Asymptotic expansion scheme

Let S o be a surface located at mid-channel between two smooth surfaces separated by a narrow gap. The curvilinear coordinate system, corresponding to this 

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have 

After substitution of the leading terms of the expanded variables into the model equations and equating coefficients of equal powers of e from their sides, they are divided by common factors to obtain the following set  

Continuity equation corresponding to the first-order terms 

Equations (5,61) and (5.62) can be used to derive a pressure potential equation applicable to thin-layer flow between curved surfaces using the following procedure. In a thin-layer flow, the following velocity boundary conditions are prescribed  

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Observe that scheme (11) belongs to this family, corresponds to the case it = 1 and has the residual ip = ut + (tus + (1 — [Pg.361]

As it should be, at (3 = 0 this formula reduces to Eq. (4.174), which was obtained for a one-dimensional case. We remark, however, that in a tilted situation ((3 / 0) the coefficient D2 acquires a contribution independent on a that assumes the leading role. This effect is clearly due to admixing of transverse modes to the set of eigenfunctions of the system, and it is just it that causes so a significant discrepancy between the zero-derivative approximation and the correct asymptotic expansion for x(3 ) curves in Figure 4.12. Evaluation of the coefficient L>4 is done according to the same scheme and requires taking into account a number of the perturbation terms that makes it rather cumbersome. 

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,... 

At high Reynolds numbers, the terms in the equation of motion (see Supplement 6) and in the continuity equation with regard to the expressions (6.1.1) and (6.1.4) can be estimated by the same scheme as for Newtonian fluids. As a result, after isolating the leading terms of the corresponding asymptotic expansions, we obtain... 

As shown in [4], the discretization error of this scheme has a perturbed asymptotic expansion, where the perturbations depend on the choice of the applied Jacobian approximation within the linearly implicit scheme. Up to... 

In addition to providing benchmarks for extrapolation schemes, Eqs. (28) and (29) give rise to some new identities of mathematical interest. First, combining Eq. (28) with the known asymptotic expansions for the total energy and its / = 0 component at the limit of [Pg.153]

The second procedure, several aspects of which are reviewed in this paper, consists of directly computing the asymptotic value by employing newly-developed polymeric techniques which take advantage of the one-dimensional periodicity of these systems. Since the polarizability is either the linear response of the dipole moment to the field or the negative of the second-order term in the perturbation expansion of the energy as a power series in the field, several schemes can be proposed for its evaluation. Section 3 points out that several of these schemes are inconsistent with band theory summarized in Section 2. In Section 4, we present the main points of the polymeric polarization propagator approaches we have developed, and in Section 5, we describe some of their characteristics in applications to prototype systems. 

Numerical calculations using Kapuy s partitioning scheme have shown that for covalent systems the role of one-particle localization corrections in many-body perturbation theory is extremely important. For good quality results several orders of one-particle perturbations have to be taken into account, although the additional computational power requirement is much less in these cases than for the two-electron perturbative corrections. Another alternative for increasing the precision of the calculations is to estimate of the asymptotic behavior of the double power series expansion (24) from the first few terms by applying Canterbury approximants [31], which is a two-variable generalization of the well-known Pade approximation method. It has also been found [6, 7] that in more metallic-like systems the relative importance of the localization corrections decreases, at least in PPP approximation. 

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