Leading order terms, asymptotic solutions

Hence we seek an asymptotic solution for the leading-order terms in the thin-film approximation, i.e., u<(>>p<(l>w<(>>, as a regular perturbation in 8, i.e.,... 

It should be remembered that (9-225) and its solution (9-240) represent only the first term in an asymptotic expansion for Pe - oo in the boundary-layer regime, and (9-203) and its solution (9-207) is the leading-order term in a corresponding expansion for the outer region. To obtain the next level of approximation, it is necessary to calculate an additional term in both of these expansions. We do not pursue this calculation here because of the... 

In particular, let us start with the nondimensionalized vorticity transport equation (10-6) and attempt to obtain an approximate solution for Re 1. We expect an asymptotic expansion with Re 1 as the small parameter, but we restrict our attention here to the leading-order term in this expansion, which we can obtain by solving the limiting form of (10-6) lor Re —> oo, namely,... 

The task is to find the asymptotic approximation of the solution as e- 0 uniformly over a long time interval 0main goal. The higher order corrections of 0(e) will be defined in order to identify both the secular terms and the time intervals of asymptotic fitness. 

The equations for the coefficients of the asymptotic solutions (4.4) are modified too. In particular, one algebraic and two differential equations determine the leading order terms... 

Let us consider the leading order terms of the various asymptotic solutions. On the first (fast) scale the third component... 

The solution will therefore be adequate if A A+ < 0.01, say. Since the product A A+ will be found to behave as exp(—2kdR), the leading order term for large KdR of Eq. [121] can be identified as the asymptotic solution for the interaction of two planar surfaces. Further analysis must await explicit expressions for coefficients A . 

Fig. 3.3.2. Profiles of bound ion concentration a(x). ———— Numerical solution, -----Leading order asymptotic term.

The analyses of Hunt, Liebovich and Richards, 1988 [287] and of Finnigan and Belcher, 2004 [189] divide the flow in the canopy and in the free boundary layer above into a series of layers with essentially different dynamics. The dominant terms in the momentum balance in each layer are determined by a scale analysis and the eventual solution to the flow held is achieved by asymptotically matching solutions for the flow in each layer. The model apphes in the limit that H/L 1. By adopting this limit, Hunt, Liebovich and Richards [287] were able to make the important simplification of calculating the leading order perturbation to the pressure held using potential how theory. This perturbation to the mean pressure, A p x, z), can then be taken to drive the leading order (i.e. 0(II/I.) ]) velocity and shear stress perturbations over the hill. 

It has been stated repeatedly that the boundary-layer and potential-flow equations apply to only the leading term in an asymptotic expansion of the solution for Re F> 1. This is clear from the fact that we derived both in their respective domains of validity by simply taking the limit Re -= oc in the appropriately nondimensionalized Navier-Stokes equations. Frequently, in the analysis of laminar flow at high Reynolds number, we do not proceed beyond these leading-order approximations because they already contain the most important information a prediction of whether or not the flow will separate and, if not, an analytic approximation for the drag. Nevertheless, the reader may be interested in how we would proceed to the next level of approximation, and this is described briefly in the remainder of this section.13... 

Now, 90/97 = 0(1) at the leading order of approximation. Hence, it follows that the dimensionless temperature gradient within the innermost region should be 0(Pr1/2). But the solution (11-69) represents only the first term in an asymptotic expansion. The condition (11-70) suggests strongly that... 

In the case of a very diluted polymer network or at high temperatures liquid crystal is ordered only in the vicinity of the polymer fibers. If the order parameter 5 is small the quadratic and cubic term in (12.53) can be neglected and an analytical solution can be expressed with the modified Bessel functions Ko r) and K r) [74] which asymptotically leads to s r) oc For a particular case, profiles are shown in Figure 12.27. 

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