Vortexing 776 INDEX

There is little error in replacing i/,(r) in the wall region by v,w for m = 1 (ideal vortex) in general for cyclones, the vortex index m is between 0.5 and 1. We can therefore write... 

For a properly designed and operated cyclone, the sharpness index is constant, typically 0.6. The cut size and apparent bypass are a function of the cyclone geometry, the volumetric feed rate, the material relative density, the feed solids concentration, and the slurry rheology. The relationship for a standard cyclone geometry, where if D. is the cylinder diameter in cm and inlet area = 0.05 D vortex finder diameter = 0.35 Dc ... 

In nematic liquid crystals, subjected to an external electric field at a certain critical voltage, a periodic distribution of the space charge Q and the electric potential appears, resulting in the corresponding periodic variations of the initial director orientation L and the hydrodynamic fiow with the velocity v. This effect, known as the electrohydrodynamic instability (EHDI), could be visualized optically as a periodic pattern of domains. Fig. 5.5. In a screen, domains become visible as black and white stripes perpendicular to the distortion plane, where periodic director deformation and vortex liquid crystal movement is observed. These stripes are caused by the periodicity of the change in the refractive index for an extraordinary ray due to variations in the director. Fig. 5.6. These spatially periodic variations of the refractive index (domains) were first detected by Zvereva and Kapustin [32]. Then Williams [33] investigated transverse domains in detail, and it is current practice to call this type of instability Williams or Kapustin-Williams [34] domains. 

The topology of the 7 vector field deserves a careful and detailed investigation. Its most interesting features are observed in the proximity of an SP at which the modulus 7 vanishes. An SP is classified in terms of topological index i [85, 86], and of a (rank, signature) label [15, 55-57, 87-90]. A continuous, open or closed, path of SPs is referred to as stagnation line (SL), consisting of either vortex points (index = + 1), or saddle points (index = —1). 

The topological index i counts the number of times that the current density vector 7 rotates completely while one walks counterclockwise around a circle of radius e, so small that 7 has no zeroes inside except the SP at its centre. The topological index i of a saddle (vortex) line is -1 (+1). Both SPs have (r, i) = (2,0). 

The splitting of a SL into several SLs is regirlated by a fundamental topological theorem proved by Gomes [55-57, 93] in the form of an index conservation constraint. Recall that, according to footnote 3, the index of a saddle (vortex) line is — 1 (+1). When an SL of index rp splits into m new lines, the sum of the indices of the SLs which emerge from the branching point must satisfy the condition... 

For instance, a vortex line may bifurcate giving rise to two new vortex lines and one saddle line. This bifurcation conserves the total index +1. 

The primary diatropic vortex SL branches out at two (0, 0) critical points (too far to be seen in Fig. 7.7) into a set of three SLs on the plane of the nuclei a central (blue) saddle SL and two (green) vortical SLs crossing the C-H bonds. These vortex lines become saddle-type at points with coordinates (x = 0, y = 1.05, z = 0.86) bohr, where the i topological index changes from -1 to 1. The saddle SLs merge at... 

Spin Right hand spins fist with index finger raised index finger makes a small vortex. Multiple viewpoints simultaneously RH imitates spin of the bolt, but not the action of a miner spinning the Kilt with a spanner. Libby watches her hand at a distance—from an analytic viewpoint of the observ er... 

As discussed in Chap. 2, the index n is 1 for loss-free, and —1 for solid-body rotation. We recall that, in the outer part of the vortex (outside the surface CS in Fig. 4.2.1 above), the swirl is close to loss-free, although some loss is caused by wall friction. As a result of measurements of the tangential velocity profiles within laboratory cyclones, n is found to have a value between 0.7 and 0.8. This is a good approximation for smooth-walled cyclones operating at low solids-to-gas mass ratios (i.e., low solids loading ), but not so for cyclones operating at high mass ratios. ... 

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