Flows on the Circle

So far we ve concentrated on the equation x = f(x), which we visualized as a vector field on the line. Now it s time to consider a new kind of differential equation and its corresponding phase space. This equation, 

However, in all other respects, flows on the circle are similar to flows on the line, so this will be a short chapter. We will discuss the dynamics of some simple oscillators, and then show that these equations arise in a wide variety of applications. For example, the flashing of fireflies and the voltage oscillations of superconducting Josephson junctions have been modeled by the same equation, even though their oscillation frequencies differ by about ten orders of magnitude  

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Co-plane-rotational two streams enter the device tangentially and counter currently with the central lines in the same plane before impingement, and then flow on the wall of the device, with streamlines of a half circle form, e.g. (d) and (f) in Fig. 2. Non-co-plane-rotational two streams enter the device tangentially and counter currently with the central lines in different planes before impingement, and then flow on the wall of the device, with streamlines of several half circle forms, e.g. (e) in Fig. 2. 

These general results are fundamentally topological in origin. They reflect the fact that X = f(x corresponds to flow on a line. If you flow monotonically on a line, you ll never come back to your starting place—that s why periodic solutions are impossible. (Of course, if we were dealing with a circle rather than a line, we could eventually return to our starting place. Thus vector fields on the circle can exhibit periodic solutions, as we discuss in Chapter 4.)... 

Here we consider (1) more abstractly, to illustrate some general features of flows on the torus and also to provide an example of a saddle-node bifurcation of cycles (Section 8,4), To visualize the flow, imagine two points running around a circle at instantaneous rates 0, 02 (Figure 8.6.1). Alternatively, we could imagine a sin-... 

It is clear that 9p/9r is proportional to the factor (1 - < /Py, hence, the radial Darcy velocity vanishes on the circle r = c. Next, at distances far away where r approaches infinity. Equation 5-59 behaves like p(r,0) - (U ,p/k) r (cos a cos 0 + sin a sin 0) that is, in Cartesian (x,y) coordinates, p(x,y) - (Uoop/k) (x cos a + y sin a). Thus, the vertical Darcy velocity satisfies - (k/p) dpidj = IXoSin a, while the horizontal velocity satisfies - (k/p) 5p/5x = UcoCos a. Hence, Equation 5-59 gives the pressure field that will produce a flow past the circle, but inclined with angle a at infinity. This flow is shown in Figure 5-5. 

Consider the conventional scheme of gas flow in the separator [6]. Dust particle flows at high velocity tangentially enters the cylindrical part of the separator, centrifugal separation effect has been on the circle. 

In essence, a guided-ion beam is a double mass spectrometer. Figure A3.5.9 shows a schematic diagram of a griided-ion beam apparatus [104]. Ions are created and extracted from an ion source. Many types of source have been used and the choice depends upon the application. Combining a flow tube such as that described in this chapter has proven to be versatile and it ensures the ions are thennalized [105]. After extraction, the ions are mass selected. Many types of mass spectrometer can be used a Wien ExB filter is shown. The ions are then injected into an octopole ion trap. The octopole consists of eight parallel rods arranged on a circle. An RF... 

The other ease is when there is too niiieh flow through the pump. The pump is operating to the right of the BEP on its eurve (Figure 9-8). The same problem oceurs, but now in the other direction. With the severe increase in velocity through the pump, the pressures tall dramatically in the H-F-G-H arc of the volute circle (Bernoulli s Law-says that as velocity goes up, pressure comes down). Now the shaft deflects, or even breaks in the opposite direction. .. at approximately 240° around the volute from the cutwater. 

Figure 9-13 shows the Ynryn P ° obtained for the first case where four regions are defined a region of complete separation, two regions where only one outlet stream is 100 % pure and a last region where neither of them is 100 % pure. The closed circles are numerical results based on the equivalence between the TMB and the SMB the thick lines connect those results. The thin line in Fig. 9-13 has two branches. The diagonal 7 -7 corresponds to zero feed flow rate therefore, 7 must be higher than Yn- The horizontal branch Ym corresponds to zero raffinate flow rate in this case, the extract flow rate is 25.09 mL min k... 

Figure 5B. Correlation of right-angle light scatter measured by fluorometry and flow cytometry. The top panel shows flow-cytometric data of side scatter of fixed, stained cells during the time course of stimulation by 1-nM (solid line, solid circles) or 0.01-nH (dashed line, open circle) FLPEP. The bottom panel shows the corresponding right-angle light-scatter data acquired pseudo-simultaneously on live cells in the fluorometer. The flow-cytometric data have been averaged, but the fluorometry data are plotted for both duplicates from one donor. Reproduced with permission from Ref. 27. Copyright 1985 Rockefeller University Press.

FIG. 2. The dependence of the carbon fraction. v = Cl/(lSi] + [C ) on the gas-flow ratio r = [CH4]/([SiH4] -I- ICH4I) for films deposited in the ASTER system [ASTI (filled circles) and AST2 (filled triangles)] and for films deposited in a similar system (ATLAS) [ATLI (open circles) and ATL2 (open triangles)]. (From R, A. C. M. M. van Swaaij, Ph.D. Thesis, Universiteit Utrecht, Utrecht, the Netherlands, 1994. with permission.)... 

Figure 18 illustrates the difference between normal hydrodynamic flow and slip flow when a gas sample is confined between two surfaces in motion relative to each other. In each case, the top surface moves with speed ua relative to the bottom surface. The circles represent gas molecules, and the length of an arrow is proportional to the drift velocity for that molecule. The drift velocity variation with distance is illustrated by the plots on the right. When the ratio of the mean free path to the separation distance between surfaces is much less than unity (Fig. 18a), collisions between gas molecules are much more frequent than collisions of the gas molecules with the surfaces. Here, we have classical fluid flow or viscous flow. If the flow were flow in tubes, Poiseuille s law would be obeyed. The velocity of gas molecules at the surface is the same as the velocity of the surface, and in the case of the stationary surface the mean tangential velocity of the gas at the surface is zero. 

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