Showing papers on "Method of matched asymptotic expansions published in 1970"
TL;DR: In this paper, a theory is proposed to describe the behavior of a class of turbulent shear flows as the Reynolds number approaches infinity, and a detailed analysis is given for simple representative members of this class, such as fully developed channel and pipe flows and two-dimensional turbulent boundary layers.
Abstract: A theory is proposed in this paper to describe the behaviour of a class of turbulent shear flows as the Reynolds number approaches infinity. A detailed analysis is given for simple representative members of this class, such as fully developed channel and pipe flows and two-dimensional turbulent boundary layers. The theory considers an underdetermined system of equations and depends critically on the idea that these flows consist of two rather different types of regions. The method of matched asymptotic expansions is employed together with asymptotic hypotheses describing the order of various terms in the equations of mean motion and turbulent kinetic energy. As these hypotheses are not closure hypotheses, they do not impose any functional relationship between quantities determined by the mean velocity field and those determined by the Reynolds stress field. The theory leads to asymptotic laws corresponding to the law of the wall, the logarithmic law, the velocity defect law, and the law of the wake.
135 citations
TL;DR: In this article, matched asymptotic expansions are applied to the fluid model of the low-pressure positive column and the expansion of the eigenvalue in the plasma balance equation is obtained to second order in plane and in cylindrical geometry, and uniformly valid expressions for charged particle densities and fluid velocity in two separate regions are indicated.
Abstract: The method of matched asymptotic expansions is applied to the fluid model of the low-pressure positive column. The expansion of the eigenvalue in the plasma balance equation is obtained to second order in plane and in cylindrical geometry, and uniformly valid expressions for charged particle densities and fluid velocity in two separate regions are indicated.The free-fall model is also examined and the scales of the transition layer and sheath layer found. Comparison is made with the results of direct numerical integration of the equations involved for both models.
129 citations
73 citations
TL;DR: In this paper, an analysis of the facilitated transport problem with steady diffusion through a finite membrane, accompanied by rapid and reversible, homogeneous chemical reaction is presented, where matched asymptotic expansions are used to analyze the structure of the diffusion field, which is shown to consist of an equilibrium core, together with boundary-layer reaction zones at the membrane boundaries.
62 citations
01 Jan 1970
TL;DR: A review of asymptotic methods of solution can be found in this paper, where the authors describe the present level of development of such methods at their current level of research.
Abstract: The main aim of investigations undertaken during the early stages of research into differential equations was the derivation of exact solutions. It was subsequently found, however, that the effective representation of the exact solution in terms of elementary functions is possible only for a limited number of special classes of differential equations. Therefore, the question of methods for the construction of approximate solutions of differential equations was recognized to be the main area of research. Work in this area proceeded along two directions: a) the development of numerical methods of solution and b) the development of the so-called asymptotic methods of solution. The aim of this review is to describe asymptotic methods at their present level of development.
59 citations
57 citations
TL;DR: In this article, matched asymptotic expansions are employed to calculate the mass transport velocity due to small amplitude oscillatory waves propagating in conditions of density and viscosity discontinuities.
Abstract: The method of matched asymptotic expansions is employed to calculate the mass transport velocity due to small amplitude oscillatory waves propagating in conditions of density and viscosity discontinuities. For progressive waves in a two-layer system, it is found that the velocity at the interface is in the direction of wave propagation; when the uppermost surface is free, the velocity there is in the direction opposite to that at the interface. If the difference in the densities is small, the calculated transport velocity associated with an internal wave can be of more importance than that associated with the surface wave as obtained from the work of Longuet-Higgins (1953).
46 citations
38 citations
37 citations
TL;DR: In this paper, matched asymptotic expansions are applied to the problem of determining the azimuthal velocity field due to the opposing slow rotations of a pair of thin coaxial disks in a homogeneous, incompressible, viscous fluid.
Abstract: The method of matched asymptotic expansions is applied to the problem of determining the azimuthal velocity field due to the opposing slow rotations of a pair of thin coaxial disks in a homogeneous, incompressible, viscous fluid. An asymptotic expansion for the velocity field, which is uniformly valid as the ratio e of the separation between the disks to the disk diameter approaches zero, is given and integrated to obtain an asymptotic expansion for the turning moment which contains the full second approximation including terms of the orders e log2e, e loge, and e. Corresponding results are given for the electrostatic condenser. The plane counterparts of the circular‐disk viscometer and condenser are also considered.
34 citations
TL;DR: In this paper, matched asymptotic expansions are used to simplify calculations of noise produced by aerodynamic flows involving small perturbations of a stream of non-negligible subsonic Mach number.
Abstract: The method of matched asymptotic expansions is used to simplify calculations of noise produced by aerodynamic flows involving small perturbations of a stream of non-negligible subsonic Mach number. This technique is restricted to problems for which the dimensionless frequency e, defined as ωb/a0, is small, ω being the circular frequency, b the typical body dimension, and a0 the speed of sound. By combining Lorentz and Galilean transformations, the problem is transformed to a space in which the approximation appropriate to the inner region is found to be incompressible flow and that appropriate to the outer, classical acoustics. This approximation for the inner region is the unsteady counterpart of the Prandtl-Glauert transformation, but is not identical to use of that transformation in a straightforward quasi-steady manner. For wings in oscillatory motion, it is the same approximation as was given by Miles (1950).To illustrate the technique, two examples are treated, one involving a pulsating cylinder in a stream, the other the impinging of plane sound waves upon an elliptical wing in a stream.
TL;DR: In this article, an asymptotic expansion in terms of the amplitude and phase of the solution is developed for determining approximate solutions to a class of differential equations characterized by resonance phenomena where resonance phenomena (arising, for example, when ƒ(x, [xdot], t) = ǫ cosω0 t) may be neglected.
Abstract: A method is presented for determining approximate solutions to a class of differential equations characterized by: where resonance phenomena (arising, for example, when ƒ(x, [xdot], t) = x cosω0 t) may be neglected. The approximation is developed from an asymptotic expansion in terms of the amplitude and phase of the solution. Three examples are considered in illustration of the application of the approximation technique, and using an integral error function, solution error is shown graphically for these examples in terms of equation parameters. An expression for the approximate solution is derived which makes it possible to determine solution accuracy for any function f(x, x˙, t) once the approximate amplitude envelope and phase relationships have been derived. Graphical solutions demonstrate the accuracy which can be maintained even up to relatively large values of the parameter µ.
TL;DR: In this article, the authors used the comparison principle and the Schauder-Tychonoff fixed point theorem for perturbing the asymptotic manifold of a nonlinear scalar equation.
Abstract: if f(t, x) is sufficiently \"small\". The present paper investigates this problem further. In particular, our results are motivated by two recent studies. Brauer and Wong [1] obtained quite general results on the asymptotic relationships between the solutions of (1) and (2). In §3 we will weaken significantly the hypotheses of some of their results. Toroselidze [5] considered the problem of perturbing the asymptotic manifold of a nonlinear scalar equation. We will extend these results to the nonlinear system (2) and consider some related questions in §4. The main result of our paper establishes that, under certain conditions, the set of all initial positions of the solutions in a certain asymptotic manifold is an open set. Our techniques are a combination of the well-known comparison principle and the Schauder-Tychonoff fixed point theorem. Fundamental in the application of the comparison principle is a scalar equation which we will denote by
TL;DR: In this paper, a technique is developed by which high-frequency trapped scalar wave problems are reduced to finding solutions for a sequence of equations which are independent of the frequency, applied to whispering gallery waves, ducted waves, edge waves and surface waves.
Abstract: A technique is developed by which high-frequency trapped scalar wave problems are reduced to finding solutions for a sequence of equations which are independent of the frequency. The technique is applied to whispering gallery waves, ducted waves, edge waves and surface waves. Several successful comparisons are made between the asymptotic expansions of exact analytic solutions and the results obtained by using the technique while one numerical comparison shows that such results can be extremely accurate even when the frequency is low. The range of validity of the solutions is discussed with particular reference to numerical solutions.
TL;DR: In this article, analytic solutions of the time-dependent Nernst-Planck equations are obtained for voltage-clamp boundary conditions in the form of asymptotic expansions, valid for small and large times and small voltageclamp values.
Abstract: Some analytic solutions of the time‐dependent Nernst–Planck equations are obtained for “voltage‐clamp” boundary conditions. The solutions are in the form of asymptotic expansions, valid for small and large times and small voltage‐clamp values.
TL;DR: In this article, an asymptotic method was used to analyze the free vibrations of a solid elastic cylinder of circular cross section, where the displacement components and the frequencies were sought as power series of the dimensionless wavenumber e, where e = 2π ω× ω × ωRADIUS/WAVELENGTH.
Abstract: An asymptotic method is used to analyze the free vibrations of a solid elastic cylinder of circular cross section. In this method, the displacement components and the frequencies are sought as power series of the dimensionless wavenumber e, where e = 2π × RADIUS/WAVELENGTH. By substituting the expansions in the displacement equations of motion and the boundary conditions, and by collecting terms of the same order en, a system of coupled second‐order inhomogeneous ordinary differential equations is obtained with the radial variable as independent variable. Subsequent integration yields the coefficients of en for the displacement components and the frequencies for all modes and for the whole range of frequencies, but in a range of real‐valued dimensionless wavenumbers 0 < e < 1.
TL;DR: In this paper, the boundary layer theory was used to obtain asymptotic expansions near caustics and shadow boundaries, where the amplitude terms v become unbounded, and modified forms of the above form are needed.
TL;DR: Power series uniformization by transformations of perturbation functions irregular part, considering compatibility problem is proposed in this paper, where the authors consider compatibility problem in the power series regularization.
TL;DR: In this article, the authors use matched asymptotic expansions to solve the governing conservation equations of a spherical diffusion flame and reveal the influence of chemical kinetics on flame structure.
Abstract: Hydrogen gas is assumed to be injected through the surface of a ‘fuel spher’ into a pure oxygen atmosphere, and so to give rise to a spherical diffusion flame. The combustion process is presumed to be sustained by a set of five reactions which involve the six species H2, O2, O, H, OH and H2O, and a proper multi-component, variable diffusion-coefficient, description of the diffusion processes is therefore adopted. Use of the method of matched asymptotic expansions to solve the governing conservation equations makes it possible to deal analytically with this problem and, by focussing attention on the dominance of certain physical phenomena in special regions of the field, reveals the influence of chemical kinetics on flame structure. For example, inner region (i.e. intense reaction zone) behaviour indicates the crucial role played on the O2-rich side of the flame by the reaction OH + H; in one case it may produce O + H2 and in the other (with the aid of a third body) it may result in H2O and so act...
TL;DR: In this article, the authors formulated the definition of the asymptotic expansion of a generalized function depending on a parameter and proved a number of theorems about the properties of such expansions and operations.
Abstract: The definition is formulated of the asymptotic expansion of a generalized function depending on a parameter. A number of theorems are proved about the properties of asymptotic expansions and operations on them, in particular, theorems on differentiation and integration. For generalized functions of the formf (x)eixt,f (x) ɛS', t → ±∞ the relation is investigated between the singularity carrierf and the carrier of coefficient functionals.
TL;DR: In this paper, the correct asymptotic behavior of an inviscid radiating gas flow near the stagnation point of a blunt body in the weakly radiating and weakly absorbing limits is given.
Abstract: The correct asymptotic behavior of an inviscid radiating gas flow near the stagnation point of a blunt body in the weakly radiating and weakly absorbing limits is given. While reflecting different physical situations, the two limits are shown to imply similar singular perturbation behavior which has been overcome through use of the method of matched asymptotic expansions. Comparison of these results with those obtained using the PLK asymptotic technique indicate the failure of the latter to correctly overcome the singular behavior.
TL;DR: In this paper, the effect of a uniform current on the velocity field of a viscous, incompressible, conducting fluid streaming past a stationary conducting ellipsoid of revolution, assuming that the Reynolds number is small, was investigated.
Abstract: The method of matched asymptotic expansions is employed for investigating the effect of a uniform current on the velocity field of a viscous, incompressible, conducting fluid streaming past a stationary conducting ellipsoid of revolution, assuming that the Reynolds number is small. It is also assumed that at infinity the velocity and uniform current are parallel to the axis of the ellipsoid. It is found that the presence of the current increases or decreases the drag coefficient, depending on whether the fluid conductivity is larger than that of the ellipsoid or vice versa. It is suggested that this effect of the current on the drag coefficient holds for all axisymmetric bodies that are also symmetric about a plane perpendicular to their axis. The case of a circular disk broadside on the undisturbed current, obtained as a special case of a planetary ellipsoid, is slightly different; when the conductivity of the disk is non-zero the electromagnetically induced flow field vanishes.
TL;DR: In this article, the majorant function that is used in connection with the comparison technique is usually assumed to be non-increasing in the dependent variable, however, properties of the asymptotic manifold are discussed here under the opposite monotonicity assumption.
Abstract: The asymptotic properties of the solutions of a linear homogeneous system of differential equations determine, under suitable restrictions, the asymptotic properties of a set of solutions of a nonlinear perturbation of this linear equation. The comparison principle is used here to generate an asymptotic manifold of the perturbed equation. The majorant function that is used in connection with the comparison technique is usually assumed to be nondecreasing in the dependent variable. However, properties of the asymptotic manifold are discussed here under the opposite monotonicity assumption, namely, that the majorant function is nonincreasing in the dependent variable. This type of majorant, function arises, for example, in certain gravitation problems. The main result on the structure of asymptotic manifolds which have an asymptotic uniformity is that solutions close to the manifold are either in the manifold or do not exist in the future.
TL;DR: In this paper, a procedure is developed for finding asymptotic expansions at high frequencies of the solutions of Helmholtz's equation subject to boundary conditions on certain guiding surfaces.
Abstract: A procedure is developed for finding asymptotic expansions at high frequencies of the solutions of Helmholtz's equation subject to boundary conditions on certain guiding surfaces. This includes surface waves along surfaces of rather general shapes, and wave-guide modes in a class of non-uniform waveguides. Guided waves have some features of both eigenvalue (mode) and radiation problems. The method of this paper combines the two techniques, finding “modes” that propagate along rays in the general waveguide region and whose amplitudes vary along the paths of propagation. The phases of these modes are found from two coupled equations, one analogous to the eiconal equation of geometrical optics, and the other analogous to the eigenvalue or “transverse resonance” equation of waveguides. The amplitudes are asymptotic series in inverse powers of the wavenumber, and the coefficients satisfy a set of ordinary differential equations that can be solved recursively. It is found that the ray paths are not only functions of the refractive index (as in “pure” radiation problems), but depend also on the local geometrical properties of the guiding surface.