Linear stability analysis

With the exceptions of Schowalter (38) and of Hickernell and Yortsos (22), all previous linear stability analyses (34-37, 39-42) have used the local volume-averaged ecjuation of continuity for an incompressible fluid, although they assumed that density was a function of concentration and therefore position and time. This is... 

Unfortunately, most studies have not taken this approach. Numerical simulations of displacement have not been preceded by linear stability analyses to define the parameter limits within which an unstable displacement could be expected. [In their nonlinear stability analysis, Perrine and Gay (43) allowed their dispersion coefficients to be functions of velocity. In his linear stability analysis, Perrine (34, 35) assumed that his dispersion coefficients were constants.]... 

A fully transient simulation that does not use commercial software, but is based on the shallow water approximation relevant to Hall cells has been reported by Zikanov et al. [89], Because of the speed of this simulation, additional features become available for investigation, such as continuing oscillations of constant amplitude, which can sometimes be detected in operating cells. This behavior, due to nonlinear interactions, is not within the capabilities of the linearized stability analyses that have formed the bulk of the literature to date. 

The Sterling-Sleicher [13] result is often used in current approaches as it incorporates aerodynamic effects. Chapter 3 provides a detailed description of the analytic technique employed for these linear stability analyses, only a final result (dispersion equation) relating the growth rate of disturbances given a certain wavenumber k) wiU be provided here. The inviscid form of Sterling and Sleicher s result may be written (see (1.4.2)) ... 

CAT+ would vary periodically in both the phases along with the ratio r= [CAT ]org/[CAT+]aq, which would be responsible for electric potential oscillations. The formation of reverse micelles in the non-aqueous phase would be responsible for the decrease in concentration of [CAT ] ,g in the non-aqueous phase. On the other hand, it would be compensated by the transfer from the aqueous phase. But this would require more H in the aqueous phase to interact with CTAB to generate CTA. The required amount of H+ would be produced by transfer of HP from the non-aqueous phase and the cycle would be repeated over and over again, giving rise to oscillations in electric potential and PH. Using relevant kinetic equations [27] and linear stability analyses, mathematical formalism has been developed. 

Instability is a nonlinear phenomenon. However, the dynamic behavior of nuclear reactors can be assumed to be linear for small perturbations around steady-state conditions. This allows the reactor stability to be studied and the threshold of instability in nuclear reactors to be predicted by using a linear model and solving linearized equations. Linear stability analyses in the frequency domain have been... 

Linear stability analyses were also carried out for bubble columns by, e.g., Shnip et al (1992), Minev et al (1999), Leon-Becerril and Line (2001), Sankaranarayanan and Sundaresan (2002), Lucas et al (2005, 2006), and Monahan and Fox (2007). The reader is also referred to the review paper by Joshi (2001). The purpose of these analyses was to learn more about the stabihty of homogeneous (or uniform) bubble flow and the transition to heterogeneous flow regimes. In most cases, the results of the analyses were again strongly dependent on the exact formulation of the fluid—bubble interaction force, particularly of its lateral component. Several of these authors pinpoint the crucial effects of virtual mass and of (the sign of) the lift force. 

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