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Assessment and development of the gas kinetic boundary condition for the Boltzmann equation

Published online by Cambridge University Press:  21 June 2017

Lei Wu Open the ORCID record for Lei Wu [Opens in a new window]  and
Henning Struchtrup
Lei Wu*
Affiliation:
James Weir Fluids Laboratory, Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK
Henning Struchtrup
Affiliation:
Department of Mechanical Engineering, University of Victoria, Victoria, British Columbia V8W 2Y2, Canada
*
†Email address for correspondence: lei.wu.100@strath.ac.uk
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Abstract

Gas–surface interactions play important roles in internal rarefied gas flows, especially in micro-electro-mechanical systems with large surface area to volume ratios. Although great progress has been made to solve the Boltzmann equation, the gas kinetic boundary condition (BC) has not been well studied. Here we assess the accuracy of the Maxwell, Epstein and Cercignani–Lampis BCs, by comparing numerical results of the Boltzmann equation for the Lennard–Jones potential to experimental data on Poiseuille and thermal transpiration flows. The four experiments considered are: Ewart et al. (J. Fluid Mech., vol. 584, 2007, pp. 337–356), Rojas-Cárdenas et al. (Phys. Fluids, vol. 25, 2013, 072002) and Yamaguchi et al. (J. Fluid Mech., vol. 744, 2014, pp. 169–182; vol. 795, 2016, pp. 690–707), where the mass flow rates in Poiseuille and thermal transpiration flows are measured. This requires that the BC has the ability to tune the effective viscous and thermal slip coefficients to match the experimental data. Among the three BCs, the Epstein BC has more flexibility to adjust the two slip coefficients, and hence for most of the time it gives good agreement with the experimental measurements. However, like the Maxwell BC, the viscous slip coefficient in the Epstein BC cannot be smaller than unity but the Cercignani–Lampis BC can. Therefore, we propose to combine the Epstein and Cercignani–Lampis BCs to describe gas–surface interaction. Although the new BC contains six free parameters, our approximate analytical expressions for the viscous and thermal slip coefficients provide useful guidance to choose these parameters.

JFM classification

Rarefied Gas Flow: Kinetic theory Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics Rarefied Gas Flow: Rarefied Gas Flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 823 , 25 July 2017 , pp. 511 - 537
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Copyright
© 2017 Cambridge University Press 

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References

Barbante, P., Frezzotti, A. & Gibelli, L. 2015 A kinetic theory description of liquid menisci at the microscale. Kinet. Relat. Models 8, 235–254.Google Scholar
Barisik, M. & Beskok, A. 2014 Scale effects in gas nano flows. Phys. Fluids 26, 052003.CrossRefGoogle Scholar
Barisik, M. & Beskok, A. 2016 ‘Law of the nano-wall’ in nano-channel gas flows. Microfluid Nanofluid 20, 46.CrossRefGoogle Scholar
Bird, G. A. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford University Press.Google Scholar
Brull, S., Charrier, P. & Mieussens, L. 2016 Nanoscale roughness effect on Maxwell-like boundary conditions for the Boltzmann equation. Phys. Fluids 28, 082004.Google Scholar
Cercignani, C. 1971 Model for gas surface interaction: comparison between theory and experiments. In 7th International Symposium on Rarefied Gas Dynamics, vol. 1, pp. 75–79. Editrice Tecnico.Google Scholar
Cercignani, C. 1988 The Boltzmann Equation and its Applications. Springer.Google Scholar
Cercignani, C. & Lampis, M. 1971 Kinetic models for gas-surface interactions. Transp. Theory Stat. Phys. 1, 101–114.CrossRefGoogle Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-uniform Gases. Cambridge University Press.Google Scholar
Dodulad, O. I., Kloss, Y. Y., Savichkin, D. O. & Tcheremissine, F. G. 2014 Knudsen pumps modeling with Lennard–Jones and ab initio intermolecular potentials. Vacuum 109, 360–367.Google Scholar
Dodulad, O. I. & Tcheremissine, F. G. 2013 Computation of a shock wave structure in monatomic gas with accuracy control. Comput. Math. Math. Phys. 53, 827–844.CrossRefGoogle Scholar
Edmonds, T. & Hobson, J. P. 1965 A study of thermal transpiration using ultrahigh-vacuum techniques. J. Vac. Sci. Technol. 2, 182–197.CrossRefGoogle Scholar
Epstein, M. 1967 A model of the wall boundary condition in kinetic theory. AIAA J. 5, 1797–1800.Google Scholar
Ewart, T., Perrier, P., Graur, I. A. & Méolans, J. G. 2007 Mass flow rate measurements in a microchannel, from hydrodynamic to near free molecular regimes. J. Fluid Mech. 584, 337–356.Google Scholar
Frezzotti, A. & Gibelli, L. 2008 A kinetic model for fluid wall interaction. Proc. IMechE, Part C: J. Mech. Eng. Science 222, 787–795.Google Scholar
Gad-el-Hak, M. 1999 The fluid mechanics of microdevices – the Freeman Scholar lecture. J. Fluids Engng 121 (1), 5–33.Google Scholar
Garcia, R. D. M. & Siewert, C. E. 2009 The linearized Boltzmann equation with Cercignani–Lampis boundary conditions: basic flow problems in a plane channel. Eur. J. Mech. (B/Fluids) 28, 387–396.Google Scholar
Graur, I. A. & Ho, M. T. 2014 Rarefied gas flow through a long rectangular channel of variable cross section. Vacuum 101, 328–332.Google Scholar
Gu, K., Watkins, C. B. & Koplik, J. 2001 Atomistic hybrid DSMC/NEMD method for nonequilibrium multiscale simulations. J. Comput. Phys. 229, 1381–1400.Google Scholar
Hadjiconstantinou, N. G., Garcia, A. L., Bazant, M. Z. & He, G. 2003 Statistical error in particle simulations of hydrodynamic phenomena. J. Comput. Phys. 187, 274–297.Google Scholar
Ishiyama, T., Yano, T. & Fujikawa, S. 2005 Kinetic boundary condition at a vapor-liquid interface. Phys. Rev. Lett. 95, 084504.CrossRefGoogle Scholar
Karniadakis, G., Beskok, A. & Aluru, N. 2005 Microflows and Nanoflows: Fundamentals and Simulation. Springer.Google Scholar
Klinc, T. & Kuěčer, I. 1972 Slip coefficients for general gas surface interaction. Phys. Fluids 15, 1018–1022.CrossRefGoogle Scholar
Knudsen, M. 1909 Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren. Ann. Phys. 333, 75–130.CrossRefGoogle Scholar
Kon, M., Kobayashi, K. & Watanabe, M. 2014 Method of determining kinetic boundary conditions in net evaporation/condensation. Phys. Fluids 26, 072003.Google Scholar
Kowalczyk, P., Palczewski, A., Russo, G. & Walenta, Z. 2008 Numerical solutions of the Boltzmann equation: comparison of different algorithms. Eur. J. Mech. (B/Fluids) 27 (1), 62–74.Google Scholar
Liang, T., Li, Q. & Ye, W. J. 2013 Performance evaluation of Maxwell and Cercignani–Lampis gas-wall interaction models in the modeling of thermally driven rarefied gas transport. Phys. Rev. E 88, 013009.Google Scholar
Liang, T. & Ye, W. J. 2014 An efficient hybrid DSMC/MD algorithm for accurate modeling of micro gas flows. Commun. Comput. Phys. 15, 246–264.Google Scholar
Loyalka, S. K. 1989 Temperature jump and thermal creep slip: rigid sphere gas. Phys. Fluids A 1, 403–408.Google Scholar
Loyalka, S. K. & Cipolla, J. W. 1971 Thermal creep slip with arbitrary accommodation at the surface. Phys. Fluids 14, 1656.Google Scholar
Maxwell, J. C. 1879 On stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
Porodnov, B. T., Kulev, A. N. & Tuchvetov, F. T. 1978 Thermal transpiration in a circular capillary with a small temperature difference. J. Fluid Mech. 88, 609–622.CrossRefGoogle Scholar
Porodnov, B. T., Suetin, P. E., Borisov, S. F. & Akinshin, V. D. 1974 Experimental investigation of rarefied gas flow in different channels. J. Fluid Mech. 64, 417–437.Google Scholar
Rojas-Cárdenas, M., Graur, I. A., Perrier, P. & Méolans, J. G. 2013 Time-dependent experimental analysis of a thermal transpiration rarefied gas flow. Phys. Fluids 25, 072002.Google Scholar
Sharipov, F. 2002 Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. I. Plane flow between two parallel plates. Eur. J. Mech. (B/Fluids) 21, 113–123.CrossRefGoogle Scholar
Sharipov, F. 2003a Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. II. Slip and jump coefficients. Eur. J. Mech. (B/Fluids) 22, 133–143.Google Scholar
Sharipov, F. 2003b Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. III. Poiseuille flow and thermal creep through a long tube. Eur. J. Mech. (B/Fluids) 22, 145–154.Google Scholar
Sharipov, F. 2011 Data on the velocity slip and temperature jump on a gas-solid interface. J. Phys. Chem. Ref. Data 40, 023101.CrossRefGoogle Scholar
Sharipov, F. & Bertoldo, G. 2009a Numerical solution of the linearized Boltzmann equation for an arbitrary intermolecular potential. J. Comput. Phys. 228, 3345–3357.Google Scholar
Sharipov, F. & Bertoldo, G. 2009b Poiseuille flow and thermal creep based on the Boltzmann equation with the Lennard–Jones potential over a wide range of the Knudsen number. Phys. Fluids 21, 067101.CrossRefGoogle Scholar
Sharipov, F. & Seleznev, V. 1994 Rarefied flow through a long tube at any pressure ratio. J. Vac. Sci. Technol. A 12, 2933–2935.Google Scholar
Sharipov, F. & Strapasson, J. L. 2012 Direct simulation Monte Carlo method for an arbitrary intermolecular potential. Phys. Fluids 24, 011703.Google Scholar
Sharipov, F. & Strapasson, J. L. 2014 Ab initio simulation of rarefied gas flow through a thin orifice. Vacuum 109, 246–252.Google Scholar
Siewert, C. E. 2003 Viscous-slip, thermal-slip, and temperature-jump coefficients as defined by the linearzied Boltzmann equation and the Cercignani–Lampis boundary condition. Phys. Fluids 15, 1696–1701.Google Scholar
Strapasson, J. L. & Sharipov, F. 2014 Ab initio simulation of heat transfer through a mixture of rarefied gases. Intl J. Heat Mass Transfer 71, 91–97.CrossRefGoogle Scholar
Struchtrup, H. 2013 Maxwell boundary condition and velocity dependent accommodation coefficient. Phys. Fluids 25, 112001.Google Scholar
Takata, S. & Funagane, H. 2011 Poiseuille and thermal transpiration flows of a highly rarefied gas: over-concentration in the velocity distribution function. J. Fluid Mech. 669, 242–259.Google Scholar
Takata, S., Yasuda, S., Kosuge, S. & Aoki, K. 2003 Numerical analysis of thermal-slip and diffusion-type flows of a binary mixture of hard-sphere molecular gases. Phys. Fluids 15, 3745–3766.Google Scholar
Venkattraman, A. & Alexeenko, A. A. 2012 Binary scattering model for Lennard–Jones potential: transport coefficients and collision integrals for non-equilibrium gas flow simulations. Phys. Fluids 24, 027101.Google Scholar
Watvisave, D. S., Puranik, B. P. & Bhandarkar, U. V. 2015 A hybrid MD-DSMC coupling method to investigate flow characteristics of micro-devices. J. Comput. Phys. 302, 603–617.Google Scholar
Weaver, A. B., Venkattraman, A. & Alexeenko, A. A. 2014 Effect of intermolecular potential on compressible Couette flow in slip and transitional regimes. Phys. Fluids 26, 107102.Google Scholar
Wu, L., Liu, H. H., Zhang, Y. H. & Reese, J. M. 2015a Influence of intermolecular potentials on rarefied gas flows: fast spectral solutions of the Boltzmann equation. Phys. Fluids 27, 082002.CrossRefGoogle Scholar
Wu, L., Reese, J. M. & Zhang, Y. H. 2014 Solving the Boltzmann equation by the fast spectral method: application to microflows. J. Fluid Mech. 746, 53–84.Google Scholar
Wu, L., White, C., Scanlon, T. J., Reese, J. M. & Zhang, Y. H. 2013 Deterministic numerical solutions of the Boltzmann equation using the fast spectral method. J. Comput. Phys. 250, 27–52.Google Scholar
Wu, L., White, C., Scanlon, T. J., Reese, J. M. & Zhang, Y. H. 2015b A kinetic model of the Boltzmann equation for nonvibrating polyatomic gases. J. Fluid Mech. 763, 24–50.Google Scholar
Wu, L., Zhang, J., Liu, H. H., Zhang, Y. H. & Reese, J. M. 2017 A fast iterative scheme for the linearized boltzmann equation. J. Comput. Phys. 338, 431–451.CrossRefGoogle Scholar
Wu, L., Zhang, J., Reese, J. M. & Zhang, Y. H. 2015c A fast spectral method for the Boltzmann equation for monatomic gas mixtures. J. Comput. Phys. 298, 602–621.CrossRefGoogle Scholar
Yamaguchi, H., Perrier, P., Ho, M. T., Méolans, J. G., Niimi, T. & Graur, I. A. 2016 Mass flow rate measurement of thermal creep flow from transitional to slip flow regime. J. Fluid Mech. 795, 690–707.Google Scholar
Yamaguchi, H., Rojas-Cárdenas, M., Perrier, P., Graur, I. & Niimi, T. 2014 Thermal transpiration flow through a single rectangular channel. J. Fluid Mech. 744, 169–182.Google Scholar