612 research outputs found
Interaction of point sources and vortices for incompressible planar fluids
We consider a new system of differential equations which is at the same time
gradient and locally Hamiltonian. It is obtained by just replacing a factor in
the equations of interaction for N point vortices, and it is interpreted as an
interaction of N point sources. Because of the local Hamiltonian structure and
the symmetries it obeys, it does possess some of the first integrals that
appear in the N vortex problem. We will show that binary collisions are easily
blown up in this case since the equations of motion are of first order. This
method may be easily generalized to the blow up of higher order collisions. We
then generalize the model further to interactions of sources and vortices.Comment: 9 page
Steady Stokes flow with long-range correlations, fractal Fourier spectrum, and anomalous transport
We consider viscous two-dimensional steady flows of incompressible fluids
past doubly periodic arrays of solid obstacles. In a class of such flows, the
autocorrelations for the Lagrangian observables decay in accordance with the
power law, and the Fourier spectrum is neither discrete nor absolutely
continuous. We demonstrate that spreading of the droplet of tracers in such
flows is anomalously fast. Since the flow is equivalent to the integrable
Hamiltonian system with 1 degree of freedom, this provides an example of
integrable dynamics with long-range correlations, fractal power spectrum, and
anomalous transport properties.Comment: 4 pages, 4 figures, published in Physical Review Letter
Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities
We examine in detail the relative equilibria in the four-vortex problem where
two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and
\Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is
that for m > 0, the convex configurations all contain a line of symmetry,
forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for
all m but the isosceles trapezoid case exists only when m is positive. In fact,
there exist asymmetric convex configurations when m < 0. In contrast to the
Newtonian four-body problem with two equal pairs of masses, where the symmetry
of all convex central configurations is unproven, the equations in the vortex
case are easier to handle, allowing for a complete classification of all
solutions. Precise counts on the number and type of solutions (equivalence
classes) for different values of m, as well as a description of some of the
bifurcations that occur, are provided. Our techniques involve a combination of
analysis and modern and computational algebraic geometry
Point vortices and classical orthogonal polynomials
Stationary equilibria of point vortices with arbitrary choice of circulations
in a background flow are studied. Differential equations satisfied by
generating polynomials of vortex configurations are derived. It is shown that
these equations can be reduced to a single one. It is found that polynomials
that are Wronskians of classical orthogonal polynomials solve the latter
equation. As a consequence vortex equilibria at a certain choice of background
flows can be described with the help of Wronskians of classical orthogonal
polynomials.Comment: 20 pages, 12 figure
Enhanced tracer transport by the spiral defect chaos state of a convecting fluid
To understand how spatiotemporal chaos may modify material transport, we use
direct numerical simulations of the three-dimensional Boussinesq equations and
of an advection-diffusion equation to study the transport of a passive tracer
by the spiral defect chaos state of a convecting fluid. The simulations show
that the transport is diffusive and is enhanced by the spatiotemporal chaos.
The enhancement in tracer diffusivity follows two regimes. For large Peclet
numbers (that is, small molecular diffusivities of the tracer), we find that
the enhancement is proportional to the Peclet number. For small Peclet numbers,
the enhancement is proportional to the square root of the Peclet number. We
explain the presence of these two regimes in terms of how the local transport
depends on the local wave numbers of the convection rolls. For large Peclet
numbers, we further find that defects cause the tracer diffusivity to be
enhanced locally in the direction orthogonal to the local wave vector but
suppressed in the direction of the local wave vector.Comment: 11 pages, 12 figure
Euler configurations and quasi-polynomial systems
In the Newtonian 3-body problem, for any choice of the three masses, there
are exactly three Euler configurations (also known as the three Euler points).
In Helmholtz' problem of 3 point vortices in the plane, there are at most three
collinear relative equilibria. The "at most three" part is common to both
statements, but the respective arguments for it are usually so different that
one could think of a casual coincidence. By proving a statement on a
quasi-polynomial system, we show that the "at most three" holds in a general
context which includes both cases. We indicate some hard conjectures about the
configurations of relative equilibrium and suggest they could be attacked
within the quasi-polynomial framework.Comment: 21 pages, 6 figure
Echoes in classical dynamical systems
Echoes arise when external manipulations to a system induce a reversal of its
time evolution that leads to a more or less perfect recovery of the initial
state. We discuss the accuracy with which a cloud of trajectories returns to
the initial state in classical dynamical systems that are exposed to additive
noise and small differences in the equations of motion for forward and backward
evolution. The cases of integrable and chaotic motion and small or large noise
are studied in some detail and many different dynamical laws are identified.
Experimental tests in 2-d flows that show chaotic advection are proposed.Comment: to be published in J. Phys.
Vortex Images and q-Elementary Functions
In the present paper problem of vortex images in annular domain between two
coaxial cylinders is solved by the q-elementary functions. We show that all
images are determined completely as poles of the q-logarithmic function, where
dimensionless parameter is given by square ratio of the
cylinder radii. Resulting solution for the complex potential is represented in
terms of the Jackson q-exponential function. By composing pairs of q-exponents
to the first Jacobi theta function and conformal mapping to a rectangular
domain we link our solution with result of Johnson and McDonald. We found that
one vortex cannot remain at rest except at the geometric mean distance, but
must orbit the cylinders with constant angular velocity related to q-harmonic
series. Vortex images in two particular geometries in the limit
are studied.Comment: 17 page
Scattering off an oscillating target: Basic mechanisms and their impact on cross sections
We investigate classical scattering off a harmonically oscillating target in
two spatial dimensions. The shape of the scatterer is assumed to have a
boundary which is locally convex at any point and does not support the presence
of any periodic orbits in the corresponding dynamics. As a simple example we
consider the scattering of a beam of non-interacting particles off a circular
hard scatterer. The performed analysis is focused on experimentally accessible
quantities, characterizing the system, like the differential cross sections in
the outgoing angle and velocity. Despite the absence of periodic orbits and
their manifolds in the dynamics, we show that the cross sections acquire rich
and multiple structure when the velocity of the particles in the beam becomes
of the same order of magnitude as the maximum velocity of the oscillating
target. The underlying dynamical pattern is uniquely determined by the phase of
the first collision between the beam particles and the scatterer and possesses
a universal profile, dictated by the manifolds of the parabolic orbits, which
can be understood both qualitatively as well as quantitatively in terms of
scattering off a hard wall. We discuss also the inverse problem concerning the
possibility to extract properties of the oscillating target from the
differential cross sections.Comment: 18 page
Kinematic studies of transport across an island wake, with application to the Canary islands
Transport from nutrient-rich coastal upwellings is a key factor influencing
biological activity in surrounding waters and even in the open ocean. The rich
upwelling in the North-Western African coast is known to interact strongly with
the wake of the Canary islands, giving rise to filaments and other mesoscale
structures of increased productivity. Motivated by this scenario, we introduce
a simplified two-dimensional kinematic flow describing the wake of an island in
a stream, and study the conditions under which there is a net transport of
substances across the wake. For small vorticity values in the wake, it acts as
a barrier, but there is a transition when increasing vorticity so that for
values appropriate to the Canary area, it entrains fluid and enhances
cross-wake transport.Comment: 28 pages, 13 figure
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