Initial instability

The experimental evidence for this scenario, mentioned at the outset of this section, is less compelling since it is harder to control this sort of dosage-sensitive STM experiment than a Monte Carlo simulation. (Specifically, it problematic to convert from CO dosage to the evolution time from initial instability the analysis would be better if the surface could be instantaneously de-oxidized.) From an earlier examination of the terrace-width distribution for Ag(110)2Y—>[001], Ozcomert et al. (1993) concluded that to a good approximation the step-step interactions were purely entropic repulsions (by finding a good fit to a free-fermion form (Jobs et al, 1991)). (But see also Pai et al (1994) for remarkable behavior under different conditions.) From the relationship (Bartelt el a/., 1992)... 

Christie and Bond ( 4) began with a linear stability analysis, but they did not construct a stability curve to define the parameter domain in which instabilities could be expected. In their nonlinear analysis, instabilities were initiated by random perturbations in the initial concentration distribution at the entrance (macroscopic perturbations). [Peters and Kasap (45) used the same method to initiate instabilities.]... 

On the other hand, Settari et al. (50) used a finite-element analysis in examining the consec[uences of both velocity-dependent and constant dispersion coefficients during a two-dimensional displacement. They found that fingers in the concentration distribution developed when the permeability was homogeneous, so long as the dispersion coefficients were sufficiently small. This was apparently the first successful use of truncation and round-off errors to play the roles of physical perturbations in initiating instabilities. Russell (51) later had a similar experience. 

The buoyancy force of an initial density instability is a function of its volume. In the absence of extension, a relatively strong overburden may thus be able to resist the growth of all but the very largest of initial instabilities. This could explain why subvolcanic granitoid complexes, which provide high-amplitude primary instabilities, may be favoured to grow into final domes (e.g. Hickman 1984 Collins 1989). 

More recently, Sgalari, Leal, and Meiberg and Klein and Leal examined the instability of simple shear to secondary flows in the x-y plane. The initial instability takes on a similar form to that predicted by the LE theory, with the appearance and subsequent... 

Another practical issue associated with the use of this simulation technique is biasing the instability of the starting point. As discussed in the section on stability, as long as the shock speed exceeds the local sound speed, the volume equation of motion Eq. (16) can either force compression or expansion of the volume. While both of these steady solutions can potentially have physical significance, the solutions we focus on in this chapter are the compressive, shock-like solutions. Therefore some technique is required for biasing the initial instability so that only compression occurs. Note that this is simply a selection of the particular type of steady solution to be simulated (compressive shock versus expansion shock) and does not represent nor require an empirical parameter or extra degree of freedom. 

The sharp increase in roughness of membranes less than 9 pm thick could be due to the initial instability of the casting film caused mainly by surface tension gradients, since the Rayleigh numbers of the membranes are much smaller, in most cases, than... 

Koschmieder and BiggerstafF [75] also observed that for liquid films with thicknesses more than 2 mm and high Ma and Ra values, there was no initial instability that would result in the formation of unstable cells, but uniform regular hexagonal cells were formed. The cell size became larger as the Ma and Ra numbers increased. 

Also of significance is that initial instability of a thin film in accordance with the above mechanism does not inevitably lead to film rupture. The analysis, like all others in this chapter, is based on linear stability theory, and hence is valid only for small amplitude perturbations. It has been observed experimentally that at low surfactant concentrations instability of a film some tens of nanometers in thickness does produce rupture. But for many surfactants it is found that above a critical concentration, the instability leads to formation of black films which are only slightly thicker than the total length of two surfactant molecules (She-ludko, 1967). These black films can be very stable and are a major factor in foam stabihty. 

Figure 14.12 Process to form hierarchical bilayer structure, (a) Polymer bilayer of PMMA and PS. The polymer-air surface is first destabilized by an electric field generated between the two Si electrodes. The initial instability results in the formation of columns spanning from the surface of the lower layer to the top electrode, (b) During the column formation process, the bottom polymer layer is deformed at the polymer-polymer-air contact line, (c) In the second instability, the deformation of the lower layer is enhanced by the electric field. The polymer moves upward, surrounding the columns and finally leading to form a mantle-like structure. (Images taken from Ref. 65.)...

The margins against parallel channel hydraulic instability in the Prototype reference design are so large that it has not been possible to initiate instabilities in the full-scale test rig when modelling the Wlnfrith SGHWR performance. 

Lemma 7.2. For the linear system described by (7.18), the initial instability boundary is defined by... 

However, the thickness, which corresponds to the minimum roughness and the nodule size, shifted towards the lower end of the range. These effects of membrane thickness on the surface morphology could be explained by considering the initial instability of the polymer solution film that is caused by the surface tension and density gradient. The concentration of the polymer used in the present study was only 0.25 wt. % and 0.5 wt. %. These solutions can be considered as dilute solutions. 

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