Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1974"

Dislocations in wave trains

Abstract: When an ultrasonic pulse, containing, say, ten quasi-sinusoidal oscillations, is reflected in air from a rough surface, it is observed experimentally that the scattered wave train contains dislocations, which are closely analogous to those found in imperfect crystals. We show theoretically that such dislocations are to be expected whenever limited trains of waves, ultimately derived from the same oscillator, travel in different directions and interfere - for example in a scattering problem. Dispersion is not involved. Equations are given showing the detailed structure of edge, screw and mixed edge-screw dislocations, and also of parallel sets of such dislocations. Edge dislocations can glide relative to the wave train at any velocity; they can also climb, and screw dislocations can glide. Wavefront dislocations may be curved, and they may intersect; they may collide and rebound; they may annihilate each other or be created as loops or pairs. With dislocations in wave trains, unlike crystal dislocations, there is no breakdown of linearity near the centre. Mathematically they are lines along which the phase is indeterminate; this implies that the wave amplitude is zero.

On three-dimensional packets of surface waves

Abstract: In this note we use the method of multiple scales to derive the two coupled nonlinear partial differential equations which describe the evolution of a three-dimensional wave-packet of wavenumber k on water of finite depth. The equations are used to study the stability of the uniform Stokes wavetrain to small disturbances whose length scale is large compared with 2π/ k . The stability criterion obtained is identical with that derived by Hayes under the more restrictive requirement that the disturbances are oblique plane waves in which the amplitude variation is much smaller than the phase variation.

The linear theory of free vibrations of a suspended cable

Abstract: A linear theory is developed for the free vibrations of a uniform suspended cable in which the ratio of sag to span is about 1:8, or less. Both in-plane and out-of-plane motion is considered. It is shown that the analysis of the symmetric in-plane modes is heavily dependent on a parameter which allows for the effects of cable geometry and elasticity. The results of simple experiments are reported which establish the validity of the theory.

Perturbations of space--times in general relativity

Abstract: A definition of perturbations of space-times in general relativity is proposed. The definition leads in a natural way to a concept of gauge invariance, and to an extension of a lemma of Sachs (i964). Coupled equations governing linearized perturbations of certain tetrad components of scalar, electromagnetic, and gravitational fields are derived by the use of Geroch, Held & Penrose's (I 973) version of the tetrad formalism of Newman & Penrose (i 962). It is shown that these perturbations are gauge invariant if and only if the unperturbed space-time is vacuum of algebraic type {22} or, equivalently, if and only if the perturbation equations decouple. Finally the maximal subclass of type {22} space-times for which the decoupled perturbation equations can be solved by separation of variables is found. This class comprises all the nonaccelerating type {22} space-times, including that of Kerr, thus elucidating earlier results of Bardeen & Press

Cosmological Models and the Large Numbers Hypothesis

Abstract: The Large Numbers hypothesis asserts that all the large dimensionless numbers occurring in Nature are connected with the present epoch, expressed in atomic units, and thus vary with time. It requires that the gravitational constant G shall vary, and also that there shall be continuous creation of matter. The consistent following out of the hypothesis leads to the possibility of only two cosmological models. One of them, which occurs if one assumes that the continuous creation is a multiplication of existing matter, is Einstein’s cylindrical closed Universe. The other, which occurs if one assumes the continuous creation takes place uniformly through the whole of space, involves an approximately flat Minkowski space with a point of origin where the Big Bang occurred.

Creep Rupture of Structures

Abstract: A study is made of the failure times of structural components which operate at temperatures sufficiently high to cause material deterioration due to creep rupture. Expressions are derived which give lower bounds on failure times and which take into consideration the different stress criteria known to affect rupture mechanisms. The formulae are used to predict failure times of a variety of components, and it is found convenient, from a practical point of view, to express the times in terms of an equivalent representative rupture stress. By using this stress, failure times are obtained directly from uniaxial stress rupture data. It is found in the examples studied that the values for the representative rupture stress are almost independent of the constants used to define the deformation and rupture processes. Experimental evidence supports the prediction of the theory; for example, copper bars in torsion show better rupture characteristics than bars of aluminium alloy. The position is reversed in notched tensile specimens, with the aluminium specimens showing better characteristics than those of copper. It can be deduced that it is the form of rupture mechanism which affects behaviour rather than ductility as might be expected, since the creep ductility of the aluminium alloy is much less than that for copper.

On the Mass, Momentum, Energy and Circulation of a Solitary Wave. II

Abstract: By accurate calculation it is found that the speed $F$ of a solitary wave, as well as its mass, momentum and energy, attains a maximum value corresponding to a wave of less than the maximum amplitude. Hence for a given wave speed $F$ there can exist, when $F$ is near its maximum, two quite distinct solitary waves. The calculation is made possible, first, by the proof in an earlier paper ($I$) of some exact relations between the momentum and potential energy, which enable the coefficients in certain series to be checked and extended to a high order; secondly, by the introduction of a new parameter $\omega $ (related to the particle velocity at the wave crest) whose range is exactly known; and thirdly by the discovery that the series for the mass $M$ and potential energy $V$ in powers of $\omega $ can be accurately summed by Pade approximants. From these, the values of $F$ and of the wave height $\epsilon $ are determined accurately through the exact relations $3V=(F^{2}-1)M$ and $2\epsilon =(\omega +F^{2}-1$. The maximum wave height, as determined in this way, is $\epsilon \_{\max}$ = 0.827, in good agreement with the values found by Yamada (1957) and Lenau (1966), using completely different methods. The speed of the limiting wave is $F$ = 1.286. The maximum wave speed, however, is $F\_{\max}$ = 1.294, which corresponds to $\epsilon =0.790$. The relation between $\epsilon $ and $F$ is compared to the laboratory observations made by Daily & Stephan (1952), with reasonable agreement. An important application of our results is to the understanding of how waves break in shallow water. The discovery that the highest solitary wave is not the most energetic helps to explain the qualitative difference between plunging and spilling breakers, and to account for the marked intermittency which is characteristic of spilling breakers.

On the Theory of Water Waves.

Abstract: The aim of this paper is to formulate a two-dimensional theory for the propagation of fairly long water waves. The approach differs from the usual in that the theory is set up via two-dimensional postulates. Subsequently, it is shown how a simple three-dimensional approximation enables us to relate the two-dimensional theory to the three-dimensional theory. The resulting equations are used to discuss the unidirectional propagation of waves. It is shown how the results obtained from the theory proposed here are related to the results of Korteweg & de Vries (1895) and to those of Benjamin, Bona & Mahony (1972).

Theory of the stabilization of colloids by adsorbed polymer

Abstract: The configurational free energy of random flight polymer chains adsorbed by one end onto a plane surface as a function of the distance from a parallel plane surface is expressed to a good approximation in simple analytic form. The result is used to discuss the stabilization of a colloid suspension by adsorbed polymer. According to this theory two types of aggregation of colloid particles may occur. If $Ll$ < $AS$/2$\pi $^{3}$NkT$, where $l$ is the link length and $L$ the contour length of a polymer chain, $A$ is the Hamaker constant, $N/S$ is the number of adsorbed polymer chains per unit area and $kT$ is the Boltzman constant multiplied by temperature, the particles adhere closely, but if $AS$/2$\pi $^{3}$kT$ < $lL$ < $AS$/$8\pi kT$ lg 2$N$ a looser association is formed. It is expected that the presence of excluded volume effects would greatly increase the stability against the looser association.

A New Interior Schwarzschild Solution

Abstract: A new interior Schwarzschild solution is presented. It is static, spherically symmetric, regular everywhere inside a sphere of coordinate radius a , and across the surface of this sphere it is joined smoothly to the exterior Schwarzschild solution. There are no radial stresses inside the sphere. The radius a is subject to the inequality a > 2 m , where m is the gravitational mass of the sphere. Under certain conditions the new solution may be interpreted as the field inside an Einstein cluster.

A constructive model potential method for atomic interactions

Abstract: A model potential method is presented that can be applied to many electron single centre and two centre systems. The development leads to a Hamiltonian with terms arising from core polarization that depend parametrically upon the positions of the valence electrons. Some of the terms have been introduced empirically in previous studies. Their significance is clarified by an analysis of a similar model in classical electrostatics. The explicit forms of the expectation values of operators at large separations of two atoms given by the model potential method are shown to be equivalent to the exact forms when the assumption is made that the energy level differences of one atom are negligible compared to those of the other.

On the Theory of Rods. II. Developments by Direct Approach

Abstract: This paper, which may be regarded as a companion to part I under the same title, is concerned with some aspects of both the nonlinear and the linear theories of elastic rods by a direct approach based on the theory of a Cosserat curve with two directors. Special attention is given to the development of the linear isothermal theory of straight isotropic rods of variable cross-section possessing two axes of symmetry. The resulting equations, which are applicable to rods of non-uniform section, separate into those appropriate for extensional, torsional and flexural modes of deformation. Application of these results to torsion and flexure of non-uniform rods are considered, and the problem of identification of constitutive coefficients for rods of uniform cross-section is dealt with at some length.

The Mutual Diffusion Coefficient of Ethanol-Water Mixtures: Determination by a Rapid, New Method

Abstract: Mutual diffusion coefficients for liquid mixtures of ethanol and water have been measured over the entire range of composition and for temperatures from 25 to 65 degrees C at a pressure of 1 bar (10$^{5}$ Pa). At the lowest temperature, the results establish the validity of a new experimental method based upon Taylor's analysis of solute dispersion in laminar flow. The method offers advantages of simplicity and speed over other techniques, and allows direct measurement of diffusion coefficients at well-defined mixture compositions. The experimental data have an estimated uncertainty of $\pm $2.5%. The results have been utilized to evaluate friction coefficients arising in the statistical mechanical theory of transport in liquid mixtures.

Surface Finish and Damage in Sliding

Abstract: The ability of rubbing surfaces to support load depends on the surface finish. Many types of surface used in engineering have a random structure; their profile may be represented by the waveform of a random signal defined by two parameters $\sigma $ and $\beta $$^{\ast}$ describing the height distribution and the autocorrelation function respectively. Surfaces covering a range of values of both $\sigma $ and $\beta $$^{\ast}$ have been prepared and for each surface the ability to withstand damage in conditions of boundary lubrication has been determined. It is shown that the values of $\sigma $ and $\beta $$^{\ast}$ taken together provide good criteria for assessing this property. For the rougher surfaces a sufficient criterion is the ratio $\sigma $/$\beta $$^{\ast}$. Very smooth surfaces, however, fail readily, and when $\beta $$^{\ast}$ is very small all surfaces fail regardless of the value of $\sigma $.

On the theory of rods I. Derivations from the three-dimensional equations

Abstract: Starting with the three-dimensional theory of classical continuum mechanics, some aspects of both the nonlinear and the linear theories of elastic rods are discussed. Detailed attention is given to the derivation of constitutive equations for the linear isothermal theory of elastic rods of an isotropic material and of variable cross-section, deduced by an approximation procedure from the three-dimensional equations. Explicit linear constitutive relations are obtained for straight isotropic circular rods of non-uniform cross-section; the explicit calculation is carried out (in terms of an approximate specific Gibbs free energy function) in four distinct parts, since the complete system of equations involved separate into those appropriate for extensional, torsional and two flexural modes of deformation. A system of displacement differential equations is derived for flexure of a beam of variable circular cross-section; they reduce to those of the Timoshenko beam theory when the radius of the cross-section is constant.

Wind tunnel studies of Martian aeolian processes

Abstract: Preliminary results are reported of an investigation which involves wind tunnel simulations, geologic field studies, theoretical model studies, and analyses of Mariner 9 imagery. Threshold speed experiments were conducted for particles ranging in specific gravity from 1.3 to 11.35 and diameter from 10.2 micron to 1290 micron to verify and better define Bagnold's (1941) expressions for grain movement, particularly for low particle Reynolds numbers and to study the effects of aerodynamic lift and surface roughness. Wind tunnel simulations were conducted to determine the flow field over raised rim craters and associated zones of deposition and erosion. A horseshoe vortex forms around the crater, resulting in two axial velocity maxima in the lee of the crater which cause a zone of preferential erosion in the wake of the crater. Reverse flow direction occurs on the floor of the crater. The result is a distinct pattern of erosion and deposition which is similar to some martian craters and which indicates that some dark zones around Martian craters are erosional and some light zones are depositional.

Trade-winds during the glacial cycles

Abstract: Two deep-sea cores (one north, the other south of the Cape Verde Islands) have been analysed for (i) size distribution of the quartz mineral grains, (ii) mineralogy and colour of carbonate-free clay, (iii) frequency of the Globorotalia menardii foraminifera, (iv) the total carbonate, and (v) the 18 O/ 16 O deviations. The size analyses establish a general law . The plot of the logarithm of the cumulative mass percentage for particles equal to or greater than diameter D against D 2 is linear above a critical particle size, which in this case is about 7 μm . The linear form can be explained by a simple winnowing theory which expresses the slope of the line in terms involving the speed of bulk flow of the trade winds, the distance along a trajectory between the coast and the core site, and a characteristic height describing the vertical distribution of dust in the air. Variations of the slope of the size distributions down the core relate to trade wind conditions during the glacial cycles. Together with the mineralogical and biological data it seems that the trades have been more ‘vigorous’ during the glacial stages.

The ignition of a thin layer of explosive by impact

Abstract: The behaviour of thin layers of solid materials under drop-weight impact is studied with the aid of high-speed photographic and pressure-measuring techniques. Photographic sequences taken with a high-speed framing camera show that explosive materials suffer large-scale deformation before initiation of explosion. The sample may undergo plastic flow in bulk, show evidence of partial fusion, and even (with PETN) melt completely. There is also evidence of Munroe jetting and instability of flow of material at the anvil/layer interfaces. The flow speed of the sample during these processes is considerable and may reach 300 m/s. When ignition of the layer occurs it does so at a small number of local hot spots, following which rapid combustion develops at speeds of 200-700 m/s. Strain-gauge measurements show that the pressures attained during drop-weight impact are typically 0.5-1 GPa (5–10 kbar) and the duration of impact 300–500 μs. In the course of impact of a thin layer of granular material a sharp pressure drop may occur, frequently from several hundred MPa down to zero. With an explosive layer, ignition occurs immediately following the instant of the pressure drop. The sudden fall in pressure is due to mechanical failure of the sample, and correlation of the two experiments shows that this is the cause of the very high flow speeds attained during impact. On the basis of these results a possible mechanism of ignition is suggested.

The Density Dependence of the Refractivity of Gases

Abstract: A sensitive interferometer is described for measuring differentially the effects of molecular interactions on the refractivity of gases. The technique involves changing the density but not the amount of the gas in the optical path. Second refractivity virial coefficients, describing the change in molar refraction due to pairwise interactions, have been determined for neon, argon, nitrogen, carbon dioxide, methane, sulphur hexafluoride and fluoroform.

The role of rapidly compressed gas pockets in the initiation of condensed explosives

Abstract: The mechanism of initiation of explosion by the rapid compression of gas spaces has been studied by means of high-speed photography at framing rates up to 10 7 frames s -1 . Single crystals of silver azide, lead azide and PETN were mounted in a water tank and gas bubbles of chosen composition and size (diameter in the range 50 μm to 1 mm) collapsed on to them by water shocks of strength about 0.1 GPa (1 kbar). The gas bubbles collapsed to minimum volumes in times of the order of 1 μs (depending on the initial bubble size). Initiation of fast reaction occurred in the azides within ca . 5 × 10 -8 s of the bubble reaching minimum volume provided the bubble made thermal contact with the explosive. During the collapse, the bubble involuted to form a jet of velocity of a few 100 ms -1 , and after reaching minimum volume, expanded giving an expansion shock. The importance of these phenomena in the initiation of explosion, as well as possible initiation by shock perturbation, was assessed in a series of experiments designed to separate the various possible mechanisms. The conclusion is that adiabatic heating of the gas in the bubble was the prime cause for initiation. Calculations, and experiments with gases such as argon and helium (high value of γ ; the ratio of the specific heats) and butane (low γ ) supported this conclusion. Finally, the relevance to other explosive situations is discussed.

Observations of Viscoelastic Behaviour of an Elastohydrodynamic Lubricant Film

Abstract: A novel rolling-contact experiment has been devised with the object of making direct observations of viscoelastic behaviour of elastohydrodynamic lubricant films. A 'point contact' disk machine has been used which permits the axes of the disks to be either tilted or skewed through small angles relative to each other. Tilt subjects the film to cyclic shear while skew subjects it to unidirectional shear. By measuring and comparing the resultant shear forces developed by the film in each case, its viscoelastic properties can be deduced. Experiments at 23 degrees C with a high-viscosity mineral oil revealed viscoelastic behaviour above a contact pressure of about 0.5 GPa. In the pressure range 1.0 to 2.0 GPa the effective elastic shear modulus of the oil was found to vary from 0.1 to 1.0 GPa. At a pressure of 0.8 GPa the oil exhibited a viscoelastic transition in the temperature range 70 to 30 degrees C. The investigation was restricted to small shear strains for which the behaviour is linear.

The Structure of the Diatomic Molecular Solids

Abstract: A simple model of the low temperature phases of the diatomic molecular solids is examined. The model consists of molecules, interacting via a Lennard-Jones atom-atom potential and quadrupole-quadrupole interactions. The internal energy of any crystallographic structure (excluding thermal effects) can then be given in terms of two dimensionless parameters, which describe the deviation of the molecular shape from a sphere and the relative importance of the quadrupole energy. The minimum energies and optimum molecular configurations in several structures are computed, for values of these dimensionless variables which span the values appropriate to the actual homonuclear diatomic molecular solids, H$\_{2}$, N$\_{2}$, O$\_{2}$, F$\_{2}$, Cl$\_{2}$, Br$\_{2}$ and I$\_{2}$. Despite its great simplicity, the model is able to explain several features of these structures. These are (i) $o-\text{H}\_{2}$ and N$\_{2}$ have the optimum quadrupole structure, Pa3; (ii) $\beta $-O$\_{2}$ is one of the optimum van der Waals' structures, R$\overline{3}$m; (iii) the monoclinic $\alpha $-F$\_{2}$ structure is the most stable structure for parameter values very close to those appropriate to F$\_{2}$; (iv) the orthorhombic Cmca structure (observed for Cl$\_{2}$, Br$\_{2}$ and I$\_{2}$ is the most stable structure for a large range of quadrupole moments which may be appropriate to these molecules. The model, is, of course, unable to take into account intermolecular bonding or spin-dependent interatomic forces. The former is important for the halogens and the latter for the (magnetic) oxygen molecule. The case of $\alpha $-O$\_{2}$ is treated in the following paper.

Adhesion between Clean Surfaces at Light Loads

Abstract: This paper describes a study of the adhesion between clean surfaces, in a vacuum of 13nPa (10$^{-10}$ Torr), the influence of hardness, ductility, chemical bonding and adsorbed vapours. The geometry was that of crossed cylinders so that the contact region could be well localized and changed from experiment to experiment. In addition, electrical resistance measurements (in the case of conducting solids) could be used to follow the formation and breaking of contact. The joining load was very small (of order 1 mN) with the result that, even for the softest metals the deformation during loading was primarily elastic. The observed adhesion, however, is found to depend markedly on the ductility of the solid. With hard elastic solids the adhesion is very small indeed: much smaller than that predicted by theory and the results suggest that this may be due to fine-scale surface irregularities scarcely larger than atomic dimensions. With metals with a limited number of slip planes the adhesion is higher but appreciably less than would be expected from the area of the junctions formed. With metals such as gold and copper the adhesion is high and equal to the tensile strength of the interfacial junctions. Adhesion of soft metals to a hard solid such as titanium carbide can be high and when the surfaces are separated fragments of metal are found attached to the harder surface. With other hard solids such as sapphire or diamond the adhesion of copper is appreciably less, indicating that the bonding between copper and these covalent materials is intrinsically weak. Experiments at room temperature show that argon and hydrogen have no detectable effect on adhesion, whereas oxygen has. If one of the pair is copper, oxygen reduces the adhesion if the exposure is relatively heavy. With iron a minute exposure of oxygen reduces the adhesion to a very small value. It is suggested that with copper an adsorbed film is formed which is relatively ineffective. By contrast, on iron, minute iron oxide nuclei are formed which reduce the amount of metal-metal contact and also interpose an interfacial region of very limited ductility.

On a Generalization of the Power-Zienau-Woolley Transformation in Quantum Electrodynamics and Atomic Field Equations

Abstract: A canonical transformation is performed on the conventional Hamiltonian for the electromagnetic radiation field and an assemblage of neutral molecules in interaction. The new Hamiltonian has interaction energies expressed in terms of the electromagnetic fields alone and these energies have direct physical significance. The terms linear in e are the multipole interactions, both electric and magnetic, and the term quadratic in e is a generalization of the elementary diamagnetic energy shift. The intermolecular Coulomb energies have cancelled with transverse polarization fields in the new Hamiltonian, although the intramolecular Coulomb potentials are left unaffected. The equations of motion that follow from the new Hamiltonian are deduced. They are the so-called atomic-field equations for the Maxwell fields and Schrodinger equation for an electron wave in a transverse electromagnetic field. The former are the microscopic analogues of Maxwell’s equations in a medium (not restricted to dipole polarization fields) and the latter are dependent on the field strengths alone (not explicit functions of the vector potential).

Remote Sounding of Atmospheric Temperature from Satellites. V. The Pressure Modulator Radiometer for Nimbus F

Abstract: A new radiometer is described for remote sounding of the temperature of the stratosphere and mesophere (40-90 km) from a satellite platform. In order to select radiation emitted from close to the centres of CO$\_{2}$ lines in the 15 $\mu $m infrared band, included in the optical system is a cell containing CO$\_{2}$ gas whose pressure can be modulated at a frequency of ca. 15 Hz. The parts of the emission lines selected can be altered by two methods known as pressure scanning and Doppler scanning. A two-channel instrument for the Nimbus F satellite is described in detail.

Boundary layer over a blunt body at high incidence with an open-type of separation

Abstract: This paper presents general methods for calculating three dimensional laminar boundary layers over inclined blunt bodies (not necessarily bodies of revolution). Complete incompressible results for a prolate spheroid at 30 degrees incidence are presented, and the computational procedures are described. The rule of the zone of dependence was observed. The results display for the first time the boundary layer structure over a non-spherical and non-conical blunt body in all details, and confirm the open-type of separation proposed earlier.

Negative temperature states of two-dimensional plasmas and vortex fluids

Abstract: The two-dimensional guiding centre plasma and a system of interacting line vortices in an ideal fluid are examples of Hamiltonian systems with bounded phase space. The statistical mechanics of such systems is investigated. An interesting feature is that they can exist in negative temperature states which show observable intrinsic characteristics, such as the formation of clusters of particles.

Fluid Forces on a Body in Shear-Flow; Experimental Use of 'Stationary Flow'

Abstract: The lift and drag forces have been measured on a sphere and a transverse cylinder immersed in an open liquid shear-flow and situated close to the lower, frictional, boundary (the bed). Two conditions were investigated: ( a ) that of zero drag, when the body was drifting with the flow, and ( b ) that when it was held against the flow. In condition ( a ) the body could be either allowed to rotate about a transverse axis subject to unavoidable pivot friction, or prevented from rotating. Marked difference was found in the magnitude of the lift force according to the applied resistance to rotation. The lift force was a maximum when rotation was prevented and small or undetectable when free rotation was allowed. In the conditions ( a ) and ( b ) the lift force decreased with increasing clearance between body and boundary, to zero when the clearance exceeded approximately one body diameter. In condition ( b ) lift, i. e. normally repulsive, forces of approximately equal magnitudes to those below were exerted as the body approached the upper free liquid surface. In the drifting condition ( a ) the considerable difficulties of observation and force measurement when a body is moving with the flow were removed by the use of a backward-moving bed boundary. By thus superimposing a reverse velocity on the whole system, the mean fluid velocity at any desired distance from the boundary can be made zero relative to the observer without appreciably affecting the internal dynamics of the flow. This device also permitted the repetition of the measurements made by using liquids of greater viscosity than water available in limited quantities. The results are interpreted with an explanation in mind of certain aspects of the motions of unsuspended solids in saltation over a stream bed.

The Numerical Solution of the Asymptotic Equations of Trailing Edge Flow

Abstract: According to Stewartson (1969, 1974) and to Messiter (1970), the flow near the trailing edge of a flat plate has a limit structure for Reynolds number $Re\rightarrow \infty $ consisting of three layers over a distance $O(Re^{-\frac{3}{8}})$ from the trailing edge: the inner layer of thickness $O(Re^{-\frac{5}{8}})$ in which the usual boundary layer equations apply; an intermediate layer of thickness $O(Re^{-\frac{1}{2}})$ in which simplified inviscid equations hold, and the outer layer of thickness $O(Re^{-\frac{3}{8}})$ in which the full inviscid equations hold. These asymptotic equations have been solved numerically by means of a Cauchy-integral algorithm for the outer layer and a modified Crank-Nicholson boundary layer program for the displacement-thickness interaction between the layers. Results of the computation compare well with experimental data of Janour and with numerical solutions of the Navier-Stokes equations by Dennis & Chang (1969) and Dennis & Dunwoody (1966).

Statistical Mechanics of the Melting Transition in Lattice Models of Polymers

Abstract: Five two-dimensional lattice models, four with rotational isomeric and excluded volume interactions and one with cross links, are used to discuss the theory of the melting transition in polymers. The models have been chosen because they are isomorphic to exactly solvable six vertex and dimer models. The orders of the thermodynamic transitions are extremely varied from model to model, including first-order, $\frac{3}{2}$ order and infinite order transitions. These models are used to test and reveal the shortcomings of the Flory-Huggins approximate theory, which is most aptly described as an infinite dimensional theory.