Stability hydrodynamic analysis

Theoretical analysis of sheeting in the drainage of thin liquid films has been conducted in [359]. Sheeting dynamics and hole formation (i.e. black spot formation) was described by non-linear hydrodynamic stability analysis based on the equilibrium oscillatory structural component of disjoining pressure. The effect of stepwise thinning, accompanied by formation of holes , was described qualitatively. It is rather arguable whether the term holes for a black spot is appropriate since in 1980 holes in NBF were described as lack of molecules. The use the same term for two different formations is at least confusing. Besides, to have a hole in a CBF is almost as to have a hole in the sea water . 

According to the theory of hydrodynamic stability analysis, infinitesimally small perturbations are superimposed on the steady-state values of... 

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. 

III. Work of Lord Rayleigh Hydrodynamic Stability Analysis. 82... 

To summarize then, Rayleigh, motivated by B6nard s experiments, deduced the conditions for the existence of stability (preconvective equilibrium) in viscous fluid layers heated from below, and laid the framework for the hydrodynamic stability analysis of such phenomena. However, as Rayleigh himself pointed out, the boundary conditions he employed corresponded neither to the experiments of Benard, nor perhaps to any physically realizable system, and it became necessary therefore to extend Rayleigh s problem to other sets of boundary conditions. 

With Eqs. (2.10)-(2.14) and b.c. (2.15)-(2.21) we have carried out a standard hydrodynamic stability analysis and the following results have been obtained ... 

Pukhnachev (Ref 26) made a stability analysis of Chapman-Jouguet detonations to clarify the development of spinning detonations. The phenomena leading to them cannot be described by solution of simple hydrodynamic and reaction-kinetic equations for flat detonation fronts. The analysis was based on previous detonation stability analyses by Shchelkin et al with constant supersonic flow postulated along the z-axis at z <0. There is a sharp discontinuity at z=0, followed by the combustion zone. 

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. 

Several attempts have been made to explain theoretically the effects of flow on the phase behavior of polymer solutions [112,115-118,123,124]. This has been done by modification of the mean-field free energy. The key point is to include properly the elastic energy of deformation produced by flow. A more rigorous approach originates from Helfand et al. [125, 126] and Onuki [127, 128] who proposed hydrodynamic theories for the dynamics of concentration fluctuations in the presence of flow coupled with a linear stability analysis. 

Individual structural elements of the foam, such as films and borders, can be under hydrostatic equilibrium and can correspond to a true metastable state. Therefore, when there is no diffusion expansion of bubbles in a monodisperse foam, its state can be regarded as metastable in the whole disperse system. Krotov [5-7] has performed a detailed analysis of the real hydrodynamic stability of polyhedral foam by solving two problems determination of... 

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... 

Now, the limiting form for this equation for Re 1 has been derived in the book on hydrodynamic stability by Chandrasekhar,5 and we follow his analysis. First, we note that, for Re <2< I, we can approximate as... 

Our analysis describes virus adsorption from the standpoint of chemical equilibrium. Since adsorption equilibrium appears to be approached closely in our systems in less than or equal to 2 hr, and since the residence time of viruses in natural water systems is greater than 2 hr for many cases (for example, lakes, groundwaters, rivers, etc.), equilibrium considerations are entirely appropriate. In other situations, where residence times of the virus in the system are small compared to expected times required for adsorption to approach equilibrium (for example, sand filters in water treatment, water distribution systems, etc.), the DLVO-Lifshitz theory may still be applied directly. The work of Fitzpatrick and Spielman (57) concerning filtration and that of Zeichner and Schowalter (58) concerning colloid stability in fiow fields demonstrate this clearly. Their developments of hydrodynamic trajectory analysis coupled to DLVO-Lifshitz considerations can be extended... 

Wang, K.H.T., Ludviksson, V., and Lightfoot, E.N., Hydrodynamic stability of Marangoni films. II. Preliminary analysis of the effect of interphase mass transfer, AIChE J., 17, 1402, 1971. 

A set of linear equations describing the infinitesimal disturbances in the mean flow variables follow from Equations 4.1-4.3 and 5.1, 5.2 with allowance made for Equation 9.1. Analysis of this linear set and of its characteristic equation should be accomplished along the well-known standard lines of the hydrodynamic stability theory which are exemplified for similar stability problems in reference [15,34]. In addition to this general formulation of the stability problem, different simplified versions of this problem can be considered, and in particular, those corresponding to the simplified fluid dynamic models discussed in Section 6. 

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. 

For the hydrodynamic instability the main state (46) is assumed to have no x-dependence as c s xq) is a slowly varying function. Because of x enters in (46) as a parameter, the stability analysis must be fulfilled for various sections x = xq. 

The authors review the theoretical analysis of the hydrodynamic stability of fluid interfaces under nonequilibrium conditions performed by themselves and their coworkers during the last ten years. They give the basic equations they use as well as the associate boundary conditions and the constraints considered. For a single interface (planar or spherical) these constraints are a Fickean diffusion of a surface-active solute on either side of the interface with a linear or an erfian profile of concentration, sorption processes at the interface, surface chemical reactions and electrical or electrochemical constraints for charged interfaces. General stability criteria are given for each case considered and the predictions obtained are compared with experimental data. The last section is devoted to the stability of thin liquid films (aqueous or lipidic films). 

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