We study experimentally statistical properties of the opening times of knots in vertically vibrated granular chains. Our measurements are in good qualitative and quantitative agreement with a theoretical model involving three random walks interacting via hard core exclusion in one spatial dimension. In particular, the knot survival probability follows a universal scaling function which is independent of the chain length, with a corresponding diffusive characteristic time scale. Both the large-exit-time and the small-exit-time tails of the distribution are suppressed exponentially, and the corresponding decay coefficients are in excellent agreement with the theoretical values.PACS: 81.05.Rm, 83.10Nn Topological constraints such as knots [1] and entanglements strongly affect the dynamics of filamentary objects including polymers [2][3][4][5] and DNA molecules [6,7]. Typically, large time scales are associated with the relaxation of such constraints [8,9]. Understanding the physical mechanisms governing the relaxation of such constraints is crucial to characterizing flow, deformation, as well as structural properties of materials consisting of ensembles of macromolecules, e.g., polymers, gels, and rubber.Scaling techniques, such as de Gennes-Edwards reptation theory, provide a powerful tool for modeling dynamics of topological constraints [8,9]. These are successful when the precise details of the interparticle interactions are secondary relative to the geometric effects. However, topological constraints are difficult to control experimentally and typically, they can be probed only using indirect methods. Here, we introduce a physical system where these difficulties are greatly reduced, thereby enabling a detailed quantitative comparison with theory. In this Letter, we study dynamics of knots in vibrated granular chains. This system has an appealing simplicity as the "molecular weight" of the chain and the driving conditions can be well controlled. Additionally, the topological constraints can be directly observed. We restrict our attention to simple knots and investigate the time it takes for a knot to open. We find that the average unknotting time τ is consistent with a diffusive behavior τ ∼ N 2 where N is the number of beads in the chain. We also show that statistical properties of opening times are well described by a one dimensional model where three random walks, representing the three exclusion points governing the knot, interact via excluded volume interactions. This model provides an excellent approximation to the knot survival probability. Furthermore, quantitative predictions of this model including fluctuations in the exit times, as well as the coefficients governing the exponential decay of the extremal tails of the distribution are in excellent agreement with the measured values.In the experiments, a simple knot was tightly tied in the center of a ball chain and placed onto a vibrating plate. In Fig. 1, we show images representative of the unknotting process starting from a tightly knotted chain,...

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A suspended fluid film with two free surfaces convects when a sufficiently large voltage is applied across it. We present a linear stability analysis for this system. The forces driving convection are due to the interaction of the applied electric field with space charge which develops near the free surfaces.Our analysis is similar to that for the two-dimensional Bénard problem, but with important differences due to coupling between the charge distribution and the field. We find the neutral stability boundary of a dimensionless control parameter R as a function of the dimensionless wave number κ. R, 1 which is proportional to the square of the applied voltage, is analogous to the Rayleigh number. The critical values R c and κ c are found from the minimum of the stability boundary, and its curvature at the minimum gives the correlation length ξ 0 . The characteristic time scale τ 0 , which depends on a second dimensionless parameter P, analogous to the Prandtl number, is determined from the linear growth rate near onset. ξ 0 and τ 0 are coefficients in the Ginzburg-Landau amplitude equation which describes the flow pattern near onset in this system. We compare our results to recent experiments.

We investigate the linear stability of a thin, suspended, annular film of conducting fluid with a voltage difference applied between its inner and outer edges. For a sufficiently large voltage, such a film is unstable to radially-driven electroconvection due to charges which develop on its free surfaces. The film can also be subjected to a Couette shear by rotating its inner edge. This combination is experimentally realized using films of smectic A liquid crystals. In the absence of shear, the convective flow consists of a stationary, azimuthally one-dimensional pattern of symmetric, counter-rotating vortex pairs. When Couette flow is applied, an azimuthally traveling pattern results. When viewed in a co-rotating frame, the traveling pattern consists of pairs of asymmetric vortices. We calculate the neutral stability boundary for arbitrary radius ratio α and Reynolds number Re of the shear flow, and obtain the critical control parameter Rc (α, Re) and the critical azimuthal mode number mc (α, Re). The Couette flow suppresses the onset of electroconvection, so that Rc (α, Re) > Rc (α, 0). The calculated suppression is compared with experiments performed at α = 0.56 and 0 ≤ Re ≤ 0.22. Fluids, 11, 3613 (1999). See also http://mobydick.physics.utoronto.ca. Physics of

We report experiments on convection driven by a radial electrical force in suspended annular smectic A liquid crystal films. In the absence of an externally imposed azimuthal shear, a stationary one-dimensional (1D) pattern consisting of symmetric vortex pairs is formed via a supercritical transition at the onset of convection. Shearing reduces the symmetries of the base state and produces a traveling 1D pattern whose basic periodic unit is a pair of asymmetric vortices. For a sufficiently large shear, the primary bifurcation changes from supercritical to subcritical. We describe measurements of the resulting hysteresis as a function of the shear at radius ratio η ∼ 0.8. This simple pattern forming system has an unusual combination of symmetries and control parameters and should be amenable to quantitative theoretical analysis.Typeset using REVT E X 1