Richard V. Craster
Richard Craster
I am a Professor
within the applied mathematics and mathematical physics
AMMP section of the Mathematics
department, and currently one of the executive editors for the Quarterly Journal of Mechanics and Applied Mathematics. (I am happy to receive electronic pdf submissions to the Journal).
This webpage contains a summary of my research
interests, other details such as a list of
publications , recent talks and presentations,
teaching related items (
M2AA2 ),
and some
useful external links can be found by
clicking on the highlighted text.
Research Interests
My research work falls into five reasonably distinct areas of Applied
Mathematics and Physics.
Wave Phenomena, Elasticity/ Fracture Mechanics,
Applied Complex Analysis,
Newtonian and Non-Newtonian fluid mechanics, Nonlinear
mathematics. If you are looking around for
a Phd place, I would be happy to discuss potential
Phd projects in these and related areas;
a couple are listed here.
Wave phenomena
- Scattering at Fluid Solid Interfaces and Diffraction Theory.
Wave-coupling involving defects or obstacles on fluid-solid interfaces
is of interest in geophysics, transducer devices, structural acoustics,
the acoustic microscope and related problems in non-destructive testing.
A useful limit is when the fluid loading is quite light, then distinctive
beaming
occurs along critical angles, and this is investigated for a variety of
canonical
problems involving surface discontinuities, subsurface cracks, and pulse
diffraction by defects.
Reciprocity and power balance theorems have also been derived.
More recently I have collaborated with Andrey Shanin on embedding techniques - a neat idea whereby one solves a single canonical problem from which the directivities of many others can be extracted.
Recent publications are :-
- Ocean wave and ice sheet interactions.
There has recently been revived interest in the interaction of
ocean waves and ice sheets, where the ice sheet is modeled as an
elastic plate. This interest arises as simpler mass-loading models
fail to predict some observed phenomena, and the mechanisms of ice sheet
break-up cannot be modeled using these simpler models. Analytic and
asymptotic techniques have been developed to solve the ocean wave and ice
sheet problem. Our article
Ocean waves and ice sheets
(co-authored with Neil Balmforth) has helped to partly re-awaken the analytical side of this subject.
Applied Complex Analysis.
- Free Boundary Problems/ Fuchsian Differential Equations
Several types of free boundary problems can be solved using conformal
mapping
techniques, these have applications in, for instance, semi-conductors,
groundwater flow, viscous film coatings, Hele-Shaw cells. In some
interesting
cases rather complicated mappings are required, and there is a connection
with Fuchsian differential equations. This arises during the conformal
mapping of a known domain to a curvilinear polygon. Several classes of
Fuchsian
equations involve free parameters and accessory parameters; thus explicit
solutions are not easy to find. New methods have been devised for their
solution.
A couple of recent publications are :-
- Effective properties
I have also worked on effective medium theories for model
composite structures. This has been primarily aimed at finding exact
solutions using complex analysis, and might be better described as a model
for obtaining effective resistivities in
electrostatics. Once again this involves conformal mappings, but this
time with doubly periodic functions.
This is related to previous work by
Yurii Obnosov; awarded a visiting EPSRC Fellowship
in 2000 (and 2002 and 2004).
The basic idea is that one wishes to know, in say electrostatics, the effective, i.e average in some sense, resistance/conductivity across a medium, such as that shown in the figure (a checkerboard medium), where each phase individually has a different constant resistance. It turns out that elegant and beautiful formulae emerge and that a result known as the Mortola and Steffe conjecture can be proved to be correct.
Some recent work is:
Newtonian and Non-Newtonian Fluid Mechanics:
- Viscoplastic and viscoelastic Flows.
This work falls into two main categories: firstly, longitudinal shear flows and
then thin-layer theories.
Longitudinal shear flows for a viscoplastic fluid form a particularly
useful
class of free boundary problems. Such a fluid is a widely used model of
pastes, paints, slurries - and everyday substances such as toothpaste.
The constitutive equations for such a material, here the Herschel-Bulkley
rheological model, are nonlinear. Nonetheless progress can be made when
the specific problems considered are mapped into a hodograph plane. Using
this analysis one is able to identify the solutions of the various problems
explicitly and identify regions of unyielded material. Mathematically the
identification of the yield surfaces forms an interesting class of
nonlinear
free boundary problems.
The problems also have interpretations as nonlinear filtration problems,
relevant to the filtration of non-Newtonian fluids, and nonlinear elastic
crack or dislocation problems. Actually, this is an old interest and I have not worked on this for 6 or 7 years.
Another (more) interesting aspect of Bingham-type materials is an apparent
inconsistency in thin-layer theory, recent work with N.J. Balmforth
has been aimed at this, and related problems; in particular aimed at lava flows.
Following this vein, an
article Visco-plastic models of
isothermal lava domes authored by Neil Balmforth, Adam Burbidge, Richard Craster,
John Salzig
and Amy Shen is aimed at
applying
thin layer theory to isothermal
lava domes (slowly evolving domes of lava). As an
aside,
the USGS have a
homepage with many interesting facts and details regarding volcanoes and
lava flows. In addition, several photographs of
lava
domes in various stages of their evolution can be found there too.
To
model the evolution of a dome we proceeded to use the aforementioned
thin-layer theory with a numerical Fortran code
based upon an
algorithm due to Blom and Zegeling dome.f.gz;
this code calculates the evolving dome shapes.
More recently, a non-isothermal theory for
lava domes (and analogue fluids) has been pursued (with Neil Balmforth and Roberto Sassi). The first article just has a simple theory of cooling, but more recently we have generated a skin/crust theory that allows for a developing cooling, rheologically distinct, skin.
Mud-flows are a related topic, as muds are often viscoplastic. We have pursued a mathematical theory for slumping domes downslope - here we obtain exact asymptotic dome shapes for large plasticity.
Other issues, such as the
stability of shear flows of viscoelastic materials, are also of interest.
Another aspect considered, again in collaboration with Neil Balmforth, is
how to utilize
recent asymptotic analyses of shear flows of Newtonian fluids
for Non-Newtonian fluids. An approximation is developed to study the continuous
spectrum associated with the elasticity of the constitutive models.
A couple of recent publications are :-
- Surfactant driven flows
Non-Newtonian effects also occur in biological fluid systems, some
recent collaborative work with Omar Matar
has been aimed at modelling surfactant transport
on thin mucus layers; this is relevant to a clinical treatment for
respiratory distress syndrome which can affect prematurely born infants.
Surfactant transport is also important in some industrial processes such
as Marangoni drying, and recent work has also been directed towards
looking at this. The pictures are (left) a simulation of a fingering instability and (right) an experiment (from Afsar-Siddiqui, Luckham, and Matar ) on surfactant spreading illustrating the instability.
A couple of recent publications are :-
- Pattern formation in thin films
- Jet and thread dynamics
Nonlinear mathematics
Two (or more) dynamical systems can, under certain circumstances, be made
to synchronize. That is, all the systems converge to the same temporal
behaviour even when the underlying dynamics are chaotic. This is not
guaranteed
to occur for all couplings, and even when it occurs may cease if some
system
parameters are varied slightly. Therefore methods by which this convergence
can be achieved have recently been of much interest. Apart from an
intrinsic
interest in synchronization, there are also applications to a wide variety
of subjects; these include cryptography, neural networks, electrical
circuits
and coupled lasers.
Other work includes the study of travelling waves in a model for
autocatalytic
reactions which have, for some parameter regimes, oscillatory
instabilities.
The instability is studied using Evan's functions and direct numerical
simulations. Just beyond the stability boundary, the unstable rocking of
the front saturates supercritically. If the order is increased still
further,
period doubling bifurcations occur, and there appears to be a transition
to chaos through intermittency after the disappearance of a period-4 orbit.
A couple of recent publications are :-
Elasticity.
- Poroelasticity.
This is primarily older work that arose from my Phd Thesis
with Colin Atkinson, we were investigating fracture in fluid
saturated porous elastic materials; the subject of Poroelasticity has a
home page .
The aim is to understand the physical processes involved when a crack
propagates,
or is initiated, in a porous material. The practical applications are to
problems in the oil industry such as hydraulic fracturing or in-situ
wellbore
stress determination; there are also geophysical applications to shear
faulting and pore pressure initiated earthquakes.
Other issues of interest are the application of invariant integrals
based upon the energy momentum tensor concepts of Eshelby, fracture along
interfaces and dynamic problems. A couple of the more recent publications are linked to here .
- Waves interacting with cracks.
This really overlaps with the earlier research interest on
waves.
Elastic waves can interact with cracks. A recent piece of
work is:
If you would like any reprints or further information feel free to email me.
r.craster (and then @imperial.ac.uk)
Richard Craster
Department of Mathematics
Imperial College
London
SW7 2BZ
U.K.
Useful links
Mathematical Software.
Numerical Recipes in Fortran.
Fortran 90/95 FAQ.
PACS Numbers.
Grammatical.
assistance
Web of Science database.
IC library.
Train times
Dictionary
Telephone Directory
Newspaper
Journals
Organizations
UK Maps
Info-Mac
Flights
Underground map
Last modified December 2006
