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Moving boundary problems for a rarefied gas: spatially one-dimensional case. (English) Zbl 1349.82059
Summary: Unsteady flows of a rarefied gas in a full space caused by an oscillation of an infinitely wide plate in its normal direction are investigated numerically on the basis of the Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation. The paper aims at showing properties and difficulties inherent to moving boundary problems in kinetic theory of gases using a simple one-dimensional setting. More specifically, the following two problems are considered: (Problem I) the plate starts a forced harmonic oscillation (forced motion); (Problem II) the plate, which is subject to an external restoring force obeying Hooke’s law, is displaced from its equilibrium position and released (free motion). The physical interest in Problem I lies in the propagation of nonlinear acoustic waves in a rarefied gas, whereas that in Problem II in the decay rate of the oscillation of the plate. An accurate numerical method, which is capable of describing singularities caused by the oscillating plate, is developed on the basis of the method of characteristics and is applied to the two problems mentioned above. As a result, the unsteady behavior of the solution, such as the propagation of discontinuities and some weaker singularities in the molecular velocity distribution function, are clarified. Some results are also compared with those based on the existing method.

82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
65Z05 Applications to the sciences
Full Text: DOI
[1] Karniadakis, G.; Beskok, A.; Aluru, N., Microflows and nanoflows: fundamentals and simulation, (2005), Springer New York · Zbl 1115.76003
[2] Bird, G. A., Molecular gas dynamics, (1976), Oxford Univ. Press Oxford
[3] Bird, G. A., Molecular gas dynamics and the direct simulation of gas flows, (1994), Oxford Univ. Press Oxford
[4] Stefanov, S.; Gospodinov, P.; Cercignani, C., Monte Carlo simulation and Navier-Stokes finite difference calculation of unsteady-state rarefied gas flows phys, Fluids, 10, 289-300, (1998)
[5] Ohwada, T.; Kunihisa, M., Direct simulation of a flow produced by a plane wall oscillating in its normal direction, (Ketsdever, A. D.; Muntz, E. P., Rarefied Gas Dynamics, (2003), AIP Melville), 202-209 · Zbl 1062.76585
[6] D.J. Radar, M.A. Gallis, J.R. Torczynski, DSMC moving-boundary algorithms for simulating MEMS geometries with opening and closing gaps, in: D.A. Levin, I.J. Wysong, A.L. Garcia (Eds.), 27th International Symposium on Rarefied Gas Dynamics, 2010, AIP Conf. Proc. 1333, AIP, Melville, 2011, pp. 760-765.
[7] Russo, G.; Filbet, F., Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics, Kinet. Relat. Models, 2, 231-250, (2009) · Zbl 1372.76090
[8] Dechristé, G.; Mieussens, L., Numerical simulation of micro flows with moving obstacles, J. Phys.: Conf. Ser, 362, 012030, (2012) · Zbl 1349.76783
[9] G. Dechristé, L. Mieussens, A moving mesh approach for the numerical simulation of gas micro flows, in: M. Mareschal, A. Santos (Eds.), 28th International Conference on Rarefied Gas Dynamics 2012, AIP Conf. Proc. 1501, AIP, Melville, 2012, pp. 366-372.
[10] C. Pekardan, S. Chigullapalli, L. Sun, A. Alexeenko, Immersed boundary method for Boltzmann model kinetic equations, in: M. Mareschal, A. Santos (Eds.), 28th International Conference on Rarefied Gas Dynamics 2012, AIP Conf. Proc. 1501, AIP, Melville, 2012, pp. 358-365.
[11] Sone, Y.; Takata, S., Discontinuity of the velocity distribution function in a rarefied gas around a convex body and the S layer at the bottom of the Knudsen layer, Transp. Theory Stat. Phys., 21, 501-530, (1992) · Zbl 0793.76078
[12] Sone, Y., Kinetic theory and fluid mechanics, (2002), Birkhäuser Boston
[13] Sone, Y., Molecular gas dynamics: theory, techniques, and applications, (2007), Birkhäuser Boston · Zbl 1144.76001
[14] 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, 511-525, (1954) · Zbl 0055.23609
[15] Welander, P., On the temperature jump in a rarefied gas, Ark. Fys., 7, 507-553, (1954) · Zbl 0057.23301
[16] Inoue, Y.; Yano, T., Propagation of strongly nonlinear plane waves, J. Acoust. Soc. Am., 94, 1632-1642, (1993)
[17] T. Tsuji, K. Aoki, Decay of an oscillating plate in a free-molecular gas, in: D.A. Levin, I.J. Wysong, A.L. Garcia (Eds.), 27th International Symposium on Rarefied Gas Dynamics, 2010, AIP Conf. Proc. 1333, AIP, Melville, 2011, pp. 140-145.
[18] Tsuji, T.; Aoki, K., Decay of a linear pendulum in a free-molecular gas and in a special Lorentz gas, J. Stat. Phys., 146, 620-645, (2012) · Zbl 1245.82066
[19] Caprino, S.; Marchioro, C.; Pulvirenti, M., Approach to equilibrium in a microscopic model of friction, Commun. Math. Phys., 264, 167-189, (2006) · Zbl 1113.82059
[20] Caprino, S.; Cavallaro, G.; Marchioro, C., On a microscopic model of viscous friction, Math. Models Methods Appl. Sci., 17, 1369-1403, (2007) · Zbl 1216.70008
[21] Cavallaro, G., On the motion of a convex body interacting with a perfect gas in the mean-field approximation, Rend. Mat. Appl., 27, 123-145, (2007) · Zbl 1134.76055
[22] Aoki, K.; Cavallaro, G.; Marchioro, C.; Pulvirenti, M., On the motion of a body in thermal equilibrium immersed in a perfect gas, Math. Model. Numer. Anal., 42, 263-275, (2008) · Zbl 1133.76046
[23] K. Aoki, T. Tsuji, G. Cavallaro, Approach to steady motion of a plate moving in a free-molecular gas under a constant external force, Phys. Rev. E 80 (2009) 016309 1-13.
[24] Chu, C. K., Kinetic-theoretic description of the formation of a shock wave, Phys. Fluids, 8, 12-22, (1965)
[25] Kim, C., Formation and propagation of discontinuity for Boltzmann equation in non-convex domains, Commun. Math. Phys., 308, 641-701, (2011) · Zbl 1237.35126
[26] Takata, S.; Funagane, H., Singular behavior of a rarefied gas on a planar boundary, J. Fluid Mech., 717, 30-47, (2013) · Zbl 1284.76333
[27] Golse, F.; Lions, P. L.; Perthame, B.; Sentis, R., Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76, 110-125, (1988) · Zbl 0652.47031
[28] Sone, Y.; Sugimoto, H., Strong evaporation from a plane condensed phase, (Meier, G. E.A.; Thompson, P. A., Adiabatic Waves in Liquid-Vapor Systems, (1990), Springer-Verlag Berlin), 293-304
[29] Aoki, K.; Sone, Y.; Nishino, K.; Sugimoto, H., Numerical analysis of unsteady motion of a rarefied gas caused by sudden changes of wall temperature with special interest in the propagation of a discontinuity in the velocity distribution function, (Beylich, A. E., Rarefied Gas Dynamics, (1991), VCH Weinheim), 222-231
[30] Takata, S.; Oishi, M., Numerical demonstration of the reciprocity among elemental relaxation and driven-flow problems for a rarefied gas in a channel, Phys. Fluid, 24, 012003, (2012)
[31] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in: B. Cockburn, C. Johnson, C.-W. Shu, E. Tadmor (Editor-in-Chief: A. Quarteroni) (Eds.), Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics 1697, Springer, Berlin, 1998, pp. 325-432. · Zbl 0927.65111
[32] T. Tsuji, K. Aoki, Numerical analysis of nonlinear acoustic wave propagation in a rarefied gas, in: M. Mareschal, A. Santos (Eds.), 28th International Conference on Rarefied Gas Dynamics 2012, AIP Conf. Proc. 1501, AIP, Melville, 2012, pp. 115-122.
[33] Wang Chang, C. S.; Uhlenbeck, G. E., On the propagation of sound in monatomic gases, (1952), University of Michigan Report
[34] Sirovich, L.; Thurber, J. K., Propagation of forced sound waves in rarefied gasdynamics, J. Acoust. Soc. Am., 37, 329-339, (1965)
[35] Kahn, D.; Mintzer, D., Kinetic theory of sound propagation in rarefied gases, Phys. Fluids, 8, 1090-1102, (1965) · Zbl 0127.22004
[36] Thomas, J. R.; Siewert, C. E., Sound-wave propagation in a rarefied gas, Trans. Theory Stat. Phys., 8, 219-240, (1979) · Zbl 0419.76065
[37] Loyalka, S. K.; Cheng, T. C., Sound-wave propagation in a rarefied gas, Phys. Fluids, 22, 830-836, (1979) · Zbl 0396.76062
[38] Garcia, R. D.M.; Siewert, C. E., The linearized Boltzmann equation: sound-wave propagation in a rarefied gas, Z. Angew. Math. Phys., 57, 94-122, (2006) · Zbl 1083.76055
[39] Sharipov, F.; Kalempa, D., Numerical modeling of the sound propagation through a rarefied gas in a semi-infinite space on the basis of linearized kinetic equation, J. Acoust. Soc. Am., 123, 1993-2001, (2008)
[40] Yano, T.; Inoue, Y., Quasisteady streaming with rarefaction effect induced by asymmetric sawtooth-like plane waves, Phys. Fluids, 8, 2537-2551, (1996) · Zbl 1027.76654
[41] Cavallaro, G.; Marchioro, C., On the approach to equilibrium for a pendulum immersed in a Stokes fluid, Math. Models Methods Appl. Sci., 20, 1999-2019, (2010) · Zbl 1402.76039
[42] Cavallaro, G.; Marchioro, C.; Tsuji, T., Approach to equilibrium of a rotating sphere in a Stokes flow, Ann. Univ. Ferrara, 57, 211-228, (2011) · Zbl 1252.76026
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