28,385 research outputs found
Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions
The second-grade fluid equations are a model for viscoelastic fluids, with
two parameters: , corresponding to the elastic response, and , corresponding to viscosity. Formally setting these parameters to
reduces the equations to the incompressible Euler equations of ideal fluid
flow. In this article we study the limits of solutions of
the second-grade fluid system, in a smooth, bounded, two-dimensional domain
with no-slip boundary conditions. This class of problems interpolates between
the Euler- model (), for which the authors recently proved
convergence to the solution of the incompressible Euler equations, and the
Navier-Stokes case (), for which the vanishing viscosity limit is
an important open problem. We prove three results. First, we establish
convergence of the solutions of the second-grade model to those of the Euler
equations provided , as , extending
the main result in [19]. Second, we prove equivalence between convergence (of
the second-grade fluid equations to the Euler equations) and vanishing of the
energy dissipation in a suitably thin region near the boundary, in the
asymptotic regime ,
as . This amounts to a convergence criterion similar to the
well-known Kato criterion for the vanishing viscosity limit of the
Navier-Stokes equations to the Euler equations. Finally, we obtain an extension
of Kato's classical criterion to the second-grade fluid model, valid if , as . The proof of all these results
relies on energy estimates and boundary correctors, following the original idea
by Kato.Comment: 20pages,1figur
Convergence of the 2D Euler- to Euler equations in the Dirichlet case: indifference to boundary layers
In this article we consider the Euler- system as a regularization of
the incompressible Euler equations in a smooth, two-dimensional, bounded
domain. For the limiting Euler system we consider the usual non-penetration
boundary condition, while, for the Euler- regularization, we use
velocity vanishing at the boundary. We also assume that the initial velocities
for the Euler- system approximate, in a suitable sense, as the
regularization parameter , the initial velocity for the limiting
Euler system. For small values of , this situation leads to a boundary
layer, which is the main concern of this work. Our main result is that, under
appropriate regularity assumptions, and despite the presence of this boundary
layer, the solutions of the Euler- system converge, as ,
to the corresponding solution of the Euler equations, in in space,
uniformly in time. We also present an example involving parallel flows, in
order to illustrate the indifference to the boundary layer of the limit, which underlies our work.Comment: 22page
Developing a site-conditions map for seismic hazard Assessment in Portugal
The evaluation of site effects on a broad scale is a critical issue for seismic hazard and risk assessment, land use planning and emergency planning. As characterization of site conditions based on the shear-wave velocity has become increasingly important, several methods have been proposed in the literature to estimate its average over the first thirty meters (Vs30) from more extensively available data. These methods include correlations with geologic-geographic defined units and topographic slope. In this paper we present the first steps towards the development of a site–conditions map for Portugal, based on a regional database of shear-wave velocity data, together with geological, geographic, and lithological information. We computed Vs30 for each database site and classified it according to the corresponding geological-lithological information using maps at the smallest scale available (usually 1:50000). We evaluated the consistency of Vs30 values within generalized-geological classes, and assessed the performance of expedient methodologies proposed in the literature
Vanishing Viscosity Limits and Boundary Layers for Circularly Symmetric 2D Flows
We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [LMN],
on the vanishing viscosity limit of circularly symmetric viscous flow in a disk
with rotating boundary, shown there to converge to the inviscid limit in
-norm as long as the prescribed angular velocity of the
boundary has bounded total variation. Here we establish convergence in stronger
and -Sobolev spaces, allow for more singular angular velocities
, and address the issue of analyzing the behavior of the boundary
layer. This includes an analysis of concentration of vorticity in the vanishing
viscosity limit. We also consider such flows on an annulus, whose two boundary
components rotate independently.
[LMN] Lopes Filho, M. C., Mazzucato, A. L. and Nussenzveig Lopes, H. J.,
Vanishing viscosity limit for incompressible flow inside a rotating circle,
preprint 2006
Compact stars with a small electric charge: the limiting radius to mass relation and the maximum mass for incompressible matter
One of the stiffest equations of state for matter in a compact star is
constant energy density and this generates the interior Schwarzschild radius to
mass relation and the Misner maximum mass for relativistic compact stars. If
dark matter populates the interior of stars, and this matter is supersymmetric
or of some other type, some of it possessing a tiny electric charge, there is
the possibility that highly compact stars can trap a small but non-negligible
electric charge. In this case the radius to mass relation for such compact
stars should get modifications. We use an analytical scheme to investigate the
limiting radius to mass relation and the maximum mass of relativistic stars
made of an incompressible fluid with a small electric charge. The investigation
is carried out by using the hydrostatic equilibrium equation, i.e., the
Tolman-Oppenheimer-Volkoff (TOV) equation, together with the other equations of
structure, with the further hypothesis that the charge distribution is
proportional to the energy density. The approach relies on Volkoff and Misner's
method to solve the TOV equation. For zero charge one gets the interior
Schwarzschild limit, and supposing incompressible boson or fermion matter with
constituents with masses of the order of the neutron mass one gets that the
maximum mass is the Misner mass. For a small electric charge, our analytical
approximating scheme valid in first order in the star's electric charge, shows
that the maximum mass increases relatively to the uncharged case, whereas the
minimum possible radius decreases, an expected effect since the new field is
repulsive aiding the pressure to sustain the star against gravitational
collapse.Comment: 23 pages, no figure
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