A Kinetic equation for granular media
A kinetic equation for granular media
In this short note we correct a conceptual error in the heuristic derivation of a kinetic equation used for the description of a one-dimensional granular medium in the so called quasi-elastic limit, presented by the same authors in reference[1]. The equation we derived is however correct so that, the rigorous analysis on this equation, which constituted the main purpose of that paper, remains unchanged.
A Langevin Description for Driven Granular Gases
The study of the fluctuations in the steady state of a heated granular system is reviewed. A Boltzmann-Langevin description can be built requiring consistency with the equations for the one- and two-particle correlation functions. From the Boltzmann-Langevin equation, Langevin equations for the total energy and the transverse velocity field are derived. The existence of a fluctuation-dissipation relation for the transverse velocity field is also...
A Roe-type scheme for two-phase shallow granular flows over variable topography
We study a depth-averaged model of gravity-driven flows made of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. The model system consists of mass and momentum balance equations for the solid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the...
Dense Granular Poiseuille Flow
We consider a dense granular shear flow in a two-dimensional system. Granular systems (composed of a large number of macroscopic particles) are far from equilibrium due to inelastic collisions between particles: an external driving is needed to maintain the motion of particles. Theoretical description of driven granular media is especially challenging for dense granular flows. This paper focuses on a gravity-driven dense granular Poiseuille flow...
Dynamics of granular fluids
Elementary preamble to a theory of granular gases
Homogeneous Cooling with Repulsive and Attractive Long-Range Potentials
The interplay between dissipation and long-range repulsive/attractive forces in homogeneous, dilute, mono-disperse particle systems is studied. The pseudo-Liouville operator formalism, originally introduced for hard-sphere interactions, is modified such that it provides very good predictions for systems with weak long-range forces at low densities, with the ratio of potential to fluctuation kinetic energy as control parameter. By numerical simulations, ...
Hydrodynamics of Inelastic Maxwell Models
An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model allows us to get exact results for different problems. First, the Navier–Stokes constitutive equations with explicit expressions for the corresponding transport coefficients are derived by applying the Chapman–Enskog...
Hyperbolic relaxation models for granular flows
In this work we describe an efficient model for the simulation of a two-phase flow made of a gas and a granular solid. The starting point is the two-velocity two-pressure model of Baer and Nunziato [Int. J. Multiph. Flow16 (1986) 861–889]. The model is supplemented by a relaxation source term in order to take into account the pressure equilibrium between the two phases and the granular stress in the solid phase. We show that the relaxation process can be made thermodynamically coherent with an...
Improved continuous models for discrete media.
Kinetic and hydrodynamic equations for granular media
In this lecture i present some open mathematical problems concerning some PDE arising in the study of one-dimensional models for granular media.
On the one-dimensional Boltzmann equation for granular flows
We consider a Boltzmann equation for inelastic particles on the line and prove existence and uniqueness for the solutions.
On the one-dimensional Boltzmann equation for granular flows
We consider a Boltzmann equation for inelastic particles on the line and prove existence and uniqueness for the solutions.
One-dimensional kinetic models of granular flows
One-dimensional kinetic models of granular flows
We introduce and discuss a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. Then, the behavior of the Boltzmann equation in the quasi elastic limit is investigated for a wide range of the rate function. By this limit procedure we obtain a class of nonlinear equations classified as nonlinear friction equations. The analysis of the cooling process shows that the nonlinearity on the relative velocity is of paramount importance...
Some recent results on the Muskat problem
We consider the dynamics of an interface given by two incompressible fluids with different characteristics evolving by Darcy’s law. This scenario is known as the Muskat problem, being in 2D mathematically analogous to the two-phase Hele-Shaw cell. The purpose of this paper is to outline recent results on local existence, weak solutions, maximum principles and global existence.
Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit
In this paper we introduce numerical schemes for a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. In particular, we study the numerical passage of the Boltzmann equation with singular kernel to nonlinear friction equations in the so-called quasi elastic limit. To this aim we introduce a Fourier spectral method for the Boltzmann equation [25, 26] and show that the kernel modes that define the spectral method have the correct...
Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit
In this paper we introduce numerical schemes for a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. In particular, we study the numerical passage of the Boltzmann equation with singular kernel to nonlinear friction equations in the so-called quasi elastic limit. To this aim we introduce a Fourier spectral method for the Boltzmann equation [CITE] and show that the kernel modes that define the spectral method have the correct...
Vertical flow of a multiphase mixture in a channel.