Regular asymptotic expansion

Thus the power expansion (16. 9) seems to be valid practically only in the case of bounded perturbation. So we are obliged to regard it as asymptotic expansion even in the case of regular perturbation. In this section we shall assume (17.1) and examine under what conditions the first several coefficients of (16. 9) are significant, and then, discuss the validity of the asymptotic expansion. 

Summing up, the series (16. 9) Is valid when interpreted as an asymptotic expansion at least to the 0-th, order for arbitrary initial state fo, to the fkst order if second order if special case of regular perturbation (17.1) is assumed. 

We saw in the previous example lor R, <approximate solution, (4-4), which we obtained by taking the limit R, —> 0 in the exact equation, (4-1), was just the first term in an asymptotic solution for Rr approximate equation and solution (4-17) that we obtain by taking the limit Rm -> oo in (4-16) will also be the first term in a formal asymptotic expansion of H for Rm y>> 1. Assuming this expansion is regular, it will take the form... 

In the previous section we demonstrated the application of asymptotic expansion techniques to obtain the high- and low-frequency limits of the velocity field for flow in a circular tube driven by an oscillatory pressure gradient. In the process, we introduced such fundamental notions as the difference between a regular and a singular asymptotic expansion and, in the latter case, the concept of matching of the asymptotic approximations that are valid in different parts of the domain. However, all of the presentation was ad hoc, without the benefit of any formal introduction to the properties of asymptotic expansions. The present section is intended to provide at least a partial remedy for that shortcoming. We note, however,... 

Examining these solutions, we see that the temperature becomes nonuniform and the velocity profile nonlinear. These results are to be expected from a qualitative point of view. In a sense, the most important conclusion is that the regular asymptotic expansion in terms of the small parameter Br provides a method to obtain an approximate solution of the highly nonlinear boundary-value problem to evaluate the influence of weak dissipation, which can clearly be applied to other problems. 

As a starting point, we recall that the limit a/R = 0 corresponds to a straight circular tube, with the flow described by the Poiseuille flow solution w = (1 — r2), u = v = 0. In the present context, we consider small, but nonzero, values of a/R, and recognize the Poiseuille flow solution as a first approximation in an asymptotic approximation scheme. In particular, if we assume that a solution exists for u in the form of a regular asymptotic expansion,... 

Now we seek a solution for w in the form of a regular asymptotic expansion ... 

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. 

The basis of our analysis is, again, a very simple variation of the regular perturbation technique in which we assume that the bubble radius can be expressed in terms of a regular asymptotic expansion of the form... 

A natural question is to what extent the overall rate of heat transfer is modified by convection for small, but nonzero, values of the Peclet number. To obtain a more accurate estimate of Nu for this case, it would appear, from what has been stated thus far, that we must calculate added terms in the regular asymptotic expansion (9-15) for 0. To attempt this, we must substitute (9-15) into (9-7) and (9-8) to obtain governing equations and boundary conditions for the subsequent terms 6 . In this section, we consider only the second approximation, 9. The governing equation and boundary conditions derived from (9-7), (9-8), and (9-15) are... 

The nondimensionalization leading to (9-7) and/or (9-24) has led us to the conclusion that convection terms should be negligible compared with conduction terms anywhere in the domain if Pe is sufficiently small. However, the estimates (9-28) show clearly that this is not true. Specifically, for any arbitrarily small Pe, we can always find a value of r 0(Pe x) such that the conduction and convection terms are of equal importance. We conclude that the nondimensionalization leading to (9-7) can be valid only in the part of the domain within a distance r < ()(l e ) from the sphere. It follows that a regular (uniformly valid) asymptotic expansion cannot exist for Pe 1. 

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. 

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. 

Suppose the expansion of r(x,e) is desired in a domain Q. If the asymptotic convergence occnrs for all x in Q, the expansion is said to be a regular asymptotic expansion. 

Generally, the occurrence of the regular asymptotic expansions is in finite (bonnded) domains. In other situations, the convergence of a particular expansion may occnr only in a snbdomain. For example, the entire domain may be made up of two separate subdomains, Q+ separated from another denoted Or, with an intersection 30 separating the two domains. This is also called the overlap region. Thus, we may have two separate expansions of the function T(x,e) as follows ... 

In the mass exchange problem for a circular cylinder freely suspended in linear shear flow, no diffusion boundary layer is formed as Pe - oo near the surface of the cylinder. The concentration distribution is sought in the form of a regular asymptotic expansion (4.8.12) in negative powers of the Peclet number. The mean Sherwood number remains finite as Pe - oo. This is due to the fact that mass and heat transfer to the cylinder is blocked by the region of closed circulation. As a result, mass and heat transfer to the surface is mainly determined by molecular diffusion in the direction orthogonal to the streamlines. In this case, the concentration is constant on each streamline (but is different on different streamlines). 

Here, the solution can be found in the form of a regular asymptotic expansion (4.8.12) in powers of the small parameter Pe l. The leading term of this series satisfies the equation derived in [238], The numeric solution leads to the following formula for the mean-volume dimensionless concentration inside the drop ... 

It should be remarked that the asymptotic expansions for spirals waves developed here are not regular at the origin. We saw the same problem in Chapter I, see Eq. (1.22). At the moment, the core of a spiral wave still eludes analysis. Another misfortune of this theory is the existence of a continuous family of spiral waves for all wavenumbers k sufficiently small. Only spiral waves of fixed wavenumber (pitch) are observed. An adequate explanation of this fact has also yet to appear. 

The cases of a sphere and slightly deformed sphere in a uniform flow field are considered first in Sections 4 and 5. The mathematical method used conventionally in these problems is the regular asymptotic expansion. The reader is introduced to this method. In Section 6, the dip coating problem under the lubrication theory approximation is examined. (The closely related slender body approximation is outlined in Problem 7.5.) A more sophisticated method of matched asymptotic expansions is used to solve this problem and its main features... 

The asymptotic expansion of the solution of problem (6.1), (6.2) consists of regular and boundary layer parts... 

The regular part of the asymptotic expansion does not generally satisfy the boundary condition (6.2). In the terminology of the paper of Vishik and Lyusternik [27], the regular terms of the asymptotics introduce a discrepancy into the boundary condition. The purpose of the boundary layer functions Il,(p, is to compensate for this discrepancy. The equality (6.5) shows that the boundary layer functions together with the regular terms must satisfy the boundary condition (6.2). 

Thus, the II-functions compensate for the discrepancy introduced by the regular part of the asymptotic expansion to the boundary condition on the side y = 0. 

Q f and P f are similar to Qf and Pf. Now, expanding the terms of (7,18) into power series in e and equating coefficients of like powers of e separately for the regular functions and for each type of boundary function, we obtain equations for the terms of the asymptotic expansion. Additional conditions are obtained by substituting (7.17) into (7.14) and (7.15). This gives... 

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