Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1982"
TL;DR: In this paper, the mechanical properties of two-dimensional cellular materials, or honeycombs, are analyzed and compared with experiments, in terms of bending, elastic buckling and plastic collapse of the beams that make up the cell walls.
Abstract: The mechanical properties (linear and nonlinear elastic and plastic) of two-dimensional cellular materials, or honeycombs, are analysed and compared with experiments. The properties are well described in terms of the bending, elastic buckling and plastic collapse of the beams that make up the cell walls.
1,567 citations
TL;DR: In this paper, a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of Langevin equations for the Fourier modes of the surface are derived.
Abstract: The problem of the surface fluctuations in a settled granular material is posed. A simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of Langevin equations for the Fourier modes of the surface are derived. These equations imply that the Fourier amplitudes behave like the velocities of a set of independent Brownian particles. We show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random, the divergence being removed by a more accurate description of the settling material, for example by having the granules fall through a sieve.
1,068 citations
TL;DR: In this paper, a new approach to defining energy-momentum and angular momentum in general relativity is presented which avoids some of the difficulties of previous definitions and which can be applied quasi-locally.
Abstract: A new approach to defining energy-momentum and angular momentum in general relativity is presented which avoids some of the difficulties of previous definitions and which can be applied quasi-locally. It depends on the construction of a twistor space T α ( S ) associated with any spacelike topological 2-sphere S . Though several problems of interpretation remain to be solved, the new definition works well at I + , reproducing the Bondi-mass-momentum as four of the ten precisely determined quantities at each cut of I + . The remaining six quantities provide a definition of angular momentum which appears to be new.
375 citations
TL;DR: In this paper, the sliding friction between two highly oriented monolayers has been studied by using molecularly smooth mica substrates in the form of contacting orthogonal cylinders.
Abstract: The sliding friction between two highly oriented monolayers has been studied by using molecularly smooth mica substrates in the form of contacting orthogonal cylinders. The monolayers in the form of various normal alipathic carboxylic acids and their soaps were deposited with the aid of the Langmuir-Blodgett technique by transfer from aqueous substrates. The normal alkyl group has been varied in length from 14 to 22 methylene repeat units. Data are reported also on the influence of partial saponification of the carboxylic acid and fluorination of the alkyl chain. Most of the investigation has been confined to two contacting single monolayers although a limited amount of data is presented for multilayers sliding over one another. The character of the sliding motion depends not only on the machine but also on the monolayers, particularly their chemistry. Most of the monolayers studied provide a continuous rate of energy dissipation. However, a small number, such as certain soaps, show discontinuous or stick-slip motion. The experimental arrangement allows simultaneous measurement of the sliding frictional force, contact area and film thickness to be made during sliding. In some experiments this friction is the monotonic sliding friction but in others it is the mean maximum value during the stick phase. The film thickness measurement is accurate to 0.2 mm which allows a precise assessment of the shear plane during sliding. In all cases the monolayers and multilayers were found to be extremely durable and shear invariably occurred at the original interface between the monolayers. The sliding friction data are presented as the dynamic specific friction force or interface shear strength, and a number of contact variables have been examined. These include the applied normal load per unit contact area or mean contact pressure, the temperature and the sliding velocity. The interface shear strength is found, to a good approximation, to increase linearly with mean contact pressure but to decrease linearly with temperature in the ranges studied. The influence of sliding velocity is more complex. In the case where intermittent motion is detected the mean maximum values decrease linearly with the logarithm of the velocity.
340 citations
TL;DR: In this paper, the nature of the microscopic stress tensor in an inhomogeneous fluid is discussed, with emphasis on the statistical mechanics of drops, and a particular radial distortion of a drop leads to statistical mechanical expressions for the pressure difference across the surface of the drop.
Abstract: The nature of the microscopic stress tensor in an inhomogeneous fluid is discussed, with emphasis on the statistical mechanics of drops. Changes in free energy for isothermal deformations of a fluid are expressible as volume integrals of the stress tensor ‘times’ a strain tensor. A particular radial distortion of a drop leads to statistical mechanical expressions for the pressure difference across the surface of the drop. We find that the stress tensor is not uniquely defined by the microscopic laws embodying the conservation of momentum and angular momentum and that the ambiguity remains in the ensemble average, or pressure tensor, in regions of inhomogeneity. This leads to difficulties in defining statistical mechanical expressions for the surface tension of a drop.
334 citations
TL;DR: The rotameric state model developed by Flory was used to describe the conformations adopted by the alkyl chain in this paper, and it was also used to predict the conformational distributions for both the isotropic and nematic phases of these mesogens; the calculated distributions are contrasted with those assumed by other models.
Abstract: The constituent molecules of thermotropic liquid crystals invariably contain an aromatic core to which is attached one and often two n -alkyl chains. Nuclear magnetic resonance investigations have revealed that the chain does not exist entirely in an all- trans configuration and that the orientational order of the methylene groups varies markedly along the chain. In this paper we develop a theory to explain these variations for a uniaxial phase at a given temperature and pressure. Our theory contains two distinct parts: one is a model for the conformational distribution of the alkyl chains while the other is for the potential of mean torque experienced by a conformer and which is responsible for its alignment with respect to the director. We use the rotameric state model developed by Flory to describe the conformations adopted by the alkyl chain. The potential of mean torque presents a more difficult problem and we assume that it may be represented by a sum of contributions from each rigid group in the molecule. The order parameters for the methylene and methyl groups in the alkyl chains of the 4- n -pentyl and 4- n -octyl-49-cyanobiphenyls predicted by the theory are found to be in good accord with those obtained experimentally. We have also used the theory to predict the conformational distributions for both the isotropic and nematic phases of these mesogens; the calculated distributions are contrasted with those assumed by other models. In addition we have obtained the ordering matrices for the various conformers adopted by the two 4- n -alkyl-49-cyanobiphenyls. The results of these calculations do not support the notion that the orientational order of such molecules can be described by a single, cylindrically symmetric ordering matrix.
255 citations
TL;DR: In this paper, it was shown that for general classes of unbounded domains, there exists no solution distinct from 0 offor any smooth f satisfying f(0) = 0, and this result is obtained by the use of new identities that solutions of semilinear elliptic equations satisfy.
Abstract: In this paper, we prove various existence and non-existence results for semilinear elliptic problems in unbounded domains. In particular we prove for general classes of unbounded domains that there exists no solution distinct from 0 offor any smooth f satisfying f(0) = 0. This result is obtained by the use of new identities that solutions of semilinear elliptic equations satisfy.
218 citations
TL;DR: In this article, the authors reported work in which synthetic diamonds containing a high concentration of single nitrogen atoms have been heated in a temperature range of 1500 to 2500°C under various pressures.
Abstract: Most natural diamonds contain nitrogen as the main impurity. In some rare diamonds (termed type lb) the nitrogen is mainly present as single substitutional atoms. However, the large majority of diamonds (termed type Ia) contain the nitrogen atoms in various forms of aggregate. The different types of aggregate are the A centre (two nitrogen atoms), the N3 centre (three nitrogen atoms) and the B centre (a larger number of nitrogen atoms). These defects give characteristic absorption spectra in the infrared except for the N3 centre which gives a characteristic absorption in the visible region. In this type of diamond there are usually platelets present in the cube planes and these defects can be examined by transmission electron microscopy and infrared absorption techniques. The paper reports work in which synthetic diamonds containing a high concentration of single nitrogen atoms have been heated in a temperature range of 1500 to 2500°C under various pressures. These heat treatments have resulted in the formation of all the types of aggregate that are found in natural type la diamonds. Also some natural diamonds have been heated up to 2700°C under pressure and the ratio of the concentration of the A centres to that of the B centres has been changed. Information has been obtained on the kinetics of the aggregation process. This information has been used to give an approximate estimate of the length of time that natural diamonds spent in the Upper Mantle prior to being ejected to the surface of the Earth. It is suggested that the type I a diamonds spent between about 200 and 2000 Ma in the Upper Mantle at temperatures of between 1000 and 1400°C. Type l b diamonds either spent a comparable time in the Upper Mantle at about 800°C or a considerably shorter period if they encountered temperatures in the same range as the type la diamonds.
215 citations
TL;DR: In this article, the acoustic radiation efficiency σ of a rectangular panel that undergoes time-harmonic transverse oscillations with sinusoidal mode shapes is estimated when the acoustic wavenumber and the platewavenumber components are large.
Abstract: The acoustic radiation efficiency σ of a rectangular panel that undergoes time-harmonic transverse oscillations with sinusoidal mode shapes is estimated when the acoustic wavenumber and the plate wavenumber components are large. The results confirm that σ has different asymptotic forms in different regions of plate wavenumber space. Results are given for each region, together with transition formulae near the various connecting boundaries. Some earlier results are confirmed, modified and extended.
195 citations
TL;DR: In this paper, the moduli and crushing strength of balsa wood were determined by the properties of the cell-wall material, and, in part, by the dimensions and shape of the cells themselves.
Abstract: The structure of balsa wood has been characterized by scanning microscopy. The moduli and crushing strengths of the wood, in three orthogonal directions, have been related to the structure and its response to load: during deformation the cell walls bend elastically and collapse plastically or by fracture. It is found that the moduli, crushing strengths and anisotropy of balsa and of other wood are determined, in part, by the properties of the cell-wall material, and, in part, by the dimensions and shape of the cells themselves. This second aspect of the deformation determines how moduli and strength depend on the density of the wood, and is treated here for the first time.
189 citations
TL;DR: In this article, the authors investigated the behavior of a flux of gravitational and/or electromagnetic radiation crossing the event horizon of a Reissner-Nordstrom black-hole.
Abstract: The behaviour, on the Cauchy horizon, of a flux of gravitational and/or electromagnetic radiation crossing the event horizon of a Reissner-Nordstrom black-hole is investigated as a problem in the theory of one-dimensional potential-scattering It is shown that the flux of radiation received by an observer crossing the Cauchy horizon, along a radial time-like geodesic, diverges for all physically reasonable perturbations crossing the event horizon, even including those with compact support
TL;DR: In this article, it was shown that a real-analytic 3-manifold with Riemannian conformal metric is naturally the conformal infinity of a germ-unique realanalytic 4-MANIFold with real-analysis Riemanian metric satisfying the self-dual Einstein equations with cosmological constant.
Abstract: It is demonstrated that a real-analytic 3-manifold with Riemannian conformal metric is naturally the conformal infinity of a germ-unique real-analytic 4-manifold with real-analytic Riemannian metric satisfying the self-dual Einstein equations with cosmological constant — 1 Moreover, this result holds if ‘Riemannian’ is replaced in the first case by ‘Lorentzian’ (i e signature + - - ) and in the second case by ‘pseudo-Riemannian with signature + + - - ’, or if ‘real-analytic’ is replaced by ‘complex-analytic’ and 9Riemannian’ is replaced by ‘holomorphic’ This provides a cosmological-constant analogue of Newman’s ℋ-space construction (Newman 1976, 1977)
TL;DR: In this article, photographic and pressure-measuring techniques have been used to study the detailed behavior of thin layers of explosive when impacted, and evidence for ignition by a variety of mechanisms including adiabatic shear of the explosive, heating of trapped gas spaces, viscous flow, friction, fracture or shear added particles, and triboluminescent discharge.
Abstract: In earlier papers we have described the development of photographic and pressure-measuring techniques to study the detailed behaviour of thin layers of explosive when impacted. Ignition, when it occurs, invariably does so at local sites where 'hot spots' have been generated. The present paper gives the evidence for ignition by a variety of mechanisms including adiabatic shear of the explosive, adiabatic heating of trapped gas spaces, viscous flow, friction, fracture or shear of added particles, and triboluminescent discharge. Some of these mechanisms have been suggested earlier though others have resulted from our recent work. The value of the new photographic evidence is that it allows us to assess the viability of each mechanism for different explosives and impact conditions.
TL;DR: In this article, the effective moduli of a two-component composite material were derived up to fourth order in the sense that the effective bulk modulus involved three microstructural parameters whereas eight parameters were needed in the bounds on the effective shear modulus.
Abstract: Based on the perturbation solution, we derive new bounds on the effective moduli of a two-component composite material which are exact up to fourth order in $\delta\mu$ = $\mu\_1$ - $\mu\_2$ and $\delta\kappa$ = $\kappa\_1$ - $\kappa\_2$, where $\mu\_i$ and $\kappa\_i$, i = 1, 2, are the shear and bulk modulus, respectively, of the phases. The bounds on the effective bulk modulus involve three microstructural parameters whereas eight parameters are needed in the bounds on the effective shear modulus. For engineering calculations, we recommend the third-order bounds on the effective shear modulus which require only two geometrical parameters. We show in detail how Hashin-Shtrikman's bounds can be extended and how Walpole's bounds can be improved using two inequalities on the two geometrical parameters that appear in the third-order bounds on the effective shear modulus. The third- and fourth-order bounds on the effective moduli are shown to be more restrictive than, or at worst, coincident with, existing bounds due to Hashin and Shtrikman, McCoy, Beran and Molyneux and Walpole.
TL;DR: In this article, it was shown that the three-vortex problem in two-dimensional hydrodynamics is integrable, whereas the motion of four identical vortices is not.
Abstract: It is shown that the three-vortex problem in two-dimensional hydrodynamics is integrable, whereas the motion of four identical vortices is not. A sequence of canonical transformations is obtained that reduces the N -degree-of-freedom Hamiltonian, which describes the interaction of N identical vortices, to one with N — 2 degrees of freedom. For N = 3 a reduction to a single degree of freedom is obtained and this problem can be solved in terms of elliptic functions. For N = 4 the reduction procedure leads to an effective Hamiltonian with two degrees of freedom of the form found in problems with coupled nonlinear oscillators. Resonant interaction terms in this Hamiltonian suggest non-integrable behaviour and this is verified by numerical experiments. Explicit construction of a solution that corresponds to a heteroclinic orbit in phase space is possible. The relevance of the results obtained to fundamental problems in hydrodynamics, such as the question of integrability of Euler’s equation in two dimensions, is discussed. The paper also contains a general exposition of the Hamiltonian and Poisson-bracket formalism for point vortices.
TL;DR: In this paper, the authors studied the wave reflexion at the contact surface separating pure gas from the dusty-gas layer in terms of the properties of the gas and the dusty gas, and the time-dependent transition properties through the shock waves, contact surfaces and rarefaction waves are found by solving the equations of motion numerically by a modified random-choice method with an operator splitting technique.
Abstract: The flow resulting from the passage of a shock wave through a dusty-gas layer is studied theoretically. On the basis of an idealized equilibrium-gas approximation, the criteria for the wave reflexion at the contact surface separating the pure gas from the dusty-gas layer are obtained in terms of the properties of the gas and the dusty gas. For the cases treated here, a shock wave is reflected at the first contact surface and a shock wave stronger than the incident one is transmitted into the dusty-air layer. Subsequently, a rarefaction wave is reflected at the second contact surface and the shock wave transmitted into the free air is weakened by this nonlinear interaction. The induced rarefaction wave reflects later at the first contact surface as a compression wave, which runs through the layer to overtake the transmitted shock wave in air. The final emergent shock wave from the dusty air has almost the same strength as the original shock wave entering the layer. The time-dependent transition properties through the shock waves, contact surfaces and rarefaction waves are found by solving the equations of motion numerically by a modified random-choice method with an operator-splitting technique.
TL;DR: The problem of numerical inversion of the Laplace transform when the inverse function is of bounded, strictly positive support has been studied in this article, where a priori knowledge of the support is shown to lead to increased resolution in the inversion, and the number of exponentials that can be recovered in given levels of noise is calculated.
Abstract: The problem of numerical inversion of the Laplace transform is considered when the inverse function is of bounded, strictly positive support. The recent eigenvalue analysis of McWhirter and Pike for infinite support has been generalized by numerical calculations of singular values. A priori knowledge of the support is shown to lead to increased resolution in the inversion, and the number of exponentials that can be recovered in given levels of noise is calculated.
TL;DR: In this article, the relation between the number of linearly independent solutions of the electro- and magnetostatic boundary value problems and topological characteristics of the underlying domain in 3-dimensional euclidean space is investigated in the framework of Hilbert space theory.
Abstract: The classically well-known relation between the number of linearly independent solutions of the electro- and magnetostatic boundary value problems (harmonic Dirichlet and Neumann vector fields) and topological characteristics (genus and number of boundaries) of the underlying domain in 3-dimensional euclidean space is investigated in the framework of Hilbert space theory. It can be shown that this connection is still valid for a large class of domains with not necessarily smooth boundaries (segment property). As an application the inhomogeneous boundary value problems of electro- and magnetostatics are discussed.
TL;DR: In this paper, a study of the mechanical deformation properties of β$-HMX, an important secondary explosive, has been made, where it is shown that under compressive loading twinning takes place on the (101)-plane and usually precedes fracture.
Abstract: A study has been made of the mechanical deformation properties of $\beta$-HMX, an important secondary explosive. It is shown that under compressive loading twinning takes place on the (101)-plane. At low loads, this twinning is elastic and usually precedes fracture. Cleavage in $\beta$-HMX takes place on the {011}-planes. The fracture surface energy of 0.06 J m$^{-2}$ has been determined by a micro-indentation technique. This compares with a value of 0.045 J m$^{-2}$ obtained for the thermodynamic surface energy from contact-angle measurements. The values suggest that there is relatively little energy loss by plastic deformation associated with crack propagation in HMX compared with, for example, the secondary explosives PETN and RDX. Despite this brittleness the twin deformation allows $\beta$-HMX to undergo large changes of shape: the significance of this in plastic-bonded explosives is commented on.
TL;DR: In this article, the authors provided a reliable theoretical foundation for the open-resonator method of measuring permittivity and loss-tangent of dielectric materials.
Abstract: The objective of this paper is to provide a reliable theoretical foundation for the open-resonator method of measuring permittivity and loss-tangent of dielectric materials. From an exact solution of Maxwell's equations obtained by means of the complex-source-point method, simplified formulae for the six Cartesian components of the electromagnetic field are obtained. These are then used in a variational formula to obtain an accurate formula for the resonant frequency of an open resonator with spherical mirrors (of equal radius of curvature) having a parallel-plane slab of dielectric material located centrally between the two mirrors. It is assumed that the mirrors are sufficiently large for diffraction effects to be negligible. In the following companion paper II, experimental results are presented which verify the accuracy, within the usual limits of practical measurement, of the simple correction formulae deduced here.
TL;DR: In this paper, an inverse transversal of a regular semigroup S is defined as an inverse subsemigroup that contains a single inverse of every element of S. A certain multiplicative property (which in the case of a band is equivalent to normality) is imposed on an inverse Transversal and a complete description of the structure of S is obtained.
Abstract: By an inverse transversal of a regular semigroup S we mean an inverse subsemigroup that contains a single inverse of every element of S. A certain multiplicative property (which in the case of a band is equivalent to normality) is imposed on an inverse transversal and a complete description of the structure of S is obtained.
TL;DR: In this paper, the authors derived canonical line elements admitting a simple KillingYano tensor f* ab and showed several close analogies between the vector field la = f * arp r along a geodesic with tangent field p r and the angular momentum l = rxpin the case of a spacelike, timelike and null tensor.
Abstract: In this paper we derive canonical line elements admitting a simple KillingYano tensor f* ab. There exist three distinct cases according to the character o f f* ab (spacelike, timelike, null). We reveal several close analogies between the vector field la = f * arp r along a geodesic with tangent field p r and the angular momentum l = rxpin the case of a spacelike Killing-Yano tensor. In particular, we show that, in consequence of the Killing-Yano tensor equations, there exists an analogue of the three-dimensional position vector field in certain hypersurfaces and th at la can be written in the form (r x p )a. Furthermore, an analogue of the ‘ equatorial plane ’ of the classical Kepler problem can be constructed intrinsically.
TL;DR: In this article, the authors investigated the nonlinear stability of the Hagen-Poiseuille flow through a pipe of circular cross section when subjected to three-dimensional disturbances of typical relative size δ large enough to yield only a vanishingly small phase shift across the critical layer.
Abstract: An investigation is described for the nonlinear stability, at large Reynolds numbers R , of the Hagen-Poiseuille flow through a pipe of circular cross section when subjected to three-dimensional disturbances of typical relative size δ large enough to yield only a vanishingly small phase shift across the critical layer. A crucial size is δ = O ( R -⅓ ) since then this small phase shift is in tune with the small phase shift produced by the viscous wall layers. The critical layer, which is fully nonlinear and three-dimensional, and the wall layers, where the disturbance is greater than the basic flow and flow reversal occurs, are discussed in detail. Neutral solutions are then found to exist in the range c 01 c 0 N = 1, where c 0 is the non-dimensional wavespeed, c 01 = 0.284 and N is the azimuthal wavenumber; there is also evidence to suggest that no similar neutral solutions exist outside that range. The amplitude-dependence of the neutral modes follows and it is such that the cut-off value c 0 = c 01 + is approached as the amplitude shrinks, whereas centre modes with c 0 → 1 - are produced as the amplitude becomes relatively large.
TL;DR: In this paper, the authors examined the dispersion of a passive contaminant in steady and oscillatory turbulent shear flow in a two-dimensional channel using a random walk technique.
Abstract: The paper examines in detail the dispersion of a passive contaminant in steady and oscillatory turbulent shear flow in a two-dimensional channel. The aim of this examination is to understand dispersion in estuaries. A new method of analysing and predicting concentration distributions has been developed from work of Sullivan ( J. Fluid Mech . 49, 551–576 (1971)). A random walk technique is used, the contaminant being represented by a large number of marked particles whose paths are tracked as they move through the fluid. The technique seeks to model the physics of dispersion more realistically than the standard diffusion equation, and results from the simulation, with input based on data taken in the Mersey, show it to be a useful and versatile method of studying dispersion in oscillatory flows.
TL;DR: In this paper, the rotational spectrum of H$\_2$O $\cdots$ HF in vibrationally excited states associated with the hydrogen bond out-of-plane and in-plane bending modes was assigned and used to decide whether the equilibrium configuration at the oxygen atom is planar or pyramidal.
Abstract: We have assigned the rotational spectrum of H$\_2$O $\cdots$ HF in vibrationally excited states associated with the hydrogen bond out-of-plane and in-plane bending modes $
u\_{\beta(o)}$ and $
u\_{\beta(i)}$ respectively. This has allowed us to decide whether the equilibrium configuration at the oxygen atom is planar (C$\_{2
u}$ symmetry) or pyramidal (C$\_S$ symmetry) by combining three types of information: vibrational spacing (from relative intensity measurements of vibrational satellites), the dependence of rotational constants on vibrational excitation (from frequency measurements), and the dependence of the electric dipole moment on the quantum numbers $v\_{\beta(o)}$ and $v\_{\beta(i)}$ (from the Stark effect). The $v\_{\beta(o)} = 1 \leftarrow 0$ separation in D$\_2$ O $\cdots$ DF has also been obtained and used in reaching the conclusion that the equilibrium configuration at oxygen is pyramidal. Under the assumption that the mode $
u\_{\beta(o)}$ is governed by a one-dimensional potential function, a quantitative analysis of all available data leads to the expression $V(\phi)/\mathrm{cm}^{-1} = 328\phi^4 - 406\phi^2,$ where $\phi$ is the angle between the O $\cdots$ HF line and the bisector of the angle HOH. This function implies a barrier height of 126 cm$^{-1}$ or 1.5 kJ mol$^{-1}$ to inversion of the configuration at oxygen. The equilibrium angle $\phi$ is now available for the series H$\_2$O$\cdots$ HF ($\phi$ = 46$^\circ$), (CH$\_2$)$\_3$O$\cdots$ HF ($\phi$ = 54$^\circ$) and (CH$\_2$)$_2$O$\cdots$ HF ($\phi$ = 72$^\circ$). The variation of $\phi$ along the series is interpreted in terms of the angle between the conventionally viewed non-bonding pairs on the oxygen atom in relation to the angle between the chemical bonds at that atom.
TL;DR: In this article, a survey of a variety of solids that show a metal-insulator transition is given, and the concept of a minimum metallic conductivity is discussed, with special reference to materials of mixed valence.
Abstract: A survey is given of a variety of solids that show a metal-insulator transition. In crystals most transitions are expected to be of first order as the composition or temperature is changed; in disordered systems this is not necessarily the case. The transition in an impurity band with change of donor concentration is described, and also with change of stress, magnetic field or gate voltage. The concept of a minimum metallic conductivity is discussed, with special reference to materials of mixed valence.
TL;DR: In this article, the authors discuss the physics of the flow in a cylindrical vortex tube as various independent parameters are varied, and present evidence to suggest a connection between vortex breakdown and the criticality and stability of the vortex core upon which it occurs.
Abstract: Referring to flow-visualization and laser-Doppler anemometer measurements of swirl and axial velocity profiles, we discuss the physics of the flow in a cylindrical vortex tube as various independent parameters are varied. Three main classes of flow occur, depending upon the location of a vortex jump within the vortex tube. We present evidence to suggest a connection between vortex breakdown and the criticality and stability of the vortex core upon which it occurs and attempt to reconcile the various explanations that have been proposed for the breakdown phenomenon. Similarities and differences between the present experiments and those of previous investigators are also pointed out. Finally, as an Appendix, we present the results of a hydraulic analogue of our vortex experiment.
TL;DR: In this article, the existence of analytic solutions of the asymptotic characteristic initial value problem for Einstein's field equations for analytic data on past null infinity and on an incoming null hypersurface was proved.
Abstract: This paper proves the existence of analytic solutions of the asymptotic characteristic initial value problem for Einstein's field equations for analytic data on past null infinity and on an incoming null hypersurface.
TL;DR: In this article, the results of two approaches to the problem of nonlinear refraction at frequencies just below the band edge in semiconductors are compared, and the physical origin of the nonlinear dispersion is taken to be direct saturation of independent, T$_2$broadened band states in the first model discussed.
Abstract: Detailed comparisons are made of the results of two approaches to the problem of nonlinear refraction at frequencies just below the band edge in semiconductors. The physical origin of the nonlinear dispersion is taken to be direct saturation of independent, T$_2$-broadened band states in the first model discussed. In the second model the refraction associated with a dynamic Burstein-Moss shift of the absorption edge is considered. Both models predict an extremely large nonlinearity, $\chi^(3) \approx$ 1 e.s.u., of importance in the field of dispersive optical bistability and consistent with experimental observations of bistability, two-beam differential gain and four-wave mixing.
TL;DR: The Cracks Microgauge as mentioned in this paper is an instrument for the detection and measurement of surface cracks in metals by using a.c. electric currents in the metal surface, which can be used to measure surface voltages in the neighborhood of surface-breaking cracks.
Abstract: This paper describes some recent studies associated with the development of an instrument, the Crack Microgauge, at University College London, for the detection and measurement of surface cracks in metals by using a.c. electric currents in the metal surface. The design of the instrument has enhanced the accuracy of measurement of surface voltages in the neighbourhood of surface-breaking cracks to the point where it is now advantageous to consider the distribution of surface voltages by mathematical analysis. This enables better use to be made of the instrument since measurements can then be more accurately interpreted in terms of crack size and geometry. In contrast, earlier applications of the method relied on the calibration of the instruments against test blocks. The first generation of users of the Crack Microgauge interpreted their readings according to a simple one-dimensional normal crack model. This, however, is an oversimplification for cracks of finite aspect ratio and it leads to underestimates of centre-line crack depth typically of the order of 30-40 when the aspect ratio is about 3. The first major contribution of the theory was therefore to provide a model for surface current flow around cracks of finite aspect ratio. For this purpose a useful unfolding theorem was deduced in cases in which the current skin depth is small compared with crack dimensions. This allows the surface field to be unfolded into a plane distribution of potential satisfying Laplace's equation, and it enables us to obtain solutions for many different crack shapes by using standard mathematical methods. An account of these developments is given in $\xi$ 2 of this paper. The unfolding of the surface field gives the global distribution of surface current around a crack, that is, on the length scale of the crack dimensions. On the length scale of the skin depth, however, the solutions need modification in the neighbourhood of the surface and interior edges of the crack. In developing the unfolding theorem it was argued that these modifications, which we describe as corner solutions, could be neglected. The purpose of this paper is to construct these corner solutions to confirm that this is so. The mathematical theory described in $\xi$ 2 was developed for the most important case in which cracks develop in planes perpendicular to the surface of the metal. It is clear from the construction of that solution, however, that the unfolding may be applied with the same result to plane cracks inclined to the surface. Thus the global solution is not able to distinguish the inclination of the crack plane to the surface. We shall find, however, that the corner solutions are sensitive to this inclination, and they are given in $\S$ 3 for arbitrary angles of inclination. These studies suggest, therefore, that information on crack inclination could be deduced from experimental investigation of these corner solutions. Since the solutions are found to possess a phase of $\frac{1}{4} \pi$, it might be possible to extract this information by measuring perturbation signals in quadrature with the main field or perhaps by increasing the skin depth so that the scale of the solutions is enhanced. For a given specimen this could be done by reducing the frequency of the a.c. field, but, if the frequency is reduced significantly, the field distributes itself through the interior of the specimen on the scale of the crack dimensions and may no longer be described as a surface distribution. Further work to describe such field distributions when the skin depth is of the order of the crack depths is in progress.