3,319 research outputs found
The rich behavior of the Boltzmann equation for dissipative gases
Within the framework of the homogeneous non-linear Boltzmann equation, we
present a new analytic method, without the intrinsic limitations of existing
methods, for obtaining asymptotic solutions. This method permits extension of
existing results for Maxwell molecules and hard spheres to large classes of
particle interactions, from very hard spheres to softer than Maxwell molecules,
as well as to more general forcing mechanisms, beyond free cooling and white
noise driving. By combining this method with numerical solutions, obtained from
the Direct Simulation Monte Carlo (DSMC) method, we study a broad class of
models relevant for the dynamics of dissipative fluids, including granular
gases. We establish a criterion connecting the stability of the non-equilibrium
steady state to an exponentially bound form for the velocity distribution ,
which varies depending on the forcing mechanism. Power laws arise in marginal
stability cases, of which several new cases are reported. Our results provide a
minimal framework for interpreting large classes of experiments on driven
granular gases
Propagation and Structure of Planar Streamer Fronts
Streamers often constitute the first stage of dielectric breakdown in strong
electric fields: a nonlinear ionization wave transforms a non-ionized medium
into a weakly ionized nonequilibrium plasma. New understanding of this old
phenomenon can be gained through modern concepts of (interfacial) pattern
formation. As a first step towards an effective interface description, we
determine the front width, solve the selection problem for planar fronts and
calculate their properties. Our results are in good agreement with many
features of recent three-dimensional numerical simulations.
In the present long paper, you find the physics of the model and the
interfacial approach further explained. As a first ingredient of this approach,
we here analyze planar fronts, their profile and velocity. We encounter a
selection problem, recall some knowledge about such problems and apply it to
planar streamer fronts. We make analytical predictions on the selected front
profile and velocity and confirm them numerically.
(abbreviated abstract)Comment: 23 pages, revtex, 14 ps file
BV solutions and viscosity approximations of rate-independent systems
In the nonconvex case solutions of rate-independent systems may develop jumps
as a function of time. To model such jumps, we adopt the philosophy that rate
independence should be considered as limit of systems with smaller and smaller
viscosity. For the finite-dimensional case we study the vanishing-viscosity
limit of doubly nonlinear equations given in terms of a differentiable energy
functional and a dissipation potential which is a viscous regularization of a
given rate-independent dissipation potential. The resulting definition of 'BV
solutions' involves, in a nontrivial way, both the rate-independent and the
viscous dissipation potential, which play a crucial role in the description of
the associated jump trajectories. We shall prove a general convergence result
for the time-continuous and for the time-discretized viscous approximations and
establish various properties of the limiting BV solutions. In particular, we
shall provide a careful description of the jumps and compare the new notion of
solutions with the related concepts of energetic and local solutions to
rate-independent systems
Flow-parametric regulation of shear-driven phase separation in two and three dimensions
The Cahn-Hilliard equation with an externally-prescribed chaotic shear flow
is studied in two and three dimensions. The main goal is to compare and
contrast the phase separation in two and three dimensions, using
high-resolution numerical simulation as the basis for the study. The model flow
is parametrized by its amplitudes (thereby admitting the possibility of
anisotropy), lengthscales, and multiple time scales, and the outcome of the
phase separation is investigated as a function of these parameters as well as
the dimensionality. In this way, a parameter regime is identified wherein the
phase separation and the associated coarsening phenomenon are not only arrested
but in fact the concentration variance decays, thereby opening up the
possibility of describing the dynamics of the concentration field using the
theories of advection diffusion. This parameter regime corresponds to long flow
correlation times, large flow amplitudes and small diffusivities. The onset of
this hyperdiffusive regime is interpreted by introducing Batchelor
lengthscales. A key result is that in the hyperdiffusive regime, the
distribution of concentration (in particular, the frequency of extreme values
of concentration) depends strongly on the dimensionality. Anisotropic scenarios
are also investigated: for scenarios wherein the variance saturates
(corresponding to coarsening arrest), the direction in which the domains align
depends on the flow correlation time. Thus, for correlation times comparable to
the inverse of the mean shear rate, the domains align in the direction of
maximum flow amplitude, while for short correlation times, the domains
initially align in the opposite direction. However, at very late times (after
the passage of thousands of correlation times), the fate of the domains is the
same regardless of correlation time, namely alignment in the direction of
maximum flow amplitude.Comment: 27 pages, 14 figure
A streamline derivative POD-ROM for advection-diffusion-reaction equations
We introduce a new streamline derivative projection-based closure modeling strategy for the numerical stabilization of Proper Orthogonal Decomposition-Reduced Order Models (PODROM). As a first preliminary step, the proposed model is analyzed and tested for advection-dominated advection-diffusion-reaction equations. In this framework, the numerical analysis for the Finite Element (FE) discretization of the proposed new POD-ROM is presented, by mainly deriving the corresponding error estimates. Numerical tests for advection-dominated regime show the efficiency of the proposed method, as well the increased accuracy over the standard POD-ROM that discovers its well-known limitations very soon in the numerical settings considered, i.e. for low diffusion coefficients.Nous introduisons une nouvelle stratégie de modélisation de type streamline derivative basée sur projection pour la stabilisation numérique de modèles d’ordre réduit de type POD (PODROM). Comme première étape préliminaire, le modèle proposé est analysé et testé pour les équations d’advection-diffusion-réaction dominées par l’advection. Dans ce cadre, l’analyse numérique de la discrétisation par éléments finis (FE) du nouveau POD-ROM proposé est présentée, en dérivant principalement les estimations d’erreur correspondantes. Des tests numériques pour le régime dominé par l’advection montrent l’efficacité de la méthode proposée, ainsi que la précision accrue par rapport à la méthode POD-ROM standard qui d´ecouvre très rapidement ses limites bien connues dans le cas des paramètres numériques considérés, c’est-à -dire pour de faibles coefficients de diffusion
Extension of Haff's cooling law in granular flows
The total energy E(t) in a fluid of inelastic particles is dissipated through
inelastic collisions. When such systems are prepared in a homogeneous initial
state and evolve undriven, E(t) decays initially as t^{-2} \aprox exp[ -
2\epsilon \tau] (known as Haff's law), where \tau is the average number of
collisions suffered by a particle within time t, and \epsilon=1-\alpha^2
measures the degree of inelasticity, with \alpha the coefficient of normal
restitution. This decay law is extended for large times to E(t) \aprox
\tau^{-d/2} in d-dimensions, far into the nonlinear clustering regime. The
theoretical predictions are quantitatively confirmed by computer simulations,
and holds for small to moderate inelasticities with 0.6< \alpha< 1.Comment: 7 pages, 4 PostScript figures. To be published in Europhysics Letter
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