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). I am also an Associate Editor of Wave Motion and submissions for it should be made through the journal editorial system.
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.
This is joint work with Dmitri Gridin, Alexander Adamou, Samuel Adams, Julia Postnova and Elizabeth Skelton aimed at generating theories for wave propagation in bent elastic waveguides, thickening waveguides or otherwise deformed elastic surfaces. There is also interest in efficient numerical methods for elastic waves in waveguide, scattering from cracks and so on.
A recently completed EPSRC grant enabled us to collaborate with the Non-destructive testing group (the final report) in the Mechanical Engineering department. Our aim is to generate both accurate and fast numerical/ semi-analytical methods for generating dispersion curves and asymptotic methods for general curvature. A notable discovery is that elastic guided waves can be trapped in regions of high curvature and criteria that determine the trapping can be found. The pictures show a typical real geometry and the trapping of a mode near high curvature.
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 :-
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.
A couple of recent publications are :-
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 :-
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 :-
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 :-
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 .