Fisher velocity

We plot in Fig. 6.5 the dimensionless front velocity vt/u vs the reaction rate r on a log-log scale. The front velocity increases with r. For the cases / = a and I = 2a, the slope is very similar, but for / = oo it is steeper. In all cases the front velocity increases as a power law of r, straight line in a log-log plot, for small and moderate values of r and saturates to 1 for larger values, the slope in the log-log plot tends to 0. This behavior is due to the fact that an increase of the reaction rate r leads to an increase of the front velocity. However, the front cannot travel faster than the jump velocity of the particles if all of them jump in the backbone direction, i.e., V < ajx. For I = a and / = 2a the transport is diffusive, and the diffusion coefficient is properly defined. If this transport is combined with a KPP reaction, a Fisher velocity is expected, i.e., in both cases v fr. Computing numerically the slope from a linear fit in Fig. 6.5 we obtain and for / = a and I = 2a, respectively. The case / oo is quite different, because the transport is anomalous. Equation (5.36) with y = 1/2 yields v while the linear fit of the numerical results yields Numerical and analytical results are in good agreement. 

The Hamilton-Jacobi formalism, on the other hand, only holds for KPP kinetics, but in contrast to singular perturbation analysis there is no need to assume either weak or smooth heterogeneities. The local velocity approach is based on the assumption that for weak and smooth heterogeneities the velocity of the front is given by the local value of the reaction rate r and the diffusion coefficient D at each spatial point, i.e., the front velocity coincides with the instantaneous Fisher velocity V 2y/r x)D x). In general, this simple-minded approach is not consistent with results from the other analytical methods or with numerical solutions. 

Fig. 3.17. Spectrum of the central region of an SO galaxy, NGC 3384, showing hydrogen, magnesium and iron spectral features used in the Lick system. The resolution is 3.1 A ( 75kms 1), compared to a line-of-sight velocity dispersion 140kms 1. After Fisher, Franx and Illingworth (1996). Courtesy Garth Illingworth.

The evaluation of stability for travelling-wave solutions is by no means a simple process and will not even be attempted here. The result that systems governed by quadratic Fisher equations tend to pick up their minimum permitted velocity will be used later. 

The quadratic Fisher result of the previous section is based on the quadratic chemical timescale tch = 1 /kqa0 if we represent the wave velocity measured in these terms as k, defined by... 

Figure 22.8 Dendrite tip velocity vs. tip radius for an Al/Cu alloy. The diffusion-limit portion of the curve is unaffected by capillarity. The capillarity limit indicates the point where the tip curvature causes the dendrite growth to stop. From Kurz and Fisher [3],...

Fig. 3.3. Oxygen capacity of Ridox at two different flow rates. A, space velocity 3000 h B, space velocity 6000 h (Adapted with permission of the copyright owner from Fisher Scientific Co., Bulletin I99B/8-02I-0I.)...

For this sample, the galactic distance scale has been set by Mathewson, Ford, and Buchhorn (MFB) using the Tully-Fisher relationship, which sets distance scales by using an observed correlation between the maximum rotation velocity of a spiral and its absolute luminosity, and so is quite distinct from Hubble-based distance determinations. Even so, the Tully-Fisher method gives an absolute scale only after calibration, and the MFB calibration gave a scale that was statistically similar to a Hubble scale using H = 85kms-1 Mpc 1. 

Figure 6.20 Comparison between experimental and computed velocity profiles during fiber spinning using Denn and Fisher s viscoelastic model [4].

Figure 1.10 Schematic wind distribution over an urban territory after Bottema, 1993 [71] and Fisher et al., 2005 [193] 1 - logarithmic portion high enough over the city 2 - decelerated and distorted velocity profile within the urban canopy 3 - transition layer (roughness sublayer).

In studying the kinetics of the homogeneous gas-phase reaction between sulfur vapor and methane, Fisher reported conversions for various space velocities. These space velocities were deflned as the volumetric flow rate in milliters per hour divided by the total volume of empty reactor in cubic centimeters. The flow rate is based on all the sulfur being considered S2 and is referred to 0°C and 1 atm pressure. 

The roots Ali2(1,2) in this model are identical with the eigenvalues, for a = D = 1, in the Fisher-Kolmogorov model. As the second pair of the roots does not lead to a generation of the sensitive states, the nature of a catastrophe for the Oregonator with diffusion is the same as for the Fisher-Kolmogorov model. In other words, the waves with the velocities v centre manifold theorem the sensitive state is associated only with the zlt z2 variables. 

This is the Fisher-Kolmogorov equation, and whereas no analytical solution is known for it, a simple phase-plane analysis allows a determination of the minimal front propagation velocity. 

A front corresponds to a traveling wave solution, which maintains its shape, travels with a constant velocity v, p x, t) = p(x - v t), and joins two steady states of the system. The latter are uniform stationary states, p(x, t) = p, where Ffp) = 0. For the logistic kinetics, the steady states are = 0 and jo2 = 1- While the logistic kinetics has only two steady states, three or more stationary states can exist for a broad class of systems in nonlinear chemistry and population dynamics with Alice effect, but a front can only connect two of them. To determine the propagation direction of the front, we need to evaluate the stability of the stationary states, see Sect. 1.2. The steady state jo is stable if P (fp) < 0 and unstable if F (jo) > 0. Let the initial particle density p x,0) be such that on a certain finite interval, p x,0) is different from 0 and 1, and to the left of this interval p(x,0) = 1, while to the right p x, 0) = 0. In this case, the initial condition is said to have compact support. Kolmogorov et al. [232] showed for Fisher s equation that due to the combined effects of diffusion and reaction, the region of density close to 1 expands to the... 

The integrand is appreciably different from zero only in the front region of the wave a smaller value of v corresponds to a narrower, i.e., steeper front. As for the Fisher equation (4.1), the steepness of the wave front depends inversely on the wave velocity V, and natural initial conditions, i.e., initial conditions that are localized or that decay faster than exponentially, relax to the front with the minimal front velocity. The minimal front velocity for the Fisher DIRW is smaller than the minimal front velocity for the Fisher equation, prd = 2 /d, see Sect. 4.1.1, and approaches the latter in the diffusive limit. 

Reasoning along the same lines as for the DIRW case, we obtain the following result In the diffusive as well as the ballistic regime, the stable state (1/2, 1/2) invades the unstable state (0, 0) in the Fisher DDRW in the form of a propagating front. The front travels at constant velocity v with v e [wddrw> K)-... 

Fig. 7.11 Plot of the results for the front velocity vs the parameter rr. The points tire obttiined from simulations of the stochastic process on the OCNs (fiill circles) tmd the Peano basin open circles). The lines correspond to theoretical values from the Fisher equation (solid) and for the Peano basin (dotted and dotted-dashed). Reprinted from [64]. Copyright 2006, with permission from Elsevier...

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