Localized states in sheared electroconvection

Electroconvection in a thin, sheared fluid film displays a rich sequence of bifurcations between different flow states as the driving voltage is increased. We present a numerical study of an annular film in which a radial potential difference acts on induced surface charges to drive convection. The film is also sheared by independently rotating the inner edge of the annulus. This simulation models laboratory experiments on electroconvection in sheared smectic liquid crystal films. The applied shear competes with the electrical forces, resulting in oscillatory and strongly subcritical bifurcations between localized vortex states close to onset. At higher forcing, the flow becomes chaotic via a Ruelle-Takens-Newhouse scenario. The simulation allows flow visualization not available in the physical experiments, and sheds light on previously observed transitions in the current-voltage characteristics of electroconvecting smectic films. Driven, dissipative nonlinear systems sometimes exhibit spatially localized structures which are analogous to solitons [1, 2]. Examples are found in systems as diverse as vegetation patterns [3], wide lasers [4], vibrated granular media [5] and ferrofluids [6]. Such states can also arise in very well-studied fluid-mechanical settings such as binary fluid convection [7, 8] (where they have been called “convectons” [9]), and in very general model equations [10–12]. In this Letter, we describe a new and unexpected instance of localized patterns: solitary vortex states in twodimensional, sheared electroconvection. We show numerically that these states can be extremely localized, consisting of only a single, isolated vortex surrounded by a uniform background state. This system has an especially simple geometry which is experimentally realized using thin free-standing films of smectic liquid crystals [13–19]. We employ a direct numerical simulation which allows us to study the spatial structure of the full velocity, charge and potential fields [19]. This approach compliments existing theory [15, 19] and experiments [13–18] on this system, which mainly consisted in observations of the total current through the thin film, without flow visualization. The simulation reveals localized states in the form of vortices which travel in the direction of the applied shear. These are preceded by lower-amplitude, extended traveling and oscillatory vortex states. At sufficiently high levels of electrical forcing, the flow becomes chaotic via a RuelleTakens-Newhouse scenario [20,21]. Our numerical study simulates a laboratory experiment shown schematically in Fig. 1. The system consists of a submicron-thick liquid crystal film freely suspended between concentric circular electrodes. The weakly conducting film is driven to convect when a sufficiently large electric potential is imposed across it. The inner edge of the annular film is held at absolute potential V with respect to infinity, while the outer edge is held at zero potential. In addition to the control parameter associated with this electrical driving force, it is possible to independently rotate the inner electrode, which imposes an azimuthal Couette shear on the film. The experimental signature of convection consists of measurements of the total current through the film, which is increased by convective flow. The film develops a surface charge configuration which is unstable to the applied voltage. This instability is closely analogous to that of Rayleigh-Bénard convection, in which an inverted density distribution is unstable to buoyancy forces. As in Rayleigh-Bénard convection, there are two important dimensionless parameters. The Rayleigh-like number R describes the ratio of the electrical forcing to the viscous and electrical dissipation; this serves as the main control parameter. The Prandtl-like number P is the ratio of charge relaxation time τc to viscous relaxation time τv. The annular geometry of the film is characterized by the radius ratio α. These parameters p-1 ar X iv :0 80 7. 07 52 v1 [ nl in .P S] 4 J ul 2 00 8 Peichun Tsai1, Stephen W. Morris1, and Zahir A. Daya2