Asymptotic Scheme Research Articles

Interactions of solitons in nonintegrable systems: Direct perturbation method and applications

Problems of approximate description of soliton interaction in nonintegrable systems are discussed. It is shown that in an arbitrary Lagrangian system in the first nontrivial order of the small parameter asymptotic scheme for a system of solitons with close velocities a Lagrangian may be introduced analogous to that for classical particles with a pair interaction potential. A classification of possible types of soliton interaction is given and criteria of the existence of bound states are formulated. A possibility of generalization of approximated description of soliton interaction in the multidimensional case is noted.

On the propagation of a three-dimensional packet of weakly non-linear internal gravity waves

The evolution of a three-dimensional packet of weakly non-linear internal gravity waves propagating obliquely at an arbitrary angle to the vertical line is considered. Two coupled non-linear equations connecting variations of a packet amplitude and induced flows are derived. three-dimensionality of the packet having been found responsible for the non-linearity of the system. Explicit formulae for the induced flow vertical component and the mean density field variation caused by packet propagation have been obtained. The plane wave is shown to be unstable at any arbitrary slope of the wave vector. The non-linear equation describing the evolution of the two-dimensional packet is derived in the subsequent order of the asymptotic scheme. It has been found possible for the packet to collapse. The collapse of internal waves packets may be one of the possible mechanisms of “blini”-shaped regions of mixed waters formation in the ocean.

Mixture theory for longitudinal wave propagation in unidirectional composites with cylindrical fibers of arbitrary cross section—I: Formulation

A binary mixture theory is developed for propagation of longitudinal waves in a unidirectional, fibrous composite containing a two-dimensional periodic array of cylindrical fibers of arbitrary cross section. The case considered concerns a class of problems for which the waves propagate in the direction of the fiber axis. Model construction is based upon an asymptotic scheme wherein the signal wavelength is assumed to be large compared to composite micro-dimension; this premise is appropriate for many applications. The resulting theory, which retains information on the displacement profile in the microstructure, is closed in the sense that all the mixture properties that it contains can be determined once the geometry of the unit cell, shape of the fiber cross section and the properties of the constituents are known.In the theory formulated herein, calculations of the mixture properties require the solution of time-independent microstructure boundary value problems (MBVPs) defined over a unit cell. In this part of the paper, the MBVPs are derived, and the relationship between their solution and mixture properties is established. The description of a finite-element procedure to solve the MBVPs is deferred to the second part of the paper, which also contains an application of mixture theory to a study of dispersion of time-harmonic waves in composites with several microstructural geometries and combination of material properties.

A mixture theory for thermal diffusion in unidirectional composites with cylindrical fibers of arbitrary cross section

A binary mixture theory is developed for heat conduction in a unidirectional, fibrous composite containing a two dimensional periodic array of cylindrical fibers of arbitrary cross section. The case considered concerns a class of problems for which conduction occurs primarily in the direction of the fiber axis. Model construction is based upon an asymptotic scheme wherein the ratio of transverse-to-longitudinal diffusion times is assumed to be small: this premise, and the problem class studied, are appropriate for many composites designed primarily for thermal protection. The resulting theory, which retains information on the temperature distribution in the microstructure, contains a mixture interaction coefficient: the latter is determined from the solution of a time-independent boundary value problem in a unit cell. A variational principle-based finite element method is proposed for the solution of this boundary value problem. Consequently, the theory is closed in the sense that no unknown coefficients exist, i.e. the theory is completely determined by the material properties and geometries of the constituents.Numerical analyses are carried out for several microstructural geometries of practical interest. The results indicate that, for achievable volume fractions, the concentric circular cylindrical approximation often used in practice provides an adequate measure of global and averaged local temperature fields for composites containing circular fibers in a hexagonal array, and square fibers arranged in a square array. It is found, however, that such an approximation may not be accurate for composites containing rectangular fibers in a similar unit cell. Here a parametric study reveals that the interaction coefficient is a strong function of the unit-cell aspect ratio.

Asymptotic analysis of the flows of viscous compressible fluids at low mach numbers II. The case of rotating heavy fluids

NON-STATIONARY flows of a viscous compressible heavy rotating fluid, whose Mach number tends to zero, are analyzed. Such an analysis enables us to obtain by a rational method the equations of Boussinesq and the conditions on the limits which must be associated with them. Also, assuming that the Rossby number is small, an asymptotic scheme is constructed which describes the so-called quasi-geostrophic model, the problem of adaptation to geostrophism and the so-called Ackerblom model for the Ekman layer.

A Continuum Model for Diffusion in Laminated Composite Media

Using a method developed for studying wave propagation problems, a continuum theory is developed for diffusion-type processes in a laminated composite with periodic micro-structure. Construction is based upon an asymptotic scheme in which a typical macrodimension is assumed large compared to a microdimension. The order of truncation of the asymptotic sequence so obtained defines a hierarchy of models. Solutions are given for the lowest-order models and compared with the results from a finite difference code. For most cases the zeroth-order “effective conductivity” theory yields good results. For exceptional problems requiring a higher-order theory, a modified version of the first-order theory is shown to suffice. For many applications these elementary equations may offer an attractive alternative to other means for obtaining solutions.

Linearized buoyant motion in a closed container

An arbitrarily-shaped, closed container completely filled with fluid is considered. It is assumed that the fluid is originally in a stably-stratified state of rest, and that at an initial instant the temperature of the container walls is impulsively changed. The ensuing unsteady laminar motion is found by solving the linearized Boussinesq equations governing buoyancy-driven flows. A ‘boundarylayer/inviscid-interior’ decomposition leads to a modified asymptotic expansion scheme of analysis. The boundary-layer concept is valid only for large values of the Rayleigh number, and, in addition, we limit the Prandtl number to order unity. It is found that the inviscid interior region heats up by means of a convection process that is driven by suction induced by the boundary layer. The inviscid, adiabatic interior responds to a special horizontal ‘average’ value of the container temperature perturbation. The boundary layer smears out, or averages, any circumferential variation in this perturbation, so that the interior, in effect, responds to an isothermal boundary in each horizontal plane. The interior temperature and vertical velocity component are expressed simply in terms of this horizontal ‘average’ container temperature. The horizontal velocity potential is governed by a Poisson equation, whose solution is developed for several specific geometries to illustrate the nature of the flow.

Nonaxisymmetric equilibria in tokamaks and Ohmically heated stellarators

An asymptotic expansion scheme is set up, which is appropriate to deal with the deviations from axisymmetry occurring in tokamaks and Ohmically heated stellarators. Solutions of the magnetohydrostatic equations are sought to all orders in the expansion parameter, which is related both to the magnitude of the deviations from axisymmetry and to the ratio of poloidal vs toroidal components of current and magnetic field. Arguments are presented for the existence of the solution to any finite order, the pressure and current profiles across the plasma column, in general, being determined by the nonaxisymmetric part of the external magnetic field. If the latter, however, has certain symmetries, then a degeneracy occurs, and the pressure and current profiles can again be given with some degree of arbitrariness. It is shown that for tokamaks and stellarators of standard design, viz., including the perturbations associated with the ’’ripples’’ in the toroidal field, the ’’gaps’’ in the copper shell, and the helical windings, the degree of arbitrariness in the pressure and current profiles is the same as in the axisymmetric theory. Thus, in such machines the above profiles are expected to be governed entirely by transport effects, a conclustion which is borne out by the experiments.

Spherical electric probe in a continuum gas

The problem of a spherical electric probe in a continuum gas without chemical reactions is re-examined when the dimensionless probe potential omega p is large and comparable to rho p, the ratio of probe radius to the ion Debye length. An asymptotic scheme for omega p to infinity shows that the ion-sheath region near the probe and the quasi-neutral region further away are separated by a single transition region which must be carefully matched to them. The properties of the transition equation are investigated in some detail, and numerical results are obtained determining the second-order (ion) current to the probe. Universal curves are given for the first- and second-order currents when the electron temperature equals the ion temperature.

A General Continuum Theory With Microstructure for Wave Propagation in Elastic Laminated Composites

A continuum theory with microstructure for wave propagation in laminated composites, proposed in previous works concerning propagation normal and parallel to the laminates, is extended herein to the general two-dimensional case. Continuum model construction is based upon an asymptotic scheme in which dominant signal wavelengths are assumed large compared to typical composite microdimensions. A hierarchy of models is defined by the order of truncation of the obtained asymptotic sequence. Particular attention is given to the lowest order dispersive theory. The phase velocity spectrum of the general theory is investigated for one-dimensional wave propagation at various propagation angles with respect to the laminates. Retention of all terms in the asymptotic sequence is found to yield the exact elasticity spectrum, while spectral collation of the lowest order dispersive theory with the first three modes of the exact theory gives excellent agreement.

Approximation for hypersonic flow past an inclined cone.

An approximate analytical solution is obtained for hypersonic flow past an inclined circular cone. The governing equations are simplified by means of the constant-density approximation, and the ensuing quadrature solution is evaluated in closed form by approximations based on hypersonic small disturbance theory. This solution corresponds to the outer solution in a matched asymptotic expansion scheme and describes the region outside the thin vortical layer that lies adjacent to the surface of the cone. The results are in good agreement with tabulated numerical solutions and are also valid for values of the hypersonic similarity parameter that lie outside the range of the tabulated results. The present approximation is also applicable to much broader flow regimes than previous analytical descriptions, such as that of the Newtonian approximation. The results of this analysis are useful whenever analytical expressions are desired and can be coupled with one of the existing inner solutions for the vortical layer to form a composite uniformly-valid approximation to first order in angle of attack, valid for a wide range of hypersonic similarity parameters.

A Continuum Theory for Wave Propagation in Laminated Composites—Case 1: Propagation Normal to the Laminates

A continuum theory is developed for wave propagation normal to the layers of a laminated composite with elastic, periodic, microstructure. Construction is based upon an asymptotic scheme in which dominant signal wavelengths are assumed large compared to typical composite microdimensions. A hierarchy of models are defined by the order of truncation of the asymptotic sequence obtained. To estimate system accuracy, the phase velocity spectrum is investigated. Retention of all terms in the asymptotic sequence is found to yield the exact spectrum of Rytov. Based upon spectral collation of the lowest-order dispersive model, accuracy superior to several existing theories is observed. In addition, treatment of transient pulse cases show good correlation with exact data. Finally, the lowest-order dispersive theory is cast in a standard mixture form.

A continuum theory for wave propagation in laminated composites/ Case 2: Propagation parallel to the laminates

A continuum theory with microstructure for wave propagation in laminated composites, proposed in a previous work concerning normal propagation, is extended herein to the case of propagation parallel to the laminates. Model construction is based upon an asymptotic scheme in which dominant signal wavelengths are assumed large compared to typical composite microdimensions. A hierarchy of models is defined by the order of truncation of the obtained asymptotic sequence. To estimate accuracy, the phase velocity spectrum is investigated. As in the previous case, retention of all terms in the asymptotic sequence is found to yield the exact pectrum of Rytov. Spectral collation of the lowest order dispersive theory with exact first mode data gives excellent agreement. Based upon asymptotic expansions, a simplified first-order theory is also developed. Transient pulse data obtained from the latter exhibits good correlation with experimental results. In addition, representative calculations of microcomponent velocity distributions and interface shear stress are carried out using the simplified theory.

Light-Cone Structures and Sum Rules as Seen in the Parton Dual-Resonance Model

We propose a simple and unique asymptotic symmetry scheme to incorporate spins, SU (3), $\mathrm{CPT}$ invariance, and the generalized Bose statistics into the parton dual-resonance model for deep-inelastic lepton-hadron reactions. The scheme embodies Harari-Rosner quark duality diagrams, adopts Chan-Paton SU (3)-symmetry factors, identifies Mandelstam, Bardakci, and Halpern's multiplicative quark model in the Bjorken limit, and generates Fritzsch and Gell-Mann's light-cone structures identically. The dynamical (parton-dual-resonance-model) approach is therefore unified with the symmetry (light-cone) approach, and the complementarity between these two is established. Light-cone sum rules are derived, and a new sum rule is suggested. As a further application of this scheme, we calculate the complete sets of bilocal light-cone structures occurring in the two-heavy-boson processes.

Critical Remarks on the Asymptotic Symmetry Scheme

The asymptotic symmetry scheme proposed by Oneda and Matsuda is analyzed rather critically on two counts. First of all, we discuss the underlying reasons given for the scheme's basic assumption, that single-particle matrix elements of the SU(3) raising operator ${V}^{K}$ are effectively not renormalized, even when the symmetry is broken, and we find that some of the arguments are not completely convincing. Second, it is shown that most of the results of the scheme can be obtained without using this basic assumption, and that it is only most of the results derived from equal-time commutators of the form $[{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}}^{K},{A}^{i}]=0$ which are necessary consequences of this assumption. These latter results include all of their intermultiplet mass sum rules together with a few other predictions such as the degeneracy of the ${\ensuremath{\Sigma}}^{0}$ and ${\ensuremath{\Lambda}}^{0}$, and, in general, these are not as well satisfied experimentally as the other results.

Further Development of Scaled Particle Theory of Rigid Sphere Fluids

A new approach to the scaled particle theory of fluids is developed which is based on detailed consideration of the reversible work required to create a fixed spherical cavity of radius r in a rigid sphere fluid. The new results depend only on the mechanical concept of work and the classical virial theorem. A cycle greatly facilitates the necessary detailed analysis. A new integral equation for G(r), the primary function in scaled particle theory, culminates the new results. Approximation schemes may be employed relatively easily in this integral equation. The series approximation G(r) = ∑ n = 0∞ Gn(r−1)n, which is asymptotically correct as r → ∞, is discussed. A new set of exact conditions on the coefficients Gn in this asymptotic expansion, the most striking of which is G3 ≡ 0 in three dimensions, is derived. These exact conditions relate to surface properties. The infinity and integral exact conditions on G(r) are shown to be truly independent. The jump condition on the second derivative of G(r) is incorporated successfully for the first time with other exact conditions in the derivation of two new expressions for G(a), both of which give rise to equations of state. One of these expressions yields the best virial coefficients thus far obtained for a rigid sphere system from any analytic equation of state not based on cluster expansions. The new approach and results are easily generalized to arbitrary dimension. Future work will center on attempts to extend the new integral equation for G(r), on possible extensions of the present method to real fluids, and on studies of the nature of the asymptotic series approximation scheme.

Radiation dominant interaction in gasdynamics

The interaction between a dominant radiation field and a flow field is investigated, where the time scale for such an unsteady interaction may be quite short compared to a characteristic “flow” time. To lowest order there results a radiative cooling process with an uncoupled flow, and an expansion in Boltzmann number then systematically introduces higher order effects. A matched asymptotic expansion scheme is required in view of the disparate characteristic lengths that exist in different spatial regions. Results are presented for the specific example of instantaneous addition of energy to a finite region with one-dimensional symmetry (i.e. planar, cylindrical and spherical). For simplicity the gas is assumed inviscid, perfect and in local thermodynamic equilibrium, and the differential approximation is used for the radiative field. A numerical scheme was developed to avoid time consuming iterations and results are compared to those for an analytic solution limited to the linear case. Both nongrey approximations and up-stream absorption effects are included. Results indicate a negligible upstream heating level, moderate nongrey but large curvature and optical thickness effects. Depending on the local balance between emission and absorption, the blast shock can be either accelerated or decelerated. Finally, comparison between differential approximation and exact formulations for a planar grey case is made. Excellent agreement between the present work and a laser experiment indicates the possible energy coupling to be inferred.

Asymptotic high-order schemes for integro-differential problems arising in markets with jumps

In this paper we deal with the numerical approximation of integro-differential equations arising in financial applications in which jump processes act as the underlying stochastic processes. Our aim is to find finite differences schemes which are high-order accurate for large time regimes.Therefore, we study the asymptotic time behavior of such equations and we define as asymptotic high-order schemes those schemes that are consistent with this behavior. Numerical tests are presented to investigate the efficiency and the accuracy of such approximations.