Archimedean spiral

The presence of scroll helicity replaces a set of concentric cylinders by a single sheet rolled upon itself (Fig. 1). Assuming that the distance between the successive rolls of the scroll is constant, its cross-section can be conveniently represented in polar coordinates by the Archimedean spiral ... 

Fig. 3.4.5 Schematic description of the two-dimensional SPIRAL-SPRITE technique. Cx and Gy are the phase encode magnetic field gradients that are amplitude cycled to traverse /(-space along an Archimedean Spiral. A single data point is acquired from the FID at a fixed encoding time tp after an rf excitation pulse. TR is the time between rf pulses.

A scroll vacuum pump uses two interleaved Archimedean spiral-shaped scrolls to pump or compress gases (see Fig. 1.9). One of the scrolls is fixed, while the other orbits eccentrically without rotating, thereby trapping and compressing gases between the scrolls and moving it towards the outlet. 

The shape of the growth spirals may be quite complex and only approximated by analytical functions such as the Archimedean spiral employed by Burton et al. [24] and shown in Fig. 12. The relationship between the velocity of a straight step and curved step may be derived using the approach of Nielsen [27]. This enables the steady-state shape of a spiral in contact with a fixed supersaturation solution to be calculated. 

Fig. 12. An approximation to a single growth spiral (an Archimedean spiral). References pp. 230-231...

The cylinders of cylindrite only grow on a macroscopic scale. Thus, one coil will contain millions of both sorts of unit cells. For all macroscopic cylinders the difference in coil length in two consecutive coils is practically equal to 2nAr, where Ar is the thickness of the increment in the Archimedean spiral. For a layer pair with the thickness Ar =... 

Fig. 3.6. A global bifurcation diagram of rotating Archimedean spirals with rotation frequency oj > 0 and normal tip velocity (7(0).

The asymptotics (3.52) for the curvature k readily identifies Archimedean spirals. Indeed, arc length parametrization of the position vector z = Z t, s) in (3.33) implies that we can write the unit vector Zs t, s) in the form... 

In particular, this also determines the asymptotic wave length in the far field to be 2nc/ij, as claimed. The minus sign in (3.57) indicates that the Archimedean spiral Z is indeed right winding in outward direction. 

To render our convergence statement more quantitative and, at the same time, exhibit the Archimedean spiral character of our solutions, we now address the asymptotics of the characteristic so( )- We claim... 

By the curvature analysis of section 3.3.2, (3.52)-(3.60) this shows that our periodic solution w t, s) represents a periodically fluctuating Archimedean spiral of time independent asymptotic (average) rotation frequency... 

It may be worth interpreting the characteristic front so t) in the Archimedean spiral geometry. Let r(t) = r(t,.so t)) denote the distance of the characteristic point on the spiral from its origin. For large s, t, we then have a radial propagation speed r of the stability zone which is given by... 

In summary, we have shown that the periodically forced eikonal flow (3.66) exhibits Archimedean spirals with very strongly stable meandering and drifting tip motions, if the normal velocity c t), the tangent tip speed G t), and the tip curvature Ko(t), are all positive [34, 43, 61]. 

In section 3.3 above we have imposed positive sign conditions c, G, kq > 0. Other combinations of signs are of course conceivable. We only mention the case c,G<0Reversing time, t i—> —t, then converts the characteristics to the previous ones of section 3.3.3 see (3.68). Therefore asymptotically Archimedean spirals still exist, as a periodic response to the periodic forcing by c, G, ko. However, they now rotate backwards compared to the previous case. Moreover, these spirals are now extremely unstable, just as they were extremely stable before. Indeed, any perturbation in the far field now propagates inwards at negative radial velocity f = (c) -t-. .., see (3.78), and destructively reaches the spiral tip after some finite time. Afterwards the Archimedean spiral disappears and the initial conditions reign. This may account for far-field break-up of spiral waves. 

Fig. 9.1. Different representations of a rigidly rotating spiral wave, (a) Involute of a hole, (b) Solution of the kinematical equation with linear velocity-curvature relationship, (c) Archimedean spiral, (d) Superposition of the three wave fronts where the dotted, dashed and solid lines correspond to (a), (b), and (c), respectively. Far from the rotation center the fronts practically coincide.

Since with increasing arc length the front curvature goes rapidly to zero, it is not necessary to use this rather complicated approach to determine the front shape far away from the rotation center. A. Winfree was the first who suggested to approximate the spiral front by an Archimedean spiral [2]... 

It is important to stress that the above three descriptions of the spiral wave practically coincide far away from the rotation center. Moreover, already at a relatively small distance ta from the rotation center, the Archimedean spiral becomes very close to the curvature affected spiral obtained from Eq. (9.3), as can bee seen in Fig. 9.1(d). In this example ta can be estimated as ss 9.0 A. Recent computations performed with the Oregonator model [40] and experiments with the BZ reaction [43] also confirm that an Archimedean spiral provides a suitable approximation of the wave front except in a relatively small region of radius A near the rotation center. Even the shape of a slightly meandering spiral waves exhibits only small oscillations around an Archimedean shape, and the amplitude of these oscillations vanishes very quickly with r [44]. Therefore, the Archimedean spiral approximation will be used below to specify the shape of the wave front. 

Fig. 9.4(b) shows the spiral tip trajectory obtained experimentally under this feedback control. After a short transient the spiral core center drifts in parallel to the line detector. The asymptotic drift trajectory reminds the resonance attractor observed under one-channel control, because a small variation of the initial location of the spiral wave does not change the final distance between the detector and the drift line. To construct the drift velocity field for this control algorithm an Archimedean spiral approximation is used again. Assume the detector line is given as a = 0 and an Archimedean spiral described by Eq. (9.5) is located at a site x,y) with a > 0. A pure geometrical consideration shows that the spiral front touches the detector each time ti satisfying the following equation ... 

To explain this amazing result we consider a circular detector of radius Ra centered at the origin of a Cartesian coordinate system. We approximate the spiral wave by an Archimedean spiral located at the point z = x + iy. Prom pure geometrical considerations (compare Fig. 9.8) one finds that the spiral wave will touch the detector at instants ti satisfying the following equation... 

Fig. 9.8. Archimedean spiral (thick solid line) touching a circular detector of radius Ra (dashed line). The thin solid line denotes the common tangent line to both curves. The figure explains the meaning of distances cr, d, and r used in the text.

If the shape of a slightly meandering wave can be approximated by a counterclockwise rotating Archimedean spiral, the first Fourier component of v z, t z) reads... 

Figure 4. SEM images of ultra-thin film structures (a) a ring with vertically aligned nanowalls, (b) bended strips (cantilevers) (c) arrays of needles (d) Archimedean spiral, spiral-like strip.

The material to be purified is enclosed in an additional tube of Pyrex or quartz glass from which air is excluded 2. The latter tube is closed off by a break joint 6. Thus, one can remove parts of the purified material without contact to the air. The Archimedean spiral 7 produces the required intermittent up and down motion of the tubes 2. After [1]. 

The spread plate technique may be automated by the use of a piece of apparatus known as the spiral plater. An agar plate is rotated on an Archimedean spiral whilst being inoculated. The volume of the sample decreases (and is therefore effectively diluted) as the spiral moves towards the outer edge of the plate. A specialised counting grid relating the area of plate to the sample volume enables colonies in the appropriate sector to be counted. An electronic colony counter travelling in the same Archimedean screw may be used. 

The Spiral configuration is the more realistic one in terms of telescope motion, and it recreates a variation of an Archimedean spiral. The simulator first calculates the number of turns of the spiral, Nturns = (bmax - bmin)/bstep- The user also defines the number of baselines to be created, Nb. The baseline vector is then created in polar coordinates as... 

We might paraphrase Theorems 2 and 4 loosely as limit cycles => plane waves spiral waves. Furthermore we might extend the conjecture of Kopell and Howard as follows if = f(x) has a stable limit cycle and D is sufficiently close to I then for K sufficiently small there exist stable rotating solutions of Eq. (10 ) which look like Archimedean spirals far from the origin. 

Before leaving the Ulam spiral, we should mention that there have been a few modifications of Ulam s spiral. For example, R. Sacks constructed the Archimedean spiral by plotting integers uniformly on the spiral, and when composite numbers have been deleted, one obtains what is known as the Sacks prime number spiral [128]. On this spiral, prime numbers that are obtained from Euler s prime number generator x - x+41 are clearly seen on a line approaching the left horizontal axis. There are modifications of the... 

Another piece of equipment useful in high-volume microbiology laboratories is the spiral plater system (Swanson et al., 1992). In essence, a known volume ofjuice or wine is automatically dispensed onto a rotating agar plate in an Archimedean spiral. Because the amount of sample decreases as the stylus moves away from the center of the plate, specific zones on the plate can be counted and then related to the population in the original sample. Although initially expensive, these instruments can save extensive amounts of media and employee time. 

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