Transport Phenomena and Conservation Laws
|Unit level:||Level 4|
|Teaching period(s):||Semester 2|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
Students are not permitted to take, for credit, MATH45122 in an undergraduate programme and then MATH65122 in a postgraduate programme at the University of Manchester, as the courses are identical.
The aim of the course is to introduce some ideas associated with transport equations and conservation laws, including linear and nonlinear wave propagation, wave steepening, shock formation, diffusion, dispersion and solitons.
Transport phenomena and conservation laws are ubiquitous and are a crucial element in all models of physical systems. As material is transported waves may form and the course starts with some examples of linear wave propagation. The effects of non-linearity are very important and lead to wave expansion, wave steepening and shock formation, at which there are discontinuous jumps in the solutions. It is shown how these can be modelled using jump conditions at propagating discontinuities and how diffusion and/or dispersion competes against wave steepening to create diffuse non-linear wave fronts and solitons. The formation of non-linear waves will be demonstrated using simple experiments during the lectures. The mathematical theory underlying these systems will be explained in detail, using many examples ranging from gas dynamics and shallow water flows, to granular, two-phase and traffic flows.
On successful completion of the course unit students will be able to
- interpret the solutions and describe the different physical properties of systems governed by the wave equation, kinematic wave equation, Burger’s equation, segregation and avalanche equations, gas dynamic and shallow water equations, irrotational water wave equations, Boussinesq equations and Korteweg-de Vries equation;
- solve ut + c(u)ux = 0 for given initial data and be able to identify the formation of shocks;
- determine the appropriate solution for the evolution and propagation of shocks in one space-dimension system and draw corresponding solution diagrams using the method of characteristics;
- adapt the method of characteristics and shock propagation techniques to two dimensional systems;
- demonstrate and apply the simple wave theorem method of characteristics to solve the gas dynamic and shallow water equations in systems similar to those studied in the course;
- non-dimensionalise and linearize the shallow water equations and the Euler irrotational equations;
- determine invariants and similarity solutions for the Korteweg-de Vries equation and similar partial differential equations;
- perform a phase plane analysis for the Korteweg-de Vries equation and related equations to identify travelling wave solutions, solitary wave solutions.
- Other - 20%
- Written exam - 80%
Assessment Further Information
- Mid-semester coursework: weighting 20%
- End of semester examination: three hours weighting 80%
1.The hyperbolic wave utt = c02Î'2u, ut + c0ux = 0; wave forms; Fourier synthesis; dispersion;
C(k) = dw/dk, group velocity; diffusion, e.g. Burger's linear equation ut + c0ux = vuxx.
2.First order wave equation ut + c(u)ux = 0; characteristics; conservation ideas; conservation forms; granular and traffic flow models. Waves in other physical systems.
3.First order equations in two-dimensions; breaking size segregation waves and their representation in terms of shocks.
4.Shallow water wave theory; the nonlinear equations; wave breaking, dam break problems, via characteristics; normal and oblique shocks in granular flows, linearisation and check against linear theory, and linear irrotational theory.
5.Irrotational water wave theory to obtain the Boussinesq equations; steady solutions of the Boussinesq equations; derivation of the Korteweg-de Vries equation from Boussinesq equations; conservation laws for K - dV; analytical solution of K - dV equation; the soliton.
- P.G. Drazin and R.S. Johnson, Solitons, An Introduction, CUP 1989.
- G.B. Whitham, Linear and Non-linear Waves, Wiley 1974.
- J. Stoker, Water Waves, Wiley Interscience 1957.
- L. Debnath, Nonlinear Water Waves, Academic Press 1994.
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
- Lectures - 22 hours
- Tutorials - 11 hours
- Independent study hours - 117 hours