340 research outputs found
Some qualitative properties of the solutions of the Magnetohydrodynamic equations for nonlinear bipolar fluids
In this article we study the long-time behaviour of a system of nonlinear
Partial Differential Equations (PDEs) modelling the motion of incompressible,
isothermal and conducting modified bipolar fluids in presence of magnetic
field. We mainly prove the existence of a global attractor denoted by \A for
the nonlinear semigroup associated to the aforementioned systems of nonlinear
PDEs. We also show that this nonlinear semigroup is uniformly differentiable on
\A. This fact enables us to go further and prove that the attractor \A is
of finite-dimensional and we give an explicit bounds for its Hausdorff and
fractal dimensions.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s10440-014-9964-
Nonlinear dynamical analysis for globally modified incompressible non-Newtonian fluids
We present the global modification of the Ladyzhenskaya equations, for incompressible non-Newtonian fluids. This modification is through a cut-off function that multiplies the
convective term of the equation and an additional artificial smoothing dissipation term as part of
the viscous term of the equation. The goal of this work is the comparative analysis between the
modified system and the non-modified system. Therefore, we show the existence and regularity
of weak solutions, the existence of global attractors, the estimation of the fractal dimension
of the global attractors, and finally, the relationship of the autonomous dynamics between the
modified system and the non-modified system
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
Dynamics of a non-autonomous incompressible non-Newtonian fluid with delay
We first study the well-posedness of a non-autonomous incompressible non-Newtonian fluid with delay. The existence of global solution is obtained by classical Galerkin approximation and the energy method. Actually, we also prove the uniqueness of solution as well as the continuous dependence on the initial value. Then we analyze the long time behavior of the dynamical system associated to the incompressible non-Newtonian fluid. Finally, we establish the existence of pullback attractors for the non-autonomous dynamical system associated to the problem.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de AndalucíaNational Natural Science Foundation of ChinaScience and Technology Commission of Shanghai MunicipalityShanghai Leading Academic Discipline Projec
On the singular times of fluids with nonlinear viscosity
Comisión Interministerial de Ciencia y TecnologíaPatronato de la Fundación Cámara de la Universidad de Sevill
On the approximation of turbulent fluid flows by the Navier-Stokes- equations on bounded domains
The Navier-Stokes- equations belong to the family of LES (Large Eddy
Simulation) models whose fundamental idea is to capture the influence of the
small scales on the large ones without computing all the whole range present in
the flow. The constant is a regime flow parameter that has the
dimension of the smallest scale being resolvable by the model. Hence, when
, one recovers the classical Navier-Stokes equations for a flow of
viscous, incompressible, Newtonian fluids. Furthermore, the
Navier-Stokes- equations can also be interpreted as a regularization of
the Navier-Stokes equations, where stands for the regularization
parameter.
In this paper we first present the Navier-Stokes- equations on
bounded domains with no-slip boundary conditions by means of the Leray
regularization using the Helmholtz operator. Then we study the problem of
relating the behavior of the Galerkin approximations for the
Navier-Stokes- equations to that of the solutions of the Navier-Stokes
equations on bounded domains with no-slip boundary conditions. The Galerkin
method is undertaken by using the eigenfunctions associated with the Stokes
operator. We will derive local- and global-in-time error estimates measured in
terms of the regime parameter and the eigenvalues. In particular, in
order to obtain global-in-time error estimates, we will work with the concept
of stability for solutions of the Navier-Stokes equations in terms of the
norm
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