Showing papers on "Method of matched asymptotic expansions published in 1987"

Heating of the solar corona by the resonant absorption of Alfven waves

Abstract: An improved method for calculating the resonance absorption heating rate is discussed and the results are compared with observations in the solar corona. To accomplish this, the wave equation for a dissipative, compressible plasma is derived from the linearized magnetohydrodynamic equations for a plasma with transverse Alfven speed gradients. For parameters representative of the solar corona, it is found that a two-scale description of the wave motion is appropriate. The large-scale motion, which can be approximated as nearly ideal, has a scale which is on the order of the width of the loop. The small-scale wave, however, has a transverse scale much smaller than the width of the loop, with a width of about 0.3-250 km, and is highly dissipative. These two wave motions are coupled in a narrow resonance region in the loop where the global wave frequency equals the local Alfven wave frequency. Formally, this coupling comes about from using the method of matched asymptotic expansions to match the inner and outer (small and large scale) solutions. The resultant heating rate can be calculated from either of these solutions. A formula derived using the outer (ideal) solution is presented, and shown to be consistent with observations of heatingmore » and line broadening in the solar corona. 34 references.« less

Singularly perturbed Volterra integral equations II

Abstract: A formal methodology is developed to obtain a uniformly valid asymptotic solution to a class of singularly perturbed Volterra integral equations. The methodology systematically determines the appropriate integral equations which govern the inner and outer solutions at each order of the perturbation analysis. Several examples are presented to illustrate the procedure.

An unsteady lifting-line theory

Abstract: A lifting-line theory is developed for wings of large aspect ratio undergoing time-harmonic oscillations, uniformly from high to low frequencies. The method of matched asymptotic expansions is used to enforce the compatibility of two approximate solutions valid far from and near the wing surface. The far-field velocity potential is expressed as a distribution of normal dipoles on the wake, and its expansion near the wing span leads to an expression for the oscillatory downwash. The near-field flow is two-dimensional. A particular solution is obtained from strip theory and a homogeneous component is added to account for the spanwise hydrodynamic interactions. The compatibility of the inner and outer solutions leads to an integral equation for the distribution of circulation along the wing span. In the zero-frequency limit it reduces to that in Prandtl's lifting-line theory, and for high frequencies it tends to the two-dimensional strip theory. Lift computations are presented for an elliptic and a rectangular wing of aspect ratio A = 4.

Comparison between two asymptotic methods

Abstract: Two complete asymptotic expansions of an integral with many simple pole singularities and a first-order, isolated, saddle point evaluated by two different methods are compared. It is shown that both expansions are exactly the same (term by term) inside and outside the transition regions.

The propagation of low‐frequency sound in a two‐dimensional duct system with T joints and right angle bends: Theory and experiment

Abstract: The method of matched asymptotic expansions is used to determine a low‐frequency approximation of the acoustic pressure in a waveguide system with a T joint. The low‐frequency approximation is compared with (a) experimental data of Lippert [Acustica 4, 313–319 (1954)]; (b) new experimental data; and (c) the first approximation of the mode expansion proposed by Miles [J. Acoust. Soc. Am. 17, 259–272 (1946)].

An asymptotic anaylsis of a transient p-n -junction model

Abstract: In this paper we carry out an asymptotic analysis of the system of differential equations describing the transient behavior of a p-n-junction device (i.e., a diode). We determine the different time scales present in the equations and investigate which of them actually occur in physical situations. We derive asymptotic expansions of the solution and perform some numerical experiments.

Triple-deck solutions for supersonic flows past flared cylinders

Abstract: Using the method of matched asymptotic expansions, the interaction between axisymmetric laminar boundary layers and supersonic external flows is investigated in the limit of large Reynolds numbers. Numerical solutions to the interaction equations are presented for flare angles α that are moderately large. If α > 0 the boundary layer separates upstream of the corner and the formation of a plateau structure similar to the two-dimensional case is observed. In contrast to the case of planar flow, however, separation can occur also if α

Analytical approximations for offshore pipelaying problems

Abstract: 1. Abstract The geometrically non-linear slender bar equation is solved for a number of problems involving suspended pipelines, related to the off-shore gas- and oil-pipe laying. The problems concern the use of a lay-barge with stinger, and the process of abandoning and recovery of a pipe. The usually stiff equation requires for a completely numerical solution considerable computer power, not always available on board. Therefore, the solutions are analytical (matched asymptotic expansions, and linear theory) to allow the results being evaluated on a small computer. It is shown that for the majority of the practical cases the two solution methods complement each other very well.

Mean drift forces on arrays of bodies due to incident long waves

Abstract: The scattering of long water waves by an array of bodies is investigated using the method of matched asymptotic expansions. Two particular geometries are considered: a group of vertical cylinders extending throughout the depth and a group of floating hemispheres. From these solutions, the low-frequency limit of the ratio of the mean drift force on a group of N bodies to that on a single body is calculated. For a wide range of circumstances this drift-force ratio is N2, which is in agreement with previous numerical work. Further drift-force enhancement is possible for certain configurations of vertical cylinders.

A systematic formula for the asymptotic expansion of singular integrals

Abstract: Asymptotic theories like the lifting-line, the slender body or the slender ship lead to lineintegrals with singular kernels. Sometimes these integrals are “improper”, that is to say that they are defined only by their Finite Part. To find asymptotic expansions of these integrals, the Matched Asymptotic Expansion Method is widely used along with other more specific methods depending on the kernel type. The first method is laborious and not systematic, and the other methods are sometimes too much specific to treat general cases. Moreover, all of them are not well adapted to deal with Finite Part integrals.

Justification of asymptotic expansions of the eigenvalues of nonselfadjoint singularly perturbed elliptic boundary value problems

Abstract: Boundary value problems for scalar elliptic equations are considered with a small parameter in front of the leading derivatives in domains with a smooth boundary. As this problem degenerates in a regular way into an elliptic problem of lower order. Neither problem is assumed to be selfadjoint. Formal asymptotic expansions of the eigenvalues and eigenfunctions of the singularly perturbed spectral problem are justified under assumptions regarding the initial problem and the limit problem.Bibliography: 16 titles.

Asymptotic expansions of solutions for the Boltzmann equation

Abstract: The Hilbert and Chapman-Enskog expansions approximate solutions of the Boltzmann equation, but each has some disadvantages: The Hilbert expansion, which results in nonlinear and linearized Euler equations, is invalid for weak shocks, weak boundary layers and long time asymptotics. The Chapman-Enskog expansion results in nonlinear Euler then Navier-Stokes then Burnett and super-Burnett equations. Navier-Stokes is correct for weak shocks, weak boundary layers and long time asymptotics, but the Burnett equations have spurious high order dispersive effects. In this paper a modified expansion is developed, which combines the best features of the two expansions. It results in nonlinear and linearized Navier-Stokes equations only and is valid in the above-mentioned regimes.

Analytical treatment of MOSFET source—Drain resistance

Abstract: Series resistance in the metal-oxide-semiconductor field-effect transistor becomes increasingly important as design rules shrink. Material properties associated with the interconnect metal, the semiconductor, and the interface separating the two regions thus assume greater importance. An analytical formulation of the resistance in terms of these material properties is thus quite desirable. We formulate and solve a two-dimensional model of current flow by the method of matched asymptotic expansions. The major utility of this solution is provided by higher order corrections to the standard transmission line model of current flow in the ohmic contact region. We find that present methods for the extraction of material properties from experimental measurements with a transmission line model analysis would be enhanced by the inclusion of higher order terms we present here.

Asymptotic expansions for a closed multiple access system

Abstract: A closed multiple access system composed of M sources and a single service facility is analyzed. The closed network is modeled as a finite source ${M / G / 1}$ queueing system. We consider systems with a large number of sources (i.e. $M \gg 1$) and we assume that the mean time required to process an individual request is short $(O({1 / M}))$. We then construct asymptotic approximations to the stationary distribution of the number of requests in the service facility by using the method of matched asymptotic expansions. We give formulas for the first and second moments of the number of requests, for all traffic intensities.

On the expansion wave problem in a long pipe with wal friction

Abstract: The method of matched asymptotic expansions is used to study the flow out of a long tube or pipeline caused by a sudden rupture. The flow in the tube generated by the expansion wave propagating into the tube from the break point is shown to be modified substantially by the presence of wall friction. This depends primarily on the fact that, when there is wall friction, the velocity derivative along the tube becomes singular at the broken end exit for all times, as long as the flow is critical at the exit.

Boundary layer solutions for some nonlinear elastic membrane problems

Abstract: The method of matched asymptotic expansions is used to describe finite axisymmetric deformations of two thin elastic membrane problems: a circular membrane with a small radial traction at the edge, an annular membrane with a small circular hole at the center. In both problems, the surface load is assumed to be an axial pressure. There is a boundary layer at the outer edge in the first problem, and at the hole in the second problem.

Normal co-ordinates and asymptotic expansions in resonance cases in celestial mechanics.

Abstract: In the presence of a single small-integer near commensurability of orbital period, the construction of a complete formal solution of the equations for the mutual perturbations in a planetary or satellite system, entirely in periodic terms, can be carried out after the use of a transformation of the variables which brings the quadratic terms of the Hamiltonian to a suitable normal form. A method for finding such a transformation is described.