70 research outputs found
Stability of Relative Equilibria of Point Vortices on a Sphere and Symplectic Integrators
This paper analyzes the dynamics of N point vortices moving on a sphere from the point of view of geometric mechanics. The formalism is developed for the general case of N vortices, and the details are provided for the (integrable) case N = 3. Stability of relative equilibria is analyzed by the energy-momentum method. Explicit criteria for stability of different configurations with generic and non-generic momenta are obtained. In each case, a group of transformations is specied, such that motion in the original (unreduced) phase space is stable modulo this group. Finally, we outline the construction of a symplectic-momentum integrator for vortex dynamics on a sphere
An approximate homogenization scheme for nonperiodic materials
AbstractRecently in [1], Briane announced a new homogenization method for certain nonperiodic materials in which the H-limit of a highly oscillatory but nonperiodic matrix Aε is obtained by comparing to a locally-periodic matrix Bε in domains whose size α(ε) → 0 as ε → 0 but slower than ϵ. The H-limit of Bε is a function of every point in the material, and so theoretically, in order to homogenize Aε, the solution to the usual periodic cell problem must be obtained for every point in the material. Computationally this is not feasible, so we approximate the homogenization method by keeping α fixed. We show that this approximation is O(α) by proving that the difference of two nearby cell solutions (within a cube of side length α) is O(α) in the H1-norm. This result requires that we show a uniform bound exists for the gradients of the periodic cell solutions in Lp. We then apply our approximate homogenization theory to the analysis of certain defects in fiber-reinforced composites. In particular, we show that when unexpected local spreading of the fibers occurs in a small region of the material, constituent stress concentrations of nearly three can arise
The geometry and analysis of the averaged Euler equations and a new diffeomorphism group
We present a geometric analysis of the incompressible averaged Euler
equations for an ideal inviscid fluid. We show that solutions of these
equations are geodesics on the volume-preserving diffeomorphism group of a new
weak right invariant pseudo metric. We prove that for precompact open subsets
of , this system of PDEs with Dirichlet boundary conditions are
well-posed for initial data in the Hilbert space , . We then use
a nonlinear Trotter product formula to prove that solutions of the averaged
Euler equations are a regular limit of solutions to the averaged Navier-Stokes
equations in the limit of zero viscosity. This system of PDEs is also the model
for second-grade non-Newtonian fluids
Reduction in principal fiber bundles: covariant Euler-Poincare equations
Let be a principal G-bundle, and let be a G-invariant Lagrangian density. We obtain the
Euler-Poincare equations for the reduced Lagrangian l defined on , the bundle of connections on P
Averaged Template Matching Equations
By exploiting an analogy with averaging procedures in fluid
dynamics, we present a set of averaged template matching equations.
These equations are analogs of the exact template matching equations
that retain all the geometric properties associated with the diffeomorphismgrou
p, and which are expected to average out small scale features
and so should, as in hydrodynamics, be more computationally efficient
for resolving the larger scale features. Froma geometric point of view,
the new equations may be viewed as coming from a change in norm that
is used to measure the distance between images. The results in this paper
represent first steps in a longer termpro gram: what is here is only
for binary images and an algorithm for numerical computation is not
yet operational. Some suggestions for further steps to develop the results
given in this paper are suggested
Dynamical elastic bodies in Newtonian gravity
Well-posedness for the initial value problem for a self-gravitating elastic
body with free boundary in Newtonian gravity is proved. In the material frame,
the Euler-Lagrange equation becomes, assuming suitable constitutive properties
for the elastic material, a fully non-linear elliptic-hyperbolic system with
boundary conditions of Neumann type. For systems of this type, the initial data
must satisfy compatibility conditions in order to achieve regular solutions.
Given a relaxed reference configuration and a sufficiently small Newton's
constant, a neigborhood of initial data satisfying the compatibility conditions
is constructed
On the existence of solutions to the relativistic Euler equations in 2 spacetime dimensions with a vacuum boundary
We prove the existence of a wide class of solutions to the isentropic
relativistic Euler equations in 2 spacetime dimensions with an equation of
state of the form that have a fluid vacuum boundary. Near the fluid
vacuum boundary, the sound speed for these solutions are monotonically
decreasing, approaching zero where the density vanishes. Moreover, the fluid
acceleration is finite and bounded away from zero as the fluid vacuum boundary
is approached. The existence results of this article also generalize in a
straightforward manner to equations of state of the form
with .Comment: A major revision of the second half of the pape
The Navier-Stokes-alpha model of fluid turbulence
We review the properties of the nonlinearly dispersive Navier-Stokes-alpha
(NS-alpha) model of incompressible fluid turbulence -- also called the viscous
Camassa-Holm equations and the LANS equations in the literature. We first
re-derive the NS-alpha model by filtering the velocity of the fluid loop in
Kelvin's circulation theorem for the Navier-Stokes equations. Then we show that
this filtering causes the wavenumber spectrum of the translational kinetic
energy for the NS-alpha model to roll off as k^{-3} for k \alpha > 1 in three
dimensions, instead of continuing along the slower Kolmogorov scaling law,
k^{-5/3}, that it follows for k \alpha < 1. This rolloff at higher wavenumbers
shortens the inertial range for the NS-alpha model and thereby makes it more
computable. We also explain how the NS-alpha model is related to large eddy
simulation (LES) turbulence modeling and to the stress tensor for second-grade
fluids. We close by surveying recent results in the literature for the NS-alpha
model and its inviscid limit (the Euler-alpha model).Comment: 22 pages, 1 figure. Dedicated to V. E. Zakharov on the occasion of
his 60th birthday. To appear in Physica
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