Spiral trajectories

In some cases there also occur semistable limit cycles (in this discussion the single term cycle is used wherever it is unambiguous or if no confusion is to be feared) characterized by stability on one side and instability on the other side. Figure 6-5(a), (b), and (c) illustrate these definitions. Physically, only stable cycles are of interest the unstable cycles play the role of separating the zones of attraction of stable cycles in the case when there are several cycles. It is seen from this definition that, instead of an infinity of closed trajectories, we have now only one such trajectory determined by the differential equation itself and the initial conditions do not play any part. In fact, the term initial conditions means just one point (x0,y0) of the phase plane as a spiral trajectory O passes through that point and ultimately winds itself onto the cycle 0, it is clear that the initial conditions have nothing to do with this ultimate closed trajectory C—the stable [Pg.329]

Particles subject to Brownian motion tend to adopt random orientations, and hence do not follow these rules. A particle without these symmetry properties may follow a spiral trajectory, and may also rotate or wobble. In general, the drag and torque on an arbitrary particle translating and rotating in an unbounded quiescent fluid are determined by three second-order tensors which depend on the shape of the body ... 

Fluctuations in speed and direction also occur in the subcritical range (down to Re = 270) (GIO). A sphere shows a rocking motion and follows a zigzag or spiral trajectory in this range (C5, Ml, P5) with wavelength about 12[Pg.114]

Figure 4 Spiral trajectory as a particle moves toward the center of rotation.

The particle moves toward O in a spiral trajectory involving many turns. 

Noting that dr/dt v, the angle between v and VR is 8, where tand = (dr/dt)/v. As the particle moves inward towards 0, the component of T along the spiral trajectory is responsible for increasing its speed. 

This confirms that v increases for an inward spiral trajectory (i.e., when... 

In recent years attempts have been made to improve the gas-liquid mass transfer by changing the design of the mechanically agitated vessel. Mann et al. (1989) evaluated the use of horizontal baffles mounted near the gas-liquid surface. Horizontal baffles prevent vortex formation, generate less shear than standard baffles, increase gas holdup, and improve gas-liquid mass transfer. The latter two results are due to the rotational flow below the baffles, which causes gas bubbles to move upward in a spiral trajectory and induces surface aeration. For a 12-inch i.d. and 18-inch-tall stirred vessel, they showed kLat to be improved by a factor of 1.6 to 2.3 with 30 to 50% lower agitation power compared to the standard vessel. 

When the hydrodynamic conditions correspond to Re = 1500 (R = 2 - 10 mm) bubbles become strongly deformed when rising. They acquire the shape of a flat ellipsoid and begin to vibrate and move on a spiral trajectory. In fact their size does not influence the velocity of rise [14]. The following relation was derived from the results on the velocity of rise of large bubbles reported in [16]... 

However, in the more general case of an asymmetric particle, the combined action of the lateral and rotational forces may lead to motion along a spatial, for instance, spiral trajectory. At the same time, a steady-state settling trajectory with helicoidal (propeller-like) symmetry remains rectilinear, notwithstanding the body rotation [179]. 

The difference between conventional and magnetron processes lies largely in the plasma environment. As will be shown later, the plasma is confined to the surface of the cathode by a magnetic field created by permanent magnets located under the target and by an electric field which is situated perpendicularly to the surface. The electrons travel in spiral trajectories and can thus carry out many ionizing collisions with the atoms of the sputter gas. Either the whole vacuum chamber is used as anode or the anode is built of metal or metal bars, which are positioned near to the chath-ode. 

The type of steady state illustrated by the Brusselator example is called a focus because it is the pivot point for the spiraling trajectories that move toward or away from it. As we will see, the existence of a focus is often the prerequisite for the existence of oscillatory solutions to the full equations of motion. In particular, we look for an unstable focus (one for which the real part of the stability eigenvalues is positive) because the trajectories that spiral away from the focus may eventually reach a stable cyclic path surrounding that focus called a limit cycle. 

For the Brusselator example, it turns out that a solution of the full equations of motion show that the outwardly spiraling trajectories do, indeed, end... 

Pig.7 Elastic collision of atom 2 with atom 1 r,tf polar coordinates % initial velocity b impact parameter r jn minimum separation between the atoms be critical value of b leading to a stationary rotation of atom 2 around atom 1 for b[Pg.56]

Vortex dryers are characterized by a spiral flow of a particle-gas suspension due to tangential entry of the gas stream into the dryer chamber. Wet material is fed directly into the drying chamber by a screw feeder or nozzle atomizer, or dispersed with the drying medium (air, inert gas, superheated steam, etc.). After performing several spiral trajectories, the particulate material leaves the dryer with an exhaust gas. Alternatively, it may be separated from the gas within the dryer and further discharged via a rotary lock. 

An ion trap is a device where gaseous ions can be formed and/or stored for periods of time, confined by electric and/or magnetic fields. There are three commercial types of ion traps in use in MS, the QIT the Orbitrap, which uses a spiral trajectory that oscillates along a ID linear axis and the ICR trap. 

If an oscillating electric field E t) is introduced in the xy plane such that its frequency co f is equal to for a given ion, the ion will absorb energy from the field and will be accelerated by it (see below). From Eq. (4) it can be inferred that the radius of the ion s trajectory will become larger as the ion is accelerated, and as shown in Fig. 3a, the ion will follow a spiral trajectory as long as it remains in the crossed magnetic and rf electric fields. 

Figure 2c. Spiral trajectories inside the hysteresis loop starting from the boundary adsorption and desorption curves.

The rotating cone was developed at the University of Twente in the Netherlands. It is also based on the solid biomass being heated up by contact with a hot surface. The reactor is essentially a cone upside down with its base open. The cmie is heated on the outside surface, and it rotates continually. Biomass is fed along with sand, and it is driven centrifugaUy, rotating in a spiral trajectory until it leaves the reactor at the top (base). It has been proposed that the biomass and sand recovered at the top should go to a burner so that the char can be burned and the sand returned to the reactor. Figure 9 shows a rotating cone. 

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