Showing papers on "Method of matched asymptotic expansions published in 1969"

On the method of matched asymptotic expansions

Abstract: The method of matched (or of ‘inner and outer’) asymptotic expansions is reviewed, with particular reference to two general techniques which have been proposed for ‘matching’; that is, for establishing a relationship between the inner and outer expansions, to finite numbers of terms, of an unknown function. It is shown that the first technique, which uses the idea of overlapping of the two expansions, can be difficult and laborious in some applications; while the second, which is the ‘asymptotic matching principle’ in the form stated by Van Dyke(13) can be incorrect. Two different sets of conditions sufficient for the validity of the asymptotic matching principle are then established, on the basis of assumptions about the structure of expressions which approximate to the desired function f(x,∈) for all relevant values of x. Finally, it is noted that in four classes of singular-perturbation problems for which complete and rigorous asymptotic theories exist, uniform approximations to the solutions have a structure which is a particular form of the general one assumed in this paper.

Lee waves in a stratified flow. Part 4. Perturbation approximations

Abstract: A two-dimensional stratified flow over an obstacle in a half space is considered on the assumptions that the upstream dynamic pressure and density gradient are constant (Long’s model). A general solution of the resulting boundary-value problem is established in terms of an assumed distribution of dipole sources. Asymptotic solutions for prescribed bodies are established for limiting values of the slenderness ratio s (heightlbreadth) of the obstacle and the reduced frequency k (inverse Froude number based on the obstacle breadth) as follows: (i) s-f 0 with k fixed; (ii) k+O with s fixed; (iii) k .+ 00 with ks fixed. The approximation (i) is deveoped to both first (linearized theory) and second order in s in terms of Fourier integrals. The approximation (ii), which constitutes a modification of Rayleigh-scattering theory, is obtained by the method of matched asymptotic expansions and depends essentially on the dipole form (which is proportional to the sum of the displaced and virtual masses) of the obstacle with respect to a uniform flow. A simple approximation to this dipole form is proposed and validated by a series of examples in an appendix. The approximation (iii) is obtained through the reduction of the original integral equation to a singular integral equation of Hilbert’s type that is solved by the techniques of function theory. A composite approximation to the lee-wave field that is valid in each of the limits (i)-(iii) also is obtained. The approximation (iii) yields an estimate of the maximum value of ks for which completely stable lee-wave formation behind a slender obstacle is possible. The differential and total scattering cross-sections and the wave drag on the obstacle are related to the power spectrum of the dipole density. It is shown that the drag is invariant under a reversal of the flow in the limits (i) and (ii), but only for a symmetric obstacle in the limit (iii). The results are applied to a semi-ellipse, an asymmetric generalization thereof, the Witch of Agnesi (Queney’s mountain), and a rectangle. The approximate results for the semi-ellipse are compared with the more accurate results obtain by Huppert & Miles (1969). It appears from this comparison that the approximate solutions should be adequate for any slender obstacle within the range of ks for which completely stable lee-wave formation is possible. The extension to obstacles in a channel of finite height is considered in an appendix.

A rational strip theory of ship motions: part i.

Abstract: The exact ideal-fluid boundary-value problem is formulated for a ship forced to heave and pitch sinusoidally in otherwise calm water. This problem is then simplified by applying three restrictions: 1) the body must be slender; 2) the motions must be small in amplitude compared with ship beam or draft; 3) the frequency of oscillation, must be high, based on the slenderness parameter. The hydrodynamic problem is then recast as a singular perturbation problem which is solved by the method of matched asymptotic expansions. Formulas are derived for the hydrodynamic heave force and pitch moment, from which added-mass and damping coefficients can be easily obtained. The latter are similar but not identical to those used in several other versions of "strip theory;" in particular, the forward-speed effects have the symmetry required by the theorem of Timman and Newman, A result which has not been realized in previous versions of strip theory. In order to calculate the coefficients by the formulas derived, it is necessary to solve numerically a set of boundary-value problems in two dimensions, namely, the problem of a cylinder oscillating vertically in the free surface, At least two practical procedures are available to this purpose.

Free convection at low Prandtl numbers

Abstract: In this paper it is shown that the free convection boundary layer approaches a singular character if the Prandtl number tends to zero. The method of matched asymptotic expansions is used to integrate the equations for this extreme case. An expression is derived for the Nusselt—Grashof relation and the results are compared with those of previous investigations which attack the problem in a different way.

Lateral motion of a slender body between two parallel walls

Abstract: The method of matched asymptotic expansions is used to determine the lateral flow of an ideal fluid past a slender body, when the flow is constrained by a pair of closely spaced walls parallel to the long axis of the body. In the absence of walls, the flow field would be nearly two-dimensional in the cross-flow plane normal to the body axis, but the walls introduce an effective blockage in the cross-flow plane, which causes the flow field to become three-dimensional. Part of the flow is diverted around the body ends, and part flows past the body in the inner cross-flow plane with a reduced 'inner stream velocity'. An integro-differential equation of identical form to Prandtl's lifting-line equation is derived for the determination of this unknown inner stream velocity in the cross-flow plane. Approximate solutions are applied to determine the added mass and moment of inertia for a accelerated body motions and the lift force and moment acting on a wing of low aspect ratio. It is found that the walls generally increase these forces and moments, but that the effect is significant only when the clearance between the body and the walls is very small.

On a Nonlinear Differential Equation of Electrohydrodynamics

Abstract: A nonlinear differential equation which governs the equilibrium of two closely spaced drops, suspended from two circular rings, and maintained at electric potentials ± V is considered. When the potential reaches a critical value V m , experiments show that the drops coalesce and the equation has, in fact, no solutions for V > V m . Two limiting cases are studied. At first we consider drops which at zero potential difference, are films of zero curvature. In this case it is shown that the equation may not have a unique solution for V m . In the second case we study drops which are nearly touching when there is no potential difference, with non-zero curvature. Using the method of matched asymptotic expansions, it is shown that when the drops are about to coalesce, the original separation distance is reduced by one-half. Thus the drops are not drawn out as might have been expected from experiments on isolated drops.

Reaction-Broadening in a Hydrogen-Oxygen Diffusion Flame

Abstract: The present paper is a continuation of the study of diffusion flames in a hydrogen-oxygen mixture which began with an investigation of equilibrium-broadening by the writer (1968). In the present model the hydrogen dissociation reaction is assumed to be so slow that it can be neglected. The effect of the remaining reaction rates on flame structure is then investigated by the method of matched asymptotic expansions, which provides information about the orders of magnitude of the various species' concentrations, and identifies simplified sets of equations which govern behaviour in various regions of the flame. The general forms of the solutions are established; attention is confined to analytical questions.

On the Use of Asymptotic Solutions to Plane Ice—Water Problems

Abstract: The paper considers one-dimensional freezing and thawing of ice–water systems for the conditions first examined by Stefan. An order-of-magnitude analysis applied to the governing equations and boundary conditions quantifies the error resulting from the neglect of various factors. Principal among these are density difference, initial superheat and variable properties. Asymptotic solutions for the temperature distribution and interface history are developed for a wide range of boundary conditions: prescribed temperature or heat flux, prescribed convection and prescribed radiation. Comparison with known results reveals the general adequacy of the asymptotic solutions and an estimate of the error incurred.

Asymptotic solutions of the Navier-Stokes equations in regions with large local perturbations

Abstract: There are many problems of the dynamics of viscous flows of liquids and gases at high Reynolds numbers for the solution of which the classical theory of the boundary layer cannot be used This applies, in particular, to all the problems with various sorts of local singularities in the stream-flows in the vicinity of corners, in regions of interaction of the boundary layer with an incident shock, flows near points of separation or attachment of the stream, etc The purpose of the present paper is to attempt the theoretical investigation of problems of this type on the basis of the general analysis of the asymptotic behavior of the solutions of the Navier-Stokes equations In order to do this, use is made of the familiar method of the construction and splicing of a combination of asymptotic expansions representing the solutions in the various characteristic regions of the stream with viscosity decreasing without bound [1] As an example, detailed consideration is given to the problem of viscous supersonic flow near a wall with large local curvature of the surface

On the method of matched asymptotic expansions: Part III: Two boundary-value problems

Abstract: The formal method of matched expansions is applied to two further examples. The first concerns the magnetic field induced by a steady current in a thin toroidal wire. The second, which involves a non-linear ordinary differential equation of the fourth order, has been chosen to resemble the problem of flow past a circular cylinder at small Reynolds numbers. The results of the formal procedure are proved in each case to be expansions of the exact solution.

An Analysis of the Time-Dependent Neutron Transport Equation with Delayed Neutrons by the Method of Matched Asymptotic Expansions

Abstract: The time-dependent neutron transport equation is treated as a problem in singular perturbation theory. The method of matched asymptotic expansions is used to find equations yielding approximate sol...

Two-variable expansions and singular perturbation problems.

Abstract: The two-variable, or two-timing, procedure is formulated in a different and more general manner than usual. The procedure is first applied to an ordinary differential equation of the type ordinarily solved by the method of matched asymptotic expansions and is then applied to two wave propagation problems described by partial differential equations. The results are compared with results obtained by the method of matched asymptotic expansions and by a coordinate stretching method.

Asymptotic methods in the theory of ordinary linear differential equations

Abstract: In this paper we consider systems of the form (1)on the semiaxis , where is a column vector with components, is an matrix, and is a parameter. We pose the problem of finding the asymptotic behavior of the solutions of equation (1) as and .

Diffraction of Plane Acoustic Waves and Pulses as a Singular Perturbation Problem

Abstract: The diffraction of plane waves and pulses by cylindrical cavities is studied as a singular perturbation problem for long waves and broad pulses. The solution, which is obtained to a given order of approximation by the method of matched asymptotic expansions, is applicable to arbitrary cavity shapes and, in the case of pulse diffraction, incident profiles of arbitrary form.

Low Reynolds number flow of a variable property gas past a heated circular cylinder

Abstract: The low Reynolds number flow of a variable property gas past an infinite heated circular cylinder is studied when the temperature difference between the cylinder and the free stream is appreciable. The velocity field (and hence the drag on the cylinder) is calculated by the method of matched asymptotic expansions. It is found that the zero-order velocity field calculated on the Stokes approximation satisfies both the no slip condition at the cylinder and the uniform stream condition at infinity which is in strong contrast with the corresponding velocity field for incompressible slow flow past an unheated cylinder where the uniform stream condition at infinity cannot be satisfied. When the temperature of the cylinder is twice the temperature at infinity it is found that the drag on the cylinder is almost twice the drag on a similar unheated cylinder.

Uniform asymptotic solution of second order linear differential equations without turning varieties

Abstract: The purpose of this paper is to initiate the study of a new kind of asymptotic series expansion for solutions of differential equations containing a parameter. We obtain uniform asymptotic solutions for certain equations of the form ê»y\" = ait, e)y , ( )' = d/dt, where n is a positive integer, t and e are real variables ranging over \\t\\ S U, 0 < e ^ €o, and a is a function infinitely differentiable on the closure of this domain. We require that a(i, e) satisfy conditions which can be regarded as generalized nonturning-point conditions. These conditions imply the absence of secondary turning points, and reduce in the simplest case to the condition a(¿, 0) 5¿ 0, but also include cases (the interesting ones) in which a(0, 0) = 0. |

Asymptotic behavior and operator mass renormalization in the ϕ4 model

Abstract: As is well known in the theory of nonlinear differential equations, straightforward perturbation theory leads to approximate solutions with the wrong asymptotic behavior. We consider a real scalar field satifying the Klein-Gordon equation with a ϕ3 self-current, in both the classical and second-quantized versions. We demonstrate that the usual, approximate solutions are asymptotically unbounded in both cases. We present a solution up to a normalization factor for the function or Heisenberg operator ϕ to first order in the coupling with the correct asymptotic behavior. In the quantum case this forces one to introduce aq-number renormalized mass. The discussion is limited to solutions periodic in a finite box. A discussion of the wave-function renormalization and the limit of infinite box volume is deferred here. The method of approximation used is a generalization of standard techniques in the theory of nonlinear differential equations, and does not require adiabatic switching of the interaction.

Approximate Solution of One-Point Reactor Kinetic Equations for Arbitrary Reactivities

Abstract: This paper describes a very simple formulation of the reactor kinetic equation for arbitrary reactivity variations which can be solved analytically. The method of matched asymptotic expansions, which is a generalization of methods used in boundary-layer analysis, is employed to estimate the neutron density and the reactor period for ramp and periodic inputs. The small amounts of error arising in individual cases are analyzed quantitatively by comparison with results obtained from difference approximation (Runge-Kutta-Merson method). The validity of the zero-prompt-lifetime approximation and the stability condition for periodic inputs are also discussed. It is confirmed that the results obtained by the present method are numerically in complete agreement with those by other methods, provided the magnitudes of bias reactivity | ρ0 |, reactivity amplitude | ρ1 | and ramp reactivity | γt | are all very small compared with β, that the angular frequency ω≪β/l*, and that, in particular, l*≪10−3.

Asymptotic methods in the theory of optimum correction for systems with slowly varying parameters

Abstract: This paper deals with the problem of optimum control for the case where the constraints are linear differential equations and the functional is of quadratic type. It is assumed that the matrix of these simultaneous equations dependsslowly on the time. The paper describes a method which permits to construct effectively asymptotic solutions of such a problem when the matrix of the characteristic equation which corresponds to Pontryagin's set of equations does not have any simple elementary divisors.