Hydrodynamic stability/instability

The discussion following Eqn. (5.1.8) imply a single Hopf bifurcation when Reynolds number increases beyond Rccr It is interesting to note that Landau (1944) talked about further instabilities following the nonlinear saturation of the primary instability mode. This is akin to Floquet analysis of the resulting time periodic system (Bender Orszag (1978)). The possibility of multiple bifurcation was also mentioned in Drazin Reid (1981) who stated that in more complete models of hydrodynamic stability we shall see that there may he further bifurcations from the solution A = 0, e.g. where the next least stable mode of the basic flow becomes unstable, and from the solution A = Ae- To the knowledge of the present authors, no theoretical analysis exist that showed multiple bifurcation before for this flow. Here,... 

For Taylor numbers exceeding Tc, the flow develops a secondary flow pattern in which ur and uz are both nonozero. A sketch of the stability criteria given by (3-86) is shown in Fig. 3 8. The reader who is interested in a detailed description of the stability analysis that leads to the criterion (3-86) is encouraged to consult Chap. 12 or one of the standard textbooks on hydrodynamic stability theory (see Chandrashekhar [1992] for a particularly lucid discussion of the instability of Couette flows).12... 

The subject of hydrodynamic stability theory is concerned with the response of a fluid system to random disturbances. The word hydrodynamic is used in two ways here. First, we may be concerned with a stationary system in which flow is the result of an instability. An example is a stationary layer of fluid that is heated from below. When the rate of heating reaches a critical point, there is a spontaneous transition in which the layer begins to undergo a steady convection motion. The role of hydrodynamic stability theory for this type of problem is to predict the conditions when this transition occurs. The second class of problems is concerned with the possible transition of one flow to a second, more complicated flow, caused by perturbations to the initial flow field. In the case of pressure-driven flow between two plane boundaries (Chap. 3), experimental observation shows that there is a critical flow rate beyond which the steady laminar flow that we studied in Chap. 3 undergoes a transition that ultimately leads to a turbulent velocity field. Hydrodynamic stability theory is then concerned with determining the critical conditions for this transition. 

This inequality is the most severe condition of the hydrodynamic stability of the liquid film [241]. Therefore, an instability and rupture of the film are considered to occur when the film thickness attains a critical value /icr, as the liquid flows out of the film. The critical thickness is determined by the condition... 

Krotov, V. V., The hydrodynamic stability of polyhedral disperse systems and their kinetics under the conditions of spontaneous breakdown. 3. Kinetics associated with the instability of films, Colloid Journal, Vol. 48, No. 6, 1986. 

In 1879 Rayleigh (see Rayleigh 1894) was the first to demonstrate by a hydrodynamic stability analysis that a liquid jet is unstable to small perturbations and breaks up into segments that, under the action of surface tension, form into individual drops. The disturbance that gives rise to the instability may be random or forced. The current procedure of choice for the production of a uniformly sized droplet stream with uniform droplet spacing is forced longitudinal vibrations in the direction of the jet flow. 

Even if the jet velocity is low enough that just capillary forces need be accounted for, the hydrodynamic stability problem is relatively simple when only the jet stability to small disturbances is considered, that is, disturbances whose amplitudes are small compared with the jet radius. When this is not the case, nonlinear mechanisms enter, which are manifest in various phenomena, including the formation of satellite droplets, which are small spherules that form between the drops (Fig. 10.4.2B). For literature on these and other nonlinear effects of jet instability, see Bogy (1979). 

DAVIS, S.H. 1987. Thermocapillary instabilities. Amt. Rev. Fluid Mech. 19, 403—435. DE GENNES, RG. 1985. Wetting statics and dynamics. Rev. Mod. Phys. 57, 827-863. DRAZIN, P.G. REID, W.H. 1981. Hydrodynamic Stability. Cambridge Cambridge Univ. Press. 

The break-up of droplets in steady flow is generally approached through consideration of the hydrodynamic stability of liquid cylinders, which are assumed to be a precursor of break-up. The critical factor for the rate of break-up of a given cylinder is the rate at which instabilities grow, whereas for the ultimate drop size, it is the wavelength of the instability which dominates Figure 14.11). 

Dussan V., Hydrodynamic stability and instability of fluid systems and interfaces. Arch. Rati. Mech. Anal., 57, 363, 1975. 

Accordingly, a hypothesis was formulated, which seeks the theoretical rationale of appearance of the specific flow structures inside the liquid layer. It is known, that under certain circumstances, the surface tension variations may lead to the flow instability and to the induction of convection cells [e.g., 5, 6]. Our preliminary theoretical analysis of the hydrodynamic stability of the system [7] indicated that it is possible, that for the Reynolds numbers exceeding the critical value, convection cells inside the hypophase can be formed. This should lead to the significant increase of the mass transfer rate. 

Chemical considerations lead to three effects that influence the stability of reactive extrusion the gel effect, the ceiling temperature and phase separation. Gel effects increase the conversion by an autocatalytic behavior. If the gel effect occurs completely in the screws, it stabilizes the process if it occurs near or in the die, it may work destabilizing. The occurrence of a ceiling temperature slows down the reaction and thus has a direct negative influence on the hydrodynamic stability. Phase separation can decrease the effective viscosity and enhance the hydrodynamic instability. It should be realized, that even in systems or at temperatures, where under stationary situations no phase separation occurs, high shear forces can change the entropy such that flow-induced phase separation is possible. 

Something should be said about the advantages and disavantages of each concept. As far as the status of physics is concerned, there is primary stability , which is the status of our understanding of the hydrodynamic stability of the plasma, and secondary stability which, usually, is less well known, but which has to do with the micro-instabilities and rotational instabilities and other finer effects. 

In the second aspect, we again picture the catalytic material itself to have spatially invariant properties, but now we ask questions about the stability of a spatially uniform reaction state to spatial perturbations. This stability question is similar to that posed in studies of hydrodynamic stability and of the other reaction-diffusion problems considered by Turing [62], Prigogine [63,64], Nicolis [63], Othmer and Scriven [65,66] and their co-workers. Prigogine and his co-workers labeled this phenomena "symmetry-breaking" instabilities. The key idea is that since there is a finite rate of transport, the complex interactions between the rate of communication by diffusive transport and the rate of chemical change may make it dynamically impossible for a spatially uniform state to be sustained. 

Diagrammatically, Figure 9.12 shows the space of all possible conditions of material properties and spinning characteristics. This space is further divided into the various regions of spinnability S, hydrodynamic stability H, cohesive fracture F, capillary breakup C, and hydrodynamic instability x-H. A system is called hydrodynamically stable, if an imposed small perturbation decays with time to either zero or some small steady value (see about the growth factor q in Eqs. 6.194... 

An alternative description of membrane stability has been based on hydrodynamic models, originally developed for liquid films in various environments [54-56]. Rupture of the film was rationalized by the instability of symmetrical squeezing modes (SQM) related to the thickness fluctuations. In the simplest form it can be described by a condition [54] d Vdis/dh < where is the interaction contribution related to the dis-... 

For the hydrodynamic instability model, Lienhard and Dhir (1973b) extended the Zuber model to the CHF on finite bodies of several kinds (see Sec. 2.3.1, Fig. 2.18). Lienhard and Hasan (1979) proposed a mechanical energy stability criterion The vapor-escape wake system in a boiling process remains stable as long as the net mechanical energy transfer to the system is negative. They concluded that there is no contradiction between this criterion and the hydrodynamic instability model. 

Hydrody namical ly, fluidized beds are considered to be stable when they are not bubbling and unstable when they are bubbling. Several researchers (Knowlton, 1977 Hoffman and Yates, 1986 Guedes de Carvalho et al., 1978) have reported that fluidized beds become smoother at elevated pressures (i.e., have smaller bubbles) and, therefore, are more stable at high pressures. There are generally two approaches as to what causes instability in fluidized beds. Rietema and co-workers (Rietema et al. 1993) forwarded the theory that the stability of the bed depends on the level of interparticle forces in the bed. However, Foscolo and Gibilaro (1984) have proposed that hydrodynamics determines whether a fluidized bed is stable. 

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

The breakup or bursting of liquid droplets suspended in liquids undergoing shear flow has been studied and observed by many researchers beginning with the classic work of G. I. Taylor in the 1930s. For low viscosity drops, two mechanisms of breakup were identified at critical capillary number values. In the first one, the pointed droplet ends release a stream of smaller droplets termed tip streaming whereas, in the second mechanism the drop breaks into two main fragments and one or more satellite droplets. Strictly inviscid droplets such as gas bubbles were found to be stable at all conditions. It must be recalled, however, that gas bubbles are compressible and soluble, and this may play a role in the relief of hydrodynamic instabilities. The relative stability of gas bubbles in shear flow was confirmed experimentally by Canedo et al. (36). They could stretch a bubble all around the cylinder in a Couette flow apparatus without any signs of breakup. Of course, in a real devolatilizer, the flow is not a steady simple shear flow and bubble breakup is more likely to take place. 

Reactive extrusion has emerged from a scientific curiosity to an industrial process. Various types of extruders can be used, all with their specific advantages and disadvantages. Further development suffers from lack of kinetic and rheological data at high conversions and from uncertainties about heat transfer and reactor stability. Nonlinear effects in the process can give rise to instabilities that are of thermal, hydrodynamical or chemical origin. 

Hydrodynamic instability, where a small increase in die pressure leads to a larger local residence time, which in turn, through conversion, results in a larger viscosity. This viscosity increase will successively increase the die pressure even further. The positive feedback will be counterbalanced by the back-flow, because an increased viscosity also increases the pressure build-up ability of the extruder. An influence on the stability may be expected if the interaction parameters and the local viscosities are such that the positive feedback dominates. 

HYDRODYNAMIC AND HYDROMAGNETIC STABILITY. S. Chandrasekhar. Lucid examination of the Rayleigh-Benard problem clear coverage of the theory of instabilities causing convection. 704pp. 5b x 8b. 64071-X Pa. 12.95... 

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