## Assessment of the ellipsoidal-statistical Bhatnagar-Gross-Krook model for force-driven Poiseuille flows.(English)Zbl 1349.82058

Summary: We investigate the accuracy of the ellipsoidal-statistical Bhatnagar-Gross-Krook (ES-BGK) kinetic model for planar force-driven Poiseuille flows. Our numerical simulations are conducted using the deterministic discrete velocity method, for Knudsen numbers ($$Kn$$) ranging from 0.05 to 10. While we provide numerically accurate data, our aim is to assess the accuracy of the ES-BGK model for these flows. By comparing with data from the direct simulation Monte Carlo (DSMC) method and the Boltzmann equation, the ES-BGK model is found to be able to predict accurate velocity and temperature profiles in the slip flow regime $$(0.01<Kn\leq 0.1)$$, for both low-speed and high-speed flows. In the transition flow regime $$(0.1<Kn\leq 10)$$, however, the model does not quantitatively capture the viscous heating effect.

### MSC:

 82C40 Kinetic theory of gases in time-dependent statistical mechanics 76N15 Gas dynamics (general theory) 82C80 Numerical methods of time-dependent statistical mechanics (MSC2010) 65C05 Monte Carlo methods

### Keywords:

kinetic theory; gas dynamics; BGK model; ES-BGK model; S model
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### References:

 [1] Karniadakis, G.; Beskok, A.; Aluru, N., Microflows and nanoflows: fundamentals and simulation, Interdisciplinary Applied Mathematics, (2005), Springer · Zbl 1115.76003 [2] Struchtrup, H., Macroscopic transport equations for rarefied gas flows: approximation methods in kinetic theory, Interaction of Mechanics and Mathematics, (2005), Springer · Zbl 1119.76002 [3] Bhatnagar, P. L.; Gross, E. P.; Krook, M., A model for collision processes in gases. I: small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94, 3, 511-525, (1954) · Zbl 0055.23609 [4] Holway, L. H., New statistical models for kinetic theory: methods of construction, Phys. Fluids, 9, 9, 1658-1673, (1966) [5] Andries, P.; Tallec, P. L.; Perlat, J.-P.; Perthame, B., The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B - Fluids, 19, 6, 813-830, (2000) · Zbl 0967.76082 [6] Andries, P.; Perthame, B., The ES-BGK model equation with correct Prandtl number, (Bartel, T. J.; Gallis, M. A., Rarefied Gas Dynamics: 22nd International Symposium, 585 (1), (2001), Springer), 30-36 [7] Graur, I.; Polikarpov, A., Comparison of different kinetic models for the heat transfer problem, Heat Mass Transfer, 46, 2, 237-244, (2009) [8] Mieussens, L.; Struchtrup, H., Numerical comparison of Bhatnagar-Gross-Krook models with proper Prandtl number, Phys. Fluids, 16, 8, 2797-2813, (2004) · Zbl 1186.76372 [9] Gallis, M. A.; Torczynski, J. R., Investigation of the ellipsoidal-statistical Bhatnagar-Gross-Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls, Phys. Fluids, 23, 3, 030601, (2011) [10] Andries, P.; Bourgat, J.-F.; le Tallec, P.; Perthame, B., Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases, Comput. Methods Appl. Mech. Eng., 191, 31, 3369-3390, (2002) · Zbl 1101.76377 [11] Giddens, D. P.; Huang, A. B.; Young, V. Y.C., Evaluation of two statistical models using the shock structure problem, Phys. Fluids, 14, 12, 2645, (1971) [12] Segal, B. M., Shock-wave structure using nonlinear model Boltzmann equations, Phys. Fluids, 15, 7, 1233, (1972) · Zbl 0247.76061 [13] Zhuk, V. I.; Rykov, V. A.; Shakhov, E. M., Kinetic models and the shock structure problem, Fluid Dyn., 8, 4, 620-625, (1973) [14] Montanero, J.; Garzó, V., Nonlinear Couette flow in a dilute gas: comparison between theory and molecular-dynamics simulation, Phys. Rev. E, 58, 2, 1836-1842, (1998) [15] Garzó, V.; Santos, A., Comparison between the Boltzmann and BGK equations for uniform shear flow, Physica A, 213, 426-434, (1995) [16] Garzó, V.; López de Haro, M., Nonlinear transport for a dilute gas in steady Couette flow, Phys. Fluids, 9, 3, 776, (1997) [17] Bird, G. A., Monte Carlo simulation of gas flows, Annu. Rev. Fluid Mech., 10, 1, 11-31, (1978) · Zbl 0403.76060 [18] Shakhov, E. M., Approximate kinetic equations in rarefied gas theory, Fluid Dyn., 3, 1, 112-115, (1968) [19] Shakhov, E. M., Generalization of the Krook kinetic relaxation equation, Fluid Dyn., 3, 5, 95-96, (1968) [20] Aoki, K.; Takata, S.; Nakanishi, T., Poiseuille-type flow of a rarefied gas between two parallel plates driven by a uniform external force, Phys. Rev. E, 65, 2, 26315, (2002) [21] Sharipov, F.; Seleznev, V., Data on internal rarefied gas flows, J. Phys. Chem. Ref. Data, 27, 657-706, (1998) [22] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical recipes: the art of scientific computing, (2007), Cambridge University Press · Zbl 1132.65001 [23] Mansour, M. M.; Baras, F.; Garcia, A. L., On the validity of hydrodynamics in plane Poiseuille flows, Physica A, 240, 255-267, (1997) [24] Taheri, P.; Torrilhon, M.; Struchtrup, H., Couette and Poiseuille microflows: analytical solutions for regularized 13-moment equations, Phys. Fluids, 21, 017102, (2009) · Zbl 1183.76503 [25] Tij, M.; Santos, A., Perturbation analysis of a stationary nonequilibrium flow generated by an external force, J. Stat. Phys., 76, 1399-1414, (1994) · Zbl 0839.76076 [26] Tij, M.; Sabbane, M.; Santos, A., Nonlinear Poiseuille flow in a gas, Phys. Fluids, 10, 1021, (1998) · Zbl 0971.76502 [27] Mouhot, C.; Pareschi, L., Fast algorithms for computing the Boltzmann collision operator, Math. Comput., 75, 1833-1852, (2006) · Zbl 1105.76043 [28] Wu, L.; White, C.; Scanlon, T. J.; Reese, J. M.; Zhang, Y., Deterministic numerical solutions of the Boltzmann equation using the fast spectral method, J. Comput. Phys., 250, 27-52, (2013) · Zbl 1349.76790
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