BGK models for inert mixtures: Comparison and applications

Sebastiano Boscarino; Seung Yeon Cho; Maria Groppi; Giovanni Russo

doi:10.3934/krm.2021029

Article Contents

2021, Volume 14, Issue 5: 895-928. Doi: 10.3934/krm.2021029

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BGK models for inert mixtures: Comparison and applications

1.
Department of Mathematics and Computer Science, University of Catania, Viale Andrea Doria 6, 95125 Catania, Italy
2.
Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju-daero 501, 52828 Jinju, Republic of Korea
3.
Department of Mathematical, Physical and Computer Sciences, University of Parma Parco Area delle Scienze 53/A, I–43124 Parma, Italy

^* Corresponding author: Seung Yeon Cho
^* Corresponding author: Seung Yeon Cho

Received: January 31, 2021

Revised: June 30, 2021

Early access: September 2021

Published: October 2021

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Abstract

Consistent BGK models for inert mixtures are compared, first in their kinetic behavior and then versus the hydrodynamic limits that can be derived in different collision-dominated regimes. The comparison is carried out both analytically and numerically, for the latter using an asymptotic preserving semi-Lagrangian scheme for the BGK models. Application to the plane shock wave in a binary mixture of noble gases is also presented.

Keywords:

BGK-type models for mixtures,
inert mixtures of noble gases,
fluid dynamic limit,
multi-velocities and multi-temperatures models.

Mathematics Subject Classification: Primary: 76P05; Secondary: 82C40.

Citation:

Full Text(HTML)

Figure 1. Time evolution of relative $ L^1 $-norm of the differences in the distribution functions $ g_1 $ between BGK models for various values of $ \varepsilon $. In (a) and (b), the $ x $-axes stand for time and the $ y $-axes are the values obtained by (30)

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Figure 2. Comparison of the three BGK models for $ \varepsilon = 10^{-2} $ (Left) and $ \varepsilon = 10^{-3} $ (Right) with initial data in (29)

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Figure 3. Comparison of the three BGK models for $ \varepsilon = 10^{-4} $ with initial data in (29)

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Figure 4. Comparison of BGK model (10) and NS equations (15) for $ \varepsilon = 10^{-2} $ with initial data in (31)

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Figure 5. Comparison of BGK model (10) and NS equations (15) for $ \varepsilon = 10^{-3} $ (Left) and $ \varepsilon = 10^{-4} $ (Right) with initial data in (31)

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Figure 6. Comparison of the scaled BBGSP model (23) and Navier-Stokes equations (24) with $ \kappa = 1 $ for $ \varepsilon = 10^{-2} $ with initial data in (31)

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Figure 7. Comparison of the scaled BBGSP model (23) and Navier-Stokes equations (24) with $ \kappa = 1 $ for $ \varepsilon = 10^{-3} $ (Left) and $ \varepsilon = 10^{-4} $ (Right) with initial data in (31)

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Figure 8. Comparison of the numerical solution of the scaled BBGSP model (23) for $ \varepsilon = \kappa = 10^{-3} $ with: (left) global velocity and temperature Euler system (15) for $ \varepsilon = 0 $ and (right) multi-velocity and multi-temperature Euler system (24) for $ \varepsilon = 0 $, $ \kappa = 10^{-3} $. We use the initial data in (6.3)

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Figure 9. Comparison of the scaled BBGSP model (23) for $ \varepsilon = \kappa = 10^{-4} $ with: (left) global velocity and temperature Euler system (15) for $ \varepsilon = 0 $ and (right) multi-velocity and multi-temperature Euler system (24) for $ \varepsilon = 0 $, $ \kappa = 10^{-4} $

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Figure 10. Comparison of the scaled BBGSP model (23) for $ \varepsilon = \kappa = 10^{-5} $ with multi-velocity and multi-temperature Euler system (24) for $ \varepsilon = 0 $, $ \kappa = 10^{-5} $

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Figure 11. BDF3-QCWENO35 for $ \varepsilon = 10^{-0} $. Neon and Argon with $ n_1 = 0.1m_1,\quad n_2 = 0.9m_2 $. Black dashed lines are reference NS solutions and solid lines are BGK solutions

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