Scale characteristic

The tip is stable if pt < and it becomes destroyed if p > a - Using a self-consistent scahng theory, explicit formulas can be obtained for the dependence of growth rates and characteristic scales of the pattern [108]. There is still a substantial need for numerical support of these results [60]. 

Now suppose we reduce our real variables x, v and t according to these characteristic scales ... 

Basically, there may be three reasons for the inconsistency between the theoretical and experimental friction factors (1) discrepancy between the actual conditions of a given experiment and the assumptions used in deriving the theoretical value, (2) error in measurements, and (3) effects due to decreasing the characteristic scale of the problem, which leads to changing correlation between the mass and surface forces (Ho and Tai 1998). 

When the contact angle is close to r/2 (for example, for the system water/steel 0 0.45 r), the ratio H/ro is small, we estimate the order of magnitude of the terms in Eqs. (9.2-9.4) and conditions (9.5-9.7). Choosing the meniscus as characteristic scales of length in the longimdinal and transversal directions H and ro, respectively, we obtain... 

Using the system (9.15-9.17) we determine the distribution of velocity, temperature and pressure within the liquid and vapor domains. We render the equations dimensionless by the following characteristic scales l,o for velocity, 7l,o for temperature, Pl,o for density, Pl,q for pressure, Pl,o Lo f " force and L for length... 

Assuming steady state in Eqs. (10.8-10.10) and (10.18-10.20), we obtain the system of equations, which determines steady regimes of the flow in the heated miero-channel. We introduce values of density p = pp.o, velocity , length = L, temperature r = Ti 0, pressure AP = Pl,o - Pg,oo and enthalpy /Jlg as characteristic scales. The dimensionless variables are defined as follows ... 

In Equation 5.2.2, it is assumed that acoustic wavelengths A are large when compared with any of the characteristic scales of the flow (A L) and that the measurement point r is far from the source region (r /l). The previous expression provides the sormd pressure in the far-field for a compact source, but it can be used indifferently for premixed or nonpremixed flames [30]. 

This definition completely coincides with the characteristic time of the probability evolution introduced in Ref. 32 from the geometrical consideration, when the characteristic scale of the evolution time was defined as the length of rectangle with the equal square, and the same definition was later used in Refs. 33-35. Similar ideology for the definition of the mean transition time was used in Ref. 30. Analogically to the MTT (5.4), the mean square d2(c,x, d) = (f2) of the transition time may also be defined as... 

Let us define the characteristic scale of time evolution of the average m.f(t) as an integral relaxation time ... 

Substituting the concrete form of H2(x) (5.94) into formula (5.129), one can obtain the characteristic scale of time evolution of any average nif(t) for arbitrary potential such that 9 ( 00) = 00 ... 

The chapter is organized as follows. Section II is devoted to materials and methods. In Section III, we show [34, 35] that the GC content displays rather regular nonlinear oscillations with two main periods of 110 20 kbp and 400 50 kbp, which are well-recognized characteristic scales of chromatin loops and loop domains involved in the hierarchical folding of chromatin fibers. 

We see that due to the smallness of the fine structure constant a a one-electron atom is a loosely bound nonrelativistic system and all relativistic effects may be treated as perturbations. There are three characteristic scales... 

Fortunately, the characteristic scales of the strong and electromagnetic interactions are vastly different, and at the large distances which are relevant for the atomic problem the influence of the proton (or nuclear) structure may be taken into account with the help of a few experimentally measurable proton properties. The largest and by far the most important correction to the atomic energy levels connected with the proton structure is induced by its finite size. 

The reduced scattering vector is given by using the characteristic scale screening of Coulombic interaction by ideal Gaussian chains, r0, as follows... 

The far-field (inlet, in the case of a finite domain) mass density px is introduced as a characteristic scaling factor, yielding... 

The distinguishing characteristics of the stagnation-flow subcases depend on the domain and on the rotation. The characteristic scales are different for the subcases, but the equations themselves are the same. The boundary conditions also differ among the subcases. Table 6.1 shows the applicable scales and nondimensional groups that apply to each of the four subcases. 

Figure 15.1 shows families of solutions to the model problem for different values of X and different initial conditions. The family of solutions can be thought of as a manifold of solutions, all of which, regardless of the initial condition, tend toward the slowly varying y = t2 + 1 solution. In chemical kinetics, the behavior illustrated in Fig. 15.1 is exhibited by certain species, like the free radicals. After initial very rapid transients such as a combustion ignition, the free-radical concentrations often vary slowly, with their behavior controlled by steady-state or partial-equilibrium conditions. The faster the characteristic scales, the more rapidly the fast-time-constant species come into equilibrium with the major species (i.e., approach a slowly varying solution). 

The modified Newton iteration, and the reason that damping is effective, can be explained in physical terms. Chemical-kinetics problems often have an enormous range of characteristic scales—this is the source of stiffness, as discussed earlier. These problems are also highly nonlinear. 

From the scaling properties of G(x, t) one can derive that S = const(d, 0)T)ft/d with d = 2dj(2 + 9) the spectral dimension of the fractal. The growth of the cluster s sizes goes on until l L where L is the whole system s size. The further growth of clusters and accumulation of particles stop because the same quantity L is the characteristic scale of a pair of different particles created in the system according to [91] there is no accumulation effect when particles are created by pairs on fractals of the Sierpinski gasket type. 

J>nv(R) is characterized by a magnitude of order a (the vibrational amplitude, whereas the characteristic scale L of the electronic wavefunction is the length of a molecular bond or the period of a crystal lattice. On this basis we obtain... 

But next, consider the characteristic scales of u7 and d these follow from noting that... 

When the block length becomes comparable with Ny distinctions between the behaviors of two copolymers practically disappear. At L > 200, one observes that T Ly with y = 4/3. In this case, the characteristic scale of the microdomain structure behaves as r Ls with <5 = 1/2. This dependence is caused by the fact that flexible chains in the melt have a Gaussian conformation, and the average size of any chain section of n units is proportional to ft1/2 [75]. Hence, for sufficiently large Vs, the spatial scale of microinhomogeneities in the system is determined only by the block size. However, the behavior of the random-block copolymer at L < 102 is more complicated. In particular, r has a minimum at L 10. 

In this particular example it should be noted that there are two clearly visible characteristic scales of segregation values, about 6 cm and 0.5 cm. These represent... 

When particles are accelerated in a gas, their motion is governed by the balance between inertial, viscous, and external forces. An important characteristic scale is the time for an accelerated particle to achieve steady motion. To find this parameter, the deceleration of a particle by friction in a stationary gas is considered. In the absence of external forces, the velocity of a particle (q) traveling in the x direction is calculated by ... 

The characteristic time t/B and the characteristic scale thus appear in the theory. The non-dimensional quantity x, defined by equation (3.16), can be interpreted as the ratio of the square of twice the characteristic scale to the mean square end-to-end distance of the macromolecule... 

Four characteristic times are involved those of reaction, tR, diffusion, tp, mixing (local motion), tjj, and macromixing (gross flow patterns), "t. The relative ratios of these characteristic times (Dai, Dan Pe) determine the reactor performance. tc is a characteristic scaling time that is chosen to be tp for "fast reactions" and tR for "slow reactions"(5). 

---