Method matched asymptotic expansions

Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Kevorkian, J., and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York (1981) and Lager-strom, P. A., Matched Asymptotic Expansions Ideas and Techniques, Springer-Verlag, New York (1988). 

Earlier modeling studies were aimed at predicting the current and temperature distributions, as the nonuniform distributions contribute to stress formation, a major technical challenge associated with the SOFC system. Flow and multicomponent transport were typically simplified in these models that focused on SOFC electrochemistry. Recently, fundamental characteristics of flow and reaction in SOFCs were analyzed using the method of matched asymptotic expansions. " ... 

This paper is one of the first applications of the asymptotic method in world scientific literature, a method which twenty years later has received widespread use. Now it is called the method of matched asymptotic expansions. Without introducing the terminology which later appeared, the author essentially made use of the full arsenal of this method, which today makes the problem studied in this article a textbook example of its application. An exposition of the general technique of the method of matched asymptotic expansions and numerous examples of its use may be found in monographs.3,4... 

The approach that has been described here is an example of a perturbation method for large Damkohler numbers and may be termed Damkohler-mimber asymptotics. It has been developed on the basis of an expansion that does not distinguish among special zones within the flame. It is possible that the Damkohler-number expansion will often be good in the hot reaction zone but poor elsewhere, while radical distributions away from the hot reaction zone have relatively little influence on the main characteristics of the flame. Under these circumstances, an approach based on matched asymptotic expansions, treating different zones differently, may be helpful. Sharper definitions of values of Q consistent with the steady-state approximation (in the zone where it is applied) might thereby be developed. 

The justification of this procedure can be accomplished by means of the method of the matched asymptotic expansions. " ... 

Poisson-Boltzmann equation. By using the method of matched asymptotic expansions. Chew and Sen obtained for a thin EDL (kR > 1) ... 

In the previous sections we have seen several examples of transport problems that are amenable to analysis by the method of regular perturbation theory. As we shall see later in this book, however, most transport problems require the use of singular-perturbation methods. The high-frequency limit of flow in a tube with a periodic pressure oscillation provided one example, which was illustrative of the most common type of singular-perturbation problem involving a boundary layer near the tube wall. Here we consider another example in which there is a boundary-layer structure that we can analyze by using the method of matched asymptotic expansions. 

This solution obviously does not satisfy no-sfip conditions on the sidewalls,

oo by use of the method of matched asymptotic expansions, which does satisfy boundary conditions on the walls. Be as detailed and explicit as possible, including actually setting up the equations and boundary conditions for the solution in the regions near the walls. 

Unlike the regular perturbation expansion discussed earlier, the method of matched asymptotic expansions often leads to a sequence of gauge functions that contain terms like Pe2 In Pe or Pe3 In Pe that are intermediate to simple powers of Pe. Thus, unlike the regular perturbation case, for which the form of the sequence of gauge functions can be anticipated in advance, this is not generally possible when the asymptotic limit is singular In the latter case, the sequence of gauge functions must be determined as a part of the matched asymptotic-solution procedure. 

Problem 9-6. Inertial Effects on the Motion of a Gas Bubble for Re bubble rises through an infinite body of fluid under the action of buoyancy. The Reynolds number associated with this motion is very small but nonzero. Assume that the bubble remains spherical, and use the method of matched asymptotic expansions to calculate the drag on the bubble, including the first correction that is due to inertia at 0(Re). You may assume that the viscosity and density of the gas are negligible compared with those of the liquid so that you can apply the boundary conditions... 

Can this be done by a regular perturbation solution, or is it necessary to use the method of matched asymptotic expansions Calculate the torque to include the first inertial correction. 

Normally, the next step in the method of matched asymptotic expansions would be to seek a second approximation in the outer region, followed by a second approximation in the... 

Many problems in transport and chemical reaction engineering are nonlinear and cannot be solved analytically. A powerful approach to solve such problems lies in the method of matched asymptotic expansions that often provide analytical expressions for the solution. The method is based on an expansion whose convergence is based on concepts somewhat different from that usually understood. An example is considered below to clarify the nature of such expansions. Such expansions can be used in the solution of nonlinear equations for limiting values of parameters associated with the problem. Several examples are available in the chemical engineering literature (Leal, 1992, 2007 Been, 1998 Varma and Morbidelli, 1997). 

Low Reynolds numbers. In [216, 382] the problem on a circular cylinder of radius a in translational flow of viscous incompressible fluid with velocity Ul at low Reynolds numbers was solved by the method of matched asymptotic expansions. The study was carried out on the basis of the Navier-Stokes equations (1.1.4) in the polar coordinates 1Z, 6. Thus, the following expression for the stream function was obtained for IZ/a 1 ... 

Peclet numbers, the problem of mass exchange between a particle of arbitrary shape and a uniform translational flow were studied by the method of matched asymptotic expansions in [62]. The following expression was obtained for the mean Sherwood number up to first-order infinitesimals with respect to Pe ... 

The approximate solution of the thermal and diffusion problem can be found by the method of matched asymptotic expansions (see Section 4.4) with the stream functions (5.10.6) one must retain only the zero and the first terms of the expansions with respect to low Peclet numbers and use the boundary conditions (5.10.5) and (5.11.2) to obtain the following values of the constant B and the force acting on the drop ... 

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