Stability Taylor number

Next we turn to the stability of Couette flow for parallel rotating cylinders. This is an important flow for various applications, and, though it is a shear flow, the stability is dominated by the centrifugal forces that arise because of centripetal acceleration. This problem is also an important contrast with the first two examples because it is a case in which the flow can actually be stabilized by viscous effects. We first consider the classic case of an inviscid fluid, which leads to the well-known criteria of Rayleigh for the stability of an inviscid fluid. We then analyze the role of viscosity for the case of a narrow gap in which analytic results can be obtained. We show that the flow is stabilized by viscous diffusion effects up to a critical value of the Reynolds number for the problem (here known as the Taylor number). 

For Taylor numbers exceeding Tc, the flow develops a secondary flow pattern in which ur and uz are both nonozero. A sketch of the stability criteria given by (3-86) is shown in Fig. 3 8. The reader who is interested in a detailed description of the stability analysis that leads to the criterion (3-86) is encouraged to consult Chap. 12 or one of the standard textbooks on hydrodynamic stability theory (see Chandrashekhar [1992] for a particularly lucid discussion of the instability of Couette flows).12... 

Figure 3-8. Stability diagram for Couette flow with 0 < . 2 / -2 i < 1, plotted as Taylor number T versus ih/ The flow is stable for T < Tc and unstable for T > Tc.

Now, following the assumption of Taylor that the principle of exchange of stabilities is satisfied, so that a = 0 at the neutral stability point, the objective of the stability analysis is to obtain nontrivial solutions of (12-147) and (12-148) with a = 0 for various values of a and then determine the minimum value of T as a function of a. This minimum of T is the critical value of the Taylor number for transition to instability. The corresponding value of a is known as the critical wave number and represents the (dimensionless) wave... 

The critical Taylor number T for the onset of Taylor vortices can be predicted by examining the stability of snail amplitude disturbances when superimposed on the basic Couette flow. The use of this linear stability analysis for concentric cylinders has been extensively reviewed by Chandrasekhar (1) and Stuart (2). All such analyses assune that the cylinders are infinitely long. In addition to T they predict an initial Taylor vortex celf axial length, . ... 

To describe the chemical reactivity in the context of DFT, there are several global and local quantities useful to understand the charge transfer in a chemical reaction, the attack sites in a molecule, the chemical stability of a system, etc. In particular, there are processes where the spin number changes with a fixed number of electrons such processes demand the SP-DFT version [27,32]. In this approach, some natural variables are the number of electrons, N, and the spin number, Ns. The total energy changes, estimated by a Taylor series to the first order, are... 

The breakup or bursting of liquid droplets suspended in liquids undergoing shear flow has been studied and observed by many researchers beginning with the classic work of G. I. Taylor in the 1930s. For low viscosity drops, two mechanisms of breakup were identified at critical capillary number values. In the first one, the pointed droplet ends release a stream of smaller droplets termed tip streaming whereas, in the second mechanism the drop breaks into two main fragments and one or more satellite droplets. Strictly inviscid droplets such as gas bubbles were found to be stable at all conditions. It must be recalled, however, that gas bubbles are compressible and soluble, and this may play a role in the relief of hydrodynamic instabilities. The relative stability of gas bubbles in shear flow was confirmed experimentally by Canedo et al. (36). They could stretch a bubble all around the cylinder in a Couette flow apparatus without any signs of breakup. Of course, in a real devolatilizer, the flow is not a steady simple shear flow and bubble breakup is more likely to take place. 

Recently, the crystal structure of S-protein complexed with the model peptide has been solved to moderate resolution (3 A) (Taylor et al., 1985). Most of the structural features envisioned in the design of the model peptide were indeed observed in the structure of the complex. The peptide is in a helical conformation, the histidine is held in a reasonable orientation for catalysis, and the complex is stabilized by nonbonded interactions between the hydrophobic cleft of S-protein and the side chains of Phe-8 and Met-13 of the peptide. There were also a number of subtle differences between the structures of the native and the model S-protein S-peptide complexes. Most notably, the N terminus of the peptide has undergone a major reorientation that prevents Glu-2 from forming a hydrogen bond with Arg-10. Further, the 8-nitrogen of the active-site... 

To quantify the increase of a due to pressure, a mean bubble diameter has been estimated using Taylor s stability theory [7] on bubble deformation and break-up in sheared emulsions. According to this theory, bubble size in a sheared emulsion results from a balance between viscosity and surface tension forces. The dimensionless number that describes the ratio of these forces is called the capillary number Q. For large bubble deformations, the maximum stable bubble diameter in a shear flow is expressed as [8] ... 

Many studies have been devoted to the Taylor-Couette problem (flow between two concentric cylinders with radii R and R2, Ri < R2, of infinite length, and rotating with angular velocities fij and 02 repectively). For instance Zielinska and Demay [74] consider the general Maxwell models with —1 <0 < 1. They show that the axisymmetric steady flow (the Couette flow) does not exist for 2dl values of parameters where the steady state exists moreover all models, except for a very close to —1, predict stabilization of the Couette flow in the spectral sense, for small enough values of the Weissenberg number. (See also [55].)... 

The situation is quite different for n- r transitions. The lone electron pair is particularly well stabilized by polar and particularly by protic solvents so it becomes energetically more difficult to excite. Figure 2.45 shows the spectrum of N-nitrosodimethylamine in different solvents. Results of calculations indicate that the negative solvatochromism of carbonyl compounds can be explained on the basis of the structural changes due to the formation of hydrogen bonds (Taylor, 1982). Molecular dynamics simulations, however, indicate that the net blue shift is primarily due to electrostatic interactions (Blair et al., 1989). A large number of water molecules around the entire formaldehyde are responsible for the total blue shift the first solvation shell only accounts for one-third of the full shift. 

In a film of infinite lateral extent, k can range from 0 to oo, so a necessary condition for instability is that AH > 2npgh. Since all wave numbers are available in a film of infinite extent, we see that this analysis predicts that the thin film will always be unstable, even with the stabilizing influence of surface tension, to disturbances of sufficiently large wavelength when van der Waals forces are present. Similarly, the Rayleigh Taylor instability that occurs when the film is on the underside of the solid surface will always appear in a film of infinite extent. In reality, of course, the thin film will always be bounded, as by the walls of a container or by the finite extent of the solid substrate. Hence the maximum wavelength of the perturbation of shape is limited to the lateral width, say W, of the film. This corresponds to a minimum possible wave number... 

An important problem is to analyze the stability of fluid flows. With the exception of the Taylor-Couette and Saffman Taylor problems, this chapter has focused on stability questions when the base state of the system was one with no motion (or rigid-body motion), so that instability addresses the conditions for spontaneous onset of flow. An equally valid question is whether a particular flow, such as Poiseuille flow in a pipe (or any of the other flows that we have analyzed in previous chapters of this book), is stable, especially to infinitesimal perturbations as linear instability determines whether the particular flow is actually realizable in experiments. This question was first mentioned back in Chapter 3 when we analyzed simple unidirectional flow problems and noted that solutions such as Poiseuille s solution for flow through a tube was a valid solution of the Navier-Stokes equations for all Reynolds numbers, even though common experience tells us that beyond some critical Reynolds number there is a transition to turbulent flow in the tube. 

The objective was the production of enriched uranium-235 from natural uranium by differential diffusion of highly reactive uranium hexafluoride through a porous membrane. A number of surface chemistry problems related to catalysis had to be solved. The work at Princeton was in support of the main effort at the SAM Laboratories of Columbia University. Aside from production of the diffusion barrier, the task was to characterize its pore distribution and stabilize it from corrosion. Taylor was associate director of the SAM Laboratories, spending his days in New York and evenings and weekends at Princeton. Turkevich, in addition to teaching both at Princeton... 

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