49 research outputs found
Long time evolution of concentrated Euler flows with planar symmetry
We study the time evolution of an incompressible Euler fluid with planar symmetry when the vorticity is initially concentrated in small disks. We discuss how long this concentration persists, showing that in some cases this happens for quite long times. Moreover, we analyze a toy model that shows a similar behavior and gives some hints on the original proble
Time evolution of concentrated vortex rings
We study the time evolution of an incompressible fluid with axisymmetry
without swirl when the vorticity is sharply concentrated. In particular, we
consider disjoint vortex rings of size and intensity of the
order of . We show that in the limit , when the density of vorticity becomes very large, the movement of each
vortex ring converges to a simple translation, at least for a small but
positive time.Comment: 24 pages. This updated version provides a new Appendix B, containing
the corrected proof of Lemma 3.1. For the sake of clarity, this proof has
already been included in arXiv:2102.07807 (where the results of this article
have been extended
Molecular Dynamics Simulation of Vascular Network Formation
Endothelial cells are responsible for the formation of the capillary blood
vessel network. We describe a system of endothelial cells by means of
two-dimensional molecular dynamics simulations of point-like particles. Cells'
motion is governed by the gradient of the concentration of a chemical substance
that they produce (chemotaxis). The typical time of degradation of the chemical
substance introduces a characteristic length in the system. We show that
point-like model cells form network resembling structures tuned by this
characteristic length, before collapsing altogether. Successively, we improve
the non-realistic point-like model cells by introducing an isotropic strong
repulsive force between them and a velocity dependent force mimicking the
observed peculiarity of endothelial cells to preserve the direction of their
motion (persistence). This more realistic model does not show a clear network
formation. We ascribe this partial fault in reproducing the experiments to the
static geometry of our model cells that, in reality, change their shapes by
elongating toward neighboring cells.Comment: 10 pages, 3 figures, 2 of which composite with 8 pictures each.
Accepted on J.Stat.Mech. (2009). Appeared at the poster session of
StatPhys23, Genoa, Italy, July 13 (2007
Computing the structured pseudospectrum of a Toeplitz matrix and its extreme points
The computation of the structured pseudospectral abscissa and radius (with
respect to the Frobenius norm) of a Toeplitz matrix is discussed and two
algorithms based on a low rank property to construct extremal perturbations are
presented. The algorithms are inspired by those considered in [SIAM J. Matrix
Anal. Appl., 32 (2011), pp. 1166-1192] for the unstructured case, but their
extension to structured pseudospectra and analysis presents several
difficulties. Natural generalizations of the algorithms, allowing to draw
significant sections of the structured pseudospectra in proximity of extremal
points are also discussed. Since no algorithms are available in the literature
to draw such structured pseudospectra, the approach we present seems promising
to extend existing software tools (Eigtool, Seigtool) to structured
pseudospectra representation for Toeplitz matrices. We discuss local
convergence properties of the algorithms and show some applications to a few
illustrative examples.Comment: 21 pages, 11 figure
From particle systems to the BGK equation
In [Phys. Rev. 94 (1954), 511-525], P.L. Bhatnagar, E.P. Gross and M. Krook
introduced a kinetic equation (the BGK equation), effective in physical
situations where the Knudsen number is small compared to the scales where
Boltzmann's equation can be applied, but not enough for using hydrodynamic
equations. In this paper, we consider the stochastic particle system
(inhomogeneous Kac model) underlying Bird's direct simulation Monte Carlo
method (DSMC), with tuning of the scaled variables yielding kinetic and/or
hydrodynamic descriptions. Although the BGK equation cannot be obtained from
pure scaling, it does follow from a simple modification of the dynamics. This
is proposed as a mathematical interpretation of some arguments in [Phys. Rev.
94 (1954), 511-525], complementing previous results in [Arch. Ration. Mech.
Anal. 240 (2021), 785-808] and [Kinet. Relat. Models 16 (2023), 269-293].Comment: 10 page