Characteristic length scales

Eqs. (74-78) contain two dimensional parameters, and D, and two dimensionless parameters, A and e. This means that any characteristic length scale i and growth velocity v of the possible structures can be presented in the form... 

The aim is to predict, for given undercooling A and anisotropy e, the type of the two-phase structure and its characteristic length scales and velocity that is, to calculate the functions/ and v in the relation (80). The results will be summarized in the morphology diagram shown in Fig. 6. As it turns out. 

The most important property of the self-organized critical state is the presence of locally connected domains of all sizes. Since a given perturbation of the state 77 can lead to anything from a trivial one-site shift to a lattice-wide avalanche, there are no characteristic length scales in the system. Bak, et al. [bak87] have, in fact, found that the distribution function D s) of domains of size s obeys the power law... 

The fact that there are no characteristic length scales immediately implies a similar lack of any characteristic time scales for the fluctuations. Consider the effect of a single perturbation of a random site of a system in the critical state. The perturbation will spread to the neighbors of the site, to the next nearest neighbors, and so on, until, after a time r and a total of / sand slides, the effects will die out. The distribution of the life-times of the avalanches, D t), obeys the power law... 

Thus, the available data related to transition in circular micro-tubes testify to the fact that the critical Reynolds number, which corresponds to the onset of such transition, is about 2,000. The evaluation of critical Reynolds number in irregular micro-channels will entail great difficulty since this problem contains a number of characteristic length scales. This fact leads to some vagueness in definition of critical Reynolds number that is not a single criterion, which determines flow characteristics. 

Equations (3.11) and (3.12) show that the friction factor of a rectangular micro-channel is determined by two dimensionless groups (1) the Reynolds number that is defined by channel depth, and (2) the channel aspect ratio. It is essential that the introduction of a hydraulic diameter as the characteristic length scale does not allow for the reduction of the number of dimensionless groups to one. We obtain... 

Kn is the ratio of two length scales, the mean free path of the gas molecules A and a characteristic length scale of the flow domain L, for example the channel diameter. For molecules interacting as hard spheres of diameter d, the mean free path is given as (see, e.g., [4])... 

Figure 10.3 Dependence of the gradient of the characteristic length scales on the irradiation intensity observed for PSAF/MMA mixtures with different compositions at room temperature. The 2D power spectra corresponding to the morphologies are indicated in the inset.

Designing Polymers with an Arbitrary Distribution of Characteristic Length Scales by the Computer-Assisted Irradiation (CAI) Method... 

If the length scales associated with changes in velocity are normalized by Vv (characteristic length scale for Stokes flow), length scales associated with changes in curvature are normalized by Ss (typical striation thickness) and velocities normalized by V (a characteristic velocity), then the normal stress condition becomes,... 

The second approach assigns thermal resistance to a gaseous boundary layer at the heat transfer surface. The enhancement of heat transfer found in fluidized beds is then attributed to the scouring action of solid particles on the gas film, decreasing the effective film thickness. The early works of Leva et al. (1949), Dow and Jacob (1951), and Levenspiel and Walton (1954) utilized this approach. Models following this approach generally attempt to correlate a heat transfer Nusselt number in terms of the fluid Prandtl number and a modified Reynolds number with either the particle diameter or the tube diameter as the characteristic length scale. Examples are ... 

The characteristic length scale, L-, taken here to be the jet penetration depth is shown by Turner (1973) for a buoyancy jet to be... 

In this section, we review models for the self-assembly of tubules and helical ribbons that are not based on chiral elastic properties. Our overall assessment is that these models have not been successful in explaining the characteristic length scale for the curvature of tubules and helical ribbons. 

It is useful to consider the transient aspect of the flame spread problem. The ratio of characteristic times for the solid and the gas phase, based on d as a characteristic length scale, is... 

This is the characteristic length scale for a line plume. 

In Chapter 11 it was shown that control volume theory for the bulk compartment smoke properties could be expressed in dimensionless solutions. The characteristic length scale involves the geometric components of wall vents as 1 = f/f0 (//0 )1 /2]2/ 5. Hence, the MQH correlation [18] leads to... 

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