Spiral Archimedean shape

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. 

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. 

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]... 

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 L Control of the spiral shape in the oscillatory Belousov-Zhabotinsky reaction. Ru(bpy)3 was used as catalyst. Ar laser beam was irradiated (illustrated by white arrow) at the core of the rotating spiral to increase the size of the core region. The morphology of spiral changed reversibly from Archimedean to logarithmic, and the wave profile from trigger-wave to phase-wave. Controlling global structure by local control of singular region is characteristic in dissipative structures.

Using these simple geometrical arguments, Wiener and Rosenblueth came to the conclusion that the excitation front which rotates around any obstacle (even with a more complicated shape than a circle) must represents its involute. They pointed that at distances that are much larger than the radius of the obstacle, the involute approaches the Archimedean spiral with the constant pitch equal to the perimeter of this hole. 

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