Stability theory nonlinear

Keywords Capillary instability of liquid jets Curvature Elongational rheology Free liquid jets Linear stability theory Nonlinear theory Quasi-one-dimensional equations Reynolds number Rheologically complex liquids (pseudoplastic, dilatant, and viscoelastic polymeric liquids) Satellite drops Small perturbations Spatial instability Surface tension Swirl Temporal instability Thermocapillarity Viscosity... 

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). 

Detonation, Nonlinear Theory of Unstable One-Dimensional. J.J. Erpenbeck describes in PhysFluids 10(2), 274-89(1969) CA 66, 8180-R(1967) a method for calcg the behavior of 1-dimensional detonations whose steady solns are hydrodynamic ally unstable. This method is based on a perturbation technique that treats the nonlinear terms in the hydro-dynamic equations as perturbations to the linear equations of hydrodynamic-stability theory. Detailed calcns are presented for several ideal-gas unimol-reaction cases for which the predicted oscillations agree reasonably well with those obtd by numerical integration of the hydrodynamic equations, as reported by W. Fickett W.W. Wood, PhysFluids 9(5), 903-16(1966) CA 65,... 

In this section, we consider these problems in some detail, although with the major simplifications of assuming that the processes are isothermal and that the liquid is incompressible. As we shall see, the governing equations for even this simplified ID problem are nonlinear, and thus most features can be exposed only by either numerical or asymptotic techniques. In fact, the problem of single-bubble motion in a time-dependent pressure field turns out to be not only practically important, but also an ideal vehicle for illustrating a number of different asymptotic techniques, as well as introducing some concepts of stability theory. It is for this reason that the problem appears in this chapter. 

As noticed above, in the case of a rotationally invariant problem, linear stability theory predicts the growth of disturbances with arbitrary directions of wavevectors. One could expect that the generation of disturbances with different orientations would produce a spatially disordered state (weak turbulence [38] or turbulent crystal [39]). We shall see however that the strong nonlinear interaction between disturbances typically leads to the selection of spatially ordered patterns. 

Summarizing, in the linear stability theory of capillary breakup of thin free liquid jets, the quasi-one-dimensional approach allows for a simple and straightforward derivation of the results almost exactly coinciding with those obtained in the framework of a rather tedious analysis of the three-dimensional equations of fluid mechanics. This serves as an important argument for further applications of the quasi-one-dimensional equations to more complex problems, which do not allow or almost do not allow exact solutions, in particular, to the nonlinear stages of the capillary breakup of straight thin liquid jets in air (considered below in this chapter). 

Linear stability theory can show definitively that a system is unstable, but it gives no information about the ultimate fate of the process as the disturbance grows. Furthermore, linear stability theory can show only conditional stability. There are two ways to attack the problem of finite disturbances. One is direct numerical simulation of the full set of nonlinear partial differential equations. This approach has become increasingly popular as computer power has grown, but a fundamental difficulty of distinguishing physical from numerical instability is always present. The other, employed less now than in the past, is to expand the nonlinear equations in... 

W. Eckhaus in Studies in Nonlinear Stability Theory, Springer, New York (1965)... 

If the process model is nonlinear, then advanced stability theory can be used (Khalil, 2001), or an approximate stability analysis can be performed based on a linearized transfer function model. If the transfer function model includes time delays, then an exact stability analysis can be performed using root-finding or, preferably, the frequency response methods of Chapter 14. A less desirable alternative is to approximate the terms and apply the Routh stability criterion. 

The basis of stability theory for systems with structurally unstable equilibrium states was developed by Lyapunov. His works and numerous subsequent studies on various aspects of stability in critical cases, as well as of bifurcation phenomena accompanying the loss of stability of equilibrium states had became the foundation on which the principal notions in the theory of nonlinear oscillations had spawned in the twenties and thirties. 

CELESTIAL MECHANICS is the study of dynamics in gravitational fields of cosmic bodies. In recent years, this has come to include gravitational statistical mechanics, galactic dynamics, nonlinear stability theory and chaos. 

The most developed and widely used approach to electroporation and membrane rupture views pore formation as a result of large nonlinear fluctuations, rather than loss of stability for small (linear) fluctuations. This theory of electroporation has been intensively reviewed [68-70], and we will discuss it only briefly. The approach is similar to the theory of crystal defect formation or to the phenomenology of nucleation in first-order phase transitions. The idea of applying this approach to pore formation in bimolecular free films can be traced back to the work of Deryagin and Gutop [71]. 

An important consequence of quantal charge transfer between ions and ion pairs (dipoles) is the appearance of non-pairwise-additive cooperative or anticooperative contributions that have no counterpart in the classical theory. These nonlinear effects strongly stabilize closed-CT systems in which each site is balanced with respect to charge transfers in and out of the site, and disfavor open-CT systems in which one or more sites serves as an uncompensated donor or acceptor. This CT cooperativity accounts for the surprising stability of cyclic (LiF) clusters, which are strongly favored compared with linear structures. 

P. Glendinning. Stability, instability and chaos an introduction to the theory of nonlinear differential equations. Cambridge University Press, 1994. 

We will consider the cold-gas-convex surface of the flame front as a curved cell of the flame which had been formed after the plane flame lost its stability. The steady state of the convex flame is a result of the nonlinear hydrodynamic interaction with the gas flow field (see Zeldovich, 1966, 1979). In the linear approximation the flame perturbation amplitude grows in time in accordance with Landau theory, but this growth is restricted by nonlinear effects. 

Such nested applications of single-parameter singular perturbation theory (i.e., the extension of the analysis of two-time-scale systems presented in Chapter 2 to multiple-time-scale systems) have been used for stability analysis of linear (Ladde and Siljak 1983) and nonlinear (Desoer and Shahruz 1986) systems in the standard form. However, as emphasized above (Section 2.3), the ODE models of chemical processes are most often in the nonstandard singularly perturbed form, with the general multiple-perturbation representation... 

---