Showing papers on "Method of matched asymptotic expansions published in 1971"
TL;DR: In this article, matched asymptotic expansions were used for radiating systems in the near zone and wave zone. But their results do not depend upon any definition of gravitational field energy.
Abstract: This paper treats the slow‐motion approximation for radiating systems as a problem in singular perturbations. By using the method of matched asymptotic expansions, we can construct approximations valid both in the near zone and the wave zone. The outgoing‐wave boundary condition applied to the wave‐zone expansion leads, by matching, to a unique and easily calculable radiation resistance in the near zone. The method is developed and illustrated with model problems from mechanics and electromagnetism; these should form a useful and accessible introduction to the method of matched asymptotic expansions. The method is then applied to the general relativistic problem of gravitational radiation from gravitationally bound systems, where a significant part of the radiation can be attributed to nonlinear terms in the expansion of the metric. This analysis shows that the formulas derived from the standard linear approximation remain valid for gravitationally bound systems. In particular, it shows that, according to general relativity, bodies in free‐fall motion do indeed radiate. These results do not depend upon any definition of gravitational field energy.
153 citations
01 Jan 1971
TL;DR: In this article, the velocity field induced in the neighborhood of a vortex core by distant portions of the vortex line is calculated for a sinusoidally perturbed vortex filament and for a vortex ring.
Abstract: The linear stability of the trailing vortex pair from an aircraft is discussed. The method of matched asymptotic expansions is used to obtain a general solution for the flow field within and near a curved vortex filament with an arbitrary distribution of swirl and axial velocities. The velocity field induced in the neighborhood of the vortex core by distant portions of the vortex line is calculated for a sinusoidally perturbed vortex filament and for a vortex ring. General expressions for the self-induced motion are given for these two cases. It is shown that the details of the vorticity and axial velocity distributions affect the self-induced motion only through the kinetic energy of the swirl and the axial momentum flux. The presence of axial velocity in the core reduces both the angular velocity of the sinusoidal vortex filament and the speed of the ring. The vortex pair instability is then considered in terms of the more general model for self-induced motion of the sinusoidal vortex. The presence of axial velocity within the core slightly decreases the amplification rate of the instability. Experimental results for the distortion and breakup of a perturbed vortex pair are presented.
152 citations
TL;DR: In this article, a method of solution for flows through small apertures in otherwise impermeable barriers was proposed for water-wave flows through a small barrier, which is an application of the method of matched asymptotic expansions.
Abstract: A method of solution is proposed for flows through small apertures in otherwise impermeable barriers. This method, which is an application of the method of matched asymptotic expansions, is used to solve a specific water-wave problem, yielding an approximate formula for the transmission coefficient.
65 citations
TL;DR: Asymptotic expansions have been obtained using two theorems due to Olver for the Jacobi polynomials and an associated function as mentioned in this paper, which are uniformly valid for complex arguments over certain regions, for large values of the order.
Abstract: Asymptotic expansions have been obtained using two theorems due to Olver for the Jacobi polynomials and an associated function. These expansions are uniformly valid for complex arguments over certain regions, for large values of the order.
54 citations
TL;DR: In this paper, a function y E C':[[t,, co), R] which satisfies (1.1) is called oscillatory if it is eventually of constant sign.
49 citations
TL;DR: In this article, the multitime method was used to obtain uniform asymptotic expansions of the solution of the damped linear oscillator with small damping, which are uniformly valid for all time.
Abstract: In this paper we consider the damped linear oscillator with small damping $\varepsilon $. We obtain uniform asymptotic expansions of the solution as $\varepsilon \to 0$ that are uniformly valid for all time $t \geqq 0$, by the multitime method. We show how to determine the expansion coefficients without resorting to intuitive arguments. This is done by considering the remainder in the expansion of the solution and by requiring that it be made small in a way that is precisely defined in the paper. This analysis also yields proofs of the uniform asymptotic convergence of the expansions. We find that there are a minimum number of time scales, namely two, that are required to obtain a uniform asymptotic expansion. For a fixed number of terms in the expansion there are a maximum number of time scales, namely three, that give uniform expansions with the smallest estimate of the remainder. Finally we show how to apply the analysis to obtain uniformasymptotic expansions of a mixed, initial boundary value problem for the damped wave equation.
40 citations
TL;DR: In this paper, the heat transport equation along the stagnation point streamline is solved by the method of matched asymptotic expansions as a first approximation of the temperature distribution directly beneath an oceanic rise.
Abstract: The heat transport equation along the stagnation point streamline is solved by the method of matched asymptotic expansions as a first approximation of the temperature distribution directly beneath an oceanic rise. Far below the ocean bottom the temperature T(Z, e) is given by the temperature calculated for adiabatic (upward) displacement. In a narrow region near the surface, the thermal boundary layer, T(Z, e) rapidly adjusts through conductive heat flow to match the surface temperature. The linear coefficient of the series expansion of the vertical velocity distribution υ(Z) can be determined from the surface heat flow near the rise crest Qcrest. Under simplifying assumptions an estimate of the maximum value of υ(Z) can be obtained. Qcrest is primarily controlled by the thickness of, and temperature at the base of, the thermal boundary layer; at greater depth, a wide range of velocity and temperature distributions can exist without materially altering the thermal structure of the boundary layer. Within the limitations of the assumed model, provided that vertical velocities at depth are comparable to horizontal spreading velocities away from the rise, observed values of Qcrest are too low by at least several factors of 2. Low values of Qcrest can be rationalized as a time-dependent effect if igneous emplacement at the axis of spreading has been at a relatively low level for the past 0.5–1 m.y. The thickness of the thermal boundary layer is probably the primary constraint on the seismic depth of spreading centers and associated transform faults.
37 citations
TL;DR: In this article, the problem of slow flow past a droplet is considered where both materials may be represented as fluids of grade 3 and where the outer fluid is of infinite extent, and a double perturbation technique and the method of matched asymptotic expansions are employed to obtain solutions to the equation of motion.
Abstract: The problem of slow flow past a droplet is considered where both materials may be represented as fluids of grade 3 and where the outer fluid is of infinite extent. Both non-Newtonian and inertial effects are included in the analysis. A double perturbation technique and the method of matched asymptotic expansions are employed to obtain solutions to the equation of motion.
Solutions, in the form of Legendre polynomial series, are obtained for the stream function (both outside and inside the droplet), for the drag force exerted on the droplet, and for the shape of the droplet.
The results obtained are in complete agreement with those obtained by other workers for the flow of a Newtonian fluid past a Newtonian droplet and for the flow of a fluid of grade 3 past a solid sphere. Droplet shape predictions are in qualitative agreement with experimentally observed shapes.
28 citations
TL;DR: In this paper, an extension of the first-order analysis is presented, in which the expansion parameter is the eigenvalue and matching of both amplitude and phase of functions are required.
25 citations
TL;DR: In this paper, the authors considered the problem of optimal lift control of a hypersonic lifting body during atmospheric entry for the drag coefficient, a function of the angle of attack, and the atmospheric density, an arbitrary function of altitude.
Abstract: This paper considers the problem of optimal lift control of a hypersonic lifting body during atmospheric entry for the drag coefficient, a function of the angle of attack, and the atmospheric density, an arbitrary function of altitude. The solution obtained is valid for entering the planetary atmosphere from the Keplerian region, as well as from low altitudes. The method of matched asymptotic expansions was employed, and separate expansions were derived for the Keplerian region and for the aerodynamic region, where the aerodynamic forces are dominant. Lagrange multipliers and state variables obtained in closed form for both expansions were matched in the overlap domain. A method for estimating the order of magnitude of multipliers in various regions was discussed and will be useful in applying singular perturbation methods to a wider class of optimal control problems. For unbounded control, the lift variation can be classified into four different programs depending on the terminal altitude. Results were compared with the numerical solutions obtained by the method of steepest descent. For bounded control, there exist 12 different sequences of arcs which reduce to those obtained in an earlier study as the drag coefficient becomes independent of angle of attack.
22 citations
01 Mar 1971
TL;DR: In this paper, the axially symmetric flow past a rotating sphere due to a uniform stream of infinity is considered and the leading terms for the flow consist of a linear superposition of a primary Stokes flow past the non-rotating sphere together with an antisymmetric secondary flow in the azimuthal plane induced by the spinning sphere.
Abstract: Keller and Rubinow(l) have considered the force on a spinning sphere which is moving through an incompressible viscous fluid by employing the method of matched asymptotic expansions to describe the asymmetric flow. Childress(2) has investigated the motion of a sphere moving through a rotating fluid and calculated a correction to the drag coefficient. Brenner(3) has also obtained some general results for the drag and couple on an obstacle which is moving through the fluid. The present paper is concerned with a similar problem, namely the axially symmetric flow past a rotating sphere due to a uniform stream of infinity. It is shown that leading terms for the flow consist of a linear superposition of a primary Stokes flow past a non-rotating sphere together with an antisymmetric secondary flow in the azimuthal plane induced by the spinning sphere. For a 3 n 2 > 6 Uv , where n is the angular velocity of the sphere, U the speed of the uniform stream, and a the radius of the sphere, there is in the azimuthal plane a region of reversed flow attached to the rear portion of the sphere. The structure of the vortex is described and is shown to be confined to the rear portion of the sphere. A similar phenomenon occurs for a sphere rotating about an axis oblique to the direction of the uniform stream but the analysis will be given in a separate paper.
TL;DR: In this article, a second-order pseudo-parabolic equation with sufficiently smooth time-independent coefficients in a cylindrical domain is considered and its pointwise values decay exponentially as t→∞ while its Dirichlet norm grows expontially as t →−∞.
Abstract: Consider a solution to a second-order pseudo-parabolic equation with sufficiently smooth time-independent coefficients in a cylindrical domain. If it vanishes on the cylindrical surface for all times and if its restriction to a fixed instant belongs toC2+a, then its pointwise values decay exponentially as t→∞ while its Dirichlet norm grows expontially as t→−∞. Similar conclusion still hold for solutions to non-homogeneous equations under non-homogeneous boundary conditions provided the free term and the boundary data posses these asymptotic behaviors.
TL;DR: In this paper, the effect of porosity on the drag coefficient of a flow past a porous sphere placed in an otherwise uniform incident stream at low Reynolds number was investigated, and it was found that porosity reduces the effective radius of the sphere by a factor (1 + k'/2a2)−1, where k' is the permeability.
Abstract: The present investigation is concerned with the problem of studying the effect of permeability on drag coefficient for the flow past a porous sphere placed in an otherwise uniform incident stream at low Reynolds number. The problem is formulated using the full Navier-Stokes equations describing the flow outside the sphere while Darcy's law governs the flow inside the sphere. The solution is, then, sought by the method of matched asymptotic expansions involving three simultaneous expansions up to an order Re. It is found that the effect of porosity on the drag is that it reduces the effective radius a of the sphere by a factor (1 + k'/2a2)−1, where k' is the permeability.
TL;DR: In this paper, a uniformly valid asymptotic solution has been constructed for three-dimensional jet-flapped wings by the method of matched asymmetric expansions for high aspect ratios.
Abstract: A uniformly valid asymptotic solution has been constructed for three-dimensional jet-flapped wings by the method of matched asymptotic expansions for high aspect ratios. The analysis assumes that the flow is inviscid and incompressible and is formulated on the thin airfoil theory in accordance with the well-established Spence (1961) theory in two dimensions. A simple method emerges in treating the bound vortices along the jet sheet which forms behind the wing with the aid of the following physical picture. Three distinct flow regions—namely inner, outer and Trefitz—exist in the problem. Close to the wing the flow approximates to that in two dimensions. Therefore, Spence's solution in two dimensions applies. In the outer region a wing shrinks to a line of singularities from which the main disturbances of flow in this region arise. In particular, we find that the shape of the jet sheet, hence the strength of vortices, is now predetermined by the strength of the singularities there. Hence a complete flow field in the outer region can now be determined first by evaluating the flow due to various degrees of singularities along this line and then adding the effect of the jet bound vortices which is now known. Far removed from the wing, the well-known Trefftz region exists in which calculations of aerodynamic forces can be most easily done. The result has been applied to various wing planforms such as cusped, elliptic and rectangular wings. The present result breaks down for rectangular wings. However, we can apply Stewartson's (1960) solution for lifting-line theory for semi-infinite rectangular wings, because, to this second-order approximation it is established that the jet sheet in the outer region makes no contribution to lift, with the direct contribution of the deflected jet at the exit being cancelled by the reduced circulation in the jet vortices. This result for the rectangular wing gives excellent agreement with the experiments made on a rectangular wing, while the result for elliptic wings underestimates them considerably.
TL;DR: Stability type results concerning continuous solutions of perturbed systems of Volterra equations as well as theorems concerning the asymptotic equivalence of such perturbed and unperturbed systems are established in this article.
Abstract: Stability type results concerning continuous solutions of perturbed systems of Volterra equations as well as theorems concerning the asymptotic equivalence of such perturbed and unperturbed systems are established.
TL;DR: In this article, matched asymptotic expansion was applied to the analysis of infrared radiative energy transfer within a planetary atmosphere, and the results showed that the temperature distribution in the lower portion of the atmosphere is independent of both concentration of absorbing gas and rotational line structure of the vibrational-rotation bands.
Abstract: The method of matched asymptotic expansions is applied to the analysis of infrared radiative energy transfer within a planetary atmosphere. The goal is to illustrate, by means of a simple mathematical model, qualitative features of atmospheric thermal structure. The matched asymptotic expansion is employed in the upper portion of the atmosphere, and the analysis illustrates that the temperature distribution in the lower portion of the atmosphere is independent of both concentration of absorbing gas and rotational line structure of the vibrational-rotation bands.
TL;DR: In this article, the authors present some results on asymptotic error expansions of the approximate solutions of differential equations by finite-difference schemes, in an effort to make the presentation self-contained, as well as discuss recent work by Engquist on linear multistep methods for initial value problems, by Kreiss on extrapolation procedures for elliptic finite difference schemes, by Pereyra and co-workers on iterated deferred-correction methods for ellic equations and by the author and Hald on the finite element method.
Abstract: In this paper we will present some results on asymptotic error expansions of the approximate solutions of differential equations by finite-difference schemes. We will present some quite well-known material, in an effort to make the presentation self-contained, as well as discuss recent work by Engquist on linear multistep methods for initial-value problems, by Kreiss on extrapolation procedures for elliptic finite-difference schemes, by Pereyra and co-workers on iterated deferred-correction methods for elliptic equations and by the author and Hald on the finite-element method. Practical aspects of the subject as well as the use of error expansions as a technical device in theoretical numerical analysis are discussed.
01 Jan 1971
01 Apr 1971
TL;DR: Linear neutral differential equation uniform asymptotic stability relation to perturbed differential equation containing bounded linear operators was studied in this paper, where the stability relation was shown to be independent of the number of linear operators.
Abstract: Linear neutral differential equation uniform asymptotic stability relation to perturbed differential equation containing bounded linear operators
TL;DR: An approximate theory for the vibration of non-homogeneous, anisotropic plates is presented in this article, where the equations are derived by use of the asymptotic integration of the elasticity equations.
Abstract: An approximate theory for the vibration of non-homogeneous, anisotropic plates is presented. The equations are derived by use of the asymptotic integration of the elasticity equations. Although use of this method leads to field equations of various orders, only the first approximation equations are derived here. The application to layered plates is also shown.
TL;DR: In this paper, conditions are established which guarantee asymptotic stability, ultimate boundedness, and convergence of solutions of (1.1) equations of the form (1).
Abstract: In this paper three new results are obtained for equations of the form (1.1). Conditions are established which guarantee asymptotic stability, ultimate boundedness, and convergence of solutions of (1.1).
01 Jan 1971
TL;DR: In this article, Brauer et al. investigated the asymptotic relationship between the solutions of a linear differential system and its perturbed system and showed that the results depend upon a known result of F. Brauer and Wong.
Abstract: We investigate the asymptotic relationship between the solutions of a linear differential system and its perturbed system. Our results depend upon a known result of F. Brauer on asymptotic equilibrium. We also study the asymptotic manifold of solutions of the nonlinear system generated by the solutions of the corresponding linear system. 1. Recently, Brauer and Wong [2] and Onuchic [5] obtained general results on the asymptotic relationship between the solutions of a linear differential system and its perturbed system. Toroselidze [6] considered the problem of perturbing the asymptotic manifold of a nonlinear scalar equation. Hallam and Heidel [3] extended the results of Toroselidze to nonlinear systems and also discussed the asymptotic relationships of solutions. We, in the present paper, wish to investigate these problems further. Our results depend upon a known basic result [1 ] on asymptotic equilibrium and consequently the proofs are short and fundamental. Our approach has thrown considerable light on the problems and at the same time improved the results of [2] and [3] by weakening the hypotheses. This is explained in detail in a remark at the end of the paper. 2. Let J denote the half-line 0
TL;DR: In this article, the authors investigated the flow past a circular cylinder moving close to a free surface at high Froude number by the method of matched asymptotic expansions, where the inner flow model is that of a non-separated non-linear gravity-free flow past the doublet, while the linear outer solution is a singularity a t the free surface.
Abstract: The flow past a circular cylinder moving close to a free surface at high Froude number is investigated by the method of matched asymptotic expansions. In contrast with the linearized solution in which the dimensionless depth of immersion h = h′g/U′2 is kept constant, in the present analysis h → 0 as Fr → ∞.The inner flow model is that of a non-separated non-linear gravity-free flow past a doublet, while the linear outer solution is that of a singularity a t the free surface. At deep submergence the solution coincides with the linearized solution. At moderate immersion depths the linearized solution is still valid, provided that the depth is replaced by an effective depth, larger than the actual one. For a body close to the free surface the non-linear solution differs significantly from the linearized solution.
TL;DR: In this paper, it is shown that an asymptotic solution to the Hamilton-Jacobi functional differential equation provides an approximate general solution to all nonlinear classical field equations.
Abstract: By employing the terminology of functional differential calculus, Hamilton-Jacobi theory is extended to apply to classical field equations. It is shown that an asymptotic solution to the Hamilton-Jacobi functional differential equation provides an asymptotic general solution to the associated nonlinear classical field equations.
TL;DR: In this paper, the matched asymptotic expansion method was used to formulate a lift-line theory for high-aspect-ratio wing with a jet flap, and the result is a simple equation for the lift coefficient.
Abstract: The method of matched asymptotic expansions, which Van Dyke used to formulate a lift- ing-line theory for high-aspect-ratio wings, is applied to a wing with jet flap. The develop- ment differs from Van Dyke's in that velocity components instead of the velocity potential are the dependent variables, thin-airfoil approximations are used throughout, and the jet flap is present. The theory is given for the case of a flat elliptic wing with spanwise-uniform jet- momentum coefficient and jet angle and the result is a simple equation for the lift coefficient. Comparison with the results of two earlier finite-aspect-ratio jet-flap theories shows close agreement. Certain approximations needed in the earlier theories to solve the integral equa- tion for the upwash induced by a semi-infinite lifting surface are avoided by consistent use of the principle that, in the limit as the aspect ratio becomes infinite, the change in lift due to induced incidence is much smaller than the lift, so that the integral equation does not have to be solved.
TL;DR: In this article, the behavior of unidirectionally reinforced composites under intense impact conditions is considered and the composite behavior can be asymptotically described in terms of effective conservation equations and associated jump conditions, which differ from the ordinary balance equations by the presence of a compaction ratio which appears as a measure of compactibility of composite materials.
Abstract: The behavior of unidirectionally reinforced composites under intense‐impact conditions is considered. Based on asymptotic expansions of the basic equations it is shown that the composite behavior can be asymptotically described in terms of effective conservation equations and associated jump conditions. In particular, it is found that such equations differ from the ordinary balance equations by the presence of a compaction ratio which appears as a measure of compactibility of composite materials. An effective caloric equation of state is also derived from the expansions as the weighted geometric mean of the constituent equations of state.
01 Jan 1971
TL;DR: In this article, the problem of diffraction of a weak shock wave near a rigid wall with a kink is examined, and a uniformly applicable solution of the problem is constructed in the first approximation.