685 research outputs found
The period of a classical oscillator
We develop a simple method to obtain approximate analytical expressions for
the period of a particle moving in a given potential. The method is inspired to
the Linear Delta Expansion (LDE) and it is applied to a large class of
potentials. Precise formulas for the period are obtained.Comment: 5 pages, 4 figure
Linear stability, transient energy growth and the role of viscosity stratification in compressible plane Couette flow
Linear stability and the non-modal transient energy growth in compressible
plane Couette flow are investigated for two prototype mean flows: (a) the {\it
uniform shear} flow with constant viscosity, and (b) the {\it non-uniform
shear} flow with {\it stratified} viscosity. Both mean flows are linearly
unstable for a range of supersonic Mach numbers (). For a given , the
critical Reynolds number () is significantly smaller for the uniform shear
flow than its non-uniform shear counterpart. An analysis of perturbation energy
reveals that the instability is primarily caused by an excess transfer of
energy from mean-flow to perturbations. It is shown that the energy-transfer
from mean-flow occurs close to the moving top-wall for ``mode I'' instability,
whereas it occurs in the bulk of the flow domain for ``mode II''. For the
non-modal analysis, it is shown that the maximum amplification of perturbation
energy, , is significantly larger for the uniform shear case compared
to its non-uniform counterpart. For , the linear stability operator
can be partitioned into , and the
-dependent operator is shown to have a negligibly small
contribution to perturbation energy which is responsible for the validity of
the well-known quadratic-scaling law in uniform shear flow: . A reduced inviscid model has been shown to capture all salient
features of transient energy growth of full viscous problem. For both modal and
non-modal instability, it is shown that the {\it viscosity-stratification} of
the underlying mean flow would lead to a delayed transition in compressible
Couette flow
Solitons in a trapped spin-1 atomic condensate
We numerically investigate a particular type of spin solitons inside a
trapped atomic spin-1 Bose-Einstein condensate (BEC) with ferromagnetic
interactions. Within the mean field theory approximation, our study of the
solitonic dynamics shows that the solitonic wave function, its center of mass
motion, and the local spin evolutions are stable and are intimately related to
the domain structures studied recently in spin-1 Rb condensates. We
discuss a rotating reference frame wherein the dynamics of the solitonic local
spatial spin distribution become time independent.Comment: 8 pages, 8 color eps figure
Ultra-discrete Optimal Velocity Model: a Cellular-Automaton Model for Traffic Flow and Linear Instability of High-Flux Traffic
In this paper, we propose the ultra-discrete optimal velocity model, a
cellular-automaton model for traffic flow, by applying the ultra-discrete
method for the optimal velocity model. The optimal velocity model, defined by a
differential equation, is one of the most important models; in particular, it
successfully reproduces the instability of high-flux traffic. It is often
pointed out that there is a close relation between the optimal velocity model
and the mKdV equation, a soliton equation. Meanwhile, the ultra-discrete method
enables one to reduce soliton equations to cellular automata which inherit the
solitonic nature, such as an infinite number of conservation laws, and soliton
solutions. We find that the theory of soliton equations is available for
generic differential equations, and the simulation results reveal that the
model obtained reproduces both absolutely unstable and convectively unstable
flows as well as the optimal velocity model.Comment: 9 pages, 6 figure
Dynamics and stability of vortex-antivortex fronts in type II superconductors
The dynamics of vortices in type II superconductors exhibit a variety of
patterns whose origin is poorly understood. This is partly due to the
nonlinearity of the vortex mobility which gives rise to singular behavior in
the vortex densities. Such singular behavior complicates the application of
standard linear stability analysis. In this paper, as a first step towards
dealing with these dynamical phenomena, we analyze the dynamical stability of a
front between vortices and antivortices. In particular we focus on the question
of whether an instability of the vortex front can occur in the absence of a
coupling to the temperature. Borrowing ideas developed for singular bacterial
growth fronts, we perform an explicit linear stability analysis which shows
that, for sufficiently large front velocities and in the absence of coupling to
the temperature, such vortex fronts are stable even in the presence of in-plane
anisotropy. This result differs from previous conclusions drawn on the basis of
approximate calculations for stationary fronts. As our method extends to more
complicated models, which could include coupling to the temperature or to other
fields, it provides the basis for a more systematic stability analysis of
nonlinear vortex front dynamics.Comment: 13 pages, 8 figure
The Universal Gaussian in Soliton Tails
We show that in a large class of equations, solitons formed from generic
initial conditions do not have infinitely long exponential tails, but are
truncated by a region of Gaussian decay. This phenomenon makes it possible to
treat solitons as localized, individual objects. For the case of the KdV
equation, we show how the Gaussian decay emerges in the inverse scattering
formalism.Comment: 4 pages, 2 figures, revtex with eps
Preventing transition to turbulence: a viscosity stratification does not always help
In channel flows a step on the route to turbulence is the formation of
streaks, often due to algebraic growth of disturbances. While a variation of
viscosity in the gradient direction often plays a large role in
laminar-turbulent transition in shear flows, we show that it has, surprisingly,
little effect on the algebraic growth. Non-uniform viscosity therefore may not
always work as a flow-control strategy for maintaining the flow as laminar.Comment: 9 pages, 8 figure
Waves and instability in a one-dimensional microfluidic array
Motion in a one-dimensional (1D) microfluidic array is simulated. Water
droplets, dragged by flowing oil, are arranged in a single row, and due to
their hydrodynamic interactions spacing between these droplets oscillates with
a wave-like motion that is longitudinal or transverse. The simulation yields
wave spectra that agree well with experiment. The wave-like motion has an
instability which is confirmed to arise from nonlinearities in the interaction
potential. The instability's growth is spatially localized. By selecting an
appropriate correlation function, the interaction between the longitudinal and
transverse waves is described
Super Stability of Laminar Vortex Flow in Superfluid 3He-B
Vortex flow remains laminar up to large Reynolds numbers (Re~1000) in a
cylinder filled with 3He-B. This is inferred from NMR measurements and
numerical vortex filament calculations where we study the spin up and spin down
responses of the superfluid component, after a sudden change in rotation
velocity. In normal fluids and in superfluid 4He these responses are turbulent.
In 3He-B the vortex core radius is much larger which reduces both surface
pinning and vortex reconnections, the phenomena, which enhance vortex bending
and the creation of turbulent tangles. Thus the origin for the greater
stability of vortex flow in 3He-B is a quantum phenomenon. Only large flow
perturbations are found to make the responses turbulent, such as the walls of a
cubic container or the presence of invasive measuring probes inside the
container.Comment: 4 pages, 6 figure
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