Spontaneous generation of spiral waves by a hydrodynamic instability

The coiling of a thin filament of viscous fluid falling onto a surface is a common and easily reproducible hydrodynamic instability. Here we report for the first time that this instability can generate regular spiral patterns, in which air bubbles are trapped in the coil and then advected horizontally by the fluid spreading on the surface. We present a simple model that explains how these beautiful patterns are formed, and how the number of spiral branches and their curvature depends on the coiling frequency, the frequency of rotation of the coiling center, the total flow rate, and the thickness of the spreading fluid film. Copyright c © EPLA, 2008 The study of spirals in Nature goes back at least to the seventeenth century, when Swammerdam was among the first to describe the beautiful forms of certain seashells [1]. The standard work on spontaneous pattern formation in Nature, D’Arcy Thompson’s On Growth and Form [1] describes a multitude of spiral patterns, including those of shells, sunflower seeds, and the helical structure of branches or leaves on growing plant stems. All these spirals are self-organized and obey rather strict mathematical rules. Shells, for example, are generally logarithmic spirals in which the distance between successive loops grows in a precisely determined way with increasing distance from the center [2]. In the case of sunflower seed spirals (phyllotaxis), Douady and Couder [3] used a clever laboratory experiment to show that the spirals form due to a selforganized growth process whereby new seeds are generated in the center at a fixed frequency and then repel each other by steric repulsion. The maximization of the distance between the seeds then leads to a special subtype of the logarithmic spiral pattern: the golden or Fibonacci spiral. The same authors showed how these ideas can be applied to plants, accounting for instance for the sunflower spirals [4]. Not all natural spirals are due to a steric repulsion between constitutive elements, however. Over the past few years, self-organized spiral waves have been studied extensively [5]. These dynamic spirals form spontaneously in excitable media [6,7] and have been observed in contexts as different as catalytic surface oxidation [8], the BelousovZhabotinsky chemical reaction [9–13], aggregating colonies of slime mold [14,15] and contracting heart tissue, where such waves are believed to be related to cardiac arrhythmia and fibrillation [16]. Here we demonstrate that spirals can also arise during the coiling of a thin “rope” of viscous fluid falling onto a solid surface (fig. 1) [17–19]. In previous papers we investigated how the frequency and radius of the coiling depends on the orifice diameter, the height of fall, the flow rate, and the fluid viscosity, and we showed that coiling traverses four different dynamical regimes as the fall height increases [20–24]. Here we report that in a limited portion of the parameter space, air bubbles become trapped between successive coils and are then advected radially away to form surprising and very regular spiral patterns. We also present a simple model that explains how these beautiful patterns are formed, and