Perturbation waves velocity

The above speculations, linking fluidization quality to perturbation-amplitude growth and decay rates, are now examined. Although bubble-related phenomena clearly imply conditions outside the linear response limit of the system, initial growth rates, obtainable from the linearized relations, can be so large in these cases that they could be expected to play a major role in subsequent developments. The linearized particle bed model delivers explicit relations for perturbation-wave velocity and amplitude growth rate, thereby enabling the above considerations to be... 

Figure 10.2 illustrates this behaviour. It shows perturbation-wave velocities, as functions of scaled wavelength Xjdp, for ambient air fluidization of 70 pm alumina particles. This is a system that switches from the stable to the unstable state at a void fraction of approximately 0.52. The figure on the left represents a stable condition, mk < md, at e = 0.44 that on the right an unstable condition, mk > d at e = 0.64. The region over which the perturbation-wave velocity differs appreciably from the limiting values of md or mk is from Xjdp values of about 1 to 100. 

Figure 10.3 shows how, for the same system, perturbation-wave velocities change as the bed is progressively expanded from the point of minimum fluidization Smf 0.4). It shows the convergence, for waves of... 

Figure 10.3 Perturbation-wave velocities as functions of wavelength air fluidization of 70 gm alumina.

Eor a linear system f (c) = if, so the wave velocity becomes independent of concentration and, in the absence of dispersive effects such as mass transfer resistance or axial mixing, a concentration perturbation propagates without changing its shape. The propagation velocity is inversely dependent on the adsorption equiUbrium constant. 

Liang, K. K., Bennett, S. D., Khuri-Yakub, B. T., and Kino, G. S. (1982). Precision measurement of Rayleigh wave velocity perturbation. Appl. Phys. Lett. 41, 1124-6. [140]... 

The propagation of a dynamic wave produces dramatic changes in local solids holdup and momentum in a gas-solid flow. Thus, to analyze the dynamic wave, one can examine the wave equation obtained for the perturbation of the momentum and mass balance equations for each of the gas and solid phases. Accordingly, the dynamic wave velocity, Vd, can be expressed by... 

The presence of the viscous damping term results in a second-order perturbation of the wave velocity and a first-order contribution to the attenuatitm. Since for most materials a < k, Equation 2.22 enables solution for the attenuation coefficient a ... 

Equation 3.63 gives an approximation for the SH plate-mode spectrum found in an unperturbed quartz plate. The presence of surface features, including transducers, perturbs the wave velocity, and hence the excitation frequency, of each mode. 

Fig. 5. Cross-sections through the P-wave velocity perturbation models obtained by inversion of delay times correeted for elevation and crustal thickness, (a), (b), and (c) show plan views of velocity perturbations at depths of 150, 200 and 300 km, respectively, (d), (e) and (f) show vertical cross-sections along profiles A-A, B-B and C-C, respectively, as shown in the horizontal sections. Surface topography is plotted at 20 times actual scale. Uppermost 50 km (shaded area) in vertical sections denotes regions where station delay time residuals are ineorporated in model calculations. Colour scale shows the velocity perturbation in percent. Colours fade to black for ray hit counts less than 10 (see Fig. 7).

Fig. 6. Cross-sections through the S-wave velocity perturbation models. (See caption for Figure 5 for a description of the cross-sections.)...

One of the fundamental aspects of the theory of nonlinear chromatography is the recognition that any perturbation (e.g., the injection of any sample of composition different from that of the mobile phase stream) caused to a column in which the mobile and stationary phases are in equilibrium results in the formation of perturbation waves that, after a time, consist in a series of concentration waves that migrate along the column. The coherence theory is based on the observation that the velocities of the waves obtained for different compounds are equal. 

The classical linear stability theory for a planar interface was formulated in 1964 by Mullins and Sekerka. The theory predicts, under what growth conditions a binary alloy solidifying unidirectionally at constant velocity may become morphologically unstable. Its basic result is a dispersion relation for those perturbation wave lengths that are able to grow, rendering a planar interface unstable. Two approximations of the theory are of practical relevance for the present work. In the thermal steady state, which is approached at large ratios of thermal to solutal diffusivity, and for concentrations close to the onset of instability the characteristic equation of the problem... 

Fluid velocity, z component Perturbation fluid velocity, z component Boundary layer coordinate along surface in streamwise direction Cartesian coordinate in direction of wave motion on free surface Cartesian coordinate parallel to direction of motion of spherical particle and translating with it Cartesian or cylindrical coordinate in direction of flow or direction of motion... 

Both the EOM-CCl and the CCLR approaches require the iterative computation of the perturbed wave functions given by Eq. (26) for each perturbation operator (/ ., p, and m) and for both positive and negative field frequencies - a total of 12 perturbed wave functions for each wavelength. In addition, the modified velocity gauge... 

Time required to compute 54 perturbed wave functions 12 for each wavelength plus 6 for the static velocity-gauge tensor. 

Similar thermally induced experiments [4] can be carried out if a molecule exists that thermally decomposes to transients on a shorter time scale than that for transient decay through bimolecular reaction. Additionally, the reactant molecule must be thermally stable under thermodynamic conditions where the source has completely dissociated. In this type of experiment, the zero time is set by shock wave passage past the observation station. In both types of experiments, pressure transducers, mounted equidistant along the shock tube, are used to measure the incident shock wave velocity. Temperature and density in the reflected shock wave regime are calculated from incident shock velocities through relations and correction procedures [8] that account for boundary layer perturbations. Since the initial composition is known, the thermodynamic state of the system is fully determined. 

Figure 9.5 Chemomechanical excitability wave ( bottleneck" shape) obtained after an acid perturbation made at the lower end of a cylindrical gel. The OSFR is fed with the CT reaction. Swollen gel diameter 2.8 mm. Wave velocity ...

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