Nonlinear theory

The linear theory has been found to be applicable to limited conditions. Nonlinear theories therefore have been developed, and this has allowed prediction of nearly all liquid surface oscillations encountered in practical applications. Majority of previous investigations were concerned with liquids in circular cylindrical tanks [1,6,10]. Abramson et al. [10] focused on nonlinear oscillations of liquid contained 

Kimura and Ohashi [16] succeeded in developing a nonlinear theory for liquids contained in axisymmetrical tanks of arbitrary geometries (see Fig. 5.3) using the calculus of variation and derived the equations governing the motions of liquid surfaces. This theory was further verified by comparing the results using the finite element method with the experimental data for a circular cylindrical tank and a spherical tank undergoing horizontal excitation [17]. 

The natural frequency, f(= /T, of liquid surface oscillations in a circular cylindrical tank and a spherical tank are presented in Figs. 5.4 and 5.5, respectively. In these figures, //l is the bath depth, D is the vessel diameter. Hi,/D is the aspect ratio, m in the parentheses denotes the wth mode of surface oscillation in the tangential direction and n is the nth mode in the radial direction. The tangential mode, i.e., rotary sloshing appears first in both tanks. Further information about this study is available elsewhere [13]. 

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Vorovich 1.1., Lebedev L.P. (1972) Existence of solutions in nonlinear theory of shallow shells. Appls. Maths. Mechs. 36 (4), 691-704 (in Russian). 

Eringen, A.C., Nonlinear Theory of Continuous Media, McGraw-Hill, New York, 1962. 

The continuum theory of deformation of elastic solids is old and well developed [65T01, 74T01], and, in its linear version, is widely applied. Nonlinear theory is of much more recent origin. Most application of nonlinear theory has been to the behavior of highly deformable materials such as rubber or to the explanation of subtle effects observed by precise ultrasonic... 

Chen et al. [76C02] and Lawrence and Davison [77L01] have placed the fully coupled nonlinear theory of uniaxial piezoelectric response in a form that is convenient for numerical solution of problems and have simulated a number of experiments in terms of this theory. An example of the results obtained is given below. 

Limit Cycles.—The new closed trajectories discovered by Poincar62 and which he called cycles limites, constitute the essence, of the nonlinear theory. 

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,... 

Nonlinear Theory of Unstable One-Dimensional Theory of Detonation. See Detonation, Nonlinear Theory of Unstable... 

The usual quantum mechanics, just like the most important physical theories of our time, is a linear theory. Still, we have the feeling that these linear theories, even if they have helped us to understand and predict an astonishing quantity of phenomena, must correspond in reality to some sort of statistical approximation of deeper and more general nonlinear theories. It would be logical to assume that these nonlinear theories must generate the usual linear theories as a particular case. Natural phenomena are certainly very complex. Therefore, the deeper one goes in the level of description of nature, the farther one stands from... 

Eringen, A. C. Nonlinear theory of continuous media. New York McGraw-HiU 1962. 

Ions- Measurement. 2. Electro-diffusion. 3. Transport theory. 4. Nonlinear theories. I. Title. II. Series. 

Nonlinear Theory. It is straightforward to generalize the above linear dynamics to cases with general network deformations [12]. This is necessary for description of dynamics in deformed gels and phase separation. First, the relative velocity should be written as... 

Detonation, nonlinear theory of unstable onedimensional 4 D460... 

F.C. Larche and J.W. Cahn. A nonlinear theory of thermomechanical equilibrium of solids under stress. Acta MetalL, 26(l) 53-60, 1978. 

Note that (139) are highly nonlinear in the scalars but become exactly the linear Maxwell equations in the fields F v and F 1V. In this sense, the Maxwell equations are the exact linearization (by change of variables, not by truncation ) of a nonlinear theory with topological properties, in which the force lines... 

The superposition approximation (SA) suggested in Refs. Ill and 259 is essentially a nonlinear theory that cannot be represented in the form of Eqs. (3.707). The same is true for the extended version of SA [260]. For this reason, we focus on two derivatives of these theories linearized near the equilibrium the linearized superposition approximation (LSA) and the linearized extended superposition approximation (LESA). It was found that LSA developed in a number of works [139,175,255,260] isinfact identical to IET (see Table VIII). They both have the same concentration-independent kernel S(j ). As for LESA, it was, strictly speaking, created for the reactions in the ground state [241,242], but can be easily extended to the case of equal lifetimes, uA = uc-... 

Although a key characteristic of the mechanical behavior of rubber-like materials is their ability to undergo large elastic deformations, we will present here some important results from the theory of linear elasticity [1], which is valid only for small deformations. These serve our present purposes better than the nonlinear theory, because of their simpler character and physical transparency. 

R has been demonstrated that attachment-line boundary layer supports instability waves, the kind predicted in linear theory (shown by very careful experiments in Pfenninger Bacon 1969 Poll 1979 Arnal, Coustols Jul-lien 1984 Hall, Malik Poll 1984 Poll, Banks Yardley 1996). However, they do not cause transition occurring at the attachment-line, as no linear or nonlinear theories have explained transition at the attachment-line. In the literature, this premature transition is referred to as the Leading Edge Contamination (LEG) problem. In Sengupta et al. (2004) and in Sengupta... 

Now, we turn our attention to late stages of spinodal decomposition. Since the phase separation in binary mixture is intrinsically a nonlinear phenomenon, a number of nonlinear theories have been put forward on the basis of statistical consideration, notably the LBM (Danger, Baron k Miller) (H) and BS (Binder k Stauffer) (12) theories. Both theories predicted the power law scheme rather than the exponential growth of the structure... 

Grzywna, Z.J., Siwy, Z., and Bashford, C.L., Nonlinear theory for ionic transport through track-etched nuclear membranes. J. Membr. Sci., 1996, 121 261-269. 

As an illustration of the strong perturbation introduced by an ion in a metal and of the theoretical description of it in nonlinear theory of screening, we show some results obtained for the interaction of Ne ions with metals. Multicharged Ne ions have been widely used in the experimental study of ion neutralization and electron emission processes at metal surfaces [14-16], We plot in Fig. 1 the electronic density induced by a Ne ion in a FEG of... 

Danov, K.D., Adsorption from micellar surfactant solutions nonlinear theory and experiment, J. Colloid Interface ScL, 183, 223, 1996. 

Weinstock, J., Nonlinear theory of gravity waves Momentum deposition, generalized Rayleigh friction, and diffusion. J Atmos Set 39, 1698, 1982. 

The analysis with disturbance quantities of the form of Eqs. (10.6.15) indicates a periodic structure in the x, z plane but the shape of the cells associated with the solution is not specified and higher order nonlinear theory is required to define a particular cellular structure. Palm (1960) has shown that in the parallel Rayleigh problem for steady buoyancy driven convection of a liquid film heated from below, the cells approach a hexagonal form as a consequence of the variation of the kinematic viscosity with temperature. 

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