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). 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.
 


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.


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    Last modified December 2006