143 research outputs found
Stability of the selfsimilar dynamics of a vortex filament
In this paper we continue our investigation about selfsimilar solutions of
the vortex filament equation, also known as the binormal flow (BF) or the
localized induction equation (LIE). Our main result is the stability of the
selfsimilar dynamics of small pertubations of a given selfsimilar solution. The
proof relies on finding precise asymptotics in space and time for the tangent
and the normal vectors of the perturbations. A main ingredient in the proof is
the control of the evolution of weighted norms for a cubic 1-D Schr\"odinger
equation, connected to the binormal flow by Hasimoto's transform.Comment: revised version, 36 page
Stationary solutions of the one-dimensional nonlinear Schroedinger equation: II. Case of attractive nonlinearity
All stationary solutions to the one-dimensional nonlinear Schroedinger
equation under box or periodic boundary conditions are presented in analytic
form for the case of attractive nonlinearity. A companion paper has treated the
repulsive case. Our solutions take the form of bounded, quantized, stationary
trains of bright solitons. Among them are two uniquely nonlinear classes of
nodeless solutions, whose properties and physical meaning are discussed in
detail. The full set of symmetry-breaking stationary states are described by
the character tables from the theory of point groups. We make
experimental predictions for the Bose-Einstein condensate and show that, though
these are the analog of some of the simplest problems in linear quantum
mechanics, nonlinearity introduces new and surprising phenomena.Comment: 11 pages, 9 figures -- revised versio
Simulating (electro)hydrodynamic effects in colloidal dispersions: smoothed profile method
Previously, we have proposed a direct simulation scheme for colloidal
dispersions in a Newtonian solvent [Phys.Rev.E 71,036707 (2005)]. An improved
formulation called the ``Smoothed Profile (SP) method'' is presented here in
which simultaneous time-marching is used for the host fluid and colloids. The
SP method is a direct numerical simulation of particulate flows and provides a
coupling scheme between the continuum fluid dynamics and rigid-body dynamics
through utilization of a smoothed profile for the colloidal particles.
Moreover, the improved formulation includes an extension to incorporate
multi-component fluids, allowing systems such as charged colloids in
electrolyte solutions to be studied. The dynamics of the colloidal dispersions
are solved with the same computational cost as required for solving
non-particulate flows. Numerical results which assess the hydrodynamic
interactions of colloidal dispersions are presented to validate the SP method.
The SP method is not restricted to particular constitutive models of the host
fluids and can hence be applied to colloidal dispersions in complex fluids
Finite-gap Solutions of the Vortex Filament Equation: Isoperiodic Deformations
We study the topology of quasiperiodic solutions of the vortex filament
equation in a neighborhood of multiply covered circles. We construct these
solutions by means of a sequence of isoperiodic deformations, at each step of
which a real double point is "unpinched" to produce a new pair of branch points
and therefore a solution of higher genus. We prove that every step in this
process corresponds to a cabling operation on the previous curve, and we
provide a labelling scheme that matches the deformation data with the knot type
of the resulting filament.Comment: 33 pages, 5 figures; submitted to Journal of Nonlinear Scienc
The Viscous Nonlinear Dynamics of Twist and Writhe
Exploiting the "natural" frame of space curves, we formulate an intrinsic
dynamics of twisted elastic filaments in viscous fluids. A pair of coupled
nonlinear equations describing the temporal evolution of the filament's complex
curvature and twist density embodies the dynamic interplay of twist and writhe.
These are used to illustrate a novel nonlinear phenomenon: ``geometric
untwisting" of open filaments, whereby twisting strains relax through a
transient writhing instability without performing axial rotation. This may
explain certain experimentally observed motions of fibers of the bacterium B.
subtilis [N.H. Mendelson, et al., J. Bacteriol. 177, 7060 (1995)].Comment: 9 pages, 4 figure
A Dissipative-Particle-Dynamics Model for Simulating Dynamics of Charged Colloid
A mesoscopic colloid model is developed in which a spherical colloid is
represented by many interacting sites on its surface. The hydrodynamic
interactions with thermal fluctuations are taken accounts in full using
Dissipative Particle Dynamics, and the electrostatic interactions are simulated
using Particle-Particle-Particle Mesh method. This new model is applied to
investigate the electrophoretic mobility of a charged colloid under an external
electric field, and the influence of salt concentration and colloid charge are
systematically studied. The simulation results show good agreement with
predictions from the electrokinetic theory.Comment: 17 pages, 8 figures, submitted to the proceedings of High Performance
Computing in Science & Engineering '1
Stability of stationary states in the cubic nonlinear Schroedinger equation: applications to the Bose-Einstein condensate
The stability properties and perturbation-induced dynamics of the full set of
stationary states of the nonlinear Schroedinger equation are investigated
numerically in two physical contexts: periodic solutions on a ring and
confinement by a harmonic potential. Our comprehensive studies emphasize
physical interpretations useful to experimentalists. Perturbation by stochastic
white noise, phase engineering, and higher order nonlinearity are considered.
We treat both attractive and repulsive nonlinearity and illustrate the
soliton-train nature of the stationary states.Comment: 9 pages, 11 figure
Geometric origin of mechanical properties of granular materials
Some remarkable generic properties, related to isostaticity and potential
energy minimization, of equilibrium configurations of assemblies of rigid,
frictionless grains are studied. Isostaticity -the uniqueness of the forces,
once the list of contacts is known- is established in a quite general context,
and the important distinction between isostatic problems under given external
loads and isostatic (rigid) structures is presented. Complete rigidity is only
guaranteed, on stability grounds, in the case of spherical cohesionless grains.
Otherwise, the network of contacts might deform elastically in response to load
increments, even though grains are rigid. This sets an uuper bound on the
contact coordination number. The approximation of small displacements (ASD)
allows to draw analogies with other model systems studied in statistical
mechanics, such as minimum paths on a lattice. It also entails the uniqueness
of the equilibrium state (the list of contacts itself is geometrically
determined) for cohesionless grains, and thus the absence of plastic
dissipation. Plasticity and hysteresis are due to the lack of such uniqueness
and may stem, apart from intergranular friction, from small, but finite,
rearrangements, in which the system jumps between two distinct potential energy
minima, or from bounded tensile contact forces. The response to load increments
is discussed. On the basis of past numerical studies, we argue that, if the ASD
is valid, the macroscopic displacement field is the solution to an elliptic
boundary value problem (akin to the Stokes problem).Comment: RevTex, 40 pages, 26 figures. Close to published paper. Misprints and
minor errors correcte
Self-binormal solutions of the Localized Induction Approximation: Singularity formation
We investigate the formation of singularities in a self-similar form of
regular solutions of the Localized Induction Approximation (also referred as to
the binormal flow). This equation appears as an approximation model for the
self-induced motion of a vortex filament in an inviscid incompressible fluid.
The solutions behave as 3d-logarithmic spirals at infinity.
The proofs of the results are strongly based on the existing connection
between the binormal flow and certain Schr\"odinger equations.Comment: 60 pages, 8 figure
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