This talk is intended to give an introduction to the mechanics of granular materials. It is written from a personal viewpoint, that is to say, it introduces those aspects which are required to understand the models that I work with in my research: it does not pretend to be an objective introduction to the subject (if such a thing exists, which I doubt, there is no consensus).
It could be said that the state of the art in formulating predictive models for the behaviour of granular materials is akin to fluid mechanics prior to the advent of the Newtonian fluid or to solid mechanics prior to the advent of Hooke's law. There are no universally accepted models and no model has proven itself so superior to others as to cause researchers to gravitate towards it.
Granular materials may exhibit solid-like, liquid-like and gaseous-like behaviour (but the use of these words can be misleading, these "phases" are purely mechanical, there is no classical thermodynamics involved in this classification) and this talk is concerned with solid-like and what we may call liquid-like behaviour.
Despite the discrete nature of granular materials, I will try to convince you that continuum models may, possibly, be applicable in certain circumstances for answering certain questions. Having taken the decision to go for continuum models, I will explain the similarity and differences between the behaviour of granular materials and ordinary solid/fluid behaviour. In particular we will consider the concepts necessary to describe the deformation and flow, namely, stress, strain, velocity strain, velocity spin and intrinsic spin. The essential properties that granular materials exhibit (rather surprisingly, in common with metals) are yield, inelastic deformation, flow and history dependence. We will present a mathematical formulation for the equations, that is to say the balance equations (applicable to all continua), yield criteria (actually, an algebraic inequality rather than an equation) and flow rules. Geotechnical engineers use a flow rule called the "plastic potential model" but this is shunned by all other researchers. Unusual names for models abound, e.g. double-shearing; double-sliding free-rotating; double-slip and double-spin. Also unusual continua may be used, e.g. Cosserat continua (not so much in the UK where they have never been popular, but certainly on the continent, where, if you want to be taken seriously, you'd better incorporate some, possibly imaginary, Cosserat efects). One of the above names is the name I give to my class of models. If you are lucky I will present the equations for my models, on the other hand I may keep them a secret...
27 Apr 2012
The acoustic radiation force on spheres
The radiation force on a body in an acoustic field is a second-order (nonlinear) effect. Potential applications include particle manipulation, for example as a possible alternative to optical tweezers.
In this talk I will discuss a method for finding the radiation force on a submerged sphere subjected to a weakly nonlinear acoustic field. It is assumed that the wavelength of the incident wave is large compared to the radius of the sphere. In this long-wavelength limit, using the example of an incident plane wave, I will work through a matched asymptotic expansions method for finding the scattered field, and hence the acoustic radiation force. I will then discuss the force due to an incident Bessel beam. This problem proves to be more interesting, since placing the sphere away from the axis of the beam results in a radiation force acting in the transverse, as well as the axial, direction.
4 May 2012
Rayleigh-Benard convection generated by a diffusion flame
We investigate the Rayleigh-Benard convection problem within the context of a diffusion flame formed in a porous channel where the fuel and oxidiser concentrations are given. When formulated in the low Mach number approximation the problem depends on two control parameters, the Rayleigh number and the Damkohler number. For an infinite Damkohler number the top half of the channel is heated from below in a similar way to the standard Rayleigh-Benard problem, but the rigid wall boundary condition for velocity applies at the lower wall which is below the flame sheet. Once the system has become unstable convection rolls form, which interact with the diffusion flame to cause cellular flames. These have been studied in the literature in the context of a variable density model but this study is the first to include the effect of buoyancy to account for the full hydrodynamics. To give a background a quick summary of the main (well known) results on the instabilities of standard Rayleigh-Benard convection and the planar diffusion flame without hydrodynamics is given. We then go on to formulate the problem and numerically compute the critical Rayleigh number (and its dependence upon the aspect ratio of the numerical domain) in the case of very high Damkohler number.
11 May 2012
An introduction to seismic imaging
Seismic imaging, or migration, is a method of estimating properties of the Earth's subsurface from reflected seismic waves. Waves are transmitted from a source such as air gun, and the reflected/scattered waves are recorded by several receivers, or geophones.
Data is recorded as a function of time and (x,y) position on the Earth's surface. We will derive one method for performing a linearised inversion of the data to form a 3D image of the subsurface. This linearisation can sometimes be unsatisfactory, leading to unphysical artefacts in the image and misplaced features. In light of this, we will move on to discuss a method of improvement (which the industry terms annihilators), forming a partially linearised inversion.