Weakly nonlinear analysis

To elaborate somewhat on the above issues and mainly to pave the ground for the treatment of the PDEs. version of the Teorell model in the next section we conclude here with the standard weakly nonlinear analysis of the vicinity of bifurcation i — ic in the model (6.2.7). 

This information is used for the following formal weakly nonlinear analysis of the bifurcating solution in the system (6.3.9)-(6.3.15). 

Clearly flow aligning behavior of the director is present and do increases linearly with the tilt angle, do. Above a threshold in the Spain rate, y 0.011, undulations in vorticity direction set in. In Fig. 14 the results of simulations for y 0.015 are shown. In Fig. 15 we have plotted the undulation amplitude obtained as a function of the shear rate. The dashed line indicates a square root behavior corresponding to a forward bifurcation near the onset of undulations. This is, indeed, what is expected, when a weakly nonlinear analysis based on the underlying macroscopic equations is performed [54], In Fig. 16 we have plotted an example for the dynamic behavior obtained from molecular dynamics simulations. It shows the time evolution after a step-type start for two shear rates below the onset of undulations. The two solid lines correspond to a fit to the data using the solutions of the averaged linearized form of (27). The shear approaches its stationary value for small tilt angle (implied by the use of the linearized equation) with a characteristic time scale t = fi/Bi. 

First, we consider a 1+1 system (2D film with ID surface) since some important features of the nonhnear evolution of the film instability can be studied in this case. In order to study the possibility of pattern formation one first performs a weakly nonlinear analysis of stationary solutions of eq.(5) near the instabihty threshold. A characteristic feature of the system described by eq.(5) is the presence of the zero mode, cr = 0, corresponding to k = 0 (see Fig.2) and associated with the conservation of mass. The nonlinear interaction between the zero mode and the unstable mode can substantially affect the system behavior near the instabihty threshold [15, 16] and must be taken into account in weakly nonlinear analysis. 

The condition (10) allows one to determine regions in the q, p)-plane corresponding to different types of pattern excitation and stability near threshold as shown in Fig.3. The straight hnes OCB and OGF correspond to Ao = 0 and the curves AB, CD, EF, GH are parts of the hyperbola sw/m + Xq = 0. It is interesting that at the intersection point O, p = 1/2, q = 3/4. Since p = 1/2 corresponds to c = 0, this means that unless the wetting potential depends on the film slope the periodic structure is always subcritical and therefore blows up. Weakly nonlinear analysis is not useful in this case. 

Numerical solutions of eq.(5) in 2D by means of a pseudospectral method with periodic boundary conditions are shown in Fig.8. One observes the formation of hexagonal arrays of dots or pits in the parameter regions predicted by the weakly nonlinear analysis. It is interesting that, similar to the 1-f1 case, the formation of two types of dots is possible cone -like and cap -like (Fig.8a,b). With the increase of the supercriticality cones transform into caps . Similarly, the formation of two types of pits is observed anticones at small supercriticality and anticaps at larger ones (Fig.8c,d). 

First, let us discuss the evolution near the instability threshold by means of weakly nonlinear analysis. Consider wqi = 1/4 — 2e, e 1, introduce the long-scale coordinate X = ex and the slow time T = and expand... 

We have discussed certain aspects of self-assembly of quantum dots from thin solid films epitaxially grown on solid substrates. We have considered two principle mechanisms of instability of a planar film that lead to the formation of quantum dots the one associated with epitaxial stress and the one associated with the anisotropy of the film surface energy. We have focused on the case of particularly thin hlms when wethng interactions between the film and the substrate are important and derived nonlinear evolution equations for the him surface shape in the small-slope approximahon. We have shown that wetting interachons between the him and the substrate damp long-wave modes of instability and yield the short-wave instability spectrum that can result in the formahon of spahally-regular arrays of islands. We have discussed the nonlinear evoluhon of such arrays analyhcally, by means of weakly nonlinear analysis, and numerically, far from the instability threshold and have shown... 

In [23] a weakly nonlinear analysis for the case of counterpropagating waves was considered. In this case coupled nonlocal complex Ginzburg-Landau equations for the amplitudes of the coimterpropagating waves were derived. We now summarize the derivation. Again, for simplicity, we consider the problem in Cartesian coordinates. Moreover, though the derivation in [23] was for the case when the effect of melting was accounted for, here we do not include melting effects. 

A free boundary model is ui d to describe frontal polymerization. Weakly nonlinear analysis is applied to investigate pulsating instabilities in two dimensions, llie analysis produces a pair of Landau equations, which describe the evolution of the linearly unstable modes. Onset and stability of spinning and standing modes is described. 

The structures are often of large scale, for example, of the same order as that of the mean flow. They have been found to be responsible for mechanisms that relate to the transport and the production in almost all commonly observed turbulent flows. Findings show the existence of the underlying orderliness in the apparent chaos of turbulent flows and have fostered many modifications to the traditional thinking of turbulence, such as the turbulent gradient transport hypothesis. Weakly nonlinear analysis has been shown to reproduce the large-scale phenomena in simple flows. 

As a main result of this analysis, it is shown that the presence of the zero mode can only be detected in very large aspect ratio boxes and for very thin fluid layers. It should be realized that the results for k = 0 obtained above provide a first step towards a (weakly) nonlinear analysis. 

When the intensity is not uniform, this leads to additional long-scale surface tension gradients affecting the evolution of the long-scale mode. Indeed, the coupling effects are most pronounced in the case when the long-and the short-scale modes have instability threshold close to each other. In the above cited paper, Golovin, Nepomnyaschy and Pismen (1994), have studied these effects analytically (via a weakly nonlinear analysis) in the vicinity of the instability thresholds. 

E. Bodenschatz, M. Kaiser, L. Kramer, W. Pesch, A. Weber, W. Zimmermann Patterns and defects in liquid crystals, in P. Coullet, P. Huerre (eds) New Trends in Nonlinear Dynamics and Pattern Farming Phenomena The Geometry of Nonequilibrium, Plenum Press, NATO ASl Series, (1990) W. Pfesch, W. Decker, Q. Rng, M. Kaiser, L. Kramer, A. Weber Weakly nonlinear analysis of pattern formation in nematic liquid crystals, in J. M. Coron, E Helein, J. M. Ghidaglia (eds) Nematics Mathematical and Physical Aspects, Kluwer Academic Publishers, Dordrecht, NATO ASl Series, p. 291 (1991)... 

Anderson DM, Worster MG (1995) Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys. J Fluid Mech 302 307... 

From the eigenvalue analysis of RW states, it has been possible to completely specify the codimension-two bifurcation to linear order. To gain a deeper understanding of the organizing center, one needs to understand the codimension-two bifurcation to higher order. For this one turns to a weakly nonlinear analysis, which is the subject of the next section. 

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