Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1978"

Self-duality in four-dimensional Riemannian geometry

Abstract: We present a self-contained account of the ideas of R. Penrose connecting four-dimensional Riemannian geometry with three-dimensional complex analysis. In particular we apply this to the self-dual Yang-Mills equations in Euclidean 4-space and compute the number of moduli for any compact gauge group. Results previously announced are treated with full detail and extended in a number of directions.

Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting

Abstract: We examine the modes of a scalar field in de Sitter space and construct quantum two-point functions. These are then used to compute a finite stress tensor by the technique of covariant point-splitting. We propose a renormalization ansatz based on the DeWitt-Schwinger expansion, and show that this removes all am biguities previously present in pointsplitting regularization. The results agree in detail with previous work by dimensional regularization, and give rise to an anomalous trace with the conventional coefficient. We describe how’ our treatment may be extended to more general situations.

Solitons as Particles, Oscillators, and in Slowly Changing Media: A Singular Perturbation Theory

Abstract: In this article, we develop a singular perturbation theory for describing the long time cumulative effects of weak perturbations on solitons. In all cases, the solitons behave in a similar fashion to either relativistic or Newtonian particles or nonlinear oscillators under the influence of external forces. We show how the ubiquitous nonlinear Schrodinger soliton can become synchronized to a periodic external field and how it moves in gradual field gradients. We examine how the kink of the sine-Gordon equation acts both as a relativistic particle and a Newtonian particle in the presence of a general impurity and demonstrate the relaxation of a kink-antikink pair to a breather under the influence of damping. Finally, we discuss the motion of a soliton of the Korteweg de Vries equation under various perturbations and discover that while the soliton remains dominant, the continuous spectrum is excited and plays a crucial role in balancing the ‘mass’ and ‘energy’ depletion rates. In each case, we briefly discuss the result in the context of a physical situation.

Impact damage in brittle materials in the elastic-plastic response régime

Abstract: The impact fracture created in the elastic-plastic response regime has been characterized in terms of its surface extension and penetration. A numerical dynamic analysis has been performed of a typical impact within this regime to indicate some of the principal characteristics of the contact behaviour and the stress field. The damage has then been analysed, by using simplified postulates based on key features of the impact dynamics and basic fracture mechanics concepts. This has enabled the primary material and target parameters affecting the impact fracture to be identified. Thereafter, some implications for strength degradation and erosion have been discussed.

The conductivity of lattices of spheres I. The simple cubic lattice

Abstract: We have extended a method devised by Lord Rayleigh to calculate the conductivity of a simple cubic lattice of conducting spheres in a conducting matrix. The extended theory is capable of including the effects of multipoles of arbitrarily high order, and yields excellent agreement with measurements on arrays of perfectly conducting spheres, even when they are close to touching. Calculations and measurements on lossy spheres are also compared. An explicit formula for the conductivity is given which takes into account poles of order $2^{7}$, and two empirical formulae are discussed.

Bifurcation Phenomena in Steady Flows of a Viscous Fluid. I. Theory

Abstract: The investigation deals with questions relating to the existence of multiple solutions to hydrodynamic problems, especially questions about bifurcations of solutions and about stability. Although the aim is to resolve some of these questions generally, particular reference is made to events observed in the Taylor experiment on Couette flow between rotating cylinders, for which a new rationale is developed. In § 2 certain results for the abstract nonlinear problem of steady motion in a bounded fluid are summarized, including a set of generic properties that can be associated with real flows. In § 3 these results are applied to the interpretation of observable phenomena, and several predictions are made which are open to experimental checks. It is shown that new insights into the Taylor experiment are gained by considering its dependence on two parameters, one the Reynolds number R and the other the length l of the flow domain. Attention is paid to the initial development of cells as R is gradually increased from small values (§3.3), to a hysteresis phenomenon accompanying morphogenesis of the cellular structure at critical values of l (§3.4), and to properties of secondary modes possible above respective critical values of R (§ 3.5). In the appendix some corresponding interpretations are noted for the Benard problem of incipient convective motion.

Black Holes and Thermal Green Functions

Abstract: This paper concerns itself with the possibility of thermal equilibrium between a black hole and a heat bath implied by Hawking's discovery of black hole emission. We argue that in an isolated box of radiation, for sufficiently high energy density a black hole will condense out. We introduce thermal Green functions to discuss this equilibrium and are able to extend the original arguments, that the equilibrium is possible based on fields interacting solely with the external gravitational field, to the case when mutual and self interactions are included.

The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics

Abstract: Calculation of the normal-mode perturbation of steep irrotational gravity waves, begun in part I, is here extended to a study of the subharmonic perturbations, namely those having horizontal scales greater than the basic wavelength (2π/k). At small wave amplitudes a , it is found that all perturbations tend to become neutrally stable; but as ak increases the perturbations coalesce in pairs to produce unstable modes. These may be identified with the instabilities analysed by Benjamin & Feir (1967) when ak is small. However, as ak increases beyond about 0.346, these modes become stable again. The maximum growth scale of this type of mode in the unstable range is only about 14 % per wave period, which value occurs at ak ≈0.32. At values of ak near 0.41 a new type of instability appears which has initially zero frequency but a much higher growth rate. It is pointed out that this type might be expected to arise at wave amplitudes for which the first Fourier coefficient in the basic wave is at its maximum value, as a function of the wave height. The corresponding wave steepness was found by Schwartz (1974) to be ak = 0.412. A comparison of the calculated rates of growth are in rather good agreement with those observed by Benjamin (1967) in the range 0.07 ak

The conductivity of lattices of spheres II. The body centred and face centred cubic lattices

Abstract: The conductivities of body-centred (b. c. c.) and face-centred (f. c. c.) cubic lattices of spheres of a conducting material in a conducting matrix are calculated by using a method originally devised by Lord Rayleigh. Measurements of the conductivity of the b. c. c. lattice of perfectly conducting spheres are presented. Good agreement between theory and experiment is obtained. Our results are shown to be in agreement with asymptotic equations derived by other authors. A formula is given for the case of a disordered array.

Random partitions in population genetics

Abstract: This paper is concerned with models for the genetic variation of a sample of gametes from a large population. The need for consistency between different sample sizes limits the mathematical possibilities to what are here called ‘partition structures Distinctive among them is the structure described by the Ewens sampling formula, which is shown to enjoy a characteristic property of non-interference between the different alleles. This characterization explains the robustness of the Ewens formula when neither selection nor recurrent mutation is significant, although different structures arise from selective and ‘charge-state’ models

Bifurcation Phenomena in Steady Flows of a Viscous Fluid. II. Experiments

Abstract: The experimental work is concerned with several phenomena studied theoretically in part I (Benjamin 1977). Observations on Taylor-vortex flows between cylinders of comparatively small but variable length are reported, revealing properties unexplained by older theories. The observed flows are classified as follows: (i) the primary mode which is uniquely possible at small values of the Reynolds number $R$, and which usually develops smoothly with increasing $R$; (ii) secondary modes which are possible only above a respective critical value of $R$, and which are shown to manifest predicted behaviour as this value is approached from above (section 4). Two novel and surprising examples of (ii) are reported. A predicted hysteresis phenomenon is confirmed, relating to morphogenesis of the primary mode between two-cell and four-cell forms as the length of the annulus is varied (section 5). The experimental results are discussed from a theoretical standpoint in section 6.

The effects of randomness on the stability of two-dimensional surface wavetrains

Abstract: A simplified nonlinear spectral transport equation, for narrowband Gaussian random surface wavetrains, slowly varying in space and time, is derived from the weakly nonlinear equations of Davey & Stewartson. The stability of an initially homogeneous wave spectrum, to small oblique wave perturbations is studied for a range of spectral bandwidths, resulting in an integral equation for the amplification rate of the disturbance. It is shown for random deep water waves that instability of the wavetrain can exist, as in the corresponding deterministic Benjamin-Feir (B-F) problem, provided that the normalized spectral bandwidth $\sigma /k_{0}$ is less than twice the root mean square wave slope, multiplied by a function of the perturbation wave angle $\phi $ = arctan (m/l). A further condition for instability is that the angle $\phi $ be less than 35.26 degrees. It is demonstrated that the amplification rate, associated with the B-F type instability, diminishes and then vanishes as the correlation length scale of the random wave field (ca. $1/\sigma)$) is reduced to the order of the characteristic length scale for modulational instability of the wave system.

Triangular Potts model at its transition temperature, and related models

Abstract: Kelland has solved a restricted ice-type model on the triangular lattice. Here it is shown that this is equivalent to a restricted six-vertex model on the Kagome lattice, and to the g-state triangular (or hexagonal) Potts model at its transition temperature T c . This enables us to obtain the free energy, internal energy and latent heat of the Potts model at T c . The relation of this work to the operator method of Temperley and Lieb is explained, and this method is used to consider a generalized triangular Potts model which includes a three-site interaction on alternate triangles. It is shown that this model is self-dual. The results for the bond percolation problem on the triangular lattice give an excellent verification of series expansion predictions.

The Measurement of Optical Radiation Pressure in Dispersive Media

Abstract: The paper describes the details of a repetition of the Jones-Richards experiment (1954) to compare the recoil of a mirror due to the radiation pressure of light striking it in an optically dense medium with the recoil of the same mirror from the same light in air. A more than tenfold improvement in precision has been possible through the development of the laser and of multilayer reflectors of high reflectivity and low absorption. The new experiment confirms, to a precision of about 0.05%, that the momentum associated with electromagnetic radiation increases directly with the refractive index of the medium into which it passes, discriminating substantially in favour of the phase velocity ratio and against the group velocity ratio. The experiment also shows that the magnitude of recoil of a mirror in oblique incidence does not vary, within the foregoing precision, when the plane of polarization is changed from perpendicular to parallel to the plane of incidence.

Quantum Coherence Effects and the Second Law of Thermodynamics

Abstract: Negative energy densities and fluxes due to quantum coherence effects in quantum field theories are discussed. Such negative energy fluxes seeming­ly lead to a breakdown of the second law of thermodynamics. It is argued that such a breakdown will not in fact occur if a negative energy flux F is constrained by an inequality of the form │ F │≲ T -2 where T is a character­istic time over which the negative energy flux occurs. Particular models in which negative energy fluxes occur are examined, and it is found that such an inequality is indeed satisfied. Quantum violations of the dominant energy condition, which requires that the local energy density be greater than or equal to the magnitude of the pressure are also discussed briefly.

Self-consistent estimates for the viscoelastic creep compliances of composite materials

Abstract: In this paper the viscoelastic creep compliances of various composites are estimated by the self-consistent method. The phases may be arbitrarily anisotropic and in any concentrations but we demand that one of the phases be a matrix and the remaining phases consist of ellipsoidal inclusions. The theory is succinctly formulated with the help of Stieltjes convolutions. In order to solve the title problem, we first solve the misfitting viscoelastic inclusion problem. Numerical solutions are given for a selection of inclusion problems and for two common composite materials, namely an isotropic dispersion of spheres, and a uni-directional fibre reinforced material.

Complexities of compression failure

Abstract: The complexities of compression failure are defined in terms of the three well known failure modes: yielding, cone failure, and axial splitting. The notion of compressive strength is shown to be useful only in the case of yielding. For compression cracking, where the stress at failure varies with geometry and manner of force application, compressive strength is not such a useful parameter. Therefore, new failure criteria are suggested for compression cracking based on the energy balance theory of fracture. These criteria explain why the force required for failure depends on the mode of cracking, on the fracture surface energy of the material, and on the geometry and elasticity of the specimen. Fracture surface energy, and not strength, is found to be the fundamental material property dictating compression cracking. These ideas are verified by detailed theoretical and experimental investigation of compression splitting in glassy materials. In particular, the effects of size on compression failure are interpreted, both platen and specimen dimensions influencing the fracture force. It is further shown that the mode of failure can change from brittle to ductile under certain conditions, for example, as the platens are widened or as the sample is reduced in size. Finally, the inhibiting effect of lateral forces on compression cracking is explained.

Twisted quantum fields in a curved space–time

Abstract: Non-trivial space–time topology leads to the possibility of twisted fields viewed as cross sections of non-product vector bundles. For globally hyperbolic space–times twisted real and complex scalar fields are especially interesting, and are in one-to-one correspondence with certain groups determined by the space–time topology. Twisted fields can be quantized and lead to results differing from the usual ones. For example, spontaneous symmetry breaking may be suppressed and regularized vacuum self-energies take on different values. Sets of twisted fields may be collected together into a type of super-multiplet whose size is determined by the space–time topology.

On the flow of polythermal glaciers. I. Model and preliminary analysis

Abstract: Many interesting phenomena have been observed in the flow of glacial ice masses. In order to establish a rational theory for the study of these phenomena, we develop here a detailed continuum model for the flow of polythermal (i.e. partially cold and partially temperate) glaciers in wide mountain valleys. As a first step in the analysis of the nonlinear double free boundary problem posed by this model, the structure and stability of solutions are studied in a particular very simple limiting situation. Further analysis is deferred to part II of the paper.

An X-Ray Photoelectron Spectroscopic Study of the Interaction of Oxygen and Nitric Oxide with Aluminium

Abstract: The interaction of oxygen and nitric oxide with clean aluminium surfaces has been investigated by X-ray photoelectron spectroscopy. Studies have been in the main confined to the temperature range 80-290 K and pressure range $10^{-6}-10^{-2}$ Pa. The Al(2p) level is shown to exhibit a shifted component at a binding energy of $\sim $ 74.5 eV after oxygen interaction at 80 K. A curve-fitting and deconvolution analysis of the shifted peak indicates that it is made up of two components, one at a binding energy of 74.0 eV and the other at 75.3 eV. The lower binding energy component (designated $\alpha $) develops preferentially at 80 K while the higher binding energy one $(\beta)$, assigned to Al$\_{2}$O$\_{3}$, is dominant at 290 K. We suggest that $\alpha $ is an incorporated oxygen structure which is a precursor to the formation of Al$\_{2}$O$\_{3}$. The initial sticking probability of oxygen at 80 K is 0.07 while at 290 K it is 0.02. The plasmon-loss features associated with the Al(2s) peak are shown to be sensitive to oxygen adsorption and therefore useful in confirming the surface monolayer. At 290 K and an oxygen pressure of $10^{4}$ Pa the oxide thickness is estimated to be about 0.9 nm. When NO was adsorbed at 80 K three distinct N(1s) peaks were observed at binding energies of about 397, 403 and 407 eV. We assign the 397 eV peak to N$\_{\text{ads}}^{\delta -}$ arising from dissociation of the molecule and the two higher binding energy peaks to NO$\_{\text{ads}}^{\delta -}$ and N$\_{2}$O$\_{\text{ads}}$. The N(1s) peak characteristic of $\text{NO}\_{\text{ads}}^{\delta -}$ is close to the lower binding energy peak of the two N(1s) peaks associated with $\text{N}\_{2}\text{O}\_{\text{ads}}$ while the O(1s) of $\text{NO}\_{\text{ads}}^{\delta -}$ overlaps with the O(1s) peak of the surface oxide at a binding energy of about 532 eV. Mass spectroscopic analysis of the gas phase indicated that on warming to 85 K, the adlayer formed at 80 K, $\text{N}\_{2}\text{O}$ was desorbed, confirming our assignment of the core-level spectra. The $\text{NO}^{\delta -}$ $\text{and}$ $\text{N}\_{2}\text{O}$ species are not observed at 290 K while at 80 K exposure of the adlayer to water vapour results in the complete removal of weakly adsorbed $\text{N}_{2}\text{O}$. By monitoring the intensities of the Al(2p), N(1s) and O(1s) peaks, estimates were made of the absolute and relative concentrations of the various species and various molecular processes delineated.

On the soliton solutions of the Davey-Stewartson equation for long waves

Abstract: The Davey-Stewartson equations describe two-dimensional surface waves on water of finite depth. In the long wave limit, it is shown that these equations belong to the class derivable from operator equations in the manner of Zakharov & Shabat. The basic underlying linear system of equations is obtained and solutions to the original nonlinear system sought from the Gelfand-Levitan equations of Inverse Scattering Theory. Single soliton and multi-soliton solutions are deduced corresponding to the one-dimensional solutions already available. The solitons so obtained are pseudo one dimensional in that they have the same form as onedimensional solitons but move at an angle to the main direction of propagation. The multi-soliton solution describes the interaction of many such solitons each propagating in different directions. For two solitons, it is shown that resonance occurs and a triple soliton structure is produced.

The Deformation of Steep Surface Waves on Water. II. Growth of Normal-Mode Instabilities

Abstract: Studies of the normal-mode perturbations of steep gravity waves (Longuet-Higgins 1978 b , c ) have suggested two distinct types of instability: at low wave steepnesses we find subharmonic instabilities with fairly low rates of growth, and at higher wave steepnesses there are apparently local (‘superharmonic’) instabilities leading directly to wave breaking. Between these two types of instability is an intermediate range of wave steepnesses where the unperturbed wave train is neutrally stable. In the present paper we employ the time-stepping method of an earlier paper (Longuet-Higgins & Cokelet 1976) to test the rate of growth of each type of instability. For the initial linear stages of each instability, the computed rates of growth are accurately confirmed, and it is verified that the local instability does indeed lead to breaking. The later nonlinear stages of the subharmonic instabilities are further investigated. In the two examples so far computed it is found that the gradual rates of growth of the subharmonic instabilities are maintained, and that ultimately every alternate crest develops a fast-growing local instability which quickly leads to breaking.

Amenability of inverse semigroups and their semigroup algebras

Abstract: If G is a group, then G is amenable as a semigroup if and only if l1(G), the group algebra, is amenable as an algebra. In this note, we investigate the relationship between these two notions of amenability for inverse semigroups S. A complete answer can be given in the case where the set Es of idempotent elements of S is finite. Some partial results are obtained for inverse semigroups S with infinite Es.

On the Existence of a Wave of Greatest Height and Stokes's Conjecture

Abstract: It is shown that there exists a solution of Nekrasov's integral equation which corresponds to the existence of a wave of greatest height and of permanent form moving on the surface of an irrotational, infinitely deep flow. It is also shown that this wave is the uniform limit, in a specified sense, of waves of almost extreme form. The question of the validity of Stokes's conjecture is reduced to one of the regularity of the solution of Nekrasov's equation in this limiting case.

Spinor fields in four dimensional space-time

Abstract: An investigation is made of the role played by spacetime topology in the construction of spinor fields. Emphasis is placed on the existence of inequivalent spinor structures arising from non-trivial space-time topologies and their effect on the Dirac lagrangian.

Combustion in heat exchangers

Abstract: The ranges of flow for which flames can stably be burnt in combustors based on extensive heat recirculation between products and reactants are examined in terms of the heat exchanger characteristics. This is important because, no matter how desirable it may be to burn poor fuels and mixtures of very low heat content efficiently, there are many applications for which the scheme would be attractive only for reasonably large throughputs and rates of energy release. A simple general theory based on the observed constant reaction temperature is shown to predict correctly the shape of the empirical curves for mixtures outside the flammability limits. Numerical correlations are obtained for a more detailed analysis of the very efficient double-spiral geometry. The results are compared with the theoretical maximum heat release rates per unit volume of flames in normally flammable mixtures. It is shown that mixtures containing only one fifth of the heat content at the normal limit of flammability in practice yield heat release rates comparable to the theoretical maximum for normal flames within the flammable range; the theoretical maximum for stoichiometric methane-air mixture should be attainable at little more than half the limit of flammability.

On the Non-Uniqueness of the Machining Process

Abstract: The elimination of a class of possible slip-line field solutions for orthogonal machining indicates that the process is not uniquely defined. The range of possible solutions for any value of tool rake angle and interfacial shear stress is shown to be associated with large variations in the curvature of the machined chip. Machining conditions are split into two types, for one of which the machined chip will always curl, while the other has the Lee & Shaffer slip-line field as a lower limit of the solution range. The extent of the solution range for any value of friction is found to decrease with increasing rake angle. The analysis is shown to be consistent with certain experimental work available.

The Linear Stability of Flat Stokes Layers

Abstract: The linear stability of a flat Stokes layer is investigated. The results obtained show that, in the parameter range investigated, the flow is stable. It is shown that the Orr-Sommerfield equation for this flow has a continuous spectrum of damped eigenvalues at all values of the Reynolds number. In addition, a set of discrete eigenvalues exists for certain values of the Reynolds number. The eigenfunctions associated with this set are confined to the Stokes layer while those corresponding to the continuous spectrum persist outside the layer. The effect of introducing a second boundary a long way from the Stokes layer is also considered. It is shown that the least stable disturbance of this flow does not correspond to the least stable discrete eigenvalue of the infinite Stokes layer when this boundary tends to infinity.

On the hydration of Portland cement

Abstract: When Portland cement is contacted with water, calcium ion is rapidly leached from the solid to form calcium hydroxide solution but only traces of silica are found in the aqueous phase. It is proposed that the hydrated, calcium-depleted surface of grains consists of low molecular mass silicic acids and that these interact with dissolved hydroxylated calcium species (principally Ca(OH) 2 ) to produce a semi-permeable membrane of ‘ calcium silicate hydrate ’ at the hydrated grain surface. Osmotic pressure within this membrane causes its rupture and hence the growth of excrescences from the grain as the contents are extruded into the surrounding calcium hydroxide solution. The interstitial solid material is best regarded as a coagulum resulting from the combination of low molecular mass silicate anions with dissolved calcium hydroxide. It is proposed that the low tensile strength of Portland cement paste results from microstructural features consequent upon this mechanism of hydration and setting.

The Boundary Layer Due to Rectilinear Vortex

Abstract: The boundary layer created by the motion of a single rectilinear vortex filament above an infinite plane wall is considered. In a frame of reference which moves uniformly with the vortex the inviscid motion is steady; however, the possibility of a corresponding steady boundary-layer solution can be ruled out and it is concluded that the boundary-layer flow is inherently unsteady for all time. To investigate the nature of the unsteady boundary-layer flow, a time-dependent problem, corresponding to the sudden insertion of the plane wall at time $t$ = 0, is considered; separation in the boundary layer is found to take place in a short period of time and the solution shows possibly explosive features as $t$ increases. It is conjectured that an eventual eruption of the boundary-layer flow is to be expected along with a major modification of the inviscid flow. The theory compares favourably with experiments on the flow induced near the ground by trailing aircraft vortices.