8,662 research outputs found
Nonlinear Dynamics of Moving Curves and Surfaces: Applications to Physical Systems
The subject of moving curves (and surfaces) in three dimensional space (3-D)
is a fascinating topic not only because it represents typical nonlinear
dynamical systems in classical mechanics, but also finds important applications
in a variety of physical problems in different disciplines. Making use of the
underlying geometry, one can very often relate the associated evolution
equations to many interesting nonlinear evolution equations, including soliton
possessing nonlinear dynamical systems. Typical examples include dynamics of
filament vortices in ordinary and superfluids, spin systems, phases in
classical optics, various systems encountered in physics of soft matter, etc.
Such interrelations between geometric evolution and physical systems have
yielded considerable insight into the underlying dynamics. We present a
succinct tutorial analysis of these developments in this article, and indicate
further directions. We also point out how evolution equations for moving
surfaces are often intimately related to soliton equations in higher
dimensions.Comment: Review article, 38 pages, 7 figs. To appear in Int. Jour. of Bif. and
Chao
Regional averaging and scaling in relativistic cosmology
Averaged inhomogeneous cosmologies lie at the forefront of interest, since
cosmological parameters like the rate of expansion or the mass density are to
be considered as volume-averaged quantities and only these can be compared with
observations. For this reason the relevant parameters are intrinsically
scale-dependent and one wishes to control this dependence without restricting
the cosmological model by unphysical assumptions. In the latter respect we
contrast our way to approach the averaging problem in relativistic cosmology
with shortcomings of averaged Newtonian models. Explicitly, we investigate the
scale-dependence of Eulerian volume averages of scalar functions on Riemannian
three-manifolds. We propose a complementary view of a Lagrangian smoothing of
(tensorial) variables as opposed to their Eulerian averaging on spatial
domains. This program is realized with the help of a global Ricci deformation
flow for the metric. We explain rigorously the origin of the Ricci flow which,
on heuristic grounds, has already been suggested as a possible candidate for
smoothing the initial data set for cosmological spacetimes. The smoothing of
geometry implies a renormalization of averaged spatial variables. We discuss
the results in terms of effective cosmological parameters that would be
assigned to the smoothed cosmological spacetime.Comment: LateX, IOPstyle, 48 pages, 11 figures; matches published version in
C.Q.
Mathematical modeling of local perfusion in large distensible microvascular networks
Microvessels -blood vessels with diameter less than 200 microns- form large,
intricate networks organized into arterioles, capillaries and venules. In these
networks, the distribution of flow and pressure drop is a highly interlaced
function of single vessel resistances and mutual vessel interactions. In this
paper we propose a mathematical and computational model to study the behavior
of microcirculatory networks subjected to different conditions. The network
geometry is composed of a graph of connected straight cylinders, each one
representing a vessel. The blood flow and pressure drop across the single
vessel, further split into smaller elements, are related through a generalized
Ohm's law featuring a conductivity parameter, function of the vessel cross
section area and geometry, which undergo deformations under pressure loads. The
membrane theory is used to describe the deformation of vessel lumina, tailored
to the structure of thick-walled arterioles and thin-walled venules. In
addition, since venules can possibly experience negative transmural pressures,
a buckling model is also included to represent vessel collapse. The complete
model including arterioles, capillaries and venules represents a nonlinear
system of PDEs, which is approached numerically by finite element
discretization and linearization techniques. We use the model to simulate flow
in the microcirculation of the human eye retina, a terminal system with a
single inlet and outlet. After a phase of validation against experimental
measurements, we simulate the network response to different interstitial
pressure values. Such a study is carried out both for global and localized
variations of the interstitial pressure. In both cases, significant
redistributions of the blood flow in the network arise, highlighting the
importance of considering the single vessel behavior along with its position
and connectivity in the network
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