Asymptotic solutions matched

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... 

Brenner (B6) pointed out that similar problems arise in obtaining Eq. (3-44) as in the low Re approximation for fluid flow. The neglected convection terms dominate far from the particle, since the ratio of convective to diffusive terms is 0[Pe(r/a)]. An asymptotic solution to Eq. (3-39) with Pe 0 was therefore obtained by the matching procedure of Proudman and Pearson discussed above. Brenner s result for the first term in a series expansion for Sh may be written ... 

For Pe 0 an asymptotic solution through the matching procedure has been obtained for all k (B6). As for solid spheres its range of applicability is limited to Pe < 1. 

To match the asymptotic solutions (A.4) and (A.6) we employ the standard semiclassical ansatz1 ... 

The same problem has been solved in an alternate way for all dimensions [42]. From this solution one can calculate the number of tracer-vacancy exchanges up to time t. In two dimensions the distribution is geometric, with mean (log t)/tt. The continuum version of this problem has been considered as well in the form of an infinite-order perturbation theory [43] the solution matches the asymptotic form of the lattice model. 

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

About 30(5 from the exciter in the downstream direction the computed disturbance profile matches with the eigen-solution corresponding to the complex wave number value (0.2798261, -0.00728702), with that obtained by the stability analysis for the TS mode. It is interesting to note that there is a local component of the receptivity solution that decays rapidly in either direction. This is called the near-field response or the local solution. Thus, the receptivity solution in this figure consists of the asymptotic solution (away from the exciter) and a local solution. 

The difference in scaling between the central core of the thin cavity (6-122) and the vicinity of the end walls (6-123) means that the asymptotic solution for s <dimensionless equations and a different form for the asymptotic expansion for e <[Pg.387]

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. 

We now seek a solution of (9 7) and (9-8) for small values of the Peclet number, Pe , by using the matched asymptotic expansion procedure that was detailed for uniform flow past a sphere in Section C. Although the reader may not immediately see that the derivation of an asymptotic solution for this new problem necessitates use of the matched asymptotic expansion technique, an attempt to develop a regular expansion for 9 for Pe 1 leads to a Whitehead-type paradox similar to that encountered for the uniform-flow problem. 

Asymptotic solution. We seek an approximate closed-form solution of problem (4.4.3)-(4.4.5) at small Peclet numbers by the method of matched asymptotic... 

We call this a partial wave expansion. To determine the coefficients a, one matches asymptotic solutions to... 

Upon matching coefficients of corresponding powers of (C - Co )> the leading terms in the asymptotic solutions are found to be of the form... 

Figure 6.23. Thin solid line - numerical solution to Eq. (6.125), thick solid lines leading-order asymptotic solutions j = /Q (upper) and j=Q (lower). Dashed curve -matched as5mptotic solutions (6.134) (/ < 1) and (6.137) (/> 1).

Matching of asymptotic solutions at the moving shock front. 

Ah Ax = A h/Abc = 0, noting that B can be set to zero by arbitrarily shifting the origin. It is then possible to match the curvature A hlAx of the asymptotic solution in Eq. 35 to the outer spherical cap solutions, which, to leading order, are static solutions of the Laplace-Young equation. Chang [17] showed that this leads to... 

In the matching process it is observed that the outer solution is contained in the inner solution, hence the latter provides the overall solution. It should be noted that the justification for the choice of asymptotic expansions, seating variables, etc., ties in the fact that the two asymptotic expansions match. Note that... 

If one takes into account the subsequent corrections of order C(e"), n = 1,2,..., then the asymptotic solutions can be matched up to order p(g( +i)/2) jjj common domain. Incidentally, similar relations to Eq. (2.6) apply for the previous trivial example. 

Thus, the formulas (2.5), with C = 1, represent an appropriate continuation of the asymptotic solution (2.4) on the long time interval (on the slow time scale). Moreover, using the matching (2.6) one can both... 

Note that only one of the two initial data obtained is needed to construct the asymptotic solution on the slow scale for example, jr (0) = 1. The additional relation y°(0) = 1 is then satisfied automatically, and this is a crucial part of the matching. 

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