Linearized stability analysis

A linear stability analysis of (A3.3.57) can provide some insight into the structure of solutions to model B. The linear approximation to (A3.3.57) can be easily solved by taking a spatial Fourier transfomi. The result for the Ml Fourier mode is... 

To pursue this question we shall examine the stability of certain steady state solutions of Che above equaclons by the well known technique of linearized stability analysis, which gives a necessary (but noc sufficient) condition for the stability of Che steady state. 

Linear stability analysis has been successfully applied to derive the critical Marangoni number for several situations. 

Bai [48] presents a linear stability analysis of plastic shear deformation. This involves the relationship between competing effects of work hardening, thermal softening, and thermal conduction. If the flow stress is given by Tq, and work hardening and thermal softening in the initial state are represented... 

At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. 

These two time constants overlap assuring that film liquid exchange by capillary pumping or suction can keep pace with the pore-wall stretching and squeezing of the film during flow through several constrictions. On the other hand, based on a linear stability analysis summarized in Appendix A for a free,... 

Appendix A - Linear Stability Analysis of a Free Lamella... 

The morphological stability of initially smooth electrodeposits has been analyzed by several authors [48-56]. In a linear stability analysis, the current distribution on a low-amplitude sinusoidal surface is found as an expansion around the distribution on the flat surface. The first order current distribution is used to calculate the rate of amplification of the surface corrugation. A plot of amplification rate versus mode number or wavelength separates the regimes of stable and unstable fluctuation and... 

Aiming at a more formal analysis, the asymptotic stability of a steady-state value S° of a metabolic system upon an infinitesimal perturbation is determined by linear stability analysis. Given a metabolic system at a positive steady-state value... 

This relates the motion of step n to velocity functions / of the widths of the terrace in front [/+(w )] and behind f- (w ., )] the moving step. A straightforward linear stability analysis of (11) around the uniform step train configuration with terrace width w shows that if... 

Liang Y. (1994) Axisymmetric double-diffusive convection in a cylindrical container linear stability analysis with applications to molten Ga0-Al203-Si02. In Double-... 

Here C is defined by the boundary value in the case of the Dirichlet conditions (3.1.3b), (3.1.3d) at one of the end points or by the space averages of the initial concentrations in the case of the Neumann conditions (3.1.3a), (3.1.3c) at both ends. In the spirit of a standard linear stability analysis consider a small perturbation of the equilibrium of the form... 

In the following, we present linear stability analysis of surfaces of uniaxial gels in the clamped case. This is appropriate because the previous theories are... 

Very interestingly, a similar surface instability was predicted by Grinfeld on uniaxially compressed solids in contact with their melt [94]. The theory has been revived by Nozieres in a general context [95] and has been applied to quantum solids [96]. The theories are limited to linear stability analysis at present. 

Let us briefly outline the main concepts of a (linear) stability analysis and refer to the situation illustrated in Figure 11-7. If we artificially keep the moving boundary morphologically stable, we can immediately calculate the steady state vacancy flux, /v, across the crystal. The boundary velocity relative to the laboratory reference system (crystal lattice) is... 

Few calculations of three-dimensional convection in CZ melts (or other systems) have been presented because of the prohibitive expense of such simulations. Mihelcic et al. (176) have computed the effect of asymmetries in the heater temperature on the flow pattern and showed that crystal rotation will eliminate three-dimensional convection driven by this mechanism. Tang-born (172) and Patera (173) have used a spectral-element method combined with linear stability analysis to compute the stability of axisymmetric flows to three-dimensional instabilities. Such a stability calculation is the most essential part of a three-dimensional analysis, because nonaxisymmetric flows are undesirable. 

Sridhar and coworkers studied the kinetics of a compressed film on a viscous substrate [30], They performed linear-stability analysis to determine the onset and maximally unstable mode of this mechanical instability as a function of misfit strain, viscous layer thickness, and viscosity. 

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. 

We write the solution as the vector X = (6,(j),u,vx,vy,i ,P,) consisting of the angular variables of the director, the layer displacement, the velocity field, the pressure, and the modulus of the (nematic or smectic) order parameter. For a spatially homogeneous situation the equations simplify significantly and the desired solution Xo can directly be found (see Sect. 3.1). To determine the region of stability of Xq we perform a linear stability analysis, i.e., we add a small perturbation Xi to... 

The theoretical framework, within which the existence of surface instabilities created by capillary waves can be predicted is the linear stability analysis [23, 24]. This model assumes a spectrum of capillary waves with wave vectors q and time constant r (Fig. 1.8a). 

Together with the ansatz Eq. (1.1), Eq. (1.5) describes the response of a liquid film to an applied pressure p. The resulting differential equation is usually solved in the limit of small amplitudes q < h hv and only terms linear in f are kept ( linear stability analysis ). This greatly simplifies the differential equation. The pressure inside the film p = Pl + Tex consists of the Laplace pressure pL = —ydxxh, minimizing the surface area of the film, and an applied destabilizing pressure pex, which does not have to be specified at this point. This leads to the dispersion relation... 

The experimental and theoretical literature on instabilities in fiber spinning has been reviewed in detail by Jung and Hyun (28). The theoretical analysis began with the work of Pearson et al. (29-32), who examined the behavior of inelastic fluids under a variety of conditions using linear stability analysis for the governing equations. For Newtonian fluids, they found a critical draw ratio of 20.2. Shear thinning and shear thickening fluids... 

White et al. (38,39) presented experimental and theoretical (isothermal linear stability analysis) results that indicate the following first, that polymer melts respond similarly to uniform elongational flow and to melt spinning second, that polymers whose elongational viscosity e) increases with time or strain result in a stable spinline, do not exhibit draw resonance, and undergo cohesive failure at high draw ratios. A prime example of such behavior is LDPE. On the other hand, polymer melts with a decreasing r + (t, e)... 

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