Asymptotic Solution Method Research Articles

Magnetic Field Effects in High‐Power Batteries: I . The Penetration of an Electric Field into a Cylindrical Conductor

The penetration of the axial component of an electric field into a cylindrical conductor is described by an asymptotic solution method for long and short times. The development of the respective current distributions allows for a mathematical comparison of the solution schemes and indicates that the current initially increases at a rate proportional to time until a time of order ε/σ, subsequently at a rate proportional to the square root of time, and finally levels off exponentially to the steady‐state value. Numerical estimates of these time frames for a bipolar lead‐acid battery with porous lead electrodes of 0.10 m radius give values of for the inner time regime and 0.1 s for the diffusion time regime. Criteria for the proper omission of the displacement current also are given.

Higher‐order mode theory of three‐dimensional irregular waveguides

The method of asymptotic solution of waveguide equations for higher‐order modes is generalized to the three‐dimensional case. Geometrical optics and full wave‐type solutions are constructed in the form of integral representations. In the geometrical optics case, two sorts of partial waves are used to construct the integral representation of the field: ray‐type waves and Gaussian beam‐type waves. In full wave solution the elementary waves of the integral representation are constructed by means of complex phase in Fresnel diffraction approximation for forward scattering. Asymptotic integral representations developed take into account coupling of the modes, describe the multiray effects and focusing of the higher‐order mode field. The method is valid for waveguides with either varying height or varying boundary conditions and dielectric properties.

A mathematical model of heat conduction in a prolate spheroidal coordinate system with applications to the theory of welding

The steady-state heat conduction equation is solved in a prolate spheroidal coordinate system. Some simple solutions are derived in the limiting cases of both low and high Peclet numbers. The analysis was carried out in such a way as to avoid the physical details of conditions inside the weld pool, so that the solutions are restricted to the solid region outside the weld pool. This procedure was specifically adopted because these conditions are difficult to gain access to experimentally, as is the precise detailed shape of the pool; the solutions obtained can be verified experimentally. The high-Peclet-number approximation is likely to be particularly useful in the case of laser welding where large translation speeds of the weld piece are of interest. The solution of the problem is given in the form of a series as well as in an asymptotic form. The asymptotic method of solution presented can be adapted to any smooth shape of weld pool with only minor alterations, since the method involves integration in the tangent plane to the weld pool and the results of such an integration are independent of the global form of the weld pool. The asymptotic result is compared with the exact solution in a number of special geometric configurations. These are prolate spheroidal weld pool geometries with various aspect ratios and cylindrical weld pool geometries with elliptical or circular cross sections. The results of these comparisons were found to be satisfactory.

A finite element analysis of viscoelastically damped sandwich plates

A finite element analysis associated with an asymptotic solution method for the harmonic flexural vibration of viscoelastically damped unsymmetrical sandwich plates is given. The element formulation is based on generalization of the discrete Kirchhoff theory (DKT) element formulation. The results obtained with the first order approximation of the asymptotic solution presented here are the same as those obtained by means of the modal strain energy (MSE) method. By taking more terms of the asymptotic solution, with successive calculations and use of the Padé approximants method, accuracy can be improved. The finite element computation has been verified by comparison with an analytical exact solution for rectangular plates with simply supported edges. Results for the same plates with clamped edges are also presented.

Resonant acoustic scattering from solid targets

An asymptotic method of solution is presented for scattering of acoustic waves from solid elastic targets. The asymptotic parameter is the ratio of the fluid density to that of the solid, and the solution is developed using the method of matched asymptotic expansions in this small quantity. The perturbations to the background rigid scattered field are regular for frequencies away from the frequencies of free vibration of the target in vacuo, but, near these frequencies, the perturbation is singular in that an asymptotically small value of the density ratio produces a change in the scattered field of order unity. By combining the regularly and singularly perturbed expansions, a solution is obtained that is uniformly correct at all frequencies. The elements in the uniform solution depend only upon the in vacuo modes and frequencies, and the Green’s function for the equivalent rigid target. At no stage is it necessary to solve the fully coupled system. An analysis of the asymptotic approximation for a spherical target shows that it is equivalent in the high-frequency limit to the approximation predicted by resonance scattering theory.

An adaptation of the decomposition method for asymptotic solutions

An adaptation of the decomposition method (Adomian, 1986) allows asymptotic solutions for differential and partial differential equations.

Propagation characteristics of elliptical dielectric waveguides: Ray-optic approach

This paper is concerned with the study of guided wave propagation in step-index elliptical dielectric waveguides by the ray- optic method. We apply the method of asymptotic solution of eigenvalue problems by Keller and Rubinow and derive formulas for the calculation of dispersion characteristics corresponding to “whispering gallery” modes and “bouncing ball” modes respectively. We also give criterions for different ellipticities of the elliptical boundary. Computed dispersion curves are presented for different ellipticities. The results of our method are in good agreement with exact results, except for low frequencies. However, our expressions are rather simple and convenient to apply. We also give the briefringe curves and the curves of constant power density.

Asymptotic elastic solution for two straight cracks of arbitrary length and location

An infinite elastic plane containing two straight cracks of arbitrary length and location is analyzed within the framework of elastostatics. The mathematical formulation is based on the stress solution for a single crack and leads to a system of singular integral equations that govern the crack surface displacement densities. The solution series in terms of the reciprocal of the crack centre distance is not suitable for cracks that are spaced too closely. It is shown by way of examples that the method of asymptotic solution is convenient for developing approximation expressions of the stress and displacement field with certain characteristics. The formulas for the stress intensity factors and crack opening are given for the case of a constant tensile load. Graphical results are given for the variations of the stress intensity factors with parameters depending on the relative positions of the cracks.

A transmission problem for the plate equation

SynopsisA scattering theory is developed for transmission problems associated with the plate equation. Asymptotic methods of solution for large time are examined as are questions concerning regularity of solution, nature of the associated spectrum and existence of appropriate wave operators. It is shown that in contrast to solutions of the wave equation, signals can propagate with an infinite dispersion velocity.

Asymptotic study of rotationally symmetric waves in a circular anisotropic waveguide

The nonreciprocal propagation of rotationally symmetric TE-modes along a circular guide, uniformly filled with latching ferrite magnetised azimuthally to remanence, is studied theoretically. The boundary-value problem is solved in terms of Kummer confluent hypergeometric functions of complex parameter and variable. An asymptotic method for solution of the characteristic equation is developed, providing a basis for numerical computation of the eigenvalue spectrum and phase characteristics of the structure with a fair degree of engineering accuracy. The normalised nonreciprocal phase characteristics and the differential phase shift of the waveguide under rotationally symmetric TE-mode excitation are presented for both latched states of ferrite remanent magnetisation. The potentialities of the structure as digital nonreciprocal phase shifter and/or bistable ferrite switch using intrinsic advantages of ferrite remanent magnetisation are discussed.

Convective diffusion to a slowly rotating spherical electrode; The effect of axial diffusion in the bulk liquid for Re 10

A complete mathematical model has been solved of the steady axially symmetric convective diffusion toward the surface of a spherical electrode of radius R rotating at an angular velocity Ω under the creeping flow conditions Re ≡ ΩR2ρ/η < 10 and Pe ≡ Ω2R4ρ/(12Dη) > 10 by the method of singular perturbations. For Pe > 300 the effect of axial diffusion has been found entirely negligible; for 10 < Pe < 300 it causes an increase of local transfer coefficients by 1-10%. For Pe < 10 the applied asymptotic method of solution, assuming Pe >> 1 is no longer applicable.

Wave propagation and dispersion in the cochlea.

Insight into cochlear mechanics can be obtained from semi-analytical and asymptotic solution methods of which the Liouville -Green (LG) method - in another context known as the WKB method - is the most important one. This paper describes the dispersion properties of fluid waves in general terms and develops the LG formulation on that basis. The eikonal equation of the LG method is shown to be identical to the dispersion relation in dispersive-wave theory. Consideration of the group velocity then leads to the derivation of the central LG formula as it has been used in an earlier paper on the LG method ( Boer , E. and Viergever , M.A. (1982): Hearing Res. 8, 131-155). The formulation appears to apply as well to dissipative and active (i.e., energy-producing) systems. Of the many possible collateral subjects two are selected for a deeper discussion: amplification, concentration and expansion of energy, and the problem of reflection of cochlear waves. In the latter context, it is shown why - and under which conditions - cochlear waves are not reflected, despite the large degree of dispersion that they show. The analysis brings to light a fundamental asymmetry of the model regarding the direction of wave travel: waves travelling in the direction opposite to the normal one are likely to undergo reflection, while waves in the normal direction are not reflected.

Analysis of Hydrocarbon Emissions From Conventional Spark-Ignition Engines

Abstract The quench layer on the cylinder wall of a spark-ignition engine is modeled for the case where a tangential or swirl velocity exists. Both asymptotic and numerical methods of solution are used, The usual two-layer structure is employed for the turbulent boundary layer and Crocco's integral is used to relate the temperature and velocity fields. The quench layer is defined as the region in which the temperature is less than an ignition temperature. In the first analysis only quenching is considered, and it is shown that the resulting concentration of hydrocarbons is much too high compared to experimental results. Next, the model is extended to allow diffusion, and it is shown that in one stroke time the majority of the mass of unburned mixture in the original quench layer has diffused out and burned. Finally, two means of replenishing the unburned mixture in the quench layer. consisting of the desorbing of fuel from the oil film and outgassing from the first ring crevice, are included in the model....

Formation of carbon monoxide in turbulent diffusive combustion

The purpose of this paper is to clarify the mechanism of formation of carbon monoxide in combustion chambers of aircraft gas-turbine engines. The problem consists of estimating deviations in the concentration of carbon monoxide and clarifying the effect of turbulence on the conditions for the occurrence of chemical reactions. Considerations of the various functions of the process, the rate of the reaction, and the thickness of the transitional layer, permits constructing a solution with the help of the method of joining asymptotic solutions. A number of estimates are made to confirm the validity of the assumptions made. It is found that the possible effect of molecular transport processes on the combustion process is related to the fact that in many cases of practical importance the thickness of the transitional layer is small, while the Reynolds number is not large enough.

A sharpening of Maslov’s method for finding asymptotic solutions

A sharpening of Maslov’s method for finding asymptotic solutions

Asymptotic method for solving problems with radial laminar jets of immiscible liquids

Two problems involving radial laminar jets of immiscible liquids are considered: a free radial slit jet and a jet on a rotating disk. An asymptotic method of solution is proposed that makes it possible to determine the flow parameters far from the source. The difference between these flows and those of homogeneous liquids is demonstrated.

Convective diffusion to a particle in a fluid with linear kinetics: PMM vol. 43, no. 1, 1979, pp. 65–74

The concentration field in the neighborhood of a solid sphere in a Stokes stream at high Péclet numbers is determined by the method of joining asymptotic solutions. A chemical reaction, whose rate is arbitrarily dependent on the concentration of the diffusing matter close to the surface, takes place at the sphere surface. Dependence of the total diffusing flux at the sphere surface on the chemical reaction rate is determined. The phenomenon of diffusion flux saturation with increasing Péclet number is present, as in the case of first order chemical reaction [1]. Modes of the chemical reaction at the sphere surface are investigated. The distribution of concentration in the diffusion wake region is determined and its structure analyzed. The problem was solved in [1] for the case of linear kinetics.

A stochastic theory of collision phenomena, distribution of observables and information entropy

In this paper a stochastic model is formulated for collision phenomena of complex molecular systems. The model assumes that the evolutions of observables during a collision process between complex molecules may be regarded as brownian motions. The probability distribution of the observables is then governed by a Fokker—Planck equation, which is solved for cases of linear and nonlinear streaming velocities. For the case of nonlinear streaming velocity an asymptotic solution method is discussed, which is similar to the WKB wavefunctions in quantum mechanics. We discuss distributions of observables in terms of the solutions to Fokker—Planck equations. In particular, we compare the Δ J-dependence predicted by the present theory with the results by Pack who used a close coupling method for the ArN 2 rotational transitions and find that there is good agreement in the Δ J-dependence. We define information entropy in the conventional form and discuss the information theory approach to collision phenomena in the light of the present theory.

A mathematical analysis of fluid movement across capillary walls

A mathematical analysis of the exchange of fluid between a blood capillary and the surrounding tissue is presented. The diameter and permeability of the capillary are assumed to vary along its length in accordance with observation, and fluid movement across the capillary wall is assumed to be governed by a generalization of Starling's law. The motion of the interstitial fluid obeys a nonlinear form of Darcy's law in which the porosity and hydrodynamic conductivity of the tissue vary with interstitial fluid pressure. An asymptotic method of solution is developed for a simplified problem in order to establish techniques applicable to the general case. The results are used to discuss some specific examples of physiological interest.

Steady heat-conduction problem for a double cylinder

An analytical solution of the problem stated in the title is obtained for various conditions at the ends of the cylinder. The method of asymptotic solution of boundary-value problems for elliptic equations in thin domains is translated to the case of boundary conditions of the third kind.