Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1986"

Limit Theorems for Fuzzy Random Variables

Abstract: A strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables, whose values are fuzzy sets with compact levels. The proofs are based on embedding theorems as well as on probability techniques in Banach space.

Evolution of Wavelike Disturbances in Shear Flows: A Class of Exact Solutions of the Navier-Stokes Equations

Abstract: New classes of exact solutions of the incompressible Navier-Stokes equations are presented. The method of solution has its origins in that first used by Kelvin (Phil. Mag. 24 (5), 188-196 (1887)) to solve the linearized equations governing small disturbances in unbounded plane Couette flow. The new solutions found describe arbitrarily large, spatially periodic disturbances within certain two- and three-dimensional 'basic' shear flows of unbounded extent. The admissible classes of basic flow possess spatially uniform strain rates; they include two- and three-dimensional stagnation point flows and two-dimensional flows with uniform vorticity. The disturbances, though spatially periodic, have time-dependent wavenumber and velocity components. It is found that solutions for the disturbance do not always decay to zero; but in some instances grow continuously in spite of viscous dissipation. This behaviour is explained in terms of vorticity dynamics.

Diffusion in absorbing media

Abstract: The problem to be discussed, which arose in connexion with the uptake of moisture by cotton bales, is that of diffusion of one substance through another in the pores of a solid body which may absorb and immobilize some of the diffusing substance Heat will be evolved by the absorption process, and this will itself diffuse through the medium, and will affect the extent to which the solid can absorb the diffusing substance We thus have two diffusion processes coupled by the mutual interaction of the diffusing “substances” when they are absorbed by the solid The pores are envisaged as a continuous network of spaces included in the solid, containing the medium (eg air) through which the diffusion takes place The solid itself may be either discontinuous, as is a bale of cotton fibres, or continuous, like a sponge For convenience we will refer to the diffusing substance in the pores as the “vapour”, though the theory is not limited to the diffusion of vapours, nor is it necessary to assume pores of larger than molecular dimensions The essential point is that some of the diffusing substance becomes immobilized, and that heat is given out in the process Thus the case of a dissolved substance diffusing through a gel would be included, and it is not necessary to suppose the diffusion limited to one phase only If heat is evolved when vapour is absorbed by the solid, it follows by thermodynamic reasoning that vapour will be set free when heat is immobilized (ie disappears) Hence the equations will be symmetrical in form Equations of the same form would be obtained, neglecting thermal effects, for the diffusion through a porous solid of two substances, each capable of replacing the other in absorption by the solid

Experimental study of Lagrangian turbulence in a Stokes flow

Abstract: Experimental studies are made of chaotic particle motion in a simple Stokes flow system. Surfaces-of-section that exhibit a generic mixture of regular and chaotic particle motions in good agreement with computer simulations are constructed experimentally. Deformations of line elements exhibit \`whorl' and \`tendril' structures of great complexity that can be correlated directly with the underlying particle dynamics and that agree well with numerical computations. In some cases, the laboratory studies are able to resolve dynamical features more accurately than the computer studies. Experiments demonstrating that the flows exhibit poor time reversal in regimes of chaotic particle motion are also performed.

Platelets and the infrared absorption of type Ia diamonds

Abstract: The B9 localized mode, infrared absorption peak (which is due to {100} platelets) and its relations with other features in the infrared absorption spectrum of type la diamonds have been investigated. It is recognized that it is possible, on the basis of their one-phonon infrared absorption spectra, to separate type la diamonds into two categories. In ‘regular’ diamonds, there is a strict proportionality between the strength of the B9 peak and the strength of the lattice absorption due to B nitrogen aggregates; this is interpreted to be indicative of an epoch of smooth, undisturbed conversion of A nitrogen aggregates (or pairs of N atoms) to B aggregates, with the concurrent nucleation and growth of platelets. For ‘irregular’ diamonds this proportionality does not hold, and such specimens are recognized as crystals in which a catastrophic degradation of some or all of the platelets has occurred. It is argued that the proportionality observed for the regular specimens implies that nitrogen cannot be the dominant atomic species in the platelets, and a mechanism is described for platelet production by the formation and aggregation of carbon interstitials. Platelets are here seen as a necessary, but incidental, product of the nitrogen aggregation sequence. The B9 peak is ascribed to carbon-carbon bond vibrations of atoms within the platelet. The presence of N atoms as a minority species in the platelets is discussed, and it is suggested that other localized mode absorptions seen in type I a spectra may be attributed to vibrations of N-C and N-N bonds of atoms in the platelets. The B9 peak strength is also found to be linearly proportional to the recently-discovered D component of the lattice absorption spectrum, for both regular and irregular specimens. The D absorption is thus attributed to platelet-stimulated lattice vibrational modes. Among regular specimens, the strength of the N3 optical absorption is proportional to the B9, and hence also B, absorption strengths. The N3 centres are considered to result from minor side reactions occurring during the epoch of conversion of A centres to B centres.

The Recently Recognized Failure of Predictability in Newtonian Dynamics

Abstract: Modern theories of dynamical systems have very clearly demonstrated the unexpected fact that systems governed by the equations of Newtonian dynamics do not necessarily exhibit the ‘predictability’ property. Indeed, very recent researches have shown that in wide classes of very simple systems satisfying those equations predictability is impossible beyond a certain definite time horizon.

Rate-dependent, finite elasto-plastic deformation of polycrystals

Abstract: Based on Hill’s method, a self-consistent averaging scheme is proposed for estimating the overall, finite deformation response of polycrystalline aggregates consisting of single crystals which undergo plastic flow by rate-dependent crystallographic slip, accompanied by elastic lattice distortion. First, constitutive relations for such single crystals are developed assuming that the slip-rate and the associated resolved shear stress are governed by: (1) a power-law relation, and (2) a viscoplastic relation. Then, Hill’s idea that the constraint imposed on a single crystal by the remaining aggregates may be represented by embedding the single crystal in a homogeneous, infinitely extended matrix having the instantaneous overall moduli, is used to formulate a completely self-consistent averaging procedure, valid for rate-dependent materials at finite strains and rotations. This method includes both the Hill and the Krӧner‒Budiansky‒Wu (K. B. W.) methods as limiting cases; when rate-effects are negligible, it reduces to Hill’s self-consistent method as formulated by Iwakuma and Nemat-Nasser for finite deformations, while it reduces to a generalized finite deformation version of the K. B. W. method for strongly rate-dependent materials. Illustrative numerical examples are presented for a plane uniaxial deformation, using a two-dimensional poly crystalline model. These examples clearly show that the rate-dependent crystallographic slip on the level of single crystals produces a more stable overall behaviour of poly crystals. This supports similar results arrived at by other investigators for single crystals and for polycrystals, by using the Taylor averaging scheme. It is shown that, while Taylor’s averaging scheme gives accurate estimates of the incremental quantities at large strains, the total overall quantities differ considerably from the ones obtained by the self-consistent method.

Some applications of Hausdorff dimension inequalities for ordinary differential equations

Abstract: Upper bounds are obtained for the Hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable. From these are derived sufficient conditions for dissipative analytic n-dimensional ω-periodic differential equations to have only a finite number of ω-periodic solutions. For autonomous equations the same conditions ensure that each bounded semi-orbit converges to a critical point. These results yield some information about the Lorenz equation and the forced Duffing equation.

Stability of Water Waves

Abstract: We apply some general results for Hamiltonian systems, depending on the notion of signature of eigenvalues, to determine the circumstances under which collisions of imaginary eigenvalue for the linearized problem about a travelling water wave of permanent form are avoided or lead to loss of stability, up to non-degeneracy assumptions A new superharmonic instability is predicted and verified

Free-Fermion, Checkerboard and Z-Invariant Lattice Models in Statistical Mechanics

Abstract: It is shown that the two-dimensional free fermion model is equivalent to a checkerboard Ising model, which is a special case of the general ‘ Z -invariant’ Ising model Expressions are given for the partition function and local correlations in terms of those of the regular square lattice Ising model Corresponding results are given for the self-dual Potts model, and the application of the methods to the three-dimensional Zamolodchikov model is discussed The paper ends with a discussion of the critical and disorder surfaces of the checkerboard Potts model

An analysis of field-induced hot-electron emission from metal-insulator microstructures on broad-area high-voltage electrodes

Abstract: High-resolution electron energy distribution measurements of individual microscopic sites have been used to develop a detailed theoretical model of the field-induced hot-electron emission mechanism responsible for pre-breakdown currents at vacuum-gap fields of 10-30 MV m$^{-1}$. The model, which is based on the formation of conducting channels in a metal-insulator-vacuum (m.i.v.) micro-emission regime, has provided experimentally verified quantitative relations for the current-voltage characteristic, spectral shape, and the field-dependence of the spectral full width at half maximum (f.w.h.m.) and shift.

An Asymptotic Investigation of the Stationary Modes of Instability of the Boundary Layer on a Rotating Disc

Abstract: An investigation of high Reynolds number stationary instabilities in the boundary layer on a rotating disc is given. It is shown that in addition to the inviscid mode at high Reynolds numbers, there is a stationary short wavelength mode. This mode has its structure fixed by a balance between viscous and Coriolis forces and cannot be described by an inviscid theory. The asymptotic structure of the wavenumber and orientation of this mode is obtained. A similar analysis is given for the inviscid mode, the expansion procedure used is capable of taking nonparallel effects into account in a self consistent manner. The results are compared to numerical calculations and experimental observations.

Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations

Abstract: We demonstrate local Lipschitz regularity for minimisers of certain functionals which are appropriately convex and quadratic near infinity. The proof employs a blow-up argument entailing a linearisation of the Euler—Lagrange equations “at infinity”. As a application, we prove that minimisers for the relaxed optimal design problem derived by Kohn and Strang [3] are locally Lipschitz.

On the topology of a polymer ring

Abstract: First we survey the literature on knots and links in theoretical physics. Next, we report a numerical study in which equilibrium configurations of ring polymers in an infinite space, or confined to the interior of a sphere, are generated. By using a new algorithm, the a priori probability for the occurrence of a knot is determined numerically. The results are compatible with scaling laws of striking simplicity. We also study the mutual entanglement of links, comparing the Gauss invariant with the Alexander polynomial.

Exact estimates of the conductivity of a binary mixture of isotropic materials

Abstract: This paper describes the set GmU of effective conductivity tensors of mixtures generated by two isotropic materials taken in prescribed proportions m1 and m2 We describe microstructures which realise any point of GmU for n-dimensional space.

A sliding law for glaciers of constant viscosity in the presence of subglacial cavitation

Abstract: A method of solution for the problem of slow flow of a Newtonian viscous glacier slipping over a rough bed is constructed, for the case where cavities form when the lubricating water film pressure reaches that of the local subglacial drainage system. The treatment of Nye (Proc. R. Soc. Lond. A 311, 445-477 (1969)) is reformulated as a Hilbert problem, and the solution presented for the particular case of a periodic bedrock with one cavity per period. For such bedrocks, it is found that the basal stress has a maximum for a finite basal velocity, and the basal stress decreases towards zero as the velocity tends to infinity, in line with the suggestion of Lliboutry (J. Glaciol. 23, 67-95 (1979)). For more complicated bedrocks, with many different obstacle sizes, direct solution appears impractical and some kind of further approximation seems advisable.

Flow past a suddenly heated vertical plate in a porous medium

Abstract: An analysis is presented for the steady free convection flow about a semi-infinite vertical flat plate that is embedded in a saturated porous medium at high Rayleigh numbers. Similarity solutions are obtained for a class of problems where the wall temperature varies as a power of the distance from the leading edge of the plate. The existence and uniqueness of the solutions are considered. The approach to this steady-state solution is also considered by investigating the temporal development of the flow when the temperature of the plate is impulsively increased from that of the surroundings. A numerical solution is presented that matches the small and large time solutions. For some temperature distributions on the plate it is found that the velocity achieves its maximum value within the boundary layer. For these the disturbance from the leading edge of the plate travels fastest within the boundary layer. An asymptotic solution valid at large times is presented and the approach of the numerical solution to this asymptotic solution is illustrated. For the situation in which the plate is impulsively heated to a constant temperature an analysis is presented for the early stages of the departure from the one-dimensional solution.

Void growth in shear

Abstract: The growth of an isolated void is analysed for a void contained in a block of material undergoing simple shearing combined with superimposed hydrostatic tension. The evolution of the size, shape and orientation of two- and three-dimensional voids in an incompressible, linearly viscous solid is first discussed. The main problem addressed is the behaviour of a two-dimensional cylindrical void in an incompressible, nonlinearly viscous solid for which the strain rate varies as the stress to a power. The growth rate of the void and its shape evolution are strong functions of the degree of material nonlinearity. Relatively simple approximate formulas are obtained for the dilatation rate of a circular void as well as for the void potential. The constitutive relation of a block of material containing a dilute distribution of circular cylindrical voids is obtained directly using the isolated void potential. The paper concludes with a summary of available results for the dilatation rates of voids and cracks under combinations of shear and hydrostatic tension.

Spatial modelling of total storm rainfall

Abstract: The spatial structure of the depth of rainfall from a stationary storm event is investigated by using point process techniques. Cells are assumed to be stationary and to be distributed in space either independently according to a Poisson process, or with clustering according to a Neyman-Scott scheme. Total storm rainfall at the centre of each cell is a random variable and rainfall is distributed around the centre in a way specified by a spread function that may incorporate random parameters. The mean, variance and covariance structure of the precipitation depth at a point are obtained for different spread functions. For exponentially distributed centre depth and a spread function having quadratically exponential decay, the total storm depth at any point in the field is shown to have a gamma distribution. The probability of zero rainfall at a point is investigated, as is the stochastic variability of model parameters from storm to storm. Data from the Upper Rio Guaire basin in Venezuela are used in illustration.

The Rigid Body Dynamics of Unidirectional Spin

Abstract: A toy consists of a boat-shaped body showing great preference for spin in one direction only. Its sophisticated rigid body dynamics is examined in some detail, and fully accounts for this curious behaviour.

The closest packing of equal circles on a sphere

Abstract: The closest packing of x circles on the surface of a sphere is examined with the use of techniques that have been developed to determine the stereochemical arrangement of atoms packed around a central atom. The technique is based on the concept that the centres of the circles repel one another, and minimizing the total ‘repulsion energy’ leads to an approximate structure from which exact solutions can be determined. A number of improved packings have been discovered for values of x in the range 20–40. Many different types of structure are found that are of lower symmetries than those previously described. The packing density p , defined as the fraction of the spherical surface that is enclosed by the circles, is found to increase as the number of circles increases, in contrast to the conclusion from previous studies. However, this increase in p is very slight and the values remain substantially below that for an infinite number of circles, or a close-packed plane.

The Use of Thermal Desorption and Electron Energy Loss Spectroscopy for the Determination of the Structures of Unsaturated Hydrocarbons Chemisorbed on Metal Single-Crystal Surfaces. I. Alk-1-Enes on Pt(111)

Abstract: Thermal desorption combined with mass spectral analysis is used to determine the elemental composition, as a function of temperature, of the adsorbed monolayers of the alk-1-enes, propene, but-1-ene, pent-1-ene and isobutene chemisorbed on the (111) face of a Pt single crystal. Vibrational electron energy loss spectroscopy (EELS) is used to assign structures to the surface species adsorbed at different temperatures. At the lowest temperatures (below 200 K) it is concluded that in each case these alk-1-enes are bonded to the metal surface in the form of di-$\sigma$-bonded non-dissociatively adsorbed species. The simplicity of the EEL spectrum from the species derived from isobutene provides additional support for an earlier assignment to the di-$\sigma$-adsorbed species for ethylene on Pt(111). At ca. 300 K the EEL spectra and thermal desorption data are consistent with the presence of alkylidyne, M$\_3$C(CH$\_2$)$\_n$CH$\_3$ (M = metal; n = 1, 2 or 3) or M$\_3$CCH(CH$\_3$)$\_2$ structures for the chemisorbed species respectively, (M = metal atom). For temperatures above 300 K the thermal desorption results show substantial further loss of hydrogen, a process which commences at lower temperatures the longer the hydrocarbon chain. Near 450 K the thermal desorption data indicate average C:H compositions of approximately C$\_3$H$\_2$ for the species derived from propene, C$\_4$H$\_2$ from but-1-ene, and C$\_4$H$_4$ from isobutene. The EEL spectra are used to indicate the remaining hydrocarbon functional groups present at this and at higher temperatures. Above 450 K closely similar spectra were obtained from all the straight-chain butenes including the cis- and trans-but-2-enes and buta-1,3-diene whose spectra are discussed in more detail in the following paper. The EEL spectra indicate the occurrence of C-C bond breaking in general above ca. 500 K, the onset temperatures being somewhat dependent on the adsorbed hydrocarbon.

Oscillatory chemical reactions in closed vessels

Abstract: A skeleton kinetic scheme representing the simplest model for oscillatory chemical reactions in a closed vessel can be built around an autocatalytic feedback step $\text{precursor decay} P \rightarrow A k\_0 p,\\ \text{uncatalysed step} A \rightarrow B k\_3 a,\\ \text{autocatalysis} A + 2B \rightarrow 3B k\_1ab^2,\\ \text{catalyst decay} B \rightarrow C k\_2 b.$ The first intermediate A is formed via the slow decay of a reactant or precursor species P, initially in large excess. A is converted to B via two routes: a slow pseudo-first order process and a step in which B acts as its own catalyst. The autocatalyst B is then capable of a simple first order decay to a stable product C. The concentrations of the various species at first change steadily, with that of P decreasing while A, B and C increase. This period is followed by the onset and growth of oscillations in the concentrations of the intermediates A and B. The behaviour at long times depends upon the uncatalysed conversion of A to B. Provided k$_3$ is not taken as zero, the oscillations finally diminish in amplitude and die out leaving a steady decay of P, A and B until everything has been converted to C. The simplicity of the model allows the first self-consistent test of the 'pool chemical approximation', an approach commonly used in the analysis of mechanisms in closed systems in which the precursor concentration is assumed to be constant and set equal to its initial value. The results presented here reveal the range of applicability of the approximation and show clearly how and why it can break down to give unphysical predictions.

Growth, motion and 1-parameter families of symmetry sets

Abstract: Associated to every plane curve there is the locus of centres of circles bitangent to that curve, the so-called symmetry set of the curve. We can view this set as the spine of our curve, which can be recovered by taking the envelope of circles of varying radii along this spine. Varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object. Fixing the symmetry set and varying the radius function can be considered to model growth crudely. In this paper we shall describe the generic changes in the curves which take place in the process of growth and motion, and outline the corresponding results for spheres centred on a space curve. We also use the idea of a stratified Morse function to describe the generic changes which occur in one parameter families of (full) bifurcation sets in the plane. Applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set (and evolute) which occur in a generic isotopy of a plane curve.

Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data

Abstract: We prove the global existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L2 and the initial density is in L2 ∩ BV. Solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi-discrete difference scheme. This allows for a rather simple analysis in which pointwise control is achieved through piecewise H1 and total variation estimates.

A new type of singularity created by colliding gravitational waves

Abstract: An exact solution is obtained for colliding plane impulsive gravitational waves accompanied by shock waves, which, in contrast to other known solutions, results in the development of a null surface which acts like an event horizon. The analytic extension of the solution across the null surface reveals the existence of time-like curvature singularities along two hyperbolic arcs in the extended domain, reminiscent of the ring singularity of the Kerr metric. Besides, the space-time, in the region of the interaction of the colliding waves, is of Petrov-type D and locally isometric to the Kerr space-time in a region interior to the ergosphere. Various other aspects of the solution are also discussed.

A remark on comparison results via symmetrization

Abstract: In this paper, we study the converse of comparison results for solutions to linear second-order elliptic equations. Namely, in the inequalities proved by G. Talenti and others, we study the case of equality and prove that “equalities are achieved only in the spherical situation”. We also present some applications of these results to semilinear elliptic equations.

On the direct initiation of a plane detonation wave

Abstract: It is assumed that energy is transferred at a rapid rate through a plane wall into a spatially uniform and initially stagnant combustible gas mixture. This action generates a shock wave, just as it does in an inert mixture, and also switches on a significant rate of chemical reaction. The Navier-Stokes equations for plane unsteady flow are integrated numerically in order to reveal the subsequent history of events. Four principal time domains are identified, namely ‘early’, ‘transitional’, ‘formation’ and ‘ZND’. The first contains a conduction-dominated explosion and formation of a shock wave; in the second interval the shock wave is responsible for the acceleration of chemical activity, which becomes intense during the ‘formation’ period. Finally a wave whose structure is in essence that of a ZND detonation wave emerges.

Failure characteristics of short anchor bolts embedded in a brittle material

Abstract: The results of an analytical and experimental investigation into the failure mechanisms caused by short anchor bolts, embedded in a brittle matrix and loaded monotonically in tension, are presented and discussed. Several engineering applications are included which describe the limita­tions of currently available design procedures. A mathematical model, based on linear elastic fracture mechanics, is used to predict crack paths, determine the stability of the crack growth, and obtain the tensile capacity of the anchor bolts. The theoretical results are consistent with those obtained from experiments conducted with the use of mortar as a matrix material.

Bubble growth in porous media and Hele–Shaw cells

Abstract: We consider the characterisation of a class of free boundary problems arising in the flow of a viscous liquid in a porous medium (or, in two dimensions, a Hele–Shaw cell). Injected air forms a bubble which grows as time increases; it is shown that three kinds of behaviour can occur. Firstly, the solution may cease to exist in finite time; secondly, the solution may exist for all time and the free boundary may have one or more limit points as t tends to infinity; and thirdly, the bubble may exist for all time and fill the whole space as t tends to infinity. Two-dimensional explicit examples arc given of all three types of behaviour, and it is proved that the only solutions of the third kind are those in which the bubble is always elliptical; the proof uses the theory of null quadrature domains. It is shown that solutions for ellipsoidal bubbles exist in three dimensions and it is conjectured that the only three-dimensional null quadrature domains with finite complement are those whose complement is an ellipsoid.