Asymptotic approximation first scale

One important class of problems for which we can obtain significant results at the first level of approximation is the motion of fluids in thin films. In this and the subsequent chapter, we consider how to analyze such problems by using the ideas of scaling and asymptotic approximation. In this chapter, we consider thin films between two solid surfaces, in which the primary physics is the large pressures that are set up by relative motions of the boundaries, and the resulting ideas about lubrication in a general sense. 

With a first-order reaction, the governing equation is linear and could thus be solved without any use of scaling or asymptotic methods. However, we could just as easily assume that the reaction rate is second order in c or add other complications that do not so easily allow an exact analytic solution. The point here is to illustrate the idea of the asymptotic approximation technique, which is easily generalizable to all of these problems. 

The accuracy of the Kirkwood superposition approximation was questioned recently [15] in terms of the new reaction model called NAN (nearest available neighbour reaction) [16-20], Unlike previous reaction models, in the NAN scheme AB pairs recombine in a strict order of separation the closest pair in an initially random distribution is removed first, then the next one and so on. Thus for NAN, the recombination distance R, e.g., the separation of the closest pair of dissimilar particles at any stage of the recombination, replaces real time as the ordering variable time does not enter at all the NAN scheme. R is conveniently measured in units of the initial pair separation. At large R in J-dimensions, NAN scaling arguments [16] lead rapidly to the result that the pair population decreases asymptotically as cR d/2 (c... 

It is attractive to seek solutions by use of activation-energy asymptotics. If there is a narrow reaction zone in the vicinity of x = 0, then by stretching the coordinate about x = 0 and excluding variations on a very short time scale, the augmented version of equation (56) becomes Xgd T/dx = — qgWi to the first approximation in the reaction zone. By use of equation (7-20), the integral of this equation across the reaction zone is seen to be expressible as... 

The necessary application of activation-energy asymptotics parallels that given in Sections 9.2, 5.3.6, and 8.2.1 and has been developed for both one-reactant [172], [173] and two-reactant [174], [175] systems. For most problems (for example, for stability analyses), gradients of the total enthalpy and of the mixture ratio of the reactants are negligible in the first approximation within the reaction sheet, so that in the scaled variables appropriate to the reaction zone, both 6 -h Y Q/iY qLq ) and Y -h y2Lej/(v2Le2) are constant. Evaluation of these constants is aided by the further result that Yi = 0 (and hence dYJd = 0) downstream from the reaction zone, at least in the first approximations, so that for the reaction-zone analysis, the expressions... 

In spite of this, we shall see that potential-flow theory plays an important role in the development of asymptotic solutions for Re i>> 1. Indeed, if we compare the assumptions and analysis leading to (10-9) and then to (10-12) with the early steps in analysis of heat transfer at high Peclet number, it is clear that the solution to = 0 is a valid first approximation lor Re y> 1 everywhere except in the immediate vicinity of the body surface. There the body dimension, a, that was used to nondimensionalize (10-1) is not a relevant characteristic length scale. In this region, we shall see that the flow develops a boundary layer in which viscous forces remain important even as Re i>> 1, and this allows the no-shp condition to be satisfied. 

Outer region, where variations of velocity are characterized by the length scale a of the body and potential-flow theory provides a valid first approximation in an asymptotic expansion of the solution for Re -> oo. 

As we see, the results of the approximate (asymptotic) analysis are different in the various scales, and cannot be interchanged with each other. The structure of the first correction and remainders allows us to see this feature in detail. The accuracy of the approximation depends on both the small parameter e and the time t. There are terms both in the first correction and in the remainders that grow like r, as t tends to infinity. Such terms, occurring in the asymptotic formulas, are sometimes called secular terms. Due to this result the first formula (2.1) is suitable for times that are not very long t lle. For long times, t-lle, the order of the remainder (et) is the same as that of the leading term. Hence, the approximation turns out to be false for t=lle. The second formula (2.2) is valid for (slow) times that are not very small t e. In this case, the order of the remainder 5 ((E/T) ) is the same as that of the leading term for small times r = e. Hence, the approximation turns out to be false for very small (slow) times t - e. 

Formulas (4.9), (4.10) provide the approximate solution on the slow time scale. From these asymptotics one can see that the second and third components tend to zero, whereas the first one tends to a nonzero constant y = (W/A) at infinity. If we use Taylor expansions of the left sides of Eqs. (4.9) and (4.10), the error estimate of the asymptotic behavior at infinity can easily be derived as follows ... 

A first approximation to Eq. (10.109) may be obtained by introducing the condition, of considerable practical importance, that the resonances in the fuel (or absorber) material are narrow on the energy scale relative to the average energy loss suffered by a neutron in a scattering collision with a moderator atom and, furthermore, are widely separated. In this case we can use the asymptotic expression for the function F which appears in the collision integral for the moderator material and note that it is also consistent to take over the interval of integration. From the analysis of Sec. 4.4b we have q u) = (2i(u) (ti), and it follows that for a normalized source... 

With this asymptotic density model (ADM) an advantage over the standard multipole expansion can be achieved, so that the MESP is well approximated not only outside the molecule, but also near the atoms. This scheme has been implemented in SINDOl for first- and second-row elements. The evolving expansion has been truncated after the cumulative dipole moments terms. The three-center integrals have been approximated on the NDDO level. In order to achieve good agreement with ab initio based MESPs, the atomic hybrid moments had to be scaled. In the application to. solvation energies the atomic electronic charges had to be scaled too. 

As in the case of linear chains the problem of dynamics cannot be solved exactly because of the presence of excluded volume forces on all scales, and we need a crude theory for the problem which contains all asymptotic limits. An approximate propagator for the dynamics of the cluster will be enough to aim for. Let us first take the simplest Brownian fractal, the linear chain, and set up the equation of motion. This is now treated in textbooks and we can be brief here. 

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