A kinetic equation for a system of elastic hard spheres or disks confined by a hard wall of arbitrary shape is derived. It is a generalization of the modified Enskog equation in which the effects of the confinement are taken into account and it is supposed to be valid up to moderate densities. From the equation, balance equations for the hydrodynamic fields are derived, identifying the collisional transfer contributions to the pressure tensor and heat flux. A Lyapunov functional, $\mathcal{H}[f]$, is identified. For any solution of the kinetic equation, $\mathcal{H}$ decays monotonically in time until the system reaches the inhomogeneous equilibrium distribution, that is a Maxwellian distribution with a the density field consistent with equilibrium statistical mechanics.
N.N. Bogolyubov discovered that the Boltzmann-Enskog kinetic equation has microscopic solutions. They have the form of sums of delta-functions and correspond to trajectories of individual hard spheres. But the rigorous sense of the product of the delta-functions in the collision integral was not discussed. Here we give a rigorous sense to these solutions by introduction of a special regularization of the delta-functions. The crucial observation is that the collision integral of the Boltzmann-Enskog equation coincides with that of the first equation of the BBGKY hierarchy for hard spheres if th…
The hydrodynamic equation governing the homogeneous time evolution of the temperature in a model of confined granular gas is studied by means of the Enskog equation. The existence of a normal solution of the kinetic equation is assumed as a condition for hydrodynamics. Dimensional analysis implies a scaling of the distribution function that is used to determine it in the first Sonine approximation, with a coefficient that evolves in time through its dependence on the temperature. The theoretical predictions are compared with numerical results obtained by the direct simulation Monte Carlo metho…
In this paper we present the molecular theory of viscosity of confined fluids in small or nano systems. This theory is also applicable to the interfacial viscosity. The basis of this research work is the Enskog kinetic theory and the Boussinesq constitutive equation. The Enskog kinetic theory is first transformed into a two-dimensional form. Then the potential energy collisional transfer part of the flux vector and the contribution to the surface pressure tensor due to collisional transfer are derived. Then the kinetic energy part of the flux vector and consequently the contribution to the sur…
The fundamental question addressed in this paper is whether the partial Chapman--Enskog expansion $P_{xy}=-\sum_{k=0}^\infty η_k ({\partial u_x}/{\partial y})^{2k+1}$ of the shear stress converges or not for a gas of inelastic hard spheres. By using a simple kinetic model it is shown that, in contrast to the elastic case, the above series does converge, the radius of convergence increasing with inelasticity. It is argued that this paradoxical conclusion is not an artifact of the kinetic model and can be understood in terms of the time evolution of the scaled shear rate in the uniform shear flo…
A collisional model of a confined quasi-two-dimensional granular mixture is considered to analyze homogeneous steady states. The model includes an effective mechanism to transfer the kinetic energy injected by vibration in the vertical direction to the horizontal degrees of freedom of grains. The set of Enskog kinetic equations for the velocity distribution functions of each component is derived first to analyze the homogeneous state. As in the one-component case, an exact scaling solution is found where the time dependence of the distribution functions occurs entirely through the granular tem…
The non-equilibrium statistical mechanics and kinetic theory for a model of a confined quasi-two-dimensional gas of inelastic hard spheres is presented. The dynamics of the particles includes an effective mechanism to transfer the energy injected in the vertical direction to the horizontal degrees of freedom. The Enskog approximation is formulated and used as the basis to investigate the temperature and the distribution function of the steady state eventually reached by the system. An exact scaling of the distribution function of the system having implications on the form of its moments is poi…
A linear stability analysis of the hydrodynamic equations of a model for confined quasi-two-dimensional granular gases is carried out. The stability analysis is performed around the homogeneous steady state (HSS) reached eventually by the system after a transient regime. In contrast to previous studies (which considered dilute or quasielastic systems), our analysis is based on the results obtained from the inelastic Enskog kinetic equation, which takes into account the (nonlinear) dependence of the transport coefficients and the cooling rate on dissipation and applies to moderate densities. As…
The hydrodynamic equations for a model of a confined quasi-two-dimensional gas of smooth inelastic hard spheres are derived from the Boltzmann equation for the model, using a generalization of the Chapman-Enskog method. The heat and momentum fluxes are calculated to Navier-Stokes order, and the associated transport coefficients are explicitly determined as functions of the coefficient of normal restitution and the velocity parameter involved in the definition of the model. Also an Euler transport term contributing to the energy transport equation is considered. This term arises from the gradie…
Transport properties in a model of confined granular mixtures at moderate densities
Universidad de Extremadura（埃斯特雷马杜拉大学）；Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura（埃斯特雷马杜拉大学物理系与先进计算科学研究所）
The homogeneous steady state of a fluid of inelastic hard spheres immersed in a bath of elastic hard spheres kept at equilibrium is analyzed by means of the first Sonine approximation to the (spatially homogeneous) Enskog--Boltzmann equation. The temperature of the granular fluid relative to the bath temperature and the kurtosis of the granular distribution function are obtained as functions of the coefficient of restitution, the mass ratio, and a dimensionless parameter $β$ measuring the cooling rate relative to the friction constant. Comparison with recent results obtained from an iterative …
The shear viscosity for a moderately dense granular binary mixture of smooth hard spheres undergoing uniform shear flow is determined. The basis for the analysis is the Enskog kinetic equation, solved first analytically by the Chapman-Enskog method up to first order in the shear rate for unforced systems as well as for systems driven by a Gaussian thermostat. As in the elastic case, practical evaluation requires a Sonine polynomial approximation. In the leading order, we determine the shear viscosity in terms of the control parameters of the problem: solid fraction, composition, mass ratio, si…
The Chapman--Enskog solution to the Enskog kinetic equation of polydisperse granular mixtures is revisited to determine the first-order contributions $\varpi_i$ to the partial temperatures. As expected, these quantities (which were neglected in previous attempts) are given in terms of the solution to a set of coupled integro-differential equations analogous to those for elastic collisions. The solubility condition for this set of equations is confirmed and the coefficients $\varpi_i$ are calculated by using the leading terms in a Sonine polynomial expansion. These coefficients are given as exp…
The collective dynamics of liquid Gallium close to the melting point has been studied using Inelastic X-ray Scattering to probe lengthscales smaller than the size of the first coordination shell. %(momentum transfers, $Q$, $>$15 nm$^{-1}$). Although the structural properties of this partially covalent liquid strongly deviate from a simple hard-sphere model, the dynamics, as reflected in the quasi-elastic scattering, are beautifully described within the framework of the extended heat mode approximation of Enskog's kinetic theory, analytically derived for a hard spheres system. The present work …
We consider a dilute gas of hard spheres in dimension $d \geq 2$ that upon collision either annihilate with probability $p$ or undergo an elastic scattering with probability $1-p$. For such a system neither mass, momentum, nor kinetic energy are conserved quantities. We establish the hydrodynamic equations from the Boltzmann equation description. Within the Chapman-Enskog scheme, we determine the transport coefficients up to Navier-Stokes order, and give the closed set of equations for the hydrodynamic fields chosen for the above coarse grained description (density, momentum and kinetic temper…
A quasi-two-dimensional system of hard spheres strongly confined between two parallel plates is considered. The attention is focussed on the macroscopic self-diffusion process observed when the system is looked from above or from below. The transport equation, and the associated self-diffusion coefficient, are derived from a Boltzmann-Lorentz kinetic equation, valid in the dilute limit. Since the equilibrium state of the system is inhomogeneous, this requires the use of a modified Chapman-Enskog expansion that distinguishes between equilibrium and non-equilibrium gradients of the density of la…
An impurity particle coupling to its host fluid via inelastic hard sphere collisions is considered. It is shown that the exact equation for its distribution function can be mapped onto that for an impurity with elastic collisions and an effective mass. Application of this result to the Enskog--Lorentz kinetic equation leads to several conclusions: 1) every solution in the elastic case is equivalent to a class of solutions in the granular case; 2) for an equilibrium host fluid the granular impurity approaches equilibrium at a different temperature, with a dominant diffusive mode at long times; …
Granular gaseous mixtures under rapid flow conditions are usually modeled by a multicomponent system of smooth inelastic hard spheres with constant coefficients of normal restitution. In the low density regime an adequate framework is provided by the set of coupled inelastic Boltzmann equations. Due to the intricacy of the inelastic Boltzmann collision operator, in this paper we propose a simpler model of elastic hard spheres subject to the action of an effective drag force, which mimics the effect of dissipation present in the original granular gas. The Navier--Stokes transport coefficients f…
The local balance equations for the density, momentum, and energy of a dilute gas of elastic or inelastic hard spheres, strongly confined between two parallel hard plates are obtained. The starting point is a Boltzmann-like kinetic equation, recently derived for this system. As a consequence of the confinement, the pressure tensor and the heat flux contain, in addition to the terms associated to the motion of the particles, collisional transfer contributions, similar to those that appear beyond the dilute limit. The complexity of these terms, and of the kinetic equation itself, compromise the …
We present numerical simulations of acoustic wave propagation in confined granular systems consisting of particles interacting with the three-dimensional Hertz-Mindlin force law. The response to a short mechanical excitation on one side of the system is found to be a propagating coherent wavefront followed by random oscillations made of multiply scattered waves. We find that the coherent wavefront is insensitive to details of the packing: force chains do not play an important role in determining this wavefront. The coherent wave propagates linearly in time, and its amplitude and width depend a…
The Navier--Stokes transport coefficients for a model of a confined quasi-two-dimensional granular binary mixture of inelastic hard spheres are determined from the Boltzmann kinetic equation. A normal or hydrodynamic solution to the Boltzmann equation is obtained via the Chapman--Enskog method for states near the local version of the homogeneous time-dependent state. The mass, momentum, and heat fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. As expected, they are given in terms of the solutions…
Applications of the kinetic theory for a model of a confined quasi-two dimensional granular mixture: Stability analysis and thermal diffusion segregation
A study of the transport coefficients of a system of elastic hard disks, based on the use of Helfand-Einstein expressions is reported. The self-diffusion, the viscosity, and the heat conductivity are examined with averaging techniques especially appropriate for the use in event-driven molecular dynamics algorithms with periodic boundary conditions. The density and size dependence of the results is analyzed, and comparison with the predictions from Enskog's theory is carried out. In particular, the behavior of the transport coefficients in the vicinity of the fluid-solid transition is investiga…
The behaviour of elastic structures undergoing large deformations is the result of the competition between confining conditions, self-avoidance and elasticity. This combination of multiple phenomena creates a geometrical frustration that leads to complex fold patterns. By studying the case of a rod confined isotropically into a disk, we show that the emergence of the complexity is associated with a well defined underlying statistical measure that determines the energy distribution of sub-elements,``branches'', of the rod. This result suggests that branches act as the ``microscopic'' degrees of…
The flow characterized by a linear longitudinal velocity field $u_x(x,t)=a(t)x$, where $a(t)={a_0}/({1+a_0t})$, a uniform density $n(t)\propto a(t)$, and a uniform temperature $T(t)$ is analyzed for dilute granular gases by means of a BGK-like model kinetic equation in $d$ dimensions. For a given value of the coefficient of normal restitution $α$, the relevant control parameter of the problem is the reduced deformation rate $a^*(t)=a(t)/ν(t)$ (which plays the role of the Knudsen number), where $ν(t)\propto n(t)\sqrt{T(t)}$ is an effective collision frequency. The relevant response parameter is…
We study a one-dimensional fluid of hard-rods interacting each other via binary inelastic collisions and a short ranged square-well potential. Upon tuning the depth and the sign of the well, we investigate the interplay between dissipation and cohesive or repulsive forces. Molecular dynamics simulations of the cooling regime indicate that the presence of this simple interparticle interaction is sufficient to significantly modify the energy dissipation rates expected by the Haff's law for the free cooling. The simplicity of the model makes it amenable to an analytical approach based on the Bolt…
We study the transport properties of a large class of locally confined Hamiltonian systems, in which neighboring particles interact through hard core elastic collisions. When these collisions become rare and the systems large, we derive a Boltzmann-like equation for the evolution of the probability densities. We solve this equation in the linear regime and compute the heat conductivity from a Green-Kubo formula. The validity of our approach is demonstated by comparing our predictions to the results of numerical simulations performed on a new class of high-dimensional defocusing chaotic billiar…
