Asymptotic expansions matched

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... 

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. 

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... 

Completion of a solution by matched asymptotic expansions entails employing matching conditions. Although there are many ways to effect matching, the most infallible approach currently available is to investigate a parametric limit in an intermediate variable [35]. Thus we consider oo with f]t held fixed, where t] = s(p)t]/t(p) = ( — o)/KPX with t(p) 0 and KP)/s(P) 00 in the limit. The general matching condition is then written as... 

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 formulation of Section 9.5.1 has served to remove the chemistry from the field equations, replacing it by suitable jump conditions across the reaction sheet. The expansion for small S/l, subsequently serves to separate the problem further into near-field and far-field problems. The domains of the near-field problems extend over a characteristic distance of order S on each side of the reaction sheet. The domains of the far-field problems extend upstream and downstream from those of the near-field problems over characteristic distances of orders from to /. Thus the near-field problems pertain to the entire wrinkled flame, and the far-field problems pertain to the regions of hydrodynamic adjustment on each side of the flame in essentially constant-density turbulent flow. Either matched asymptotic expansions or multiple-scale techniques are employed to connect the near-field and far-field problems. The near-field analysis has been completed for a one-reactant system with allowance made for a constant Lewis number differing from unity (by an amount of order l/P) for ideal gases with constant specific heats and constant thermal conductivities and coefficients of viscosity [122], [124], [125] the results have been extended to ideal gases with constant specific heats and constant Lewis and Prandtl numbers but thermal conductivities that vary with temperature [126]. The far-field analysis has been... 

The analysis of the corner singularity is delicate. We refer to the recent works of Hinch [36] and Renardy [37,38], who have contructed a matched asymptotic expansion for the steady solution to a Maxwell fluid flow near the corner. 

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) ... 

---