PHYSICAL REVIEW E 91, 052912 (2015) Propagation of spiral waves pinned to circular and rectangular obstacles Malee Sutthiopad, 1 Jiraporn Luengviriya, 2, 3 Porramain Porjai, 1 Metinee Phantu, 1 Jarin Kanchanawarin, 3 Stefan C. M¨ uller, 4 and Chaiya Luengviriya 1 , * 1 Department of Physics, Kasetsart University, 50 Phaholyothin Road, Jatujak, Bangkok 10900, Thailand 2 Department of Industrial Physics and Medical Instrumentation, King Mongkut’s University of Technology North Bangkok, 1518 Pibulsongkram Road, Bangkok 10800, Thailand 3 Lasers and Optics Research Group, King Mongkut’s University of Technology North Bangkok, 1518 Pibulsongkram Road, Bangkok 10800, Thailand 4 Institute of Experimental Physics, Otto-von-Guericke University Magdeburg, Universit¨ atsplatz 2, D-39106 Magdeburg, Germany (Received 5 March 2015; published 15 May 2015) 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 ∼ 1 are similar to Archimedean spirals. When w/h increases, deformations of the spiral shapes are observed. For extremely thin rectangles with w/h ≫ 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. DOI: 10.1103/PhysRevE.91.052912 PACS number(s): 05.45.−a, 05.65.+b, 82.40.Ck, 82.40.Qt I. INTRODUCTION Propagating spiral waves have been discovered in various reaction-diffusion systems such as CO oxidation on platinum surfaces [1], cell aggregation in slime mold colonies [2], electrical wave propagation in cardiac tissues [3], and concen- tration waves in the Belousov-Zhabotinsky (BZ) reaction [4,5]. In the heart, electrical spiral waves are connected with cardiac tachycardia and life-threatening fibrillations [6,7]. Such spiral waves may cease when their tip hits the boundary of the medium. However, they will survive much longer if they are pinned to anatomical inhomogeneities or obstacles, e.g., veins or scars [3]. Unexcitable disks have been widely taken as model obsta- cles to study the effects of obstacle size on the properties of spiral waves pinned to them. Tyson and Keener’s theoretical work [8] predicted that a spiral wave rotating around a circular hole has period and velocity that increase when the hole is enlarged. Tanaka et al. [9] proposed a formula which showed that the spiral wave velocity at the periphery of the circular obstacle increases with the obstacle radius. Simulations by Fu et al. [10] revealed that both unexcitable and partially excitable circles cause the period of spiral waves to increase with their radii. Similarly, Cherubini et al. [11] showed that the wavelength and the period also increase linearly with the obstacle radius in cardiac model systems, regardless of whether the elasticity of the medium was included in the simulations. For spiral waves in cardiomyocytes, their velocity and wavelength were found to increase linearly with the circumference of the circular obstacle [12]. Experiments using thin layers of the photosensitive ruthenium-catalyzed BZ reaction [13] have demonstrated that * Corresponding author: fscicyl@ku.ac.th wave period, wavelength, and velocity of a spiral wave increased from 26 s, 1.3 mm, and 49.6 µms −1 to 49 s, 3.4 mm, and 74.3 µms −1 , respectively, after an artificial circular core was created by a laser spot of 1.2 mm in diameter. A scroll ring (i.e., a spiral structure in three dimensions) has been often observed to contract and even- tually self-annihilate [14,15]. However, the contraction was suppressed when the scroll ring was pinned to spherical plastic beads [16,17]. In this article, we present an investigation of the dynamics of pinned spiral waves in BZ media. We chose two different simple forms of obstacles: circles and rectangles. Circles are symmetric objects which were used in many studies of pinned spiral waves in experiments and simulations, whereas rectangles have the advantage that their width and height are adjustable to obtain different circumferences while the area can be fixed to a constant value. We confirmed our experimental results by numerical simulations using the Oregonator model [18,19]. II. EXPERIMENTS A. Experimental methods We prepared the Belousov-Zhabotinsky (BZ) solutions from NaBrO 3 ,H 2 SO 4 , malonic acid (MA), and ferroin, all purchased from Merck. Stock solutions of NaBrO 3 (1 M) and MA (1 M) were freshly produced by dissolving powder in deionized water (conductivity ∼ 0.056 µS cm −1 ), whereas stock solutions of H 2 SO 4 (2.5 M) and ferroin (25 mM) were commercially available. To prevent any hydrodynamic perturbation, the reaction was embedded in a 1.0% w/w agarose gel (Sigma). Appropriate volumes of the stock solutions were mixed and diluted in deionized water to form BZ solutions with initial concentrations: [H 2 SO 4 ] = 160 mM, 1539-3755/2015/91(5)/052912(8) 052912-1 ©2015 American Physical Society