484 research outputs found
Viscous evolution of point vortex equilibria: The collinear state
When point vortex equilibria of the 2D Euler equations are used as initial
conditions for the corre- sponding Navier-Stokes equations (viscous), typically
an interesting dynamical process unfolds at short and intermediate time scales,
before the long time single peaked, self-similar Oseen vortex state dom-
inates. In this paper, we describe the viscous evolution of a collinear three
vortex structure that cor- responds to an inviscid point vortex fixed
equilibrium. Using a multi-Gaussian 'core-growth' type of model, we show that
the system immediately begins to rotate unsteadily, a mechanism we attribute to
a 'viscously induced' instability. We then examine in detail the qualitative
and quantitative evolution of the system as it evolves toward the long-time
asymptotic Lamb-Oseen state, showing the sequence of topological bifurcations
that occur both in a fixed reference frame, and in an appropriately chosen
rotating reference frame. The evolution of passive particles in this viscously
evolving flow is shown and interpreted in relation to these evolving streamline
patterns.Comment: 17 pages, 15 figure
A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears
Variational principle for Kolmogorov-Petrovsky-Piskunov (KPP) minimal front
speeds provides an efficient tool for statistical speed analysis, as well as a
fast and accurate method for speed computation. A variational principle based
analysis is carried out on the ensemble of KPP speeds through spatially
stationary random shear flows inside infinite channel domains. In the regime of
small root mean square (rms) shear amplitude, the enhancement of the ensemble
averaged KPP front speeds is proved to obey the quadratic law under certain
shear moment conditions. Similarly, in the large rms amplitude regime, the
enhancement follows the linear law. In particular, both laws hold for the
Ornstein-Uhlenbeck process in case of two dimensional channels. An asymptotic
ensemble averaged speed formula is derived in the small rms regime and is
explicit in case of the Ornstein-Uhlenbeck process of the shear. Variational
principle based computation agrees with these analytical findings, and allows
further study on the speed enhancement distributions as well as the dependence
of enhancement on the shear covariance. Direct simulations in the small rms
regime suggest quadratic speed enhancement law for non-KPP nonlinearities.Comment: 28 pages, 14 figures update: fixed typos, refined estimates in
section
Chemotactic Collapse and Mesenchymal Morphogenesis
We study the effect of chemotactic signaling among mesenchymal cells. We show
that the particular physiology of the mesenchymal cells allows one-dimensional
collapse in contrast to the case of bacteria, and that the mesenchymal
morphogenesis represents thus a more complex type of pattern formation than
those found in bacterial colonies. We finally compare our theoretical
predictions with recent in vitro experiments
Apoptotic changes in the myocardium in the course of experimentally-induced pleurisy
The secreted proinflammatory interleukins IL-1, IL-6 and TNF in the course of
experimentally-induced pleurisy can be the cause of pathological changes in the
ultrastructure of cardiac muscle and of apoptosis. The pleurisy was induced in rats
by means of carrageenin. The scraps of cardiac muscle obtained during the inflammatory
reaction in the pleura were analysed by means of an electron microscope.
The scraps were also stained with the TUNEL method in order to find the
apoptotic foci. It was proved by the experiment that the inflammatory process
affected mitochondria in the cardiomyocytes, enhanced collagen fibre synthesis
and contributed to the formation of apoptotic foci in the cardiac muscle
Equation-free implementation of statistical moment closures
We present a general numerical scheme for the practical implementation of
statistical moment closures suitable for modeling complex, large-scale,
nonlinear systems. Building on recently developed equation-free methods, this
approach numerically integrates the closure dynamics, the equations of which
may not even be available in closed form. Although closure dynamics introduce
statistical assumptions of unknown validity, they can have significant
computational advantages as they typically have fewer degrees of freedom and
may be much less stiff than the original detailed model. The closure method can
in principle be applied to a wide class of nonlinear problems, including
strongly-coupled systems (either deterministic or stochastic) for which there
may be no scale separation. We demonstrate the equation-free approach for
implementing entropy-based Eyink-Levermore closures on a nonlinear stochastic
partial differential equation.Comment: 7 pages, 2 figure
Coercivity and stability results for an extended Navier-Stokes system
In this article we study a system of equations that is known to {\em extend}
Navier-Stokes dynamics in a well-posed manner to velocity fields that are not
necessarily divergence-free. Our aim is to contribute to an understanding of
the role of divergence and pressure in developing energy estimates capable of
controlling the nonlinear terms. We address questions of global existence and
stability in bounded domains with no-slip boundary conditions. Even in two
space dimensions, global existence is open in general, and remains so,
primarily due to the lack of a self-contained energy estimate. However,
through use of new coercivity estimates for the linear equations, we
establish a number of global existence and stability results, including results
for small divergence and a time-discrete scheme. We also prove global existence
in 2D for any initial data, provided sufficient divergence damping is included.Comment: 29 pages, no figure
Vanishing viscosity limit for an expanding domain in space
We study the limiting behavior of viscous incompressible flows when the fluid
domain is allowed to expand as the viscosity vanishes. We describe precise
conditions under which the limiting flow satisfies the full space Euler
equations. The argument is based on truncation and on energy estimates,
following the structure of the proof of Kato's criterion for the vanishing
viscosity limit. This work complements previous work by the authors, see
[Kelliher, Comm. Math. Phys. 278 (2008), 753-773] and [arXiv:0801.4935v1].Comment: 23 pages, submitted for publicatio
The analytic structure of 2D Euler flow at short times
Using a very high precision spectral calculation applied to the
incompressible and inviscid flow with initial condition , we find that the width of its analyticity
strip follows a law at short times over eight decades. The
asymptotic equation governing the structure of spatial complex-space
singularities at short times (Frisch, Matsumoto and Bec 2003, J.Stat.Phys. 113,
761--781) is solved by a high-precision expansion method. Strong numerical
evidence is obtained that singularities have infinite vorticity and lie on a
complex manifold which is constructed explicitly as an envelope of analyticity
disks.Comment: 19 pages, 14 figures, published versio
Absence of squirt singularities for the multi-phase Muskat problem
In this paper we study the evolution of multiple fluids with different
constant densities in porous media. This physical scenario is known as the
Muskat and the (multi-phase) Hele-Shaw problems. In this context we prove that
the fluids do not develop squirt singularities.Comment: 16 page
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