We present an experimental study of the motion of a solid sphere falling through a wormlike micellar fluid. While smaller or lighter spheres quickly reach a terminal velocity, larger or heavier spheres are found to oscillate in the direction of their falling motion. The onset of this instability correlates with a critical value of the velocity gradient scale Γc ∼ 1 s −1 . We relate this condition to the known complex rheology of wormlike micellar fluids, and suggest that the unsteady motion of the sphere is caused by the formation and breaking of flow-induced structures.PACS numbers: 47.50.+d, 83.50.Jf, 83.60.Wc A sphere falling through a viscous Newtonian fluid is a classic problem in fluid dynamics, first solved mathematically by Stokes in 1851 [1]. Stokes provided a formula for the drag force F experienced by a sphere of radius R when moving at constant speed V 0 though a fluid with viscosity µ: F = 6πµRV 0 . The simplicity of the falling sphere experiment has meant that the viscosity can be measured directly from the terminal velocity V 0 , using a modified Stokes drag which takes into account wall effects [2]. The falling sphere experiment has also been used to study the viscoelastic properties of many polymeric (nonNewtonian) fluids [3,4,5]. In general, a falling sphere in a non-Newtonian fluid always approaches a terminal velocity, though sometimes with an oscillating transient [6,7,8]. In this paper we present evidence that a sphere falling in a wormlike micellar solution does not seem to approach a steady terminal velocity; instead it undergoes continual oscillations as it falls, as shown in Fig. 1.A wormlike micellar fluid is an aqueous solution in which amphiphilic (surfactant) molecules self-assemble in the presence of NaSal into long tubelike structures, or worms [9]; these micelles can sometimes be as long as 1µm [10]. Most wormlike micellar solutions are viscoelastic, and at low shear rates their rheological behavior is very similar to that of polymer solutions. However, unlike polymers, which are held together by strong covalent bonds, the micelles are held together by relatively weak entropic and screened electrostatic forces, and hence can break under shear. In fact, under equilibrium conditions these micelles are constantly breaking and reforming, providing a new mechanism for stress relaxation [11].The nonlinear rheology of these micellar fluids can be very different from standard polymer solutions [11,12,13]. Several observations of new phenomena have been reported, including shear thickening [14,15], a stress plateau in steady shear rheology [16,17], and flow instabilities such as shear-banding [18]. Bandyopadhyay et al. have observed chaotic fluctuations in the stress when a wormlike micellar solution is subjected to a step shear rate above a certain critical value (in the plateau region of stress-shear rate curve) [19]. A shear-induced transition from an isotropic to a nematic micellar ordering has also been observed [20]. There is increasing experimental evidence relating the onset of...

The amplitude equation for an unstable electrostatic wave is analyzed using an expansion in the mode amplitude A(t). In the limit of weak instability, i.e., γ→0+ where γ is the linear growth rate, the nonlinear coefficients are singular and their singularities predict the dependence of A(t) on γ. Generically the scaling |A(t)|=γ5/2r(γt) as γ→0+ is required to cancel the coefficient singularities to all orders. This result predicts the electric field scaling |Ek|∼γ5/2 will hold universally for these instabilities (including beam-plasma and two-stream configurations) throughout the dynamical evolution and in the time-asymptotic state. In exceptional cases, such as infinitely massive ions, the coefficients are less singular and the more familiar trapping scaling |Ek|∼γ2 is recovered.

Using numerical integrations of the Boussinesq equations in rotating cylindrical domains with realistic boundary conditions, we have computed the value of the largest Lyapunov exponent 1 for a variety of aspect ratios and driving strengths. We study in particular the domain chaos state, which bifurcates supercritically from the conducting fluid state and involves extended propagating fronts as well as point defects. We compare our results with those from Egolf et al., ͓Nature 404, 733 ͑2000͔͒, who suggested that the value of 1 for the spiral defect chaos state of a convecting fluid was determined primarily by bursts of instability arising from short-lived, spatially localized dislocation nucleation events. We also show that the quantity 1 is not intensive for aspect ratios ⌫ over the range 20Ͻ⌫Ͻ40 and that the scaling exponent of 1 near onset is consistent with the value predicted by the amplitude equation formalism.

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The nonlinear evolution of a unstable electrostatic wave is considered for a multi-species Vlasov plasma. From the singularity structure of the associated amplitude expansions, the asymptotic features of the electric field and distribution functions are determined in the limit of weak instability, i.e. γ → 0 + where γ is the linear growth rate. The asymptotic electric field is monochromatic at the wavelength of the linear mode with a nonlinear time dependence. The structure of the distibutions outside the resonant region is given by the linear eigenfunction but in the resonant region the distribution is nonlinear. The details depend on whether the ions are fixed or mobile; in either case the physical picture corresponds to the single wave model originally proposed by O"Neil, Winfrey, and Malmberg for the interaction of a cold weak beam with a plasma of fixed ions.