Abstract
The dynamics of a system of hard spheres enclosed between two parallel plates separated a distance smaller than two particle diameters is described at the level of kinetic theory. The interest focuses on the behavior of the quasi-two-dimensional fluid seen when looking at the system from above or below. In the first part, a collisional model for the effective two-dimensional dynamics is analyzed. Although it is able to describe quite well the homogeneous evolution observed in the experiments, it is shown that it fails to predict the existence of non-equilibrium phase transitions, and in particular, the bimodal regime exhibited by the real system. A critical revision analysis of the model is presented , and as a starting point to get a more accurate description, the Boltzmann equation for the quasi-two-dimensional gas has been derived. In the elastic case, the solutions of the equation verify an H-theorem implying a monotonic tendency to a non-uniform steady state. As an example of application of the kinetic equation, here the evolution equations for the vertical and horizontal temperatures of the system are derived in the homogeneous approximation, and the results compared with molecular dynamics simulation results.
Highlights
In the last years, a particular geometry has attracted interest in the study of confined systems.It is a quasi-two-dimensional system of spherical particles placed between two large parallel plates separated by a distance h smaller than two particle diameters
As an example of application of the kinetic equation, here the evolution equations for the vertical and horizontal temperatures of the system are derived in the homogeneous approximation, and the results compared with molecular dynamics simulation results
A particular geometry has attracted interest in the study of confined systems
Summary
A particular geometry has attracted interest in the study of confined systems. Due to the inelasticity of collisions between macroscopic particles, a permanent injection of energy is needed to reach a steady state This is done by vibrating the two parallel walls confining the system [1,2,3,4]. An interesting phenomenological macroscopic approach has been presented in [7], focussed on explaining the existence of a bimodal regime characterized by a single dense cluster surrounded by a gas of quite agitated particles that is present in some experiments. The description incorporates both a temperature parameter in the horizontal plane and another temperature associated with the vertical motion. It shows the way to proceed from the kinetic equation in order to get evolution equations describing the macroscopic dynamics of the system
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