Asymptotic expansions gauge functions

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. 

In the so-called inner region nearest the sphere, the sphere radius is appropriate as the characteristic length scale, and thus equation (9-75) is applicable. To solve this equation for Re <asymptotic expansion, but unlike for (9-77), we do not presuppose any knowledge of the gauge functions, so that... 

However, rather than simply accepting Stokes solution as the first approximation for Re <inner region, we will show how it is obtained in the present framework of matched asymptotic expansions. To do this, we note that the governing equation for fo in the expansion (9-93) is simply the Stokes equation for any choice of the gauge function fo (Re), namely,... 

Now, with A determined, our solution for the first term in the outer expansion is completed, and we can turn to the problem of obtaining a second term in the asymptotic expansion for 6 in the inner region. In view of the fact that the mismatch between the first terms in the inner and outer expansions has been shown in the previous paragraph to be 0(Pe1/2), it is clear that the gauge function for the second term in the inner expansion (9-164) must be... 

Over the years, users of perturbation methods have evolved a shorthand language to express ideas. This reduces repetition and allows compact illustration. We first present the gauge functions, which are used to compare the size of functions, and then we present the order concept, which is convenient in expressing the order of a function (i.e., the speed it moves when e tends small). Finally, we discuss asymptotic expansions and sequences, and the sources of nonuniformity, which cause the solution for 0 to behave differently from the base case. 

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