Self-similar profile

The above recipe can be repeated indefinitely, and the mathematical result would be what is called a self-similar profile. That is, successive magnifications of a section (in this case by factors of 3) would give magnified jagged lines which could be superimposed exactly on the original one. In this limit, one finds for this case that... 

A first necessary condition for the existence of the ID analog of Eq. (25) is the existence of a self-similar (asymptotic) velocity profile (itself equivalent to the existence of a ID equation for the flow field). This self-similar profile depends only on the wall Reynolds (Rew) number and has the following form (planar slit geometry) ... 

The concentration in the adsorbed layer decreases away from the surface as (f) (z/b) for exponent 0.588. This power law concentration profile in an adsorbed polymer layer was proposed by de Gennes and is called the de Gennes self-similar carpet. This profile of adsorbed polymer can be described by a set of layers ot correlation blobs with their size of order of their distance to the surface z (see Fig. 5.11). The self-similar concentration profile starts at the adsorption blob ads in the first layer. The self-similar profile ends either at the correlation length of the surrounding solution if it is semidilute or at the chain size Rf bN if the surrounding solution is dilute. 

If we now consider two velocity profiles, one at time /, and the other at time /2, with % > i> we might expect them to reduce to a single universal (self-similar) profile if we were to scale the distance from the wal I v with the length scale = (/ ) of the diffusion process, that is,... 

Analytical expressions for dimensionless self-similar profiles of transient velocity for various n... 

In [39], for some values of n < 1, the constants Cn are calculated and closed-form expressions are obtained for the dimensionless self-similar profiles of velocity fields in terms of elementary functions in the new variables... 

A self-similar profile decaying as is thus coherent with the experimental data. As in the first case, we do not observe the transverse fluctuations of the layer. With the experimental values y = 2.13 A and R = 1.38 A, we obtain the three quantities ... 

A self-similar profile is found where the radius of the drop increases with time as... 

FIG. 8 (a) The schematic density profile for the case of adsorption from a semidilute solution we distinguish a layer of molecular thickness Z a where the pol5nner density depends on details of the interaction with the substrate and the monomer size, the proximal region a < z < D where the decay of the density is governed hy a universal power law (which cannot he obtained within mean-field theory), the central region for D < z < with a self-similar profile, and the distal region for < z, where the polymer concentration relaxes exponentially to the bulk volume fraction (b) The density profile for the case of depletion, where the concentration decrease close to the wall (j>g relaxes to its hulk value at a distance of the order of the bulk correlation length... 

Wool [32] has considered the fractal nature of polymer-metal and of polymer-polymer surfaces. He argues that diffusion processes often lead to fractal interfaces. Although the concentration profile varies smoothly with the dimension of depth, the interface, considered in two or three dimensions is extremely rough [72]. Theoretical predictions, supported by practical measurements, suggest that the two-dimensional profile through such a surface is a self-similar fractal, that is one which appears similar at all scales of magnification. Interfaces of this kind can occur in polymer-polymer and in polymer-metal systems. 

Various means of constructing self-similar surfaces are known.33 Some of them do not allow one to produce different realizations of surface profiles, for example, by making use of the Weierstrass function. These methods should be avoided in the current context because it would be difficult to make statistically meaningful statements without averaging over a set of statistically independent simulations. An appropriate method through which to construct self-similar surfaces i s to use a representation of the height profile h(x) via its Fourier transform h(q). 

The above operation is iterated at various segment sizes. Finally, the self-similar fractal dimension of the profile embedded by the two-dimensional space is given by... 

Figure 9 demonstrates the dependence of the scaled length SL on the segment size SS obtained from the self-affine fractal profiles in Figure 7 by using the triangulation method for the Euclidean two-dimensional space. The linear relation was clearly observed for all the self-affine fractal curves, which is indicative of the self-similar scaling property of the curves. 

Figure 5.7 Profiles of the time-averaged concentration at four downstream locations. The profiles are self-similar and the solid line corresponds to a Gaussian profile shape.

The release location influences the vertical distribution of the time-averaged concentration and fluctuations. For a bed-level release, vertical profiles of the time-averaged concentration are self-similar and agreed well with gradient diffusion theory [26], In contrast, the vertical profiles for an elevated release have a peak value above the bed and are not self-similar because the distance from the source to the bed introduces a finite length scale [3, 25, 37], Additionally, it is clear that the size and relative velocity of the chemical release affects both the mean and fluctuating concentration [4], The orientation of the release also appears to influence the plume structure. The shape of the profiles of the standard deviation of the concentration fluctuations is different in the study of Crimaldi et al. [29] compared with those of Fackrell and Robins [25] and Bara et al. [26], Crimaldi et al. [29] attributed the difference to the release orientation, which was vertically upward from a flush-mounted orifice at the bed in their study. 

The above self-similar velocity profiles exists only for a Re number smaller than a critical value (e.g. 4.6 for a circular pipe). The self-similar velocity profiles must be found from the solution of the Navier-Stokes equations. Then they have to be substituted in Eq. (25) which must be solved to compute the local Nusselt number Nu z). The asymptotic Nusselt number 7Vm is for a pipe flow and constant temperature boundary condition is given by Kinney (1968) as a function of Rew and Prandtl (Pr) numbers. The complete Nu(z) curve for the pipe and slit geometries and constant temperature or constant flux boundary conditions were given by Raithby (1971). This author gave /Vm is as a function of Rew and fluid thermal Peclet (PeT) number. Both authors solved Eq. (25) via an eigenfunction expansion. 

We assume stationarity and radiative equilibrium for the energy balance because the radiative timescales are short in respect to the hydrodynamic timescales soon after the initial increase in luminosity. Spherical symmetry is assumed. According to detailed numerical models (Falk and Arnett, 1977 Muller, personal communication, 1987 Nompto, 1987 Nomoto et aL, 1987) and also analytical solutions for strong shock waves in spherical expanding enveloped (Sedov, 1959) density profiles are taken which are given by the self-similar expansion of an initial structure i.e. 

The estimates of the climatic mean annual parameters of the MRC in the sections normal to the northeastern coast of the Black Sea presented in [17] yielded a distance of its core from the shore about 40 km, a full width of the current (with respect to velocity values of 0.02 m s-1) of 75 km, a penetration depth of 275 m, a maximal geostrophic velocity of 0.31 ms-1, and a volume transport of 1.3 x 106 m3 s x. These estimates are of the same order of magnitude as shown in Fig. 4a within the velocity interval to 0.20 m s x. This allows us to suggest a certain geographical universality (self-similarity) of the MRC profile normal to the coast. 

Conversely, the relationship (7.2) expresses a time-scale invariance (selfsimilarity or fractal scaling property) of the power-law function. Mathematically, it has the same structure as (1.7), defining the capacity dimension dc of a fractal object. Thus, a is the capacity dimension of the profiles following the power-law form that obeys the fundamental property of a fractal self-similarity. A fractal decay process is therefore one for which the rate of decay decreases by some exact proportion for some chosen proportional increase in time the self-similarity requirement is fulfilled whenever the exact proportion, a, remains unchanged, independent of the moment of the segment of the data set selected to measure the proportionality constant. 

So far, we have been talking about the stability of zero pressure gradient flows. It is possible to extend the studies to include flows with pressure gradient using quasi-parallel flow assumption. To study the effects in a systematic manner, one can also use the equilibrium solution provided by the self-similar velocity profiles of the Falkner-Skan family. These similarity profiles are for wedge flows, whose external velocity distribution is of the form, 11 = k x . This family of similarity flow is characterized by the Hartree parameter jSh = 2 1 the shape factor, H =. Some typical non-dimensional flow profiles of this family are plotted against non-dimensional wall-normal co-ordinate in Fig. 2.7. The wall-normal distance is normalized by the boundary layer thickness of the shear layer. 

Near the point where the two streams first meet the chemical reaction rate is small and a self-similar frozen-flow solution for Yp applies. This frozen solution has been used as the first term in a series expansion [62] or as the first approximation in an iterative approach [64]. An integral method also has been developed [62], in which ordinary differential equations are solved for the streamwise evolution of parameters that characterize profile shapes. The problem also is well suited for application of activation-energy asymptotics, as may be seen by analogy with [65]. The boundary-layer approximation fails in the downstream region of flame spreading unless the burning velocity is small compared with u it may also fail near the point where the temperature bulge develops because of the rapid onset of heat release there,... 

The profiles obtained were also inspected for the presence of self-similarity in the external boundary layer. Inhomogeneous EPRs (3.43) allow analytical estimations like (3.41) in terms of Bessel functions. 

Consider multichain adsorption in a 0-solvent, of dilute chains with AT monomers of size b, and with monomer surface interaction of 6kT. Calculate the density profile of the de Gennes self-similar carpet. Calculate the thickness ads of the adsorbed layer and the coverage P. 

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