Intermediate asymptotics

G. I. Barenblatt and Ya. B. Zel dovich, Self-similar solutions as intermediate asymptotics, Armu. Rev. Fluid Mech. 4, 285 (1972). 

G. I. Barenblatt, Similarity, Self-Similarity and Intermediate Asymptotics, Consultants Bureau, New York, 1979. 

Note that its asymptotics (2.2.34) gives essentially slower decay than (2.1.77) observed for the Poisson initial distribution. In the d = 2 case the integral in (2.2.33) should be cut off at km n oc S x/2 where S is surface square practically complete disappearance of particles takes place after t S/D. Consider briefly the applicability of (2.2.34). Use of (2.2.34) for the initial fluctuation spectrum continues infinitely similar particle aggregation takes a very prolonged time. Under finite excitation time, the peculiarity (2.2.31) at small k is not pronounced, (2.2.34) is not universal and it plays the role of the intermediate asymptotics and holds at t [Pg.93]

The intermediate asymptotic exponent could be also defined as... 

This intermediate asymptotic exponent is useful to demonstrate the formation in time of a new asymptotic law shown in Fig. 6.3. In a given temporal interval a(t) approaches its asymptotic value close to a = d/2. 

Direct establishment of the asymptotic reaction law (2.1.78) requires performance of computer simulations up to certain reaction depths r, equation (5.1.60). In general, it depends on the initial concentrations of reactants. Since both computer simulations and real experiments are limited in time, it is important to clarify which values of the intermediate asymptotic exponents a(t), equation (4.1.68), could indeed be observed for, say, r 3. The relevant results for the black sphere model (3.2.16) obtained in [25, 26] are plotted in Figs 6.21 to 6.23. The illustrative results for the linear approximation are also presented there. 

Finally, comparatively recently, Ya.B., A. P. Aldushin, and S. I. Khudyaev (23) completed a theory of flame propagation which considers the most general case of a mixture in which the chemical reaction occurs at a finite rate at the initial temperature as well. In this work the basic idea is followed through with extraordinary clarity flame propagation represents an intermediate asymptote of the general problem of a chemical reaction occurring in space and time. At the same time, the relation between the two types of solutions (KPP and ZFK) is completely clarified. 

The parameter , which enters into the downstream boundary conditions, can actually be interpreted as = T/,/71,., where T), - L2/D is the hydraulic diffusion time scale. When 1, there exists an intermediate asymptotic behavior corresponding to the zero-flux solution at z = L. When ( -C 1, the solution is essentially the membrane solution. 

These formulas coincide with those derived in Ref. 137 and explain the intermediate asymptotics, 40 = const(o>) confirmed there experimentally. The limiting value of the phase is less than half that for rigid dipoles /(( xB —> 00) = 90°. 

Furthermore, an increase in the space dimensionality results in restricting the time interval where an intermediate asymptotic limit (2.1.106) is true. This happens at the expense of a shift in its lower boundaty towards larger values of r, while the upper limit of the interval with J 1 does not depend on the space dimensionality. Indeed, with d I, w d /4 and the inequality (5.2.43) it takes the form... 

Barenblatt G1 (1996) Scaling, Self Similarity, and Intermediate Asymptotics. Cambridge Cambridge University Press... 

Two linear dimensions, the minimum length Iq (the lattice constant) and the correlation length play a key role in the behavior of a percolation system. For real systems, an intermediate asymptotic region exists such that... 

In the domain > Iq, that is in the domain of intermediate asymptotics, on the interval l all characteristics of the cluster are similar to its characteristics at the most critical point. On all scales / > , the system is homogeneous, and the self-similarity disappears. 

The correlation length , of the connecting set is confined to the range of intermediate asymptotics, which may be defined as... 

If P2/PI > 10 -r 30, curves of Pr/pi have one more intermediate asymptote corresponding to a nonconducting surrounding medium. In other words, there is such a time interval when intensity of currents is still negligible in more resistive surrounding medium, but within the formation they are distributed uniformly along the 2 -axis. 

The crucial difference of this formula in comparison to Eqs. (86) and (87) is a very slow decrease of the interaction potential within this interval of R. The existence of this intermediate asymptotic behavior of U (R) leads to crucial consequences for the correlation properties of the ion ensemble. Sect. 2.1.11.5. 

Wetting and Spreading, Fig. 11 Contact line region of a spreading drop, ahead of which a thin front running precursor film exists. The intermediate asymptotic region... 

In this paragraph, the authors are trying to make an intermediate asymptotic approximation of viscosity at different levels. 

These intermediate asymptotics are sometimes used to express the... 

It comes, of course, as no surprise that introducing a molecular cut-off and applying a kinetic slip condition to the first molecular layer resolves the notorious singularities of hydrodynamic description. The hydrodynamic singularities are eliminated, however, only at molecular distances, and are still felt in sharp interface curvatures at microscopic distances identified here as the intermediate asymptotic region. The computations are eased considerably when non-physical divergence of both viscous stress and attractive Lennard-Jones potential beyond the cut-off limit... 

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