Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1955"

On kinematic waves II. A theory of traffic flow on long crowded roads

Abstract: This paper uses the method of kinematic waves, developed in part I, but may be read independently. A functional relationship between flow and concentration for traffic on crowded arterial roads has been postulated for some time, and has experimental backing (§2). From this a theory of the propagation of changes in traffic distribution along these roads may be deduced (§§2, 3). The theory is applied (§4) to the problem of estimating how a ‘hump’, or region of increased concentration, will move along a crowded main road. It is suggested that it will move slightly slower than the mean vehicle speed, and that vehicles passing through it will have to reduce speed rather suddenly (at a ‘shock wave’) on entering it, but can increase speed again only very gradually as they leave it. The hump gradually spreads out along the road, and the time scale of this process is estimated. The behaviour of such a hump on entering a bottleneck, which is too narrow to admit the increased flow, is studied (§5), and methods are obtained for estimating the extent and duration of the resulting hold-up. The theory is applicable principally to traffic behaviour over a long stretch of road, but the paper concludes (§6) with a discussion of its relevance to problems of flow near junctions, including a discussion of the starting flow at a controlled junction. In the introductory sections 1 and 2, we have included some elementary material on the quantitative study of traffic flow for the benefit of scientific readers unfamiliar with the subject.

The Influence of Solid Boundaries upon Aerodynamic Sound

Abstract: An extension is made to Lighthill's general theory of aerodynamic sound, so as to incorporate the influence of solid boundaries upon the sound field. This influence is twofold, namely (i) reflexion and diffraction of the sound waves at the solid boundaries, and (ii) a resultant dipole field at the solid boundaries which are the limits of Lighthill's quadrupole distribution. It is shown that these effects are exactly equivalent to a distribution of dipoles, each representing the force with which unit area of solid boundary acts upon the fluid. A dimensional analysis shows that the intensity of the sound generated by the dipoles should at large distances x be of the general form I$\propto $ $\rho \_{0}$ U$\_{0}^{6}$a$\_{0}^{-3}$ L$^{2}$x$^{-2}$, where U$\_{0}$ is a typical velocity of the flow, L is a typical length of the body, a$\_{0}$ is the velocity of sound in fluid at rest and $\rho \_{0}$ is the density of the fluid at rest. Accordingly, these dipoles should be more efficient generators of sound than the quadrupoles of Lighthill's theory if the Mach number is small enough. It is shown that the fundamental frequency of the dipole sound is one half of the frequency of the quadrupole sound.

On kinematic waves I. Flood movement in long rivers

Abstract: In this paper and in part II, we give the theory of a distinctive type of wave motion, which arises in any one-dimensional flow problem when there is an approximate functional relation at each point between the flow q (quantity passing a given point in unit time) and concentration k (quantity per unit distance). The wave property then follows directly from the equation of continuity satisfied by q and k. In view of this, these waves are described as 'kinematic', as distinct from the classical wave motions, which depend also on Newton's second law of motion and are therefore called 'dynamic'. Kinematic waves travel with the velocity $\partial $q/$\partial $k, and the flow q remains constant on each kinematic wave. Since the velocity of propagation of each wave depends upon the value of q carried by it, successive waves may coalesce to form 'kinematic shock waves'. From the point of view of kinematic wave theory, there is a discontinuous increase in q at a shock, but in reality a shock wave is a relatively narrow region in which (owing to the rapid increase of q) terms neglected by the flow-concentration relation become important. The general properties of kinematic waves and shock waves are discussed in detail in section 1. One example included in section 1 is the interpretation of the group-velocity phenomenon in a dispersive medium as a particular case of the kinematic wave phenomenon. The remainder of part I is devoted to a detailed treatment of flood movement in long rivers, a problem in which kinematic waves play the leading role although dynamic waves (in this case, the long gravity waves) also appear. First (section 2), we consider the variety of factors which can influence the approximate flow-concentration relation, and survey the various formulae which have been used in attempts to describe it. Then follows a more mathematical section (section 3) in which the role of the dynamic waves is clarified. From the full equations of motion for an idealized problem it is shown that at the 'Froude numbers' appropriate to flood waves, the dynamic waves are rapidly attenuated and the main disturbance is carried downstream by the kinematic waves; some account is then given of the behaviour of the flow at higher Froude numbers. Also in section 3, the full equations of motion are used to investigate the structure of the kinematic shock; for this problem, the shock is the 'monoclinal flood wave' which is well known in the literature of this subject. The final sections (section section 4 and 5) contain the application of the theory of kinematic waves to the determination of flood movement. In section 4 it is shown how the waves (including shock waves) travelling downstream from an observation point may be deduced from a knowledge of the variation with time of the flow at the observation point; this section then concludes with a brief account of the effect on the waves of tributaries and run-off. In section 5, the modifications (similar to diffusion effects) which arise due to the slight dependence of the flow-concentration curve on the rate of change of flow or concentration, are described and methods for their inclusion in the theory are given.

The creep of polycrystalline ice

Abstract: Polycrystalline blocks of ice have been tested under compressive stresses in the range from 1 to 10 bars at temperatures from —13°C to the melting-point. Under these conditions ice creeps in a manner similar to that shown by metals at high temperatures; there is a transient creep component and also a continuing or quasi-viscous component. The relation between the minimum observed flow rate έ, the applied stress σ and the absolute temperature T is έ = B exp ( - Q / RT ) σ n , where R is the gas constant, and B , n and Q are constants; the value of n is about 3.2, that of Q is 32 kcal/mole, and that of B is 7 x 10 24 if the stress is measured in bars and the strain rate in years -1 . At the higher stresses a third, accelerating stage of creep was observed; on the basis of the appearance and behaviour of sections cut from the specimens, this acceleration was attributed to recrystallization. The effect of changing the load during a test has also been studied; for large reductions creep recovery was observed. The results of these tests are discussed in connexion with previous work on metals and ice, and also with measurements of glacier flow.

Continuous Distributions of Dislocations: A New Application of the Methods of Non-Riemannian Geometry

Abstract: When describing a crystal containing an arbitrary distribution of dislocation lines it is often convenient to treat the distribution as continuous, and to specify the state of dislocation as a function of position. Formally, however, there is then no ‘good crystal’ anywhere, and difficulties arise in defining Burgers circuits and the dislocation tensor. The dislocated state may be defined precisely by relating the local basis at each point to that of a reference lattice. The dislocation density may then be defined; it is important to distinguish this from the local dislocation density. The geometry of the continuously dislocated crystal is most conveniently analyzed by treating the manifold of lattice points in the final state as a non-Riemannian one with a single asymmetric connexion. The coefficients of connexion may be expressed in terms of the generating deformations relating the dislocated crystal to the reference lattice. The tensor defining the local dislocation density is then the torsion tensor associated with the asymmetric connexion. Some properties of the connexion are briefly discussed and it is shown that it possesses that of distant parallelism, in conformity with the requirement that the dislocated lattice be everywhere unique.

Regenerative Stochastic Processes

Abstract: A wide class of stochastic processes, called regenerative, is defined, and it is shown that under general conditions the instantaneous probability distribution of such a process tends with time to a unique limiting distribution, whatever the initial conditions. The general results are then applied to 9S.M.-processes’, a generalization of Markov chains, and it is shown that the limiting distribution of the process may always be obtained by assuming negative exponential distributions for the ‘waits’ in the different ‘states’. Lastly, the behaviour of integrals of regenerative processes is considered and, amongst other results, an ergodic and a multi-dimensional central limit theorem are proved.

The Frequency Response of a Defocused Optical System

Abstract: The response of a defocused aberration-free optical system to line-frequencies in the object is studied analytically. Curves are given showing the response as a function of line-frequency for a range of values of defect of focus. A comparison is made with the results to be expected from geometrical optics. A tolerance for defect of focus is obtained, which accords well with published experimental results. Both circular and rectangular apertures are considered.

The exact calculation of long-range forces between atoms by perturbation theory

Abstract: A general sum rule is described which has many variants; it permits, for example, the exact calculation of the long-range forces between a proton and a hydrogen atom using conventional perturbation theory. The method is exemplified by the calculation of the second-order term and a precise assessment of the significance of the continuum states is made. The basic identity for a given function g is (r$|$g$|$s)/(E$\_{r}$ - E$\_{s}$) $\equiv $ (r$|$f$|$s), where the function f satisfies a certain differential equation (13). The limitation of the method is the difficulty which may arise in solving the equation. The sum rule technique appears to have applications to many fields.

The diffusion of electrolytes in a cation-exchange resin membrane I. Theoretical

Abstract: An equation for the flux of electrolyte through a water-swollen cation-exchange resin membrane separating two solutions of the same electrolyte at different concentrations is derived on the basis of several assumptions regarding the physical nature of a swollen resinous exchanger. The complete flux equation contains three terms, one determined by the concentration difference across the membrane, another determined by the variation of the activity coefficient of the electrolyte with concentration in the membrane and a third concerned with the rate of osmotic or hydrostatic flow through the membrane. If ions in the resin are transported entirely in an internal aqueous phase, the mobilities required for the flux equation can be related to mobilities in aqueous solution and to the volume fraction of resin in the swollen membrane. The treatment is readily extended to anion exchangers.

Configuration interaction in orbital theories

Abstract: A systematic method is developed for estimating or calculating corrections for configuration interaction in atomic, molecular and nuclear wave-function calculations. Solutions of the Hartree-Fock equations for a single Slater determinant or approximate Hartree-Fock solutions obtained by Roothaan’s iterative procedure have special properties which are used to simplify the matrix of the many-particle Hamiltonian. A restricted self-consistent field method is proposed for treating states of low symmetry. This method avoids the off-diagonal Lagrange multipliers encountered in previous methods and is adapted to configuration interaction calculations.

On the partition of energy between elastic waves in a semi-infinite solid

Abstract: Expressions are derived for the power radiated in the compressional, shear and surface waves set up by a circular disk vibrating normally to the free surface of a semi-infinite isotropic solid. The total radiated power is also calculated independently by integrating the displacement velocity over the area of the source. The theory is extended to a general type of multi-element radiator in the form of an array of elements on the circumference of a circle. The calculation of the total power here involves a ‘mutual admittance’ function, a table of which is given for the case when the Poisson’s ratio of the medium is equal to ¼. The theory is applied to a three-element radiator of a type used in a recent geophysical investigation, and it is shown that the efficiency of radiation in the compressional mode can be varied between wide limits by varying the distance between the elements. Finally, an approach is suggested for problems in which the most suitable idealized boundary condition is one of known displacement under the radiator, the stress being zero elsewhere on the free surface. It is shown that the stress under the radiator satisfies an integral equation whose kernel is derived from the mutual admittance function.

Effects of Temperature on the Plastic Properties of Aluminium Crystals

Abstract: By changing the temperature of deformation, during tensile experiments on single crystals of aluminium, it has proved possible to separate reversible changes of flow stress with temperature from irreversible ones. Below 130 degrees K the reversible change of flow stress is large; above room temperature it is hardly more than the change of elastic constants with temperature. These reversible changes are highly reproducible and, over a wide range of cold-worked states, proportional to the flow stress itself. The principal irreversible effect is a sharp yield drop, accompanied by a Luders band, which appears when a crystal is strained plastically at a high temperature (e.g. 300 degrees K) immediately after being heavily cold-worked at a low temperature (e.g. 90 degrees K). Intermediate annealing treatments reduce or eliminate this yield drop. Other experiments have proved that the yield drop is not a strain-ageing phenomenon but an extreme example of work softening, and that work hardening induced by preliminary deformation at the low temperature becomes unstable during plastic deformation at the high temperature and is rapidly removed. An explanation of the yield drop is suggested, based on recent theories of work hardening. It is proposed that, under the combined influence of stress and temperature, sessile dislocations at the heads of piled-up groups of dislocations become unlocked; the piled-up groups then partly collapse and many dislocations are released for slip. The reversible changes of flow stress are interpreted in terms of the intersection of dislocation lines.

The lattice expansion of iron

Abstract: The lattice spacings of high-purity iron have been measured accurately between 20 and 1502°C. The results, together with previous low-temperature data, give data between the absolute zero and the melting-point (1534°C). The thermal expansion of the lattice and of bulk metal are in good agreement over the temperature range for which the latter has been studied accurately. At the A 2 point, there is no discontinuity in the lattice spacing. In contrast to the work of some previous investigators, we have found no observ­able difference between the coefficients of expansion of the lattice immediately above and below the A 2 point, and the results suggest that the forces responsible for ferromagnetism do not affect the interatomic distances by more than 1 part in 10000. At the A 3 point, the α → γ transformation is accompanied by an increase in the interatomic distance which, when expressed as a percentage change, is slightly less than the decrease in interatomic distance during the γ → δ change at the A 4 point. The change in bond length calculated by the Pauling equation for the change (f. c. c.⇌b. c. c.) lies between the observed values for the A 3 and A 4 points, and the accuracy of the equation is confirmed.

The electronic structure of an icosahedron of boron atoms

Abstract: The electronic structure of a regular icosahedron of boron atoms is investigated theoretically by the method of molecular orbitals. It is found that thirteen bonding orbitals are available for holding the icosahedron together, besides the twelve outward-pointing equivalent orbitals of the separate atoms. The results are used to interpret the stability of boron carbide, to question a recently proposed crystal structure for elementary boron, and to discuss the chance of existence of a hydride B$\_{12}$H$\_{12}$.

The Effects of Hydraulic Resistance in the Dam-Break Problem

Abstract: When resistance is neglected, the solution of the simple dam-break problem is readily obtained on the basis of shallow-water theory, and the results are well known. However, near the head of the wave, where the water surface meets the ground, resistance effects cannot be neglected; there is, in fact, a type of boundary layer near the wave-front. In this paper, the Pohlhausen method (which is used in conventional boundary-layer problems) is applied to a study of the effect of this ‘boundary layer’. In particular, the retardation of the wave-front behind the position predicted by the simple theory is found.

Surface interaction between elastically loaded bodies under tangential forces

Abstract: This paper describes an experimental investigation of the micro-displacement between two bodies in contact under the action of a tangential force less than that of limiting friction. Experiments have been performed for the unlubricated contact of a hard steel ball with a hard steel flat, under a range of normal loads, using balls of varying diameter. Measure­ments of micro-displacement have been made under the action of both steady and oscillating tangential forces. In the latter case vibrational energy is dissipated at the interface and fretting of the surfaces occurs. The mechanism of these processes has been studied and is discussed. The quantitative results of the experiments provide considerable support for Mindlin’s theoretical elastic analysis. For small tangential forces the displacements are almost entirely elastic and the compliance of the two bodies is given exactly by the elastic theory. As the tangential force approaches the value to cause slipping, the displacement exhibits a non­-elastic component arising from the relief of an infinite shear stress which ideally elastic behaviour predicts at the boundary of the circle of contact. Observation of the surfaces after subjection to a sustained oscillating force showed that intimate metal-to-metal contact occurs only at the crests of the surface asperities as sug­gested by Bowden & Tabor. Fretting takes place over an annular area round the boundary of the contact circle and spreads radially inwards with increasing tangential force. Measure­ments of the energy loss suggest that for small amplitudes of oscillating force the theoretic­ally infinite stress is accommodated by predominantly elastic distortion of the surface asperities. Under larger forces, however, the asperities are deformed plastically through large strains, which leads to eventual fatigue and fretting of the surfaces over the annular region of high stress. The amount of energy so dissipated agrees quantitatively with that calculated by Mindlin, showing that the degree of plastic strain occurring at the interface is governed by the stress distribution in the elastic hinterland.

Dynamical variables in the Bethe-Salpeter formalism

Abstract: It is shown that a knowledge of the behaviour of the propagators around their singularities enables one to determine not only the masses of bound states, but also the matrix element of any dynamical variable between two bound states. One is thus enabled to find such a matrix element, to any order in the coupling constant, by the integration of certain expressions over the corresponding Bethe-Salpeter wave-functions. As a consequence, it is possible to find normalization and orthogonality properties of these wave-functions, which in turn lead to the condition which must be imposed on their singularities a t the origin. More light is thus shed on Goldstein’s difficulty concerning the existence of a continuous infinity of bound states. The formalism is extended to scattering states in which some of the particles may be composite—in particular, an expression for the S -matrix is obtained

The theory of generalized functions

Abstract: This paper gives an introductory account of the construction and properties of generalized functions f(x) of real variables x$\_{1}$,x$\_{2}$,$\cdots $,x$\_{n}$. These are defined so as to ensure that (i) any generalized function f(x) possesses its full complement of generalized partial derivatives D$^{p}$f(x) of all orders; (ii) any convergent sequence of generalized functions {f$\_{n}$(x)} has a generalized limit, f(x), which is also a generalized function; (iii) the derived sequence {D$^{p}$f$_{n}$(x)} converges to D$^{p}$f(x). The construction of these generalized functions ensures that any continuous function possesses derivatives which are generalized functions, so that the delta functions of Dirac are included in the theory. The representation of generalized functions by Fourier series and integrals is considered as an example of the simplicity and generality of the theory.

On the plastic flow of metals under conditions of axial symmetry

Abstract: This paper is concerned with the axially symmetric plastic flow of a rigid-plastic nonhardening material which obeys the Tresca yield criterion of constant maximum shearing stress and the associated flow rule. A general discussion of the basic equations is given. The discussion shows that the hypothesis of Haar and von Karman is likely to be of great importance in the solution of axially symmetric problems. This conclusion is substantiated by the remainder of the work which considers problems in which the hypothesis is satisfied, i.e. problems in which the circumferential stress is equal to one of the principal stresses in the meridional planes. Possible plastic velocity fields in a circular cylindrical bar stressed to yielding in compression or tension are obtained in section 3. Section 4 examines plastic stress fields in the neighbourhood of stress-free conical surfaces. In the final sections of the paper, the plastic stress field and a permissible deformation mode for the problem of the indentation of the plane surface of a semi-infinite body by a circular flat-ended rigid punch are obtained. It is shown that the plastic stress field near the punch can be extended into the rigid region without violating the yield criterion.

The colours and magnetic properties of hydrated iron group salts, and evidence for covalent bonding

Abstract: The typical colours of paramagnetic salts containing [M(H$\_{2}$O)$\_{6}$], M = 3d$^{n}$, complexes, are generally assumed to arise from optical transitions between the orbital energy levels of the 3d ion which are split by the crystalline electric field due to the surrounding water dipoles. The available optical absorption data are analyzed using this ionic model and the associated crystal field theory, and it is shown that while experiment and theory agree fairly well, there are systematic discrepancies. In addition, the orbital level separations given by the optical spectrum are found to be smaller than those predicted from magnetic data. It is shown how these discrepancies can be accounted for by introducing weak covalent bonds into the [M(H$\_{2}$O)$\_{6}$] complex, so that there is charge transfer between the paramagnetic ion and the attached water molecules.

On the Hyperfine Structure of Paramagnetic Resonance: The s-Electron Effect

Abstract: Unpaired s -electrons play an important part in hyperfine spectra, even when the nominal spectroscopic configuration contains no unpaired s -electrons. This situation occurs in paramagnetic resonance and optical spectra. A survey of the experimental evidence for the effect is given in relation to the paramagnetic ions and the neutral atoms of the 3 d transition elements. It appears that the central density of unpaired spin is nearly the same in all the ions of the group for which experimental data are available, while for the neutral atoms it is more variable, but of the same general magnitude. A calculation of the magnitude of the effect is attempted from first principles, starting from the Hartree–Fock self-consistent wave functions as a first approximation, and adding configurations in which 3 s -, 2 s - and 1 s -electrons are promoted. The promotion of a 3 s -electron is described by an integro-differential equation, which has been solved numerically in one particular case. The contribution turns out of the right sign but ten times smaller than the observed value. Promotion of 2 s - and l s -electrons yield similar equations, which, however, have not been solved, owing to the excessive labour involved. There is no reason to believe that they would not give smaller contributions still. The full explanation of the s -electron effect is thus still an open question.

A neutron-diffraction study of the ferroelectric transition of potassium dihydrogen phosphate

Abstract: Neutron-diffraction measurements demonstrate that the hydrogen atoms in the ferroelectric form of potassium dihydrogen phosphate are in ordered positions, in accordance with Slater’s theory and the polar orthorhombic structure. The maintenance of a crystal as a single ferroelectric domain by application of an electric field along the Z -axis is illustrated, and the diffraction spectra show that reversal of the field direction produces a shift of the hydrogen atoms along the O—H—O bonds from one set of ordered positions to the other. A detailed study is made of the thermal motion of the hydrogen atoms at 77, 132 and 293° K, and it is found that the motion within the potential well defined by the KPO framework is practically independent of temperature and is due to zero-point energy. Parameters are determined for all the atoms at each temperature and discussed with reference to other published data.

The Stability of the Flow of an Electrically Conducting Fluid between Parallel Planes under a Transverse Magnetic Field

Abstract: The stability under small disturbances is investigated of the two-dimensional laminar motion of an electrically conducting fluid under a transverse magnetic field. It is found that the dominating factor is the change in shape of the undisturbed velocity profile caused by the magnetic field, which depends only on the Hartmann number M . Curves of wave number against Reynolds number for neutral stability are calculated for a range of values of M ; for large values of M the calculations are similar to those which determine the stability of ordinary boundary-layer flow. The critical Reynolds number is found to rise very rapidly with increasing M , so that a transverse magnetic field has a powerful stabilizing influence on this type of flow.

A Macroscopic Theory of Interference and Diffraction of Light from Finite Sources. II. Fields with a Spectral Range of Arbitrary Width

Abstract: The results of part I of this investigation are generalized to stationary fields with a spectral range of arbitrary width. For this purpose it is found necessary to introduce in place of the mutual intensity function of Zernike a more general correlation function $\hat{\Gamma}$(x$\_{1}$, x$\_{2}$, $\tau $) = $\langle \hat{V}$(x$\_{1}$, t + $\tau $) $\hat{V}^{\ast}$(x$\_{2}$, t)$\rangle $, which expresses the correlation between disturbances at any two given points P$\_{1}$(x$\_{1}$), P$\_{2}$(x$\_{2}$) in the field, the disturbance at P$\_{1}$ being considered at a time $\tau $ later than at P$\_{2}$. It is shown that $\hat{\Gamma}$ is an observable quantity. Expressions for $\hat{\Gamma}$ in terms of functions which specify the source and the transmission properties of the medium are derived. Further, it is shown that in vacuo the correlation function obeys rigorously the two wave equations $ abla \_{s}^{2}\hat{\Gamma}$ = $\frac{1}{c^{2}}\frac{\partial ^{2}\hat{\Gamma}}{\partial \tau ^{2}}$ (s = 1, 2), where $ abla \_{s}^{2}$ is the Laplacian operator with respect to the co-ordinates (x$\_{s}$, y$\_{s}$, z$\_{s}$) of P$\_{s}$(x$\_{s}$). Using this result, a formula is obtained which expresses rigorously the correlation between disturbances at P$\_{1}$ and P$\_{2}$ in terms of the values of the correlation and of its derivatives at all pairs of points on an arbitrary closed surface which surrounds P$\_{1}$ and P$\_{2}$. A special case of this formula (P$\_{2}$ = P$_{1}$, $\tau $ = 0) represents a rigorous formulation of the generalized Huygens principle, involving observable quantities only.

Thermal Conduction in Artificial Sapphire Crystals at Low Temperatures. I. Nearly Perfect Crystals

Abstract: In order to obtain a detailed verification of the theory of thermal conduction in dielectric crystals, measurements have been made on a number of artificial sapphire crystals between 2° and 100° K. In the region of the maximum there are variations in conductivity between crystals from different sources. The highest conductivities measured are about 140 W/cm deg., which suggests that estimates of several hundred watts for the maxima of ideal sapphire crystals are not unreasonable. At sufficiently low temperatures the conductivity of a very perfect, long crystal with rough surfaces is observed, in agreement with Casimir’s theory of boundary scattering, to be proportional to T 3 and to the radius; the phonon mean free path is then nearly equal to the crystal diameter. Imperfect crystals show some anomalous effects. The extension of Casimir’s theory to apply to short specimens has been verified. Perfect crystals with smooth surfaces exhibit some specular reflexion of phonons; a statistical description of the surface is proposed which leads to the observed variation of this effect with temperature and is compatible with the results of interferometric examination of the surface.

The Mechanism of Rolling Friction. I. The Plastic Range

Abstract: This paper describes a study of the friction and surface damage produced when a hard steel sphere rolls between flat parallel surfaces of a softer metal. When rolling first starts, even at small loads there is marked plastic flow of the surfaces and a grooved track is formed. The resistance to rolling is primarily due to the plastic displacement of metal ahead of the ball and may be calculated in terms of the area of the grooved track and the yield pressure of the metal. In the first traversal the ball is supported on the front half of the circle of contact. On rolling back along this track there is no such support for the ball unless it sinks further into the surfaces. This leads to an increase in the width of the groove and further displacement of metal. With successive traversals there is a slow increase in groove width and a gradual decrease in rolling resistance. For metals such as tin which do not work-harden appreciably these changes may be explained quantitatively on the assumption that the area supporting the load remains substantially the same throughout the series of traversals. The rolling resistance throughout this stage is due primarily to plastic displacement and depends little on lubrication. A study of the deformation produced during rolling shows that the centre of the grooved track is very heavily sheared in the direction of rolling, and with repeated traversals this may lead to rapid fatigue of the metal. Here again lubricants have little effect on the rate of fatiguing. As rolling proceeds there is an increase in track width, a slight change in track curvature and, in general, appreciable work-hardening of material within and around the track. As a result of these factors an equilibrium stage is gradually reached (if fatigue failure does not occur first) at which the whole of the load is borne by the ellipse of contact formed by elastic deformation within the width of the existing track. No further appreciable increase in track width occurs, and the rolling friction remains substantially constant for repeated traversals of the track. The rolling friction at this stage is again scarcely affected by lubricants.

Waves due to a floating sphere making periodic heaving oscillations

Abstract: The paper gives a discussion of the fluid motion due to a sphere, floating half-immersed in water, which is made to describe small heaving oscillations. The velocity potential is obtained as a series for which the unknown coefficients are given by an infinite set of equations. These are solved approximately so as to obtain curves showing the variation with frequency of the virtual inertia coefficient and of the equivalent damping parameter.

The axisymmetric boundary layer on a long thin cylinder

Abstract: The laminar boundary layer in axial flow about a long thin cylinder is investigated by two methods. One (section 2) is a Pohlhausen method, based on a velocity profile chosen to represent conditions near the surface as accurately as possible. The other (section 3) is an asymptotic series solution, valid far enough downstream from the nose for the boundary-layer thickness to have become large compared with the cylinder radius. Another series solution (due to Seban, Bond and Kelly) is known, valid near enough to the nose for the boundary layer to be thin compared with the cylinder radius. The Pohlhausen solution shows good agreement with both series, near and far from the nose, and enables an interpolation to be made (section 4) between them in the extensive range of distances from the nose for which neither is applicable. The final recommended curves, for the variation along the cylinder of skin friction, boundary-layer displacement area and momentum defect area, are displayed in graphical and tabular form (figure 1 and table 1) and are expected to be correct to within about 2%. The velocity near the wall is closely proportional to the logarithm of the distance from the axis; this is the profile used in the Pohlhausen method. The analogy with the distribution of mean velocity in turbulent flow over a flat plate is discussed at the end of section 2.

The Instability of a Layer of Fluid Heated below and Subject to Coriolis Forces. II

Abstract: This paper is devoted to examining the stability of a horizontal layer of fluid heated below, subject to an effective gravity ( g ) acting (approximately) in the direction of the vertical and the Coriolis force resulting from a rotation of angular velocity Ω about a direction making an angle ϑ with the vertical. It is shown that the effect of the Coriolis force is to inhibit the onset of convection, the extent of the inhibition depending on the value of the non-dimensional parameter T = 4 d 4 Ω 2 cos 2 ϑ/ v 2 , where d denotes the depth of the layer and v is the kinematic viscosity. Tables of the critical Rayleigh numbers ( R c ) for the onset of convection are provided for the three cases ( a ) both bounding surfaces free, ( b ) both bounding surfaces rigid and ( c ) one bounding surface free and the other rigid. In all three cases R c →constant x T 2/3 as T →∞ ; the corresponding dependence of the critical temperature gradient (— β c ) for the onset of convection, on v and d , is gαβ c = constant x ĸ (Ω 4 cos 4 ϑ/ d 4 v ) 1/2 ( ĸ is the coefficient of thermometric conductivity and α is the coefficient of volume expansion). The question whether thermal instability can set in as oscillations of increasing amplitude (i.e. as 9overstability’) is examined for case ( a ), and it is shown that if ĸ/v ĸ/v >1.478, over-stability is the first type of instability to arise for all T greater than a certain determinate value. It further appears that these latter possibilities should be considered in meteorological and astrophysical applications of the theory.

The effect of interfacial stabilizing films on the elastic and viscous properties of emulsions

Abstract: The investigation in a previous paper (Oldroyd 1953) of the elastic and viscous properties of a dilute emulsion of one incompressible viscous liquid in another, when subjected to small variable rates of strain, is extended to include the possibility of an interfacial film being present everywhere between the two components, which resists deformation as a result of internal friction or of elasticity. When the film is purely viscous its presence has no effect on the type of elastico-viscous behaviour of the emulsion, although the values of the viscosity, relaxation time and retardation time of the system are altered. When the film is ideally elastic, the viscosity in slow steady flow is the same as that of a suspension of solid particles, and two relaxation times and two retardation times are required to describe the elastico-viscous behaviour of the emulsion completely. When the interfacial film shows more complicated elastic and viscous properties combined, these can be described by means of two operators (expressible in terms of d/d t ) taking the place of elastic moduli. A viscosity operator describing the macroscopic elastic and viscous properties of the emulsion is calculated in terms of the elasticity operators of the film in the general case.