Absolute and Convective Instabilities of a Viscous Film Flowing Down a Vertical Fiber C. Duprat, 1 C. Ruyer-Quil, 1 S. Kalliadasis, 2 and F. Giorgiutti-Dauphine ´ 1 1 CNRS, Univ Pierre et Marie Curie, Univ Paris-Sud, Lab FAST, Bat 502, Campus Univ, Orsay 91405, France 2 Department of Chemical Engineering, Imperial College London, London SW7 2AZ, United Kingdom (Received 14 December 2006; published 15 June 2007) The stability of a viscous film flowing down a vertical fiber under the action of gravity is analyzed both experimentally and theoretically. At large or small film thicknesses, the instability is convective, whereas an absolute instability mode is observed in an intermediate range of film thicknesses for fibers of small enough radius. The onset of the experimental irregular wavy regime corresponds precisely to the theoretical prediction of the threshold of the convective instability. DOI: 10.1103/PhysRevLett.98.244502 PACS numbers: 47.20.Ma, 68.03.Kn, 68.15.+e The general problem of instability and pattern formation involving propagating modes has attracted much theoreti- cal and experimental interest (see, e.g., [1]). Instabilities can be either convective (disturbances grow in time only in a moving frame) or absolute (disturbances grow in time even at a fixed position). This concept was first developed in the context of plasma physics [2] and later on applied to optics and hydrodynamics [3]. In particular, transitions between different wave regimes in open-flow hydrodynamic systems can be understood within the framework of absolute or convective instabil- ities. Convectively unstable flows behave as spatial ampli- fiers of the incoming perturbations, whereas absolutely unstable flows display intrinsic self-sustained dynamics or global modes. The transition between these two classes of flows has been experimentally evidenced in numerous situations, e.g., wakes or hot jets [4]. Viscous liquid films falling down inclined planes are an example of convectively unstable open flows [5,6]. On the other hand, changing the geometry from planar to cylin- drical, such as with free jets, can make the flow absolutely unstable [7]. In this Letter, we demonstrate for the first time that a thin viscous layer, coating the outside of a vertical cylinder, and flowing under the action of gravity can also display both convective and absolute instabilities. While the dynamics of the flow on a cylinder has re- ceived considerable attention over the last two decades (see, e.g., [8] for a review), all previous works focused either on a temporal stability analysis or the fully nonlinear wave regime [9 –12]. Here we examine in detail, both experimentally and theoretically, the onset of instability with an emphasis on its spatial growth. Unlike jets where the absolute instability breaks the flow into drops, in the present case the flow is always continuous. At the same time, the Reynolds number is much smaller than for jets, which makes the problem amenable to theoretical analysis. Moreover, since the base flow is strictly parallel, a situation that is quite exceptional (e.g., jets in microgravity or sheared interfaces in confined geometries [13]), this prob- lem offers an excellent opportunity to study experimentally the development of nonlinear global modes [4]. The ex- periments can achieve a wide range of flow rates and in a certain regime of the parameter space the flow can be controlled relatively easily. A sketch of the experimental setup is shown in Fig. 1(a). A Rhodorsil silicon oil v50 (density   963 kg=m 3 , kine- matic viscosity   50  10 6 m 2 =s and surface tension   20:8  10 3 N=m at 25  C) flows axisymmetrically on Nylon fibers of 1.35 m length and of various radii R (0.23, 0.25, 0.32, 0.35, 0.47 and 1.5 mm). Aweight tied to the fiber ensures its verticality. The inlet flow rate q is controlled by varying the gap separating the two cone- shaped parts of the entrance valve. This design ensures the axisymmetry of the base flow and limits the entrance noise (thickness fluctuations of 10 3 %; estimation based on the computation of the maximum spatial growth rate in the convective regime). The possible range of flow rates is 0:01 g=s <q< 3g=s, which corresponds to a range of Scale (a) entrance valve (b) (c) FIG. 1 (color online). (a) Experimental setup. (b) and (c) Snap- shots of the liquid film at the top of the fiber and at 10 cm from the entrance valve, respectively, showing a nonlinear regular wave train (R  0:23 mm, q  0:0348 g=s and h N  0:48 mm). PRL 98, 244502 (2007) PHYSICAL REVIEW LETTERS week ending 15 JUNE 2007 0031-9007= 07=98(24)=244502(4) 244502-1  2007 The American Physical Society