Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 2000"

Phase singularities in isotropic random waves

Abstract: The singularities of complex scalar waves are their zeros; these are dislocation lines in space, or points in the plane. For waves in space, and waves in the plane (propagating in two dimensions, o...

Water vapour, sonoluminescence and sonochemistry

Abstract: Sonoluminescence is the production of light from acoustically forced bubbles; sonochemistry is a related chemical processing technique. The two phenomena share a sensitive dependence on the liquid phase. The present work is an investigation of the fate and consequences of water vapour in the interior of strongly forced argon micro–bubbles. Due to the extreme nonlinearity of the volume oscillations, excess water vapour is trapped in the bubble during a rapid inertial collapse. Water vapour is prevented from exiting by relatively slow diffusion and non–equilibrium condensation at the bubble wall. By reducing the compression heating of the mixture and through primarily endothermic chemical reactions, the water vapour reduces the temperatures within the bubble significantly. The quantity and disposition of hydroxyl radicals produced within the bubble are studied in some detail, as this is of keen interest in sonochemistry. It was recently shown by Moss and co–workers that light emission from a sonoluminescence bubble depends sensitively on the water–vapour content. The quantity of trapped water vapour determined in the present analysis is in excellent agreement with the amount found by Moss and co–workers to match photon yields and pulse widths of recent experiments.

Spirals on the sea

Abstract: Spiral eddies were first seen in the sunglitter on the Apollo Mission 30 years ago; they have since been recorded on synthetic aperture radar (SAR) images and in the infrared. We present a small sample of images. The spirals are broadly distributed over the world's oceans, 10–25 km in size and overwhelmingly cyclonic. Under light winds favourable to visualization, linear surface features with high surfactant density and low surface roughness are of common occurrence. The linear features are wound into spirals in vortices associated with horizontal shear instability, modified by rotation, in regions where the shear is comparable with the Coriolis frequency. Two models for concentrating shear are presented: a softened version of the classical sharp Margules front, and the time–dependent Lagrangian model of Hoskins & Bretherton. Horizontal shear instabilities and both frontal models favour cyclonic shear and cyclonic spirals, but for different reasons.

Linear elastic contact of the Weierstrass profile

Abstract: A contact problem is considered in which an elastic half–plane is pressed against a rigid fractally rough surface, whose profile is defined by a Weierstrass series. It is shown that no applied mean pressure is sufficiently large to ensure full contact and indeed there are not even any contact areas of finite dimension — the contact area consists of a set of fractal character for all values of the geometric and loading parameters. A solution for the partial contact of a sinusoidal surface is used to develop a relation between the contact pressure distribution at scale n − 1 and that at scale n . Recursive numerical integration of this relation yields the contact area as a function of scale. An analytical solution to the same problem appropriate at large n is constructed following a technique due to Archard. This is found to give a very good approximation to the numerical results even at small n , except for cases where the dimensionless applied load is large. The contact area is found to decrease continuously with n , tending to a power–law behaviour at large n which corresponds to a limiting fractal dimension of (2 − D ), where D is the fractal dimension of the surface profile. However, it is not a ‘simple’ fractal, in the sense that it deviates from the power–law form at low n , at which there is also a dependence on the applied load. Contact segment lengths become smaller at small scales, but an appropriately normalized size distribution tends to a limiting function at large n . † The authors dedicate this paper to the memory of Dr J. F. Archard, 1918–1989.

Information flow in entangled quantum systems

Abstract: All information in quantum systems is, notwithstanding Bell9s theorem, localized. Measuring or otherwise interacting with a quantum system S has no effect on distant systems from which S is dynamically isolated, even if they are entangled with S . Using the Heisenberg picture to analyse quantum information processing makes this locality explicit, and reveals that under some circumstances (in particular, in Einstein–Podolsky–Rosen experiments and in quantum teleportation), quantum information is transmitted through ‘classical’ (i.e. decoherent) information channels.

Solitary shock waves and other travelling waves in a general compressible hyperelastic rod

Abstract: In the literature, it has been conjectured that solitary shock waves can arise in incompressible hyperelastic rods. Recently, it has been shown that this conjecture is true. One might guess that when compressibility is taken into account, such a wave, which is both a solitary wave and a shock wave, can still arise. One of the aims of this paper is to show the existence of this interesting type of wave in general compressible hyperelastic rods and provide an analytical description. It is difficult to directly tackle the fully nonlinear rod equations. Here, by using a non–dimensionalization process and the reductive perturbation technique, we derive a new type of nonlinear dispersive equation as the model equation. We then focus on the travelling–wave solutions of this new equation. As a result, we obtain a system of ordinary differential equations. An important feature of this system is that there is a vertical singular line in the phase plane, which leads to the appearance of shock waves. By considering the equilibrium points and their relative positions to the singular line, we are able to determine all qualitatively different phase planes. Those paths in phase planes which represent physically acceptable solutions are discussed one by one. It turns out that there is a variety of travelling waves, including solitary shock waves, solitary waves, periodic shock waves, etc. Analytical expressions for all these waves are obtained. A new phenomenon is also found: a solitary wave can suddenly change into a periodic wave (with finite period). In dynamical systems, this represents a homoclinic orbit suddenly changing into a closed orbit. To the authors9 knowledge, such a bifurcation has not been found in any other dynamical systems.

Random matrix theory and the derivative of the Riemann zeta function

Abstract: Random matrix theory is used to model the asymptotics of the discrete moments of the derivative of the Riemann zeta function, ζ( s ), evaluated at the complex zeros ½; + iγ n . We also discuss the probability distribution of ln |ζ′(1/2 + iγ n )|, proving the central limit theorem for the corresponding random matrix distribution and analysing its large deviations.

Eigenvalue solution of thermoelastic instability problems using Fourier reduction

Abstract: A › nite-element method is developed for determining the critical sliding speed for thermoelastic instability of an axisymmetric clutch or brake. Linear perturbations on the constant-speed solution are sought that vary sinusoidally in the circumferential direction and grow exponentially in time. These factors cancel in the governing thermoelastic and heat-conduction equations, leading to a linear eigenvalue problem on the two-dimensional cross-sectional domain for the exponential growth rate for each Fourier wavenumber. The imaginary part of this growth rate corresponds to a migration of the perturbation in the circumferential direction. The algorithm is tested against an analytical solution for a layer sliding between two half-planes and gives excellent agreement, for both the critical speed and the migration speed. Criteria are developed to determine the mesh re› nement required to give an adequate discrete description of the thermal boundary layer adjacent to the sliding interface. The method is then used to determine the unstable mode and critical speed in geometries approximating current multi-disc clutch practice.

Quantum Probability from Decision Theory

Abstract: In a recent paper, Deutsch [1] claims to derive the “probabilistic predictions of quantum theory” from the “non-probabilistic axioms of quantum theory” and the “non-probabilistic part of classical decision theory.” We show that his derivation fails because it includes hidden probabilistic assumptions.

Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity

Abstract: We consider the semilinear heat equation with critical power nonlinearity. Using formal. arguments based on matched asymptotic expansion techniques, we give a detailed description of radially symmetric sign-changing solutions, which blow-up at x = 0 and t = T < ∞, for space dimension N = 3,4,5,6. These solutions exhibit fast blow-up; i.e. they satisfy lim(t up arrowT)(T - t)(1/(p-1))u(0, t) = ∞. In contrast, radial solutions that are positive and decreasing behave as in the subcritical case for any N ≥ 3. This last result is extended in the case of exponential nonlinearity and N = 2.

The oscillatory channel flow with large wall injection

Abstract: In the presence of smallamplitude pressure oscillations, the linearized NavierStokes equations are solved to obtain an accurate description of the timedependent field in a channel having a rectangu

Simulating quantum mechanics by non-contextual hidden variables

Abstract: No physical measurement can be performed with infinite precision. This leaves a loophole in the standard no–go arguments against non–contextual hidden variables. All such arguments rely on choosing special sets of quantum–mechanical observables with measurement outcomes that cannot be simulated non–contextually. As a consequence, these arguments do not exclude the hypothesis that the class of physical measurements in fact corresponds to a dense subset of all theoretically possible measurements with outcomes and quantum probabilities that can be recovered from a non–contextual hidden–variable model. We show here by explicit construction that there are indeed such non–contextual hidden–variable models, both for projection–valued and positive–operator–valued measurements.

On droplet formation from capillary waves on a vibrating surface

Abstract: The formation of sprays from a liquid film on a vibrating surface is used by ultrasonic atomizers for applications ranging from humidification to metal–powder manufacturing. The received opinion in the literature is that droplets are formed periodically from the apexes of an orderly pattern of standing capillary waves, with a wavelength that can be related to vibration frequency by stability analysis. It is described how this assumption may be incorrect in that, after droplet formation commences, the orderliness of the standing–wave pattern is lost due to one or more secondary instability phenomena. These phenomena, which lead to disorderliness, are investigated by using high–speed imaging techniques and a low–frequency vibrating film to model the high–frequency case, because of the difficulty of penetrating clouds of small droplets in the latter case. Different modes of droplet formation are identified and the flow patterns responsible for these modes are discussed. Physical mechanisms are proposed from which it is deduced that only a proportion of standing–wave crests can eject droplets, for a given wall–vibration period, and the identity of these ejecting waves should vary from period to period. The model thus developed demonstrates an apparently random ejection of droplets from one wave cell, even though the model itself is deterministic. The disorder of the capillary waves and the occurrence of several droplet–formation routes are sufficient to explain the range of droplet sizes that is produced by ultrasonic atomization.

The choice of objective rates in finite elastoplasticity: general results on the uniqueness of the logarithmic rate

Abstract: A Eulerian rate formulation of finite elastoplasticity is a composite one composed of a rate equation for elastic behaviour and a flow rule for plastic behaviour as well as an evolution equation for hardening behaviour, in which objective Eulerian tensor rates are used. Among a large variety of objective rates (actually infinitely many), how to choose suitable ones has been one of the crucial points in finite elastoplasticity. It is realized that the foregoing composite formulation of elastoplasticity should fulfil certain consistency criteria in order to avoid inconsistencies or contradictions. These criteria narrow the choice of objective rates. These authors (Bruhns and co–workers and Xiao and co–workers) have recently introduced the self–consistency criterion : in a composite formulation of elastoplasticity, the rate equation intended for characterization of elastic behaviour should be exactly integrable to deliver an elastic relation. It has been demonstrated in the work of the aforementioned authors that if a composite formulation of elastoplasticity with objective rates is required to fulfil the just–stated self–consistency criterion, as it should be, then the newly discovered logarithmic rate is the only possible choice among all objective corotational rates, including the Zaremba–Jaumann rate and the Green–Naghdi rate, etc. In the afore–mentioned result, non–corotational rates, including Oldroyd rates, Cotter–Rivlin rate and Truesdell rate, etc., are not taken into consideration in general. It is the main goal of this paper to further establish the uniqueness of the logarithmic rate among all corotational and non–corotational objective rates. Essential to the attainment of this goal is the use of the yielding–stationarity criterion : the vanishing of the stress rate implies that the yield function is stationary. It is shown that the just–stated criterion is necessary for a composite Eulerian rate formulation of finite elastoplasticity to be consistent and means that the stress rate must be a corotational rate. The main goal of this article is, thus, attained by combining the just–stated result and the established result stated before.

Growth and uniqueness in thermoelasticity

Abstract: A uniqueness theorem is proved for two theories of thermoelasticity capable of admitting finite speed thermal waves, the theories having been proposed by Green & Naghdi. Uniqueness is proved under ...

Eigenvalue problems for doubly periodic elastic structures and phononic band gaps

Abstract: We consider the problem of elastic waves propagating in a two–dimensional array of circular cavities, taking rigorous account of coupling between shear and dilational waves. A technique, originally due to Rayleigh, is derived that involves an elegant identity between the singular and non–singular components of the stress fields in the array. This leads to an infinite linear system which can be truncated and solved in order to determine the complete structure of the propagating modes. Of particular interest is the possibility of exhibiting phononic band gaps, i.e. domains of frequency for which all propagating vibration in the material is suppressed.

Modular solutions to equations of generalized Halphen type

Abstract: Solutions to a class of differential systems that generalize the Halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group. These functio...

Development of a lifting wavelet representation for surface characterization

Abstract: This paper reviews the existing numerical analysis methods and their problems in surface metrology. Based on the requirements of functional analysis of surfaces, this paper proposes a lifting wavelet representation for extraction of different components of a surface. The theory of the lifting wavelet is introduced and a fast algorithm is developed. Different frequency components of the surface can be separated, extracted and then reconstructed according the intended requirements of functional analysis. The surface textures can be highlighted and multi-scalar topographical features can be identified and clearly recovered. In order to verify the behaviour of the new model, a computer simulation based on sinusoidal and triangular waveforms is used. Case studies are conducted using a series of typical surfaces of engineering and bioengineering, such as planes, cylinders and curves, measured by contact (stylus) and non-contact (phase-shifting interferometry) instruments, to demonstrate the feasibility and applicability of using the lifting wavelet model in the analysis of these surfaces.

Wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations

Abstract: Linear diffusion is an established model for spatial spread in biological systems, including movement of cell populations. However, for interacting, closely packed cell populations, simple diffusion is inappropriate, because different cell populations will not move through one another: rather, a cell will stop moving when it encounters another cell. In this paper, I introduce a nonlinear diffusion term that reflects this phenomenon, known as contact inhibition of migration. I study this term in the context of two competing cell populations, one of which has a proliferative advantage over the other; this is motivated by the very early stages of solid tumour growth. I focus in particular on travelling–wave solutions, corresponding to moving interfaces between the two cell populations. Numerical simulations indicate that there are wavefront solutions for wave speeds above a critical minimum value, and I present linear analysis that explains the selection of wave speeds by initial conditions. I obtain an approximation to the shape of these waves for high speeds, and show that the minimum speed arises via quite new behaviour in the travelling–wave equations, with the proportion of cells of each type approaching a step function as the wave speed decreases towards the minimum. Exploiting this structure, I use singular perturbation theory to investigate the wave shape for speeds close to the minimum.

Explicit observability estimate for the wave equation with potential and its application

Abstract: By means of the Carleman-type estimate, we obtain an explicit observability estimate for the wave equation with a potential. As its application, we get the exact internal controllability of the semilinear wave equations in any space dimensions.

Eshelby's problem for two-dimensional piezoelectric inclusions of arbitrary shape

Abstract: Eshelby's problem for piezoelectric inclusions of arbitrarily shaped crosssection remains a challenging topic. In this paper, a simple method is presented to obtain an analytic solution for Eshelby...

Symmetries and first integrals of ordinary difference equations

Abstract: This paper describes a new symmetry-based approach to solving a given ordinary difference equation. By studying the local structure of the set of solutions, we derive a systematic method for determining one-parameter Lie groups of symmetries in closed form. Such groups can be used to achieve successive reductions of order. If there are enough symmetries, the difference equation can be completely solved. Several examples are used to illustrate the technique for transitive and intransitive symmetry groups. It is also shown that every linear second-order ordinary difference equation has a Lie algebra of symmetry generators that is isomorphic to sl(3). The paper concludes with a systematic method for constructing first integrals directly, which can be used even if no symmetries are known.

A Ternary Lattice Boltzmann Model for Amphiphilic Fluids

Abstract: A lattice Boltzmann model for amphiphilic fluid dynamics is presented. It is a ternary model that is distinguished from prior models in three respects: first, it employs three order parameters, in that it conserves mass separately for each chemical species present (water, oil, amphiphile). Second, it maintains a vector–valued orientational degree of freedom for the amphiphilic species. Third, it models fluid interactions at the microscopic level by introducing self–consistent forces between the particles, rather than by positing a Landau free energy functional. This combination of characteristics fills an important need in the hierarchy of models currently available for amphiphilic fluid dynamics, enabling efficient computer simulation and furnishing new theoretical insight. Several computational results obtained from this model are presented and compared with existing lattice–gas model results. In particular, it is noted that lamellar structures, which are precluded by the Peierls instability in two–dimensional systems with kinetic fluctuations, are not observed in lattice–gas models, but are easily found in the corresponding lattice Boltzmann models. This points out a striking difference in the phenomenology accessible to each type of model.

Two-scale convergence for nonlinear Dirichlet problems in perforated domains

Abstract: The aim of the present paper is to adapt the method of two-scale convergence to the homogenization of a pseudomonotone Dirichlet problem in perforated domains with periodic structure. The limit problem and a corrector result are obtained.

The paradox of Parrondo's games

Abstract: We introduce Parrondo's paradox that involves games of chance. We consider two fair games, A and B, both of which can be made to lose by changing a biasing parameter. An apparently paradoxical situ...

Quasi–periodic and periodic solutions for coupled nonlinear Schrödinger equations of Manakov type

Abstract: We consider travelling periodic and quasi–periodic wave solutions in coupled nonlinear Schrodinger equations. In fibre optics these equations can be used to model single–mode fibres with strong birefringence, and two–mode optical fibres. Recently these equations appear as a model describing pulse–pulse interactions in wavelength–division–multiplexed channels of optical fibre transmission systems. In some cases this model reduces to the integrable Manakov system (IMS). Two–phase quasi–periodic solutions for the IMS are given in terms of two–dimensional Kleinian functions. The reduction of quasi–periodic solutions to elliptic functions is discussed. New solutions are found in terms of generalized Hermite polynomials, which are associated with two–gap Treibich–Verdier potentials.

Electromagnetic plane–wave response characteristics of non–axially excited slabs of dielectric thin–film helicoidal bianisotropic mediums

Abstract: Thin–film helicoidal bianisotropic mediums (TFHBMs) are rotationally non–homogeneous mediums whose microstructure consists of parallel helical columns. We characterize here the plane–wave response of non–axially excited dielectric TFHBM slabs. The constitutive relations for the dielectric TFHBMs are substituted into the time–harmonic Maxwell curl equations, and a 4 × 4 matrix ordinary differential equation is obtained. The piecewise homogeneity approximation method is shown to be reasonably robust to solve the differential equation. Bragg reflection by non–axially excited dielectric TFHBM slabs is investigated, and transmission characteristics are examined. The effects of several key illumination and constitutive parameters on the response of non–axially excited dielectric TFHBM slabs are studied to elucidate trends in functional relationships. A TFHBM bilayer is presented as an example of a device based on non–axially excited dielectric TFHBMs.

Dynamics of the diffusive Nicholson's blowflies equation with distributed delay

Abstract: In this paper we study the diffusive Nicholson's blowflies equation. Generalizing previous works, we model the generation delay by using an integral convolution over all past times and results are obtained for general delay kernels as far as possible. The linearized stability of the non-zero uniform steady state is studied in detail, mainly by using the principle of the argument. Global stability both of this state and of the zero state are studied by using energy methods and by employing a comparison principle for delay equations. Finally, we consider what bifurcations are possible from the non-zero uniform state in the case when it is unstable.

Combined optical and X–ray interferometry for high–precision dimensional metrology

Abstract: The requirement for calibrating transducers having subnanometre displacement sensitivities stimulated the development of an instrument in which the displacement is measured by a combination of optical and X–ray interferometry. The need to combine both types of interferometry arises from the fact that optical interferometry enables displacements corresponding to whole numbers of optical fringes to be measured very precisely, but subdivision of an optical fringe may give rise to errors that are significant at the subnanometre level. The X–ray interferometer is used to subdivide the optical fringes. Traceability to the meter is achieved via traceable calibrations of the lattice parameter of silicon and of the laser frequency. Polarization encoding and phase modulation allow the optical interferometer to be precisely set on a specific position of the interference fringe—the null point setting. The null point settings in the interference fringe field correspond to dark or bright fringes. Null measurement ensures maximum possible noise rejection. However, polarization encoding makes the interferometer nonlinear, but all nonlinearity effects are effectively zero at the fringe set point. The X–ray interferometer provides the means for linear subdivision of optical fringes. Each X–ray fringe corresponds to a displacement that is equal to the lattice parameter of silicon, which is ca .0.19 nm for the (220) lattice planes. For displacements up to 1 m the measurement uncertainties at 95% confidence level are ± 30 pm, and for displacements up to 100 m and 1 mm the uncertainties are ± 35 and ± 170 pm, respectively. Important features of the instrument, which is located at the National Physical Laboratory, are the silicon monolith interferometer that both diffracts X–rays and forms part of the optical interferometer, a totally reflecting parabolic collimator for enhancing the usable X–ray flux and the servo–control for the interferometers.

On normal form calculations in impact oscillators

Abstract: Normal form calculations are useful for analysing the dynamics close to bifurcations. However, the application to non-smooth systems is a topic for current research. Here we consider a class of imp ...