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The Burnett equations in cylindrical coordinates and their solution for flow in a microtube. (English) Zbl 1329.76311

Summary: The Burnett equations constitute a set of higher-order continuum equations. These equations are obtained from the Chapman-Enskog series solution of the Boltzmann equation while retaining second-order-accurate terms in the Knudsen number \(Kn\). The set of higher-order continuum models is expected to be applicable to flows in the slip and transition regimes where the Navier-Stokes equations perform poorly. However, obtaining analytical or numerical solutions of these equations has been noted to be particularly difficult. In the first part of this work, we present the full set of Burnett equations in cylindrical coordinates in three-dimensional form. The equations are reported in a generalized way for gas molecules that are assumed to be Maxwellian molecules or hard spheres. In the second part, a closed-form solution of these equations for isothermal Poiseuille flow in a microtube is derived. The solution of the equations is shown to satisfy the full Burnett equations up to \(Kn \leqslant 1.3\) within an error norm of \(\pm 1.0 \%\). The mass flow rate obtained analytically is shown to compare well with available experimental and numerical results. Comparison of the stress terms in the Burnett and Navier-Stokes equations is presented. The significance of the Burnett normal stress and its role in diffusion of momentum is brought out by the analysis. An order-of-magnitude analysis of various terms in the equations is presented, based on which a reduced model of the Burnett equations is provided for flow in a microtube. The Burnett equations in full three-dimensional form in cylindrical coordinates and their solution are not previously available.

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35Q20 Boltzmann equations
35Q30 Navier-Stokes equations
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References:

[1] Chapman, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases (1970)
[2] DOI: 10.1016/j.vacuum.2012.04.043
[3] DOI: 10.1017/S0022112007006374 · Zbl 1117.76300
[4] DOI: 10.1063/1.1729249
[5] DOI: 10.1007/s10404-008-0344-y
[6] DOI: 10.1016/j.physa.2004.10.034
[7] DOI: 10.1007/s10404-010-0604-5
[8] DOI: 10.1103/PhysRevE.85.041202
[9] Bobylev, Dokl. Akad. Nauk SSSR 27 pp 29– (1982)
[10] DOI: 10.1017/jfm.2012.546 · Zbl 1284.76331
[11] DOI: 10.1080/108939599199864
[12] DOI: 10.1016/j.fluiddyn.2008.03.003 · Zbl 1143.76057
[13] DOI: 10.1063/1.1397256 · Zbl 1184.76018
[14] Agrawal, Intl J. Microscale Nanoscale Therm. Fluid Transp. Phenomena 3 pp 125– (2012)
[15] DOI: 10.1063/1.1597472 · Zbl 1186.76504
[16] DOI: 10.1088/0960-1317/17/8/036
[17] Sreekanth, Rarefied Gas Dynamics (1968)
[18] DOI: 10.1016/j.compfluid.2014.03.032 · Zbl 1391.76685
[19] DOI: 10.1007/s10404-013-1224-7
[20] Pong, Proceedings of Application of Microfabrication to Fluid Mechanics, ASME Winter Annual Meeting, Chicago pp 51– (1994)
[21] Landau, Fluid Mechanics (1958)
[22] DOI: 10.1002/andp.19093330106 · JFM 40.0825.02
[23] Karniadakis, Microflows and Nanoflows: Fundamentals and Simulation (2005)
[24] DOI: 10.1023/A:1010365123288 · Zbl 1019.82018
[25] DOI: 10.1017/S0022112002002203 · Zbl 1037.76030
[26] DOI: 10.2514/3.26657
[27] Yang, Phys. Fluids 21 (2009)
[28] DOI: 10.1080/10893950390150430
[29] DOI: 10.1017/jfm.2013.615
[30] DOI: 10.1103/PhysRevE.60.4063
[31] DOI: 10.1016/0042-207X(93)90342-8
[32] DOI: 10.1017/S002211200900768X · Zbl 1183.76850
[33] DOI: 10.1002/cpa.3160020403 · Zbl 0037.13104
[34] DOI: 10.1016/j.physrep.2008.04.010
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