The Method of Matched Asymptotic Expansion

There are a number of variations for the perturbation technique. Among them, the method of matched asymptotic expansion is the easiest to apply. The method is useful for obtaining expansions from separate domains of validity. 

Let us consider the problem in the last section (Eq. 6.48). The straightforward expansion we obtained is the outer solution (Eq. 6.55) that is, it is valid over most of the domain, except close to the origin (see Fig. 6.1). The outer solution is 

The previous section clearly indicated that this outer solution is not valid near the origin. So there is a sharp change in the solution behavior as the trajectory moves away from the outer region. To describe this sharp change, we need to magnify the variable as follows 

Note that n is unknown at this stage, since we have no preconceived notion of the thickness of the boundary. Next, we assume that yix e) has the following asymptotic expansion (called inner solution) 

Since the second order derivative did not appear in the zero order equation of the outer expansion (Eq. 6.51), we must ensure that it appears in the leading order equation of the inner solution (i.e., the zero order inner solution must be second order). Observing the differential equation, we have two possibilities. One is that the second order derivative term balances with the first order term, and the second possibility is that the second order derivative term balances with the remaining third term. 

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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... 

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 ... 

The Method of Matched Asymptotic Expansion 201 Hence, the inner solution that has an error of order of is... 

We have shown the basic steps for the method of matched asymptotic expansion. The matching principle is based on the use of the overlapping region and an intermediate variable in that region. In what follows is another simple method of matching for a class of problems, which takes the following form... 

We have shown, using the method of matched asymptotic expansions, that in the outer domain there is an adjustable variable, and in the inner region there is another such variable. The composite solution is, therefore, a function of these two variables. Exploitation of this function is the essential idea behind the Multiple Time Scale method. Interested readers should refer to Nayfeh (1973) for exposition of this technique. 

It is possible to inclnde in a model a small bnt nonzero volume of the interfacial region in analyzing transport phenomena snch as mass transfer across or to an interface. Mathematical construction can now be performed to divide the domain into a small interfacial region, called the iimer region, where all surface effects apply. The bulk is the outer region where usual convective-diffusive mass transfer (and possibly reactions as in micellization-demicellization) dominate. This is exactly the method of matched asymptotic expansions... 

In Ref [114], an approach to the dynamics of ionic surfactant adsorption was developed, which is simpler as both concept and application, but agrees very well with the experiment. Analytical asymptotic expressions for the dynamic surface tension of ionic surfactant solutions are derived in the general case of nonstationary interfacial expansion. Because the diffusion layer is much wider than the EDL, the equations contain a small parameter. The resulting perturbation problem is singular and it is solved by means of the method of matched asymptotic expansions [115]. The derived general expression for the dynamic surface tension is simplified for two important special cases, which are considered in the following section. 

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