Propagation of spiral waves pinned to circular and rectangular obstacles

We present an investigation of spiral waves pinned to circular and rectangular obstacles with different circumferences in both thin layers of the Belousov-Zhabotinsky reaction and numerical simulations with the Oregonator model. For circular objects, the area always increases with the circumference. In contrast, we varied the circumference of rectangles with equal areas by adjusting their width $w$ and height $h$. For both obstacle forms, the propagating parameters (i.e., wavelength, wave period, and velocity of pinned spiral waves) increase with the circumference, regardless of the obstacle area. Despite these common features of the parameters, the forms of pinned spiral waves depend on the obstacle shapes. The structures of spiral waves pinned to circles as well as rectangles with the ratio $w/h\sim 1$ are similar to Archimedean spirals. When $w/h$ increases, deformations of the spiral shapes are observed. For extremely thin rectangles with $w/h\gg 1$, these shapes can be constructed by employing semicircles with different radii which relate to the obstacle width and the core diameter of free spirals.

Figure 1

Spiral waves in the BZ reaction: (a) A free spiral wave (no obstacle) with a spiral core of 1.0 mm diameter (black circle), and spiral waves pinned to (b) a circle with diameter 2.8 mm and to rectangles with dimensions (c) 2.3 mm × 2.6 mm and (d) 6.5 mm × 0.9 mm.

Figure 2

Properties of spiral waves as a function of the obstacle circumference $l$ in the BZ reaction: (a) obstacle area A, (b) wavelength λ, (c) wave period T, [(d) and (e)] velocities $s_{\mathrm{obs}}$ and $s$ of waves adjacent to and far from the obstacles, respectively, and (f) percentage difference Δ$s$ between $s_{\mathrm{obs}}$ and $s$. Filled circles: no physical obstacles (spiral core diameter 1.0 mm); open circles: circular obstacles; open rectangles: rectangular obstacles.

Figure 3

Image overlays of counterclockwise rotating spiral waves pinned to (a) a circle with diameter 2.8 mm and to rectangles with dimensions (b) 2.3 mm × 2.6 mm and (c) 6.5 mm × 0.9 mm in the BZ reaction. Dashed curves in (a) and (b) are Archimedean spirals with origins located at the obstacle centers.

Figure 4

Spiral waves in the Oregonator model: (a) a free spiral wave (no obstacle, spiral core diameter 0.9 s.u.) and spiral waves pinned to (b) a circle with diameter of 1.5 s.u., (c) a rectangle with dimensions 8.5 s.u. × 0.2 s.u. and (d) a circle with diameter of 5.0 s.u.

Figure 5

Properties of spiral waves as a function of the obstacle circumference $l$ in the Oregonator model: (a) obstacle area A, (b) wavelength λ, (c) wave period T, [(d) and (e)] velocities $s_{\mathrm{obs}}$ and $s$ of waves adjacent to and far from the obstacles, respectively, and (f) percentage difference Δ$s$ of $s_{\mathrm{obs}}$ and $s$. Filled circles: no physical obstacles (spiral core diameter 0.9 s.u.); open circles: circular obstacles; open rectangles: rectangular obstacles.

Figure 6

Image overlays of counterclockwise rotating spiral waves pinned to (a) a circle with diameter 5.0 s.u. mm and to rectangles with dimensions (b) 1.3 s.u. × 1.3 s.u. and (c) 17.0 s.u. × 0.1 s.u. in the Oregonator model. Dashed curves in (a) and (b) are Archimedean spirals with origins located at the obstacle centers.

Figure 7

Estimation of the structure of a spiral wave pinned to a very thin rectangle in the Oregonator model. Some wave fronts in (a) the upper and (b) the lower sections of the overlaid image in Fig. 6 are compared to semicircles (dashed curves), the centers of which are located close to the right and the left edges of the obstacle. [(c)–(e)] When the wave front turns by an angle of 180° at the right boundary, a semicircular front appears, as if it is produced from a point source [the center of the dashed circle in (e)].

Figure 8

Graphical description of the structure of a spiral wave pinned to a very thin rectangle (black bar) using semicircles. (a) Three consecutive plots of the counterclockwise rotating spiral with the end tracing the upper boundary of the obstacle. The spiral structure at some instant of time can be estimated as a curve that alternately connects upper and lower semicircles of different sizes (corresponding halves of the dashed circles ${\mathrm{C}}_0$ to ${\mathrm{C}}_3)$. (b) Estimation of the radii $r_0$ to $r_3$ of the circles ${\mathrm{C}}_0$ to ${\mathrm{C}}_3$ having different centers (two small filled circles) close to the left and the right boundary of the obstacle. For the purpose of illustration, the distance δ is enlarged to a scale different from that of the width $w$ and the radii $r_0$ to $r_3$. The vertical dashed lines represent the contact positions of semicircle pairs as indicated by the labels.