Linearized stability theory

For the case of a thread at rest, the initial growth of a disturbance can be relatively well characterized by linear stability theory. In the initial stages, the deformation of the thread follows the growth of the fastest growing disturbance (Tomotika, 1935). Eventually the interfacial tension driven flow becomes nonlinear, leading to the formation of the smaller satellite drops (Tjahjadi et al., 1992). 

Although linear stability theory does not predict the correct number and size of drops, the time for breakup is reasonably estimated by the time for the amplitude of the fastest growing disturbance to become equal to the average radius (Tomotika, 1935) ... 

Linear stability theories have also been applied to analyses of liquid sheet breakup processes. The capillary instability of thin liquid sheets was first studied by Squire[258] who showed that instability and breakup of a liquid sheet are caused by the growth of sinuous waves, i.e., sideways deflections of the sheet centerline. For a low viscosity liquid sheet, Fraser et al.[73] derived an expression for the wavelength of the dominant unstable wave. A similar formulation was derived by Li[539] who considered both sinuous and varicose instabilities. Clark and DombrowskF540 and Reitz and Diwakar13161 formulated equations for liquid sheet breakup length. 

Meister and Scheele (MIO) examined phenomena determining the jet length, Ljet. For 2 somewhat greater than 2jet jet can be predicted from the linearized stability theory as the distance required for a symmetric disturbance to grow to an amplitude equal to the jet radius. For the apparatus and conditions... 

It may be due to the equation of state for which the deviation of the adiabatic exponent from 4/3 is small, i.e., e = 4/3-y < 0.05. However, we have to show that this proximity to N=3 should be practically be kept even during the collapse. For the structure close to N=3 the linearized stability theory for stellar structure has a wider applicability, since the... 

Contemporary understanding of liquid film rupture is based on the Linear Stability Theory and the concept of existence of fluctuational waves on liquid surfaces [81]. According to this model the film is ruptured by unstable waves, i.e. waves the amplitudes of which increase with time. The rupture occurs at the moment when the amplitude Ah or the root mean... 

Linear stability theory results match quite well with controlled laboratory experiment for thermal and centrifugal instabilities. But, instabilities dictated by shear force do not match so well, e.g. linear stability theory applied to plane Poiseuille flow gives a critical Reynolds number of 5772, while experimentally such flows have been observed to become turbulent even at Re = 1000- as shown in Davies and White (1928). Couette and pipe flows are also found to be linearly stable for all Reynolds numbers, the former was found to suffer transition in a computational exercise at Re = 350 (Lundbladh Johansson, 1991) and the latter found to be unstable in experiments for Re > 1950. Interestingly, according to Trefethen et al. (1993) the other example for which linear analysis fails include to a lesser degree, Blasius boundary layer flow. This is the flow which many cite as the success story of linear stability theory. 

It is also important to note that the linear stability theory studies a particular class of problems where the disturbances decay as one moves away in the wall-normal direction from one of the boundaries. Thus, the developed theory is mainly for disturbances that originate at the wall. The problem of destabilizing a shear layer by disturbances outside the shear layer has not received sufficient attention or adequately tackled in the past. This is also one of the major focus of this monograph, as we discuss it in chapter 2. 

Despite the reasons cited for the success of linear stability theory in predicting transition, it is important to underscore its limitation as well. This should help one to look for hitherto unknown mechanism(s) that may play a bigger role in transition prediction than it might have been suspected. For example, the envelope method does not require any information about the frequency content of the background disturbances and always predicts... 

The Reynolds number based on displacement thickness of the undisturbed flow at the outflow of the computational domain is 472. Thus, the flow is fully sub-critical in the computational domain (with respect to linear stability theory criticality). 

Schmid, P.J. (2000). Linear stability theory and bypass transition in shear flows. Phys. Plasmas. 7, 1788. 

This is very important, because the interface is stable in the presence of the diffusion-limited current, that is, the maximum current available, flowing across the interface. In other words, the flow of current or ions across the interface is not directly responsible for the interfacial turbulences, which in fact makes the strong contrast of the electrochemical instability with the instability associated with interfacial chemical reactions of the type treated within the framework of the linear stability theory [9,10,30]. 

McD] N. MacDonald (1989), Biological Delay Systems Linear Stability Theory. Cambridge Cambridge University Press. 

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... 

All of the preceding chapters seek solutions for various transport and fluid flow problems, without addressing the stability of the solutions that are obtained. The ideas of linear stability theory are very important both within the transport area and also in a variety of other problem areas that students are likely to encounter. Too often, it is not addressed in transport courses, even at the graduate level. The purpose of this chapter is to introduce students to the ideas of linear stability theory and to the methods of analysis. The problems chosen are selected because it is possible to make analytic progress and because they are of particular relevance to chemical engineering applications. The one topic that is only hghtly covered is the stability of parallel shear flows. This is primarily because it is such a subtle and complicated subject that one cannot do justice to it in this type of presentation (it is the subject of complete books all by itself). 

We begin with capillary instability of a liquid thread. This is a problem that was discussed qualitatively already in Chap. 2. It is a problem with a physically clear mechanism for instability and thus provides a good framework for introducing the basic ideas of linear stability theory. This problem is one of several examples in which the viscosity of the fluid plays no role in determining stability, but only influences the rate of growth or decay of the infinitesimal disturbances that are analyzed in a linear theory. 

The infinitely long cylinder with no motion of the interface or of the fluid within the cylinder is, of course, a possible equilibrium configuration, in the sense that it is a surface of constant curvature so that the stationary constant-radius fluid satisfies all of the conditions of the problem, including the Navier-Stokes and continuity equations (trivially), as well as all of the interface boundary conditions including especially the normal-stress balance, which simply requires that the pressure inside the cylinder exceed that outside by the factor 1//a. The question for linear stability theory is whether this stationary configuration is stable to infinitesimal perturbations of the velocity, the pressure, or the shape of the cylinder. 

The governing equations for the linear stability theory are the same as for the Rayleigh-Benard problem, namely (12-215), except that it is customary to drop the buoyancy terms because these are of secondary importance for very thin fluid layers where Marangoni instabilities are present but Ra <neutral state. Assuming that... 

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