Matched asymptotics

We call this a partial M/ave expansion. To detennine tire coefficients one matches asymptotic solutions to the radial Scln-ddinger equation with the corresponding partial wave expansion of equation (A3.11.106). It is customary to write the asymptotic radial Scln-ddinger equation solution as... 

Lagerstrom, P. A. Matched Asymptotic Expansions Ideas and Techniques, Springer-Verlag (1988). 

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 Refs. 32, 170, and 180. 

Figure 4.14. Flow-field parameter distributions in front of an expanding piston. Soiution by matched asymptotic expansions by Guirao et al. compared to exact similarity solutions for various piston Mach numbers.

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. " ... 

Relaxing the restriction of low Reynolds number, Rimmer (1968,1969) used a matched asymptotic expansion technique to develop a solution in terms of Pe and the Schmidt number Sc (or Prandtl number Pr for heat transfer), where Sc = v/D.j and Pr = v/a in which v is the kinematic viscosity of the flowing fluid. His solution, valid for Pe < 1 and Sc = 0(1), is... 

As noted in Chapters 2 and 3, deformation of fluid particles is due to inertia effects. For low Re and small deformations, Taylor and Acrivos (T3) used a matched asymptotic expansion to obtain, to terms of order We /Re,... 

Theoretical attempts to explain lift have concentrated on flow at small but nonzero Re, using matched asymptotic expansions in the manner of Proudman and Pearson for a nonrotating sphere (see Chapter 3). In the absence of shear, Rubinow and Keller (R6) showed that the drag is unchanged by rotation. With... 

The proper singular perturbation treatment has thus to take care of this initial stage. Probably the simplest way to do this is via a matched asymptotic expansion procedure, with the outer solution of the type (5.2.13), (5.2.14), valid for t = 0(1), matched with an initial layer solution that has an internal layer at x = 0. 

A few remarks are due about this feature. The nonuniformity above is a formal expression of breakdown of the local electro-neutrality assumption in concentration polarization, described in the previous chapter. Essentially, this reflects the failure of a description based upon assuming the split of the physical region into a locally electro-neutral domain and an equilibrium double layer where all of the space charge is concentrated. The source of this failure, reflected in the nonuniformity of the corresponding matched asymptotic expansions, is that the local Debye length at the interface tends to infinity as the voltage increases. In parallel a whole new type of phenomena arises, which is not reflected in the simplistic picture above. The... 

Here C is another integration constant to be determined from matching with the transition layer solution (5.5.20). This matching may be achieved, following the standard prescription of matched asymptotics [15], by introducing the intermediate variable... 

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 results for the thickness of the condensed layer agree with matched asymptotic expansions by Ramanathan and co-workers [33, 34, 60], In comparison, numerical solutions to PB indicate that the thickness of the condensed layer goes as RM = (a/k)1 2 [60, 61]. In summary, the analysis provides [60] (a) an analytic solution that is globally accurate, (b) information about the structure of the condensed fraction, and (c) the density profile of the uncondensed ions. 

Ploehn and Russel (1989) developed a matched asymptotic solution of Eq. (73) equivalent to a two-eigenfunction approximation for Gc. Near the surface, interchain interactions distort chain configurations so that the characteristic length is /. Far from the surface, chains are ideal and the characteristic length scales as nl/2l l. The widely separated length scales enable the inner ground state solution to be matched asymptotically to an outer solution, yielding a uniform approximation for all z. 

Fig. 21. Ellipsometric thickness as a function of chain length plotted on a log-log scale. The points (squares and crosses) are the data of Takahashi et al. (1980) for PS adsorbing onto chrome from cyclohexane or CC14. Curves A and B are calculated using the SCF in Eqs. (71) and (72) curves C-F utilize the SCF of Eq. (70). Curves A-D result from the matched asymptotic solution, while curves E and F are groundstate solutions. Other parameters include y, = 1 and

Prom Eqn. (2.6.65), it is apparent that this is a singular perturbation problem (as the highest derivative term is multiplied by the small parameter) and then one can use matched asymptotic expansion to obtain (f> by describing the solution in terms of outer and inner solutions. 

From equation (66) it is seen that as p becomes large the reaction term in equation (46) becomes very small ( exponentially small, since p appears inside the exponential) unless t is near unity. Hence for 1 — t of order unity, there is a zone in which the reaction rate is negligible and in which the convective and diffusive terms in equation (46) must be in balance. In describing this conyectiye-diffusive zone, an outer expansion of the form T = Tq(0 + H (p)zi( ) + H2(P)t2(0 + may be introduced, following the formalism of matched asymptotic expansions [35]. Here... 

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