scholarly journals The Random Gas of Hard Spheres

J ◽  
2019 ◽  
Vol 2 (2) ◽  
pp. 162-205 ◽  
Author(s):  
Rafail Abramov
Keyword(s):  
Boltzmann Equation ◽  
Stokes Equations ◽  
Hard Spheres ◽  
Mass Density ◽  
Navier Stokes ◽  
Enskog Equation ◽  
Hard Sphere Model ◽  
Collision Dynamics ◽  
Rigid Spheres ◽  
Navier Stokes Equations

The inconsistency between the time-reversible Liouville equation and time-irreversible Boltzmann equation has been pointed out by Loschmidt. To avoid Loschmidt’s objection, here we propose a new dynamical system to model the motion of atoms of gas, with their interactions triggered by a random point process. Despite being random, this model can approximate the collision dynamics of rigid spheres via adjustable parameters. We compute the exact statistical steady state of the system, and determine the form of its marginal distributions for a large number of spheres. We find that the Kullback–Leibler entropy (a generalization of the conventional Boltzmann entropy) of the full system of random gas spheres is a non-increasing function of time. Unlike the conventional hard sphere model, the proposed random gas system results in a variant of the Enskog equation, which is known to be a more accurate model of dense gas than the Boltzmann equation. We examine the hydrodynamic limit of the derived Enskog equation for spheres of constant mass density, and find that the corresponding Enskog–Euler and Enskog–Navier–Stokes equations acquire additional effects in both the advective and viscous terms.

scholarly journals From Boltzmann to incompressible Navier–Stokes in Sobolev spaces with polynomial weight

2018 ◽  
Vol 17 (01) ◽  
pp. 85-116 ◽  
Author(s):  
Marc Briant ◽  
Sara Merino-Aceituno ◽  
Clément Mouhot
Keyword(s):  
Boltzmann Equation ◽  
Knudsen Number ◽  
Exponential Decay ◽  
Sobolev Spaces ◽  
Stokes Equations ◽  
Linear Part ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Cauchy Problem ◽  
Global Equilibrium

We study the Boltzmann equation on the [Formula: see text]-dimensional torus in a perturbative setting around a global equilibrium under the Navier–Stokes linearization. We use a recent functional analysis breakthrough to prove that the linear part of the equation generates a [Formula: see text]-semigroup with exponential decay in Lebesgue and Sobolev spaces with polynomial weight, independently of the Knudsen number. Finally, we prove well-posedness of the Cauchy problem for the nonlinear Boltzmann equation in perturbative setting and an exponential decay for the perturbed Boltzmann equation, uniformly in the Knudsen number, in Sobolev spaces with polynomial weight. The polynomial weight is almost optimal. Furthermore, this result only requires derivatives in the space variable and allows to connect solutions to the incompressible Navier–Stokes equations in these spaces.

scholarly journals Fluctuating Navier-Stokes equations for inelastic hard spheres or disks

2011 ◽  
Vol 83 (4) ◽  
Author(s):  
J. Javier Brey ◽  
P. Maynar ◽  
M. I. García de Soria
Keyword(s):  
Stokes Equations ◽  
Hard Spheres ◽  
Navier Stokes ◽  
Navier Stokes Equations

The Burnett equations in cylindrical coordinates and their solution for flow in a microtube

2014 ◽  
Vol 751 ◽  
pp. 121-141 ◽  
Author(s):  
Narendra Singh ◽  
Amit Agrawal
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Hard Spheres ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Higher Order ◽  
Navier Stokes ◽  
Burnett Equations ◽  
Cylindrical Coordinates ◽  
Navier Stokes Equations

AbstractThe Burnett equations constitute a set of higher-order continuum equations. These equations are obtained from the Chapman–Enskog series solution of the Boltzmann equation while retaining second-order-accurate terms in the Knudsen number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Kn}$. The set of higher-order continuum models is expected to be applicable to flows in the slip and transition regimes where the Navier–Stokes equations perform poorly. However, obtaining analytical or numerical solutions of these equations has been noted to be particularly difficult. In the first part of this work, we present the full set of Burnett equations in cylindrical coordinates in three-dimensional form. The equations are reported in a generalized way for gas molecules that are assumed to be Maxwellian molecules or hard spheres. In the second part, a closed-form solution of these equations for isothermal Poiseuille flow in a microtube is derived. The solution of the equations is shown to satisfy the full Burnett equations up to $\mathit{Kn} \leq 1.3$ within an error norm of ${\pm }1.0\, \%$. The mass flow rate obtained analytically is shown to compare well with available experimental and numerical results. Comparison of the stress terms in the Burnett and Navier–Stokes equations is presented. The significance of the Burnett normal stress and its role in diffusion of momentum is brought out by the analysis. An order-of-magnitude analysis of various terms in the equations is presented, based on which a reduced model of the Burnett equations is provided for flow in a microtube. The Burnett equations in full three-dimensional form in cylindrical coordinates and their solution are not previously available.

scholarly journals Hydrodynamics of random-organizing hyperuniform fluids

2019 ◽  
Vol 116 (46) ◽  
pp. 22983-22989 ◽  
Author(s):  
Qun-Li Lei ◽  
Ran Ni
Keyword(s):  
Stokes Equations ◽  
Acoustic Resonance ◽  
Smooth Transition ◽  
Hydrodynamic Theory ◽  
Navier Stokes ◽  
General Mechanism ◽  
Hard Sphere Model ◽  
Navier Stokes Equations ◽  
Long Wavelength ◽  
Large Length

Disordered hyperuniform structures are locally random while uniform like crystals at large length scales. Recently, an exotic hyperuniform fluid state was found in several nonequilibrium systems, while the underlying physics remains unknown. In this work, we propose a nonequilibrium (driven-dissipative) hard-sphere model and formulate a hydrodynamic theory based on Navier–Stokes equations to uncover the general mechanism of the fluidic hyperuniformity (HU). At a fixed density, this model system undergoes a smooth transition from an absorbing state to an active hyperuniform fluid and then, to the equilibrium fluid by changing the dissipation strength. We study the criticality of the absorbing-phase transition. We find that the origin of fluidic HU can be understood as the damping of a stochastic harmonic oscillator in q space, which indicates that the suppressed long-wavelength density fluctuation in the hyperuniform fluid can exhibit as either acoustic (resonance) mode or diffusive (overdamped) mode. Importantly, our theory reveals that the damping dissipation and active reciprocal interaction (driving) are the two ingredients for fluidic HU. Based on this principle, we further demonstrate how to realize the fluidic HU in an experimentally accessible active spinner system and discuss the possible realization in other systems.

Steady-state analytical solutions for the lattice boltzmann equation

Open Physics ◽  
2003 ◽  
Vol 1 (3) ◽  
Author(s):  
Gábor Házi
Keyword(s):  
Steady State ◽  
Boltzmann Equation ◽  
Analytical Solutions ◽  
Poiseuille Flow ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Lattice Boltzmann Equation ◽  
Two Dimensional ◽  
Navier Stokes Equations

AbstractA general class of analytical solutions of the lattice Boltzmann equation is derived for two-dimensional, steady-state unidirectional flows. A subset of the solutions that verifies the corresponding Navier-Stokes equations is given. It is pointed out that this class includes, e.g., the Couette and the Poiseuille flow but not, e.g., the basic Kolmogorov flow. For steady-state non-unidirectional flows, first and second order solutions of the lattice Boltzmann equation are derived. Practical consequences of the analysis are mentioned. Differences between the technique applied here and those used in some earlier works are emphasized.

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