Showing papers on "Method of matched asymptotic expansions published in 1982"
TL;DR: In this article, an equation for the propagation of the discontinuity surface for arbitrary flame shapes in general fluid flows is derived, where the structure of the flame is considered to consist of a boundary layer in which the chemical reactions occur, located inside another boundary layer, in which transport processes dominate.
Abstract: Early treatments of flames as gasdynamic discontinuities in a fluid flow are based on several hypotheses and/or on phenomenological assumptions. The simplest and earliest of such analyses, by Landau and by Darrieus prescribed the flame speed to be constant. Thus, in their analysis they ignored the structure of the flame, i.e. the details of chemical reactions, and transport processes. Employing this model to study the stability of a plane flame, they concluded that plane flames are unconditionally unstable. Yet plane flames are observed in the laboratory. To overcome this difficulty, others have attempted to improve on this model, generally through phenomenological assumptions to replace the assumption of constant velocity. In the present work we take flame structure into account and derive an equation for the propagation of the discontinuity surface for arbitrary flame shapes in general fluid flows. The structure of the flame is considered to consist of a boundary layer in which the chemical reactions occur, located inside another boundary layer in which transport processes dominate. We employ the method of matched asymptotic expansions to obtain an equation for the evolution of the shape and location of the flame front. Matching the boundary-layer solutions to the outer gasdynamic flow, we derive the appropriate jump conditions across the front. We also derive an equation for the vorticity produced in the flame, and briefly discuss the stability of a plane flame, obtaining corrections to the formula of Landau and Darrieus.
677 citations
TL;DR: In this article, a matched asymptotic expansion of a small parameter L/a, where a is the particle radius and L is the length scale characteristic of the physical interaction between solute and particle surface, was used to obtain an expression for particle velocity.
Abstract: When a particle is placed in a fluid in which there is a non-uniform concentration of solute, it will move toward higher or lower concentration depending on whether the solute is attracted to or repelled from the particle surface. A quantitative understanding of this phenomenon requires that the equations representing conservation of mass and momentum within the fluid in the vicinity of the particle are solved. This is accomplished using a method of matched asymptotic expansions in a small parameter L/a, where a is the particle radius and L is the length scale characteristic of the physical interaction between solute and particle surface. This analysis yields an expression for particle velocity, valid in the limit L/a → 0, that agrees with the expression obtained by previous researchers. The result is cast into a more useful algebraic form by relating various integrals involving the solute/particle interaction energy to a measurable thermodynamic property, the Gibbs surface excess of solute Γ. An important result is that the correction for finite L/a is actually O(Γ/C∞ a), where C∞ is the bulk concentration of solute, and could be O(1) even when L/a is orders of magnitude smaller.
426 citations
TL;DR: In this article, the generalized Einstein equations derived from the Lagrangian were investigated and an approximate solution without singularity was constructed using the method of matched asymptotic expansions.
Abstract: We investigate the generalized Einstein equations derived from the Lagrangian which is an arbitrary function ofR. The importance of the saturation phenomenon is underlined, which may replace the role of a cosmological constant. The spherically symmetric homogeneous model is analyzed in more detail, and an approximate solution without singularity is constructed using the method of matched asymptotic expansions.
197 citations
01 Nov 1982
TL;DR: In this article, a matched asymptotic expansion of the inner, intermediate and outer layers of the turbulent pipe flow is analyzed by the method of matched expansion of millikan's argument leading to two overlap domains where velocity distribution is logarithmic but their slopes could be different.
Abstract: The fully developed mean turbulent pipe flow is analysed at large Reynolds number by the method of matched asymptotic expansions. From the study of various limiting processes, in the sense of Kaplun, a crucial intermediate limit is identified whose transverse dimension is of the order of geometric mean of the transverse dimensions of the classical inner and outer layers. The asymptotic expansions in the three layers (inner, intermediate and outer) are matched by the Millikan's argument leading to two overlap domains where velocity distribution is logarithmic but their slopes could be different. The measurements show that the sustantial log regions do in fact exist in the two overlap domains and the ratio of their slopes is 2.03. The present theory describes the velocity profile over a greater range when compared to the classical theory. The predictions of Reynolds stress and turbulent energy production are in remarkably good argreement with the data for almost entire turbulent flow region from the beginning of the buffer layer to the axis oj pipe.
94 citations
TL;DR: In this article, an asymptotic theory of the Navier-Stokes equations at large Reynolds numbers is presented, where the boundary value problem is reduced to an integrodifferential equation for the distribution of the friction.
Abstract: The two-dimensional flow of a viscous incompressible fluid near the leading edge of a slender airfoil is considered. An asymptotic theory of this flow is constructed on the basis of an analysis of the Navier—Stokes equations at large Reynolds numbers by means of matched asymptotic expansions. A central feature of the theory is the region of interaction of the boundary layer and the exterior inviscid flow; such a region appears on the surface of the airfoil in a definite range of angles of attack. The boundary-value problem for this region is reduced to an integrodifferential equation for the distribution of the friction. This equation has been solved numerically. As a result, closed separation regions are constructed, and the angle of attack at which separation occurs is found.
59 citations
TL;DR: In this paper, the matched asymptotic expansions were applied to the flow between partially submerged, counter-rotating rollers, a representative problem from this class, and the lubrication approximation was found to generate the first term of an outer expansion of the problem solution.
Abstract: A difficulty in applying the lubrication approximation to flows where a liquid/air interface forms lies in supplying boundary conditions at the point of formation of the interface that are consistent with the lubrication approximation. The method of matched asymptotic expansions is applied to the flow between partially submerged, counter-rotating rollers, a representative problem from this class, and the lubrication approximation is found to generate the first term of an outer expansion of the problem solution. The first term of an inner expansion describes the two-dimensional flow in the vicinity of the interface, and approximate results are found by the finite-element method. Matching between the inner and outer solutions determines boundary conditions on the pressure and the pressure gradient at the point of formation of the interface which allow the solution to the outer, lubrication flow to be completed.
58 citations
TL;DR: In this paper, the authors analyzed the spreading of a drop on a perfectly smooth solid surface using the method of matched asymptotic expansions, including the effect of intermolecular forces near the contact line both on surface diffusion of adsorbed molecules and on flow within the liquid phase itself near the surface line.
50 citations
TL;DR: In this article, asymptotic expansions for the density functions of the TSLS and LIML estimates of coefficients in a simultaneous equation system when the sample size increases and the effect of the exogenous variables increases along the sample length is derived.
45 citations
TL;DR: In this article, the capacitance of two parallel circular disks separated by a dielectric slab is derived in the limit of small separation, where terms of order in terms of δ(n) are given.
Abstract: The solution of the potential around two parallel circular disks separated by a dielectric slab is obtained by using the method of matched asymptotic expansions, asymptotic formula for the capacitance has been derived in the limit of small separation $2\delta $. The formula obtained includes terms of order $\delta $ as well. The mixed boundary value problem is solved by dividing the space around the parallel plates into three regions; the exterior region, the edge region, and the interior region. The solution of the edge region incorporating diegectric effects is obtained by using the Wiener–Hopf technique. The exterior solution of the circular disk problem is obtained by using Hankel transforms. The Hankel transform representation of the exterior solution facilitates the easy derivation of its edge expansion from the Lipschitz–Hankel integrals. The solutions are comiared with Shaw’s result for the free-space case [Phys. Fluids, 13 (1970), pp. 1935–1947] and her errors are corrected. Improvement of her ap...
38 citations
TL;DR: In this article, the forces on two spherical particles moving in a fluid are investigated by the method of matched asymptotic expansions in the small Reynolds number, for the case when the particles are within each other's inner region of expansion.
Abstract: The forces on two spherical particles moving in a fluid are investigated by the method of matched asymptotic expansions in the small Reynolds number, for the case when the particles are within each other's inner region of expansion. The particular case in which the distance l between the sphere centres is very much larger than the sphere radii a and b is studied in detail. The asymptotic expansion of the force on one of the spheres for small a/l and b/l is obtained. Some properties of the force, not to be expected from the Stokes equation, are revealed.
34 citations
TL;DR: In this paper, the viscous flow induced by two spheres oscillating in a direction parallel to their central line was studied analytically at high frequency by the method of matched asymptotic expansions.
Abstract: This paper deals with the problem of the viscous flow induced by two spheres oscillating in a direction parallel to their central line. The effect of the hydrodynamic interaction of the spheres on the steady streaming has been studied analytically at high frequency by the method of matched asymptotic expansions. It has been found the structure of the steady streaming and the forces acting on the spheres in this complicated flow, depending on the values of the parameters of the problem. Typical flow fields and forces acting on the spheres are shown graphically.
TL;DR: In this article, the steady and pulsating modes of flame propagation through a premixed combustible mixture are studied for the case in which the flame is characterized by the sequential production and depletion of a significant intermediate species.
Abstract: Steady and pulsating modes of flame propagation through a premixed combustible mixture are studied for the case in which the flame is characterized by the sequential production and depletion of a significant intermediate species. We employ the method of matched asymptotic expansions to derive a model valid for large activation energies, and show that the pulsating solution is the result of a supercritical Hopf bifurcation from the steadily propagating solution (which becomes unstable). Through a nonlinear bifurcation analysis, we calculate the pulsation amplitude and other characteristics of the flame along the bifurcated branch. It is shown that the average thickness of the pulsating flame, by which we mean the average effective separation distance between production and depletion of the intermediate species, is greater than that predicted by a steady-state theory. In addition, we find that the mean propagation speed is less than that of the steadily propagating solution, but that the instantane...
TL;DR: It is proved that under reasonable conditions the solution of the full system converges to that of the reduced in the distributional sense, and the order of the limiting distribution is computed.
TL;DR: In this article, an asymptotic method due to "Achenbach" was used to analyze the longitudinal and circumferential modes of wave propagation in a piezoelectric solid circular cylinder of crystal class (6 mm) or ceramics (∞m).
Abstract: An asymptotic method due to ’’Achenbach’’ is used to analyze the longitudinal and circumferential modes of wave propagation in a piezoelectric solid circular cylinder of crystal class (6 mm) or ceramics (∞m). Information obtained in this method is useful for the frequency spectrum at long wavelengths. In this method the displacement components, electric potential, and the frequency are expressed as power series of the dimensionless wavenumber, e = 2π× radius/wavelength. Substituting these expansions in the field equations and the boundary conditions, a system of coupled second‐order inhomogeneous ordinary differential equations with radial coordinate as the independent variable is obtained by collecting the terms of same order em. Integration of such systems of differential equations yields the various terms in the series expansions for the above modes and for the whole range of frequencies, when the real valued dimensionless wavenumber is less than unity (0
TL;DR: In this article, the authors apply matched asymptotic expansions to the low Reynolds number flow past two parallel circular cylinders, and show that the results show fairly good agreement with Taneda's experimental data.
Abstract: Jeffery's solution in bi-polar co-ordinates of the two-dimensional Stokes equations cannot be applied to the low-Reynolds-number flow past two parallel circular cylinders because of severe mathematical difficulties. These difficulties can be overcome by considering the flow field far from the cylinders and then modifying the solution near the cylinders so that it becomes the inner expansion for an application of the method of matched asymptotic expansions. After the calculation of the drag, lift and moment coefficients of two adjacent equal circular cylinders to O(1) in the Reynolds number R, the analysis is extended to incorporate partially the effects of fluid inertia of order R. The results show fairly good agreement with Taneda's experimental data.
TL;DR: In this article, the authors considered the vertical flow of an internally heated Boussinesq fluid in a vertical channel with viscous dissipation and pressure work, and they obtained two solutions; the expected solution with no flow and the second adiabatic solution with temperatures less than the wall temperature and a large downward velocity.
TL;DR: In this paper, Lagrange-like and Euler-like expansions of the perturbation parameter are considered for partial differential equations to be solved on a domain that is also perturbed.
Abstract: Expansions in powers of a perturbation parameter are considered for partial differential equations to be solved on a domain that is also perturbed. Two kinds of expansions—Lagrange-like and Euler-like—are developed and shown to be equivalent to one another under a certain transformation of dependent variables. This equivalence is used to justify the hierarchy of partial differential equations produced by the Euler-like expansion.In addition to partial differential equations, integrals extended over a perturbed domain are also considered. Such integrals suffer perturbations due to the domain perturbation. A formula for these perturbations is given that involves the domain-mapping function only at the boundary of the unperturbed domain. This contrasts with the usual change-of-domain formula, which involves the Jacobian of the domain-mapping function throughout the unperturbed domain. The formula in question is compatible with Euler-like expansions (and the usual change-of-domain formula is compatible with L...
TL;DR: In this article, a methodology to obtain an approximate solution of a singularly perturbed nonlinear differential game is presented, where the outcome of the game with approximate strategies, defined as extended value, is related to the saddle-point value.
Abstract: A methodology to obtain an approximate solution of a singularly perturbed nonlinear differential game is presented. The outcome of the game with approximate strategies, defined as extended value, is related to the saddle-point value of the game. In an example of a simple pursuit-evasion game, it is shown that the proposed methodology leads to an easily implementable feedback form solution with fairly accurate results. This approach seems to be attractive for analyzing realistic air-combat models without solving a two-point boundary-value problem.
TL;DR: In this paper, the dual integral equations generated by contact problems for half-spaces and half-planes inhomogeneous with depth were considered, and an approximate method for their solution was proposed.
TL;DR: In this paper, a method for improvement of the numerical solution of differential equations by incorporation of asymptotic approximations is investigated for a class of singular perturbation problems, and uniform error estimates are derived for this method when implemented in known difference schemes and applied to linear second order O.D.E.'s.
Abstract: A method for improvement of the numerical solution of differential equations by incorporation of asymptotic approximations is investigated for a class of singular perturbation problems.
Uniform error estimates are derived for this method when implemented in known difference schemes and applied to linear second order O.D.E.'s. An improvement by a factor of?n+1 can be obtained (where ? is the "small" parameter andn is the order of the asymptotic approximation) for a small amount of extra work. Numerical experiments are presented.
TL;DR: In this paper, double-diffusive counterbuoyant boundary layers may possess the classical self-similar structure only within two distinct and disconnected subdomains of the physical parameter space.
TL;DR: Asymptotic expansions for the error probabilities of repeated significance tests about a normal mean are developed in this paper, and the expansions appear to result in substantially improved numerical accuracy, when compared to the use of the leading term, at least in some important special cases.
Abstract: Asymptotic expansions for the error probabilities of repeated significance tests about a normal mean are developed. Use of the expansions appears to result in substantially improved numerical accuracy, when compared to the use of the leading term, at least in some important special cases. The expansions are sufficiently refined to show the effect of some simple modifications of the basic procedure, such as requiring an initial sample size.
TL;DR: In this paper, asymptotic expansions for transient electromagnetic (EM) fields are derived for layered grounds, axisymmetric structure, and two-dimensional (2-D) structures.
Abstract: Asymptotic expansions may be derived for transient electromagnetic (EM) fields. The expansions are valid when σμ0l2/t is less than about 0.1. Here l, σ, μ0, and t are the respective lengths, conductivities, permeabilities of free space and time. Cases for which asymptotic expansions are presented include (1) layered grounds, (2) axisymmetric structure, and (3) two‐dimensional (2-D) structures. In all cases the transient voltage eventually approaches that of the host medium alone, the ratio of anomalous response to the half‐space response being proportional to 1/tν. Here v is equal to 0.5 for layered structures and 1.0 for 2-D or 3-D structures.
01 Apr 1982
TL;DR: In this article, the authors consider a system of functional differential equations and obtain conditions on a Liapunov functional to insure the uniform asymptotic stability of the zero solution.
Abstract: We consider a system of functional differential equations x'(t) = F(t, x,) and obtain conditions on a Liapunov functional to insure the uniform asymptotic stability of the zero solution.
TL;DR: In this article, an asymptotic theory for differential equations with large coefficients that are like powers of the independent variable is developed for fourth-order equations with general coefficients and a critical case is identified that forms a borderline between situations where all solutions have asymPTotically a certain exponential character in terms of the coefficients and where only two solutions have this character.
Abstract: A recently formulated asymptotic theory for differential equations with large coefficients that are like powers of the independent variable is developed here for fourth-order equations with general coefficients. In this general approach a critical case is identified that forms a borderline between situations where all solutions have asymptotically a certain exponential character in terms of the coefficients and where only two solutions have this character. The asymptotic forms of a fundamental set of solutions are obtained in the critical case.
TL;DR: In this article, the authors compared the results obtained from the exact solution, from those found by Reiss and from the matched asymptotic expansion solution developed here, show that the last is far more accurate than the second.
Abstract: Traditional singular perturbation methods are employed to develop a solution to a differential equation considered by Reiss [SIAM J. Appl. Math., 39 (1980), pp. 440–455] which models an elementary chemical process. The results are compared with those found by Reiss, who used a novel asymptotic method to construct solutions which exhibit rapid transient behavior. It is shown that Reiss’ jump solution corresponds to the asymptotic (large time) representation of the more complete solution found from a formal matched asymptotic expansion procedure. A comparison of results in the rapid transition region obtained from the exact solution, from those found by Reiss and from the matched asymptotic expansion solution developed here, show that the last is far more accurate than the second.
TL;DR: In this paper, a variational approach is described to compute singular sensitivity functions of finite dimensional systems with respect to changes in system structure, and a relation between the present method and singularly perturbed optimal control theory is established.
Abstract: A new variational approach is described to compute singular sensitivity functions of finite dimensional systems with respect to changes in system structure. The reduced nominal model and its adjoint system are used to define singular sensitivity functions. A relation between the present method and singularly perturbed optimal control theory is established. It is shown that the present variational approach gives a computationally effective method in singular sensitivity analysis. The method is applied to singularly perturbed linear ordinary differential equations and singular optimal control problem.