Weakly nonlinear

Note that we have used the fluid velocity U to describe convection of particles, which is valid for small Stokes number. In most practical applications, / is a highly nonlinear function of c. Thus, in a turbulent flow the average nucleation rate will depend strongly on the local micromixing conditions. In contrast, the growth rate G is often weakly nonlinear and therefore less influenced by turbulent mixing. 

We now turn to the quantum version of these results. In this case, the analogous cumulant expansion gives exactly the same equations for the centroids as above, while the equations for the higher cumulants are different. We can again investigate whether a trajectory limit exists. Localization holds in the weakly nonlinear case if the classical condition above is satisfied. In the case of strong nonlinearity, the inequality becomes... 

Alternative methods of analysis have been examined and evaluated. Shokoohi and Elrod[533] solved the Navier-Stokes equations numerically in the axisymmetric form. Bogy15271 used the Cosserat theory developed by Green.[534] Ibrahim and Linl535 conducted a weakly nonlinear instability analysis. The method of strained coordinates was also examined. In spite of the mathematical or computational elegance, all of these methods suffer from inherent complexity. Lee15361 developed a 1 -D, nonlinear direct-simulation technique that proved to be a simple and practical method for investigating the nonlinear instability of a liquid j et. Lee s direct-simulation approach formed the... 

FIGURE 10.18 Illustration of the different types of possible peaks (1) the perturbation peak, (2) the mass peak, and (3) the plateau perturbation peaks, on three concentration plateaus. A single Langmnir model was assumed with a=2.0 and b=0.100. (a) A linear plateau, C=0.05mM. (b) A weakly nonlinear platean, C=0.5mM. (c) A clearly nonlinear plateau, C=5mM. The chromatogram shows the result of an analytical injection of a mixture of labeled and unlabeled molecules on a concentration plateau of unlabeled molecules. The solid line shows the perturbation peak (left scale), the dashed-dotted line shows the plateau perturbation peaks (left scale), and the dotted line shows the mass peak (right scale). Here (mM) is the concentration of unlabeled molecules, Q is the concentration of labeled molecules, and the x axis is time. The mean retention times,, and calculated... 

To elaborate somewhat on the above issues and mainly to pave the ground for the treatment of the PDEs. version of the Teorell model in the next section we conclude here with the standard weakly nonlinear analysis of the vicinity of bifurcation i — ic in the model (6.2.7). 

This information is used for the following formal weakly nonlinear analysis of the bifurcating solution in the system (6.3.9)-(6.3.15). 

On the other hand, most weak nonlinearities can be associated with the dependence of specific heats (Cp) upon temperature, or of diffusion coefficients upon concentrations, etc. 

Equation (4.36) makes the steady-state conversion Xsss weakly nonlinear for varying feed rates Cxr This is so since Cxf appears both in the leading linear term b/2a and under the square root sign of XsS3 (Cxf) in equation (4.36). 

Clearly flow aligning behavior of the director is present and do increases linearly with the tilt angle, do. Above a threshold in the Spain rate, y 0.011, undulations in vorticity direction set in. In Fig. 14 the results of simulations for y 0.015 are shown. In Fig. 15 we have plotted the undulation amplitude obtained as a function of the shear rate. The dashed line indicates a square root behavior corresponding to a forward bifurcation near the onset of undulations. This is, indeed, what is expected, when a weakly nonlinear analysis based on the underlying macroscopic equations is performed [54], In Fig. 16 we have plotted an example for the dynamic behavior obtained from molecular dynamics simulations. It shows the time evolution after a step-type start for two shear rates below the onset of undulations. The two solid lines correspond to a fit to the data using the solutions of the averaged linearized form of (27). The shear approaches its stationary value for small tilt angle (implied by the use of the linearized equation) with a characteristic time scale t = fi/Bi. 

The orbit does not necessarily diffuse over the whole area of the frequency space within practical computational time, because of slow diffusion under weak nonlinearity and coupling strength. Instead of waiting for a long time steps, we set the initial state in the product of primary stochastic region of each standard maps, and we fix totally iterated time steps. 

Second, we investigate the dependence of residence time distributions on nonlinearity of elements, as shown in Fig. 10. We get power-law decay of the distribution again. The exponent a is 3/2 for weak nonlinearity, and it approaches 2 for stronger nonlinearity. 

Figure 10. Dependence of residence time distributions on the strength of nonlinearity of each element (K = 0.5, 0.8, 0.9, 1.0, 1.2). Distributions decay with a power law (p(t) t ct). The exponent is a = 3/2 with weak nonlinearity, and it approaches a = 2 with stronger nonlinearity, coi — 1/2, b = 0.002, and T — 1010 steps. The data are shifted by some values along the horizontal...

The motion along the one-dimensional resonance line called Arnold diffusion is prominent at lower-order resonances when nonlinearity is weak. In fact, the motion with the residence time distribution of the power 3/2 is observed for low-order resonance with weak nonlinearity. On the other hand, overlapped resonances allow the motion across resonances which leads to Brownian motion at a two-dimensional region. Indeed, the distribution with the power 2 is observed at higher-order resonances, and it is more frequently observed with stronger nonlinearity. Hence, one can distinguish clearly the Arnold diffusion from the motion induced by resonance overlaps by the power of the residence time distribution at each resonance condition. 

To simplify the analysis of the problem, we used hypothesis (3.94) that the function e = e(t) is linear. It is, in fact, weakly nonlinear with an increasing parabolic... 

Composite Sorption Magnitude and Isotherm Nonlinearity. Accurate assessment of the extent to which the global isotherm for a system is nonlinear is important for accurate portrayal of sorption processes in that system. From a practical point of view, the extrapolation of linear approximations of weakly nonlinear or near-linear sorption isotherms to concentration ranges beyond which they are valid can result in significant errors in projections of contaminant fate and transport (1). From a conceptual point of view, observations of isotherm nonlinearity over specific concentration ranges may be employed in conjunction with models such as the DRM to probe and evaluate the extent to which multiple sorption mechanisms are operative in a particular system. 

D. D. Weiner, J. F. Spina, Sinusoidal Analysis and Modelling of Weakly Nonlinear Systems, Van Nostrand Reinhold, New York, NY, 1980. 

A small fraction of the energy flow into a component of the wave spectrum flows into wave components with small wave numbers respectively frequencies due to weak nonlinear interaction between wave components of different wave numbers. These energy flow forms the low-frequency part of the spectrum and generates the kernels of the wave spectrum that are able to grow when the wind speed increases. This mechanism adjusts the wave spectrum to varying wind velocities. 

The elementary example above reveals a more general truth There are going to be (at least) two time scales in weakly nonlinear oscillators. We ve already met this phenomenon in Figure 7.6.1, where the amplitude of the spiral grew very slowly compared to the cycle time. An analytical method called two-timing builds in the fact of two time scales from the start, and produces better approximations... 

The same steps occur again and again in problems about weakly nonlinear oscillators. We can save time by deriving some general formulas. 

Consider the equation for a general weakly nonlinear oscillator ... 

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