### Phd Projects

Here is a brief list of my areas of interest, each with a short sample
illustrative project. There is so much to do, that in each area one could
list many problems of personal interest, and each is open to discussion. By their very nature some of these projects will probably go out of date as they get done.
If you are interested and you would like to have a chat about any of these
feel free to email me at
r.craster and then @ma.ic.ac.uk.
Or contact me by post at:

Richard Craster

Department of Mathematics

Imperial College

London

SW7 2BZ

U.K.

Thin layer flows:
I have a longstanding collaboration with a colleague (Omar Matar) in Chemical Engineering and we have been looking at surfactant spreading on thin layers and the development of fingering instabilities in experiments. See for instance
Fingering phenomena associated with surfactant spreading on thin liquid films for the theory. We have other related projects on thin layer flows, these involve a mix of numerics, asymptotics and analysis together with modelling the underlying physical and chemical processes.

Waves:
There are many interesting and varied problems and projects that could be
undertaken in this area, and depending upon one's interests either a
numerical or analytical, or a combination of techniques could be used. A
sample project, that is not exclusive, many others on say, ocean waves,
or time dependent pulses, inhomogeneous media and so forth could be
equally, or more, interesting, is:
The use and development of a recent theory for diffraction, called embedding, this offers the possibility that many computations and bits of analysis could be replaced by a conceptually simpler problem. The solution of this simpler problem could then be manipulated to create the required information about the harder problem!
(see for instance Embedding formulae for diffraction by rational wedge
and angular geometries). This is quite an analytically based project, with some room for numerics if so desired.

I currently have many projects involving guided elastic waves arising from the recent asymptotic theories that we have developed (see for instance
Trapped modes in curved elastic plates,). There is considerable interest in guiding waves along elastic plates and bars, this involves a nice mix of numerical, asymptotic and theoretical work. Much of this work is in collaboration with the Non-destructive testing group in Mechanical Engineering at Imperial.

Non-Newtonian fluid mechanics:
Here my interests have mainly been in Bingham materials,
that is, the material is
a slurry-like material which has a yield stress, so it remains as a rigid
plug, until this stress is exceeded thereafter it flows as a liquid. Many
materials have this type of behaviour and it is important across a range
of geophysical (landslides) and engineering (flows of processed material,
foodstuffs and pastes) areas. There are interesting connections with geophysics and evolving lava flows (see for instance ``Dynamics of cooling domes of viscoplastic fluid''). How does a crust or surface skin develop and evolve? There are many interesting projects here too!