scholarly journals Entropy production for ellipsoidal BGK model of the Boltzmann equation

2016 ◽  
Vol 9 (3) ◽  
pp. 605-619 ◽  
Author(s):  
Seok-Bae Yun
Keyword(s):  
Boltzmann Equation ◽  
Entropy Production ◽  
Bgk Model ◽  
The Boltzmann Equation

THE CONTRIBUTION OF THE BHATNAGAR–GROSS–KROOK MODEL TO THE DEVELOPMENT OF RAREFIED GAS DYNAMICS IN THE EARLY YEARS OF THE SPACE AGE

2013 ◽  
Vol 25 (01) ◽  
pp. 1340025
Author(s):  
RODDAM NARASIMHA
Keyword(s):  
Boltzmann Equation ◽  
Gas Dynamics ◽  
Rarefied Gas ◽  
Rarefied Gas Dynamics ◽  
Early Years ◽  
Plane Shock ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Nonequilibrium Flows ◽  
Space Age

The advent of the space age in 1957 was accompanied by a sudden surge of interest in rarefied gas dynamics (RGD). The well-known difficulties associated with solving the Boltzmann equation that governs RGD made progress slow but the Bhatnagar–Gross–Krook (BGK) model, proposed three years before Sputnik, turned out to have been an uncannily timely, attractive and fruitful option, both for gaining insights into the Boltzmann equation and for estimating various technologically useful flow parameters. This paper gives a view of how BGK contributed to the growth of RGD during the first decade of the space age. Early efforts intended to probe the limits of the BGK model showed that, in and near both the continuum Euler limit and the collisionless Knudsen limit, BGK could provide useful answers. Attempts were therefore made to tackle more ambitious nonlinear nonequilibrium problems. The most challenging of these was the structure of a plane shock wave. The first exact numerical solutions of the BGK equation for the shock appeared during 1962 to 1964, and yielded deep insights into the character of transitional nonequilibrium flows that had resisted all attempts at solution through the Boltzmann equation. In particular, a BGK weak shock was found to be amenable to an asymptotic analysis. The results highlighted the importance of accounting separately for fast-molecule dynamics, most strikingly manifested as tails in the distribution function, both in velocity and in physical space — tails are strange versions or combinations of collisionless and collision-generated flows. However, by the mid-1960s Monte-Carlo methods of solving the full Boltzmann equation were getting to be mature and reliable and interest in the BGK waned in the following years. Interestingly, it has seen a minor revival in recent years as a tool for developing more effective algorithms in continuum computational fluid dynamics, but the insights derived from the BGK for strongly nonequilibrium flows should be of lasting value.

scholarly journals Convergence of a Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

2012 ◽  
Vol 50 (3) ◽  
pp. 1111-1135 ◽  
Author(s):  
Giovanni Russo ◽  
Pietro Santagati ◽  
Seok-Bae Yun
Keyword(s):  
Boltzmann Equation ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Lagrangian Scheme

The study of the Boltzmann equation of solid-gas two-phase flow with three-dimensional BGK model

2018 ◽  
Author(s):  
Chang-jiang Liu ◽  
Song Pang ◽  
Qiang Xu ◽  
Ling He ◽  
Shao-peng Yang ◽  
...  
Keyword(s):  
Boltzmann Equation ◽  
Two Phase Flow ◽  
Three Dimensional ◽  
Phase Flow ◽  
Bgk Model ◽  
Two Phase ◽  
The Boltzmann Equation

scholarly journals A Study of the Entropy Production in Physical Processes from a New Perspective of the Energy Structure

2020 ◽  
Vol 8 (6) ◽  
Author(s):  
Saeed Shahsavari ◽  
Mehran Moradi
Keyword(s):  
Boltzmann Equation ◽  
Entropy Production ◽  
Physical Process ◽  
Structure Equation ◽  
Dissipated Energy ◽  
Energy Structure ◽  
Physical Processes ◽  
Physical Systems ◽  
The Boltzmann Equation ◽  
Energy Components

When a physical process is performed, identifying the generated entropy can be used to investigate the irreversibility. But for this mean, from the perspective of the Boltzmann equation, both all microstates and macrostates must be studied. In fact, it is needed that all particles energy level to be investigated. Therefore, to investigate entropy in configurationally systems using the Boltzmann equation, a very large volume of calculations is required. In this study, we try to extract a way to investigate entropy production without the need to study all particles (or sub-structures). For this purpose, at first, a macroscopic energy structure equation “as an equation that shows the energy components of the system activated in the performed process as well as their dependence” is presented. As a study on the irreversibility (or entropy production) in physical systems, its structure and components are studied. Writing equations in the energy space of the system makes it possible to study the structure of irreversibility. Then using a new macroscopic quasi-statistical approach, the irreversibility and its structure in physical processes are investigated. Macro energy components of the system are used for this investigation and energy structure is studied base on them. Finally, a new macroscopic definition of the generated entropy is extracted using a new energy structure equation as well as dependent and independent macroscopic energy component concepts. Also, why and what entropy can be generated, from the perspective of the presented macroscopic energy structure equation are studied. In fact, this paper investigates the generated entropy structure in physical systems using macroscopic system energy components and takes a new approach to why and what irreversibility is occurred during the physical process. Therefore, presented equations can be used for investigating the irreversibility in configurationally physical systems without the need to study all its sub structures. Also, from the extracted equations, it can be concluded that entropy is generated because of the existence of the dependent energy components in the energy structure equation of the system, and this generated entropy depends on the variation of these components as well as the amount of the applied energy to the system and its conditions. Due to the kinematic theory of dissipated energy, these results are in the same line with the different formulations of the second law of thermodynamics.  

scholarly journals Steepest Entropy Ascent Models of the Boltzmann Equation: Comparisons With Hard-Sphere Dynamics and Relaxation-Time Models for Homogeneous Relaxation From Highly Non-Equilibrium States

2013 ◽  
Author(s):  
Gian Paolo Beretta ◽  
Nicolas G. Hadjiconstantinou
Keyword(s):  
Relaxation Time ◽  
Boltzmann Equation ◽  
Power Law ◽  
Collision Frequency ◽  
Equilibrium States ◽  
Space Distribution ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Far From Equilibrium ◽  
Non Equilibrium

We present a family of steepest entropy ascent (SEA) models of the Boltzmann equation. The models preserve the usual collision invariants (mass, momentum, energy), as well as the non-negativity of the phase-space distribution, and have a strong built-in thermodynamic consistency, i.e., they entail a general H-theorem valid even very far from equilibrium. This family of models features a molecular-speed-dependent collision frequency; each variant can be shown to approach a corresponding BGK model with the same variable collision frequency in the limit of small deviation from equilibrium. This includes power-law dependence on the molecular speed for which the BGK model is known to have a Prandtl number that can be adjusted via the power-law exponent. We compare numerical solutions of the constant and velocity-dependent collision frequency variants of the SEA model with the standard relaxation-time model and a Monte Carlo simulation of the original Boltzmann collision operator for hard spheres for homogeneous relaxation from near-equilibrium and highly non-equilibrium states. Good agreement is found between all models in the near-equilibrium regime. However, for initial states that are far from equilibrium, large differences are found; this suggests that the maximum entropy production statistical ansatz is not equivalent to Boltzmann collisional dynamics and needs to be modified or augmented via additional constraints or structure.

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