Wave dynamics

The Hydrodynamic Theory of fluidized bed stability was proposed by Foscolo and Gibilaro who adapted the stability principle of Wallis. They postulated that a fluidized bed is composed of two interpenetrating fluids. One fluid is the gas phase, and the solids phase is also considered as a continuous fluid phase. In this theory, voidage disturbances in the bed propagate as dynamic and kinetic waves. The stability of the fluidized bed depends upon the relative velocities of these two waves. The velocities of the kinetic wave (ue) and the dynamic wave (nj are ... 

Wallis (1969) defined the dynamic wave in one-phase flow as being that which occurs whenever there is a net force on the flowing medium produced by a concentration gradient. For a two-phase flow, i.e., gas-solid flow, the flow medium refers to the gas phase and the concentration refers to the solids holdup. Thus, to analyze dynamic waves, one can examine the wave equation obtained from the perturbation of the momentum and mass balance equations for the gas and solid phases. The analyses given later for both 6.5.2.2 and 6.5.2.3 follow those of Rietema (1991). 

In order to obtain an expression for the dynamic wave speed, consider a perturbation imposed on a steady-state flow yielding a, u, and wp, as... 

The first two terms establish the basic wave equation describing the dynamic wave propagation with velocity Vj expressed by... 

Rep Particle Reynolds number, based vd Dynamic wave velocity... 

Dynamic wave registry of four separate time intervals. Attempts at CTO recanalization have been decreased. Abbreviation CTO, chronic total occlusion. 

It can be readily seen by comparison with Eq. (50) that the left-hand side of Eq. (61) is the elastic wave or dynamic wave velocity of Gibilaro Foscolo and co-workers. 

Thus, Eq. (61) forms the basis of comparison of dynamic wave velocity... 

Equation (61) is the transition criterion provided the conditions given by equation (52) are satisfied. From Table I it can be seen that these conditions are satisfied only in the case of gas-solid fluidized beds and in some cases of solid-liquid fluidized beds where ps Pl- Therefore, for other multiphase dispersions [such as gas-liquid (bubble columns) and solid-liquid fluidized beds (where pl is not negligible)] the comparison of dynamic wave velocity with continuity wave velocity is not valid for deciding the bed stability. Further, the above analysis holds for transition from region I to II (point P in Fig. 1) and not for III to II (point Q). Therefore, the criterion does not hold for bubble columns and dilute dispersions. 

Flows where fluid dynamic wave propagation within the fluid are important. 

Let us now turn to the problem of breaking the stronger H-bond of the system, while leaving the weaker one intact. Since the previous IR laser pulse was designed to drive the asymmetric stretching vibration and create a dynamical wave packet that oscillates with the largest amplitude away from the equilibrium posi-... 

Alfven waves - Very low frequency waves which can exist in a plasma in the presence of a uniform magnetic field. Also called magnetohydro-dynamic waves. 

FIGURE 4.1 Depiction of the space-time elementary retarded path coimecting two events characterized by their dynamic wave-functions. 

STABILITY ANALYSIS IN RELATION TO KINEMATIC AND DYNAMIC WAVES, 346... 

Another approach for analyzing the stability of the flow is based on wave-theory. In deriving the characteristics of kinematic and dynamic waves in two-component flow, Wallis has shown that the relations between the velocities of these two classes of waves govern the stability of the two stratified layers [74]. It has been shown that the condition of equal kinematic and dynamic waves velocities corresponds to marginal stability. Following this approach, Wu et al. determined the stratified/ nonstratified transition in horizontal gas-liquid flows [38]. The relations between the dispersion equation. Equation 16, and stability criteria Equation 33 on one hand, and the characteristics of kinematic and dynamic waves on the other hand, (for = 0), was shown in Brauner and Moalem Maron [45]and Crowley et al. [47]. 

The relation of the general dispersion Equations 34 to dynamic waves is derived here by recalling that a pure dynamic wave occurs whenever the net force on the flowing fluids is produced only by concentration gradients (and is independent of the insitu concentration, Wallis [74]). In this case, the quasi-steady shear stress terms on the rhs of the combined momentum equation, which are functions of the insitu concentration, are ignored, whereby AF is considered as identically zero. However, the dynamic interfacial shear stress term, which is proportional to the concentration gradient, evolves from the Reynolds shear stresses in the turbulent field and is retained. The general dispersion Equation 34, with V = 0, becomes ... 

Equation 36 yields the dynamic wave velocity relative to the weighted mean velocity, c. ... 

A stable dynamic wave is obtained provided (Vg > V ) whereby ... 

Note, however, that a pure dynamic wave may be physically realized in inviscid flows. In viscid flow systems, the wave characteristics may be related to those of pure dynamic and kinematic waves by introducing - cj from Equation 36... 

In view of Equation 39, it is easily shown that at neutral stable conditions the wave velocity is equal to both the kinematic and dynamic waves velocities, c = C = Cj. Substituting c = c + ic, nto Equation 39 results in the wave frequency and wave amplification in identical forms to those derived by Wallis [74] ... 

Equations 40 indicate that the locus for which the kinematic wave velocity is equal to that of the dynamic wave, = C, represents neutral stable wave modes. 

Indeed, equating from Equation 37 to C (= C ), again renders the condition derived for neutral stability in Equation 33. Stable modes are obtained for c < c d whereas for unstable modes to exist it is required that c > c > c (since V, >0). Hence, it is the relation between kinematic and dynamic wave velocities which essentially determines the stability, as c > c corresponds to unstable modes, whereas modes with cj > c are attenuated. 

The identity between condition (41) and condition (38) for stable dynamic wave indicates that the region of well-posedness coincides with that of stable dynamic waves, c > 0. The region of c < 0, corresponds to unstable waves and their evolution, as formulated by the initial value set of equations, is ill-posed. As the stability condition for inviscid flows (obtained with = 0) is equivalent to that of stable dynamic waves, the well-posedness condition (with = 0, = 1) is... 

As a corollary, it can be stated that the condition of unstable dynamic waves or ill-posedness is sufficient to indicate instability, whereas the condition of c < 0, or well-posedness, is necessary but insufficient to ensure stability. The above ideas and interpretations as detailed above with regards to the horizontal system of Figure 8 also prevail basically in inclined flows, although limiting stability and well-posedness boundaries may demonstrate entirely different structures (Brauner and Moalem Maron [45]). 

Epstein, I.R., Pojman, J.A. Introduction to Nonlinear Chemical Dynamics Waves, Patterns... 

Katagawa, T, Kato, N. (1974). The exact dynamical wave fields for a crystal with a constant strain gradient on the basis of the Takagi-Taupin equations. Acta Cryst. A 30, 830-836. 

In estimating wave forces for cases (a)-(c), the pressures on structures of dynamic waves as well as static waves should be considered. 

We can apply the results for monochromatic waves to investigate the approximate dynamic wave setup for a simple irregular wave case. Consider a bichromatic wave system with wave heights Hi and H2 Hi > H2) and a small frequency difference between the two components. If the resulting wave group varies so slowly that static conditions occur within the surf zone, Eq. (1.7) applies and is written as... 

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