MA3305: Fluid Dynamics

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA3305
External Subject Code 100400
Number of Credits 20
Level L6
Language of Delivery English
Module Leader Dr Aikaterini Kaouri
Semester Spring Semester
Academic Year 2022/3

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 introduces students to 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. Appropriate approximations of the Navier-Stokes equations will be developed, including lubrication theory and boundary layer theory. At the end an introduction to waves will be given. The course will further develop the students’ mathematical modelling skills and real-life problem solving.

Prerequisite Modules: MA1300 Mechanics I

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
  • Know how to derive the Navier-Stokes equations and construct simple solutions
  • apply appropriate approximations that simplify the Navier-Stokes equations and gain insight into complex flows such as those encountered in lubrication and boundary layers;
  • appreciate the role of hydrodynamics and develop an understanding of waves
  • appreciate the wide variety of physical behaviour that may be modelled and extracted using the equations of fluids.  

How the module will be delivered

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

  • Weekly face to face classes (e.g. labs, lectures, exercise classes)
  • Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills that will be practised and developed

Skills:

The ability to obtain, verify and interpret 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 physical and industrial processes that involve spatial and temporal variation. Communication skills will also be enhanced as the students will be trained in providing detailed, coherent reasoning about their solutions in writing and verbally.

How the module will be assessed

 

 

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Spring Semester 100 Fluid Dynamics (20 Credit Module) 3

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.
  • Forces acting on a body.
  • Definition of the stress tensor.
  • Symmetry of the stress tensor.
  • Derivation of the Navier-Stokes equations.
  • Non-dimensionalisation, Reynold’s number, Dynamic similarity.
  • Exact solutions: Flow in pipes
  • Approximate solutions: Boundary layer theory, Lubrication theory, Squeezing flow
  • Introduction to wave theory

Copyright Cardiff University. Registered charity no. 1136855