Showing papers on "Method of matched asymptotic expansions published in 1968"
289 citations
TL;DR: In this article, a boundary layer formed adjacent to the free surface, and a classical boundary-layer analysis is applied to find the flow there (for moderate distances downstream); the influence of this boundary layer on the flow in the core of the jet is then investigated.
Abstract: This paper is concerned with the jet of liquid, open to the atmosphere, that emerges from a two-dimensional channel in which there is Poiseuille flow far upstream, the flow being driven by an applied pressure gradient. The problem is discussed with the aid of the method of matched asymptotic expansions; the small parameter involved is the inverse Reynolds number. A boundary layer forms adjacent to the free surface, and a classical boundary-layer analysis is applied to find the flow there (for moderate distances downstream); the influence of this boundary layer on the flow in the core of the jet is then investigated. Higher-order boundary-layer effects, such as indeterminacy and eigensolutions, are also discussed. The first few terms are found of an asymptotic expansion for the equation of the free surface, and considerations of momentum balance are applied to find the asymptotic contraction ratio of the jet.
69 citations
TL;DR: In this article, a perturbation scheme was developed from the Stokes solution and its perturbations in the form of co-ordinate expansions to take account of the inertia effects.
Abstract: This paper is concerned with the steady, symmetric, two-dimensional flow of a viscous, incompressible fluid issuing from an orifice and falling freely under gravity. A Reynolds number is defined and considered to be small. Due to the apparent intractability of the problem in the neighbourhood of the orifice, interest is confined to the flow region below the orifice, where the jet is bounded by two free streamlines. It is assumed that the influence of the orifice conditions will decay exponentially, and so the asymptotic solutions sought have no dependence upon the nature of the flow at the orifice. In the region just downstream of the orifice, it is expected that the inertia effects will be of secondary importance. Accordingly the Stokes solution is sought and a perturbation scheme is developed from it to take account of the inertia effects. It was found possible only to express the Stokes solution and its perturbations in the form of co-ordinate expansions. This perturbation scheme is found to be singular far downstream due to the increasing importance of the inertia effects. Far downstream the jet is expected to be very thin and the velocity and stress variations across it to be small. These assumptions are used as a basis in deriving an asymptotic expansion for small Reynolds numbers, which is valid far downstream. This expansion also has the appearance of being valid very far downstream, even for Reynolds numbers which are not necessarily small. The method of matched asymptotic expansions is used to link the asymptotic solutions in the two regions. An extension of the method deriving the expansion far downstream, to cover the case of an axially-symmetric jet, is given in an appendix.
57 citations
TL;DR: Theoretical results for forced heat convection from a circular cylinder at low Reynolds numbers are given in this paper, where the analysis in each case is based upon the matched asymptotic expansions.
Abstract: Theoretical results are obtained for forced heat convection from a circular cylinder at low Reynolds numbers. Consideration is given to the cases of a moderate and a large Prandtl number, the analysis in each case being based upon the method of matched asymptotic expansions. Comparison between the moderate Prandtl number theory and known experimental results indicates excellent agreement; no relevant experimental work has been found for comparison with the large Prandtl number theory.
42 citations
TL;DR: In this article, a simplified scheme of four chemical reactions is chosen to represent the kinetics of the hydrogen-oxygen system; m particular, this scheme includes the influence of the hydroxyl radical.
Abstract: A simplified scheme of four chemical reactions is chosen to represent the kinetics of the hydrogen-oxygen system; m particular, this scheme includes the influence of the hydroxyl radical. The diffusion flame supported by this set of reactions is assumed to form behind a (planar, two-dimensional) body of parabolic meridian profile with downstream-pointing vertex. The body initially separates the oxygen and hydrogen streams, which are assumed to have equal speeds and pressures far upstream. (The pressure is subsequently assumed to be constant everywhere.) For pressures of about one atmosphere it is found that nett reaction rates can be treated as infinitely fast, the four reactions then yield four chemical equilibrium equations whose behaviour is dominated by the largeness of the equilibrium constant for the (thermal) dissociation-recombination reaction of hydrogen. The flame-sheet model emerges as the limiting solution when the reciprocal of this large quantity is allowed to vanish. The method of matched asymptotic expansions is used to investigate the structure of the flame which results from a relaxation of this limit. The results bear a satisfactory resemblance to some experimental measurements which, although made in other gas mixtures, exemplify the behaviour of the type of diffusion flames considered.
25 citations
24 citations
TL;DR: In this paper, the perturbation expansions are derived by a technique which does not assume that convergent expansions exist, and criteria are developed to determine if a finite number of terms underestimates or overestimates the exact result for sufficiently small values of the coupling constant.
Abstract: The perturbation expansions are derived by a technique which does not assume that convergent expansions exist. The theory is shown to be asymptotic, and criteria are developed to determine if a finite number of terms underestimates or overestimates the exact result for sufficiently small values of the coupling constant.
24 citations
23 citations
TL;DR: A new boundary-value technique is proposed for the treatment of initial-boundary-value problems for linear and mildly non-linear wave equations.
Abstract: A new boundary-value technique is proposed for the treatment of initial-boundary-value problems for linear and mildly non-linear wave equations. Several illustrative examples are offered to demonstrate the ease with which the method can be applied.
20 citations
01 Jun 1968
19 citations
TL;DR: In this article, matched asymptotic expansions are employed to derive formulae for viscous corrections occurring in small amplitude oscillatory disturbances in a nonhomogeneous fluid, which depend on the value of the viscosity at rigid bounding surfaces and not at all on its actual distribution.
Abstract: The method of matched asymptotic expansions is employed to derive formulae for viscous corrections occurring in small amplitude oscillatory disturbances in a nonhomogeneous fluid. Such formulae are given for quite general variations with depth of the equilibrium density and viscosity distributions in the following cases: (i) two-dimensional disturbances in a fluid of finite depth bounded above by a free surface, (ii) two-dimensional disturbances in fluid bounded above and below by two fixed horizontal planes. The derived expressions are seen to depend on the value of the viscosity at rigid bounding surfaces and not at all on its actual distribution. An additional correction for viscosity is given for case (ii), when the density has an exponential variation with depth and the kinematic viscosity is constant. DOI: 10.1111/j.2153-3490.1968.tb00392.x
TL;DR: In this paper, the asymptotic properties of a class of nonlinear boundary value problems are studied, and very simple approximate analytic solutions are obtained, and their accuracy is illustrated by showing their good agreement with the exact numerical solution of the problem.
Abstract: The asymptotic properties of a class of nonlinear boundary‐value problems are studied. For large values of a parameter, the differential equation is of the singular‐perturbation type, and its solution is constructed by means of matched asymptotic expansions. In two special cases, very simple approximate analytic solutions are obtained, and their accuracy is illustrated by showing their good agreement with the exact numerical solution of the problem.
TL;DR: In this article, a matched asymptotic expansion for the eigenvalues arising in perturbations about the Blasius solution was proposed, and good agreement was obtained with the numerical results of Libby.
Abstract: A further term is found in the asymptotic expansion suggested by Stewartson for the eigenvalues arising in perturbations about the Blasius solution. The method employed is that of matched asymptotic expansions, and good agreement is obtained with the numerical results of Libby.
TL;DR: In this paper, the problem of radiative transport of thermal energy between parallel, black, diffusely-emitting walls is solved by the method of matched asymptotic expansions when the intervening grey gas is optically thick.
TL;DR: In this article, the method of multiple scales is used to obtain the asymptotic solution of the Orr-Sommerfeld equation for the special case of a linear velocity profile, the solution so obtained agrees well with an approximation of the exact solution which is known.
Abstract: The method of multiple-scales is used to obtain the asymptotic solution of the Orr–Sommerfeld equation. For the special case of a linear velocity profile, the solution so obtained agrees well with an approximation of the exact solution which is known. For the general case, transformations on both the dependent and independent variables are introduced to obtain a zeroth-order equation which differs from the inner equation studied so far. On the ground of the favourable comparison for the special case, the asymptotic solution constructed is expected to be uniformly valid.
TL;DR: In this paper, an approximate solution of the steady and shallow free-surface flow toward a well in a layer of infinite extent is obtained by expanding the velocity potential in a small parameter power series.
Abstract: An approximate solution of the steady and shallow free-surface flow toward a well in a layer of infinite extent is obtained by expanding the velocity potential in a small parameter power series This expansion is shown to be valid only in the vicinity of the well and is, therefore, called the inner expansion An outer expansion, which solves the flow problem at large distance from the well, is derived by using the method of matched asymptotic expansions The Dupuit approximation coincides with the zero order term of the potential outer expansion The derivation of a second order outer term makes possible the discussion of the validity of the Dupuit approximation, which tends asymptotically toward the exact solution In the outer zone, the streamlines are parabolic and are not orthogonal to the equipotentials The method is illustrated by two numerical examples
TL;DR: In this paper, matched asymptotic expansions are applied to the determination of the damping of gravity waves propagating in turbulent conditions, where the effect of the turbulence is introduced by a general system of coefficients of eddy viscosity, whilst the turbulence itself is supposed to be confined to boundary layers adjacent to a rigid impermeable bottom and the free surface.
Abstract: The method of matched asymptotic expansions is applied to the determination of the damping of gravity waves propagating in turbulent conditions. The effect of the turbulence is introduced by a general system of coefficients of eddy viscosity, whilst the turbulence itself is supposed to be confined to boundary layers adjacent to a rigid impermeable bottom and the free surface. The lowest order damping in the system is found to be independent of surface turbulence and computations are made for a physically meaningful distribution of eddy viscosity in the lower boundary layer. DOI: 10.1111/j.2153-3490.1968.tb00375.x
TL;DR: In this paper, an asymptotic expansion procedure is applied to the analysis of a typical bay in a hydrostatically loaded, ring-reinforced, non-circular cylinder.
Abstract: : An asymptotic expansion procedure is applied to the analysis of a typical bay in a hydrostatically loaded, ring-reinforced, non-circular cylinder. The existence of two distinct characteristic lengths is used in the construction of two asymptotic series which, when combined, form the total solution. The procedure is similar to that proposed by E. Reissner for circular cylinders. The leading terms in the expansions for all stress resultants and deflections are obtained explicitly, and numerical results are compared to available theoretical and experimental data. (Author)
01 Jun 1968
TL;DR: In this paper, the authors apply matched asymptotic expansions to the Radiative Transport of Thermal Energy between concentric spheres (RTE) problem and obtain uniformly valid results for emissive power, temperature slip at the walls, and heat flux.
Abstract: : Radiative transport of thermal energy between concentric spheres is treated by the method of matched asymptotic expansions when the intervening grey gas is optically thick. Adjacent to each wall is a thin thermal layer in which the Rosseland diffusion approximation is invalid. A unique feature of this analysis is the direct application of matched asymptotic expansions to an integral equation. A close connection is thereby established between the structure of the wall layers and the Milne problem of astrophysics. Using the asymptotic approach, uniformly valid results are obtained for the emissive power, temperature slip at the walls, and heat flux. The asymptotic results are compared with the Rosseland approximation and with a numerical solution. A comparison is also made with the plane parallel layer problem to demonstrate the effect of a changed geometry. (Author)
TL;DR: Asymptotic solutions of linear difference equations, discussed applications to eigenvalue problems of free oscillations of model galaxies as discussed by the authors, and discussed the applications of eigen value problems of the model galaxies.
TL;DR: In this article, a class of optimal servo problems with random input and bounded control with singular perturbation is discussed, where the problem is to find a control u which minimizes the mean-square error and to calculate this minimum.
Abstract: This paper discusses a class of optimal servo problems with random input and with bounded control. The problem is to find a control u which minimizes the mean-square error and to calculate this minimum. Such problems can often be reduced to solving a non-linear partial differential equation. Since this equation is not amenable to an exact solution, a method of successive approximation based on singular perturbation is used to obtain asymptotic solutions for a simple system. The method is applicable for the case of small input, small disturbance, and a relatively large bound on the control u . Computational results for the mean-square error are shown, and the difficulty of this method is discussed.
TL;DR: In this paper, the low Reynolds number flow of a gas past an infinite heated circular cylinder is studied for the case of variable property flow when the temperature difference between the cylinder and free stream is small.
Abstract: The low Reynolds number flow of a gas past an infinite heated circular cylinder is studied for the case of variable property flow when the temperature difference between the cylinder and free stream is small. The perturbation to the corresponding incompressible velocity field is calculated by the method of matched asymptotic expansions developed by Kaplun, and Proudman and Pearson. When the temperature difference between the cylinder and free stream is appreciable, a solution for the velocity field is found on the assumption that the viscosity and thermal conductivity of the gas are invariant. This solution possesses the interesting property that the zero‐order solution for the velocity field near the cylinder is a uniformly valid solution of the problem.
TL;DR: In this article, an elementary linearized solution to the nonlinear potential equation with matched asymptotic expansions was proposed and used to derive an elementary second-order slender body theory.
Abstract: Supersonic flow past slowly oscillating pointed bodies of revolution is studied. Starting from the complete nonlinear potential equation an elementary linearized solution is discussed and it is shown how this solution together with the method of matched asymptotic expansions can be used to derive an elementary second-order slender body theory. This approach is further demonstrated for the oscillating cone and its range of validity is evaluated by comparison with other theoretical methods.
TL;DR: In the nonoscillation theory of ordinary differential equations the asymptotic behaviour of the solutions has often been described by exhibiting asymPTotic expansions for these solutions as mentioned in this paper.
Abstract: In the nonoscillation theory of ordinary differential equations the asymptotic behaviour of the solutions has often been described by exhibiting asymptotic expansions for these solutions. A fundamental illustration of this technique may be found in Hille [6] wherein the linearly independent solutions of a second order homogeneous differential equation were described by single termed asymptotic expansions. For the dominant solution, this result was successively extended by Waltman in [8] for a second order equation and in [9] for an n th order equation. A further generalization of these results appears in [3] where a complete n th order nonhomogeneous nonlinear differential equation was considered; again, asymptotic representations were given to describe the behaviour of the solutions of the differential equation. Moore and Nehari [7], Wong [10], [11], and Hale and Onuchic [2], also use asymptotic representations in discussing the behaviour of the solutions of certain differential equations. All of the above results are essentially perturbation problems with the unperturbed linear differential equation having the form y ( n ) = h ( t ) for some n and h ( t ).
TL;DR: In this paper, the plane or axisymmetric flow of a viscous and heat conducting fluid far away from a body moving at sonic speed is considered, and a description of the shock wave structure is obtained, being an extension of the previously known nondissipative Guderley-Frankl solutions.
Abstract: The plane or axisymmetric flow of a viscous and heat conducting fluid far away from a body moving at sonic speed, is considered. The parameter characterizing the dissipative phenomena is assumed very small (highReynolds number), and in order to determine approximatively the flow the method of matched asymptotic expansions with respect to this parameter is applied. As a result, a description of the shock wave structure is obtained, being an extension of the previously known nondissipativeGuderley-Frankl solutions. Some restrictions concerning the region of validity of the obtained results are deduced from analysing the basic assumptions underlying the method of approximation.
TL;DR: In this article, a transport calculation of the lattice diffusion length, yielding the gross decay of the asymptotic flux in a lattice, is made using the method of K. M. Case.
Abstract: A transport calculation of the lattice diffusion length, yielding the “gross” decay of the asymptotic flux in a lattice, is made using the method of K. M. Case. Refinements over the diffusion calcu...