fluids. We will extract a few simple solutions of these equations and discuss how they can be interpreted. To do this we will need to introduce various fundamental notions such as: particle paths; rates of change following the fluid flow (so-called material derivatives); mass and momentum conservation equations; and vorticity, which leads to an important distinction between two possible types of flow.

Prerequisite Modules: MA1300 Mechanics I

Corequisite Modules: MA2301 Vector Calculus

On completion of the module a student should be able to

  • understand the derivation of the Euler equations for an inviscid incompressible flow.
  • understand how the effects of fluid viscosity can be modelled using the Navier-Stokes equations.
  • know how to derive and interpret some simple solutions of the Euler equations and the Navier-Stokes equations.
  • appreciate the wide variety of physical behaviour that may be modelled and extracted using these equations.  

How the module will be delivered

27 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Skills that will be practised and developed

Skills:

The ability to obtain and verify simple solutions of the governing partial differential equations for an incompressible fluid flow.

Transferable Skills:

Mathematical modelling, in particular the use of partial differential equations to model configurations that involve both spatial and temporal variation.

How the module will be assessed

Formative assessment is carried out by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Spring Semester 100 Elementary Fluid Dynamics 2

Syllabus content

  • Particle paths.
  • Rate of change following the fluid.
  • Mass conservation and incompressibility. Pressure forces.  
  • Euler equations and their derivation.
  • Vorticity: rotational and irrotational flow.
  • Bernoulli theorems.
  • Vorticity evolution equations.
  • Simple potential flows.
  • Simple viscous flows.
  • No-slip conditions.
  • Derivation of governing equation for viscous unsteady shear flow.
  • Statement of the Navier-Stokes equations for incompressible viscous flow.
  • The Reynolds number and its interpretation.
  • Shear flow solutions of the Navier-Stokes equations.

Essential Reading and Resource List

Not applicable.

Background Reading and Resource List

Elementary Fluid Dynamics, Acheson, D.J., OUP


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