Available online
Description
The Victoria Falls and the whirlpool which occurs at the La Rance barrage are spectacular examples of flows, but is it possible to analyse their behaviour mathematically? It turns out that only a v...ery few types of flow problem are reasonably easy to model, and hence analyse completely. One such type is the linear flow, where the velocity V = Ax. The programme -begins by posing the problem of describing some naturally occurring flows and then shows how some simpler flows can be simulated under laboratory conditions, several of which turn out to be linear; the source, sink, circular and spiral flows. Next, the programme asserts that all linear flows are variations of thirteen basic ones, obtained by analysing the possible combinations of eigenvalue and eigenvector for the matrix A in the linear equation V = Ax, and reducing A to its simplest form. In particular, this means taking all the eigenvectors along the axes, where appropriate. The cases correspond to situations where the eigenvalues are zero, non-zero, repeated, real and complex, and where A is singular and non-singular, so building up a comprehensive catalogue of all possible shapes of flow. Computer animations are used to illustrate each type of flow as it arises. The flows demonstrate among other things that flow is inwards when an eigenvalue is negative and outwards when it is positive. At the end of the programme the catalogue of flows is used to describe the flow for a particular example.
The Victoria Falls and the whirlpool which occurs at the La Rance barrage are spectacular examples of flows, but is it possible to analyse their behaviour mathematically? It turns out that only a v...ery few types of flow problem are reasonably easy to model, and hence analyse completely. One such type is the linear flow, where the velocity V = Ax. The programme -begins by posing the problem of describing some naturally occurring flows and then shows how some simpler flows can be simulated under laboratory conditions, several of which turn out to be linear; the source, sink, circular and spiral flows. Next, the programme asserts that all linear flows are variations of thirteen basic ones, obtained by analysing the possible combinations of eigenvalue and eigenvector for the matrix A in the linear equation V = Ax, and reducing A to its simplest form. In particular, this means taking all the eigenvectors along the axes, where appropriate. The cases correspond to situations where the eigenvalues are zero, non-zero, repeated, real and complex, and where A is singular and non-singular, so building up a comprehensive catalogue of all possible shapes of flow. Computer animations are used to illustrate each type of flow as it arises. The flows demonstrate among other things that flow is inwards when an eigenvalue is negative and outwards when it is positive. At the end of the programme the catalogue of flows is used to describe the flow for a particular example.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 04B; 1987 |
| First transmission date: | 06-09-1987 |
| Published: | 1987 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:03 |
| + Show more... | |
| Producer: | Pip Surgey |
| Contributors: | David Crowe; Peter Strain-Clark |
| Publisher: | BBC Open University |
| Keyword(s): | Determinant; Differential equation; Eigenvalue; La Rance; Linear Flow; Victoria Falls |
| Subject terms: | Eigenvectors; Matrices |
| Master spool number: | HOU5857 |
| Production number: | FOUM285B |
| Videofinder number: | 922 |
| Available to public: | no |
