Scroll filament

Wherever scroll filaments approach one another or a mirror (no-flux boundary) to within less than two core diameters ... 

It is natural to introduce a new coordinate system s,p, q) using the scroll filament as one axis and the normal and binormal directions as the other two (x, y,z)R(s) -I- pN(s) -I- qB s). This representation is locally orthogonal, but if the curve R has nontrivial curvature, the representation is unique only in a neighborhood of the centerline R. For distances more than the radius of curvature away from the curve R, the coordinate system has singularities. 

Since the scroll filament R(s, t) can be reconstructed at any time from its curvature and torsion, it can be useful to know the equations of motion for k s, t) and t(s, t). Keener [32] has shown from the Frenet-Serret equations that... 

Fig. 6. The curvature-torsion plane for helical scroll filaments. Helices evolve along semicircles, Equation (33), according to a law (34) for the evolution of torsion with time. Stationary helices exist along the nullcline (dashed curve) given by a(/c, r) = 0. (a) The leading order terms in a(/c, r). Equation (35), predict a parabolic nullcline. Straight, twisted filaments (k = 0, lu = r < 0) are stable if o (0, r) > 0 and unstable if a(0, r) < 0. (b) By including higher order terms in o (/c, r), e.g. Equation (38), we change the shape of the nullcline so that straight filaments with small twist are stable whereas ones with large twist are unstable.

The most common morphology observed in current mesophase carbon fibers of moderate modulus (55 to 75 Mpsi, 379 to 517 GPa) is a cylindrical filament with a random-structured core and a radial rim (12) Given the fracture section of Figure 3, with its scroll-like features, the core appears to be an array of +2ir and -ir disclinations. The radial rim of heavily wrinkled layers usually constitutes half or more of the cross section. 

One question arises if the observed tubules are scroll-type filaments or perfect cylinders. Scroll-type filaments should have edge overlaps on their surfaces. During our extensive STM survey along the tube surfaces on the atomic scale, we did not observe any edges due to incomplete carbon layers. Therefore we conclude that the tubes are complete graphitic cylinders. 

We first comment on several aspects of eikonal meanders see section 3.3.3. Specifically we mention the relation to tip control experiments [38, 84, 85, 87, 88], discuss the occurrence and interpretation of shock waves, the role of the signs of G and kq, possible generalizations to dynamics of scroll wave filaments in three dimensions, and the role of diffusion and viscous regularization in the sense of section 3.3.2. We then return to the reaction-diffusion view point of section 3.2 and discuss the chances of a reduced... 

Figure 6.11 Simulation of a scroll ring. (Reprinted from Winfree, A. T. Strogatz, S. H. 1983. Singular Filaments Organize Chemical Waves in Three Dimensions. II. Twisted Waves, Physica 9D, 65-80, with kind permission from Elsevier Science, The Netherlands.)...

Equation (3)]. The period of plane waves at this shorter wavelength. A, can be read from the dependence of ID wave propagation speed or period on the interval of time or distance separating consecutive waves (called the dispersion curve). The argument is that this must be the rotation period of the scroll if the filament s twist becomes stationary then its rotation period must be the same as in remote regions where we see only a periodic train of plane waves [42]. 

It was appreciated two decades ago when scroll rings were first measured and timed in Belousov-Zhabotinsky media [24] that they contract and vanish, possibly faster the greater the curvature it was supposed that they would collapse in time 0(diameter / >) [43, p. 255]. But it was not until one decade ago that Yakushevitch [68] and Panfilov and Pertsov [69] noticed and confirmed numerically (in the case of equal diffusion of all reactants, and radius of curvature/wavelength large and slowly varying or constant along the filament) that the reaction-diffusion equation prescribes such motion strictly in proportion to curvature, with coefficient equal to the diffusion coefficient ... 

Fig. 1. Scroll wave filaments (dashed curves) move slowly through space as the scroll rotates, (a) An elongated spiral becomes symmetric, and (b) an elongated ring becomes circular and then disappears (after Winfree [10]). (c) A scroll ring shrinks and disappears, and (d) a figure-eight ring splits into two circular rings which then shrink and disappear (after Welsh [17]).

To understand the motion of scroll wave filaments, we first review the differential geometry of space curves [22, 23]. We suppose that a space curve, in this case the filament of a scroll wave, is given by a position vector R(s) = (x(s),y(s), z(s)), 0 < s < L, where the independent variable s is taken to be arc length. To each point s on the curve R(s) we attach an orthogonal coordinate system defined by the unit vectors T(s), N(s), B(s), where... 

To fully describe a scroll wave we must add to the filament curve some specification of the local phase of the scroll. Restricted to the normal plane (the plane spanned by N and B), a scroll wave appears as a two-dimensional spiral wave rotating around the point where the filament pierces the normal plane (Figure 2). The phase of this spiral can be specified as the angle between some unit vector V, rotating rigidly with the spiral, and some suitable local reference direction. 

For some small positive number s, define v(s) to be the family of vectors v(s) = V(s), where the tail of v(s) lies on the filament R(s) at the coordinate point s. Then the surface 7v(s), 0<7< l,0[Pg.97]

In addition to the scroll ribbon , it is sometimes convenient to speak of the Frenet ribbon , which is the ribbon unfurled from the filament along the unit vector B instead of along V. The Frenet ribbon is uniquely defined... 

That is, the twist rate of a scroll ribbon is the sum of the torsion of the filament and the twist rate of the scroll wave with respect to the Frenet frame. It follows that the total twist of a scroll ribbon is... 

For example, a planar scroll ring offixed radius a has R(s) = (acos(s/a), a sin(s/a), 0). Clearly, k= fa, and r = 0. The Frenet ribbon has zero twist. As long as the filament is a planar closed curve without self intersection, the scroll ribbon must have an integral value of total twist. 

Consider a helical filament supporting a scroll ribbon that is untwisted with respect to the Frenet frame. Let = 7t/2 so that V = B and the scroll ribbon is identical to the Frenet ribbon. In particular, the twist of the scroll ribbon is identical to the torsion of the filament. The ribbon is wrapped flat... 

Undaunted by this lack of information, we use the known solution of the two-dimensional problem to solve the three-dimensional problem. Following the derivation by Keener [31 ], we assume that a scroll wave can be constructed by stacking spiral solutions of the two-dimensional problem, allowing them to vary slowly in the third dimension. We assume that, at any moment of time, there is a locus in space, called the filament, about which the spirals in each cross-section of the scroll are rotating (Figure 2). It is important to realize that there is nothing chemically distinct about this locus the reaction term F u) is independent of space. 

The simplest example of a scroll wave filament is a planar scroll ring. An initially planar filament remains planar for all time if Rt B is independent of s, i.e., if K.S = Ws = 0, see Equation (16c). Thus, in general, an untwisted or uniformly twisted circular filament is the only filament that will remain... 

If the filament is a perfect circle then the radius of the circle satisfies the differential equation dr/dt — -D/r, with solution = ro(f) - 2Dt, where tq is the initial radius of the ring. In other words, a circular scroll ring should collapse and disappear in the finite time T = Tq/ID, and a plot of as a function of time should be a straight line with slope -2D. 

The equation of motion = >kN does not require filaments to be perfectly circular it should hold for irregular, planar shapes as well, provided V = DI. Panfilov et al. [35] have suggested a convenient way to check this law of motion. Let A be the area in the plane contained inside a region with all or part of its boundary a planar scroll wave filament, then... 

To confirm this equation, Panfilov et al. [35] made measurements on initially noncircular scroll rings in BZ reagent. Their results are reproduced in Figure 5, where the area contained inside a closed planar filament (A0 = 27t) is plotted as a function of time. The plot shows a straight line with slope that corresponds to a diffusion coefficient of 0.12 mm /min. 

Fig. 5. The area inside the filament of a scroll ring decreases linearly with time with slope —2ttD, ZJ = 0.12 mm /min (from Panfilov et al. [35]). A is measured in mm and t in min.

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