A BGK Model for Gas Mixtures of Polyatomic Molecules Allowing for Slow and Fast Relaxation of the Temperatures

Abstract

Kinetic models for polyatomic gases have two temperatures for the two different types of degrees of freedom, the translational and the internal energy degrees of freedom. Therefore, in the case of BGK models one expects two types of relaxations, a relaxation of the distribution function to a Maxwell distribution and a relaxation of the two temperatures to an equal value. The speed for the first type of relaxation may be faster or slower than the second type of relaxation. Models found in the literature often allow only for one of these two cases. We believe that a model should allow for both cases. That is why we derive a new multi-species polyatomic BGK model which allows for both regimes. For this new model we prove conservation properties, positivity of the temperatures, the H-theorem and characterize the equilibrium as a Maxwell distribution with equal temperatures. Moreover, we prove the convergence rate to equilibrium and that we can actually capture both regimes of the relaxation processes.

A Review on BGK Models for Gas Mixtures of Mono and Polyatomic Molecules

“…This model satisfies the following asymptotic behaviour in the space-homogeneous case proven in [52] for f = f (4) . Theorem 7.…”

“…Now, for the gas mixture case, we will present different models [52][53][54] combining different ansatzes from the one-species polyatomic case and the mixture modelling.…”

“…For this model one can prove conservation of the number of particles, momentum and total energy, and also an entropy inequality. Additionally, the equilibrium can be characterized by a Maxwell distribution with equal temperatures T equ = T tr = T int , for details see [48,52]. The existence of a unique mild solution can be proven similar to the existence in the momatomic case [16].…”

BGK model of the multi-species Uehling-Uhlenbeck equation

“…Examples include the model of Gross and Krook [29], the model of Hamel [32], the model of Greene [27], the model of Garzo, Santos and Brey [26], the model of Sofonea and Sekerka [57], the model by Andries, Aoki and Perthame [1], the model of Brull, Pavan and Schneider [13], the model of Klingenberg, Pirner and Puppo [40], the model of Haack, Hauck, Murillo [31] and the model of Bobylev, Bisi, Groppi, Spiga [10]. BGK models have also been extended to ES-BGK models, polyatomic molecules or chemically reactive gas mixtures; see for example [8,9,14,28,38,39,41,49,60]. BGK models are often used in applications because they give rise to efficient numerical computations as compared to models with Boltzmann collision terms [5,6,16,17,24,48,53,54].…”

A Consistent BGK Model with Velocity-Dependent Collision Frequency for Gas Mixtures

“…There are many BGK models for gas mixtures proposed in the literature [1,5,10,12,[14][15][16]23,28], many of which satisfy these basic requirements and, in addition, are able to match some prescribed relaxation rates and/or transport coefficients that come from more complicated physics models or from experiment. Many of these approaches have been extended to accommodate ellipsoid statistical (ES-BGK) models, polyatomic molecules, chemical reactions or quantum gases; see for example [3,4,13,[24][25][26][27]31].…”