Abstract
A set of linearized 26 moment equations, along with their wall boundary conditions, are derived and used to study low-speed gas flows dominated by Knudsen layers. Analytical solutions are obtained for Kramers’ defect velocity and the velocity-slip coefficient. These results are compared to the numerical solution of the BGK kinetic equation. From the analysis, a new effective viscosity model for the Navier–Stokes equations is proposed. In addition, an analytical expression for the velocity field in planar pressure-driven Poiseuille flow is derived. The mass flow rate obtained from integrating the velocity profile shows good agreement with the results from the numerical solution of the linearized Boltzmann equation. These results are good for Knudsen numbers up to 3 and for a wide range of accommodation coefficients. The Knudsen minimum phenomenon is also well captured by the present linearized 26-moment equations.
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References
Cercignani C.: The Boltzmann Equation and its Applications. Springer-Verlag, New York (1988)
Bird G.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Claredon Press, Oxford (1994)
Chapman S., Cowling T.G.: The Mathematical Theory of Non-uniform Gases. Cambridge University Press, Cambridge (1970)
Muller I., Ruggeri T.: Extended Thermodynamics. Springer-Verlag, New York (1993)
Struchtrup H.: Macroscopic Transport Equations for Rarefied Gas Flows. Springer-Verlag, Berlin (2005)
Grad H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (1949)
Levermore C. D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065 (1996)
Xu K.: A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method. J. Comput. Phys. 171, 289–335 (2001)
Shan X., Yuan X.-F., Chen H.: Kinetic theory representation of hydrodynamics: A way beyond the Navier–Stokes equation. J. Fluid Mech. 550, 413–441 (2006)
Struchtrup H., Torrihon M.: Regularization of Grad’s 13 moment equations: derivation and linear analysis. Phys. Fluids 15, 2668–2680 (2003)
Gu X.J., Emerson D.R.: A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions. J. Comput. Phys. 225, 263–283 (2007)
Torrilhon M., Struchtrup H.: Boundary conditions for regularized 13-moment-equations for micro-channel-flows. J. Comput. Phys. 227, 1982–2011 (2008)
Gu X.J., Barber R.W., Emerson D.R.: How far can 13 moments go in modelling microscale gas phenomena?. Nano. Microscale Thermophy. Eng. 11, 85–97 (2007)
Struchtrup H., Torrihon M.: Higher order effects in rarefied channel flows. Phys. Rev. E 78, 046301 (2008)
Taheri P., Torrilhon M., Struchtrup H.: Couette and Poiseuille microflows: analytical solutions for regularized 13-moment equations. Phys. Fluids 21, 017102 (2009)
Struchtrup H.: Linear kinetic transfer: moment equations, boundary conditions, and Knudsen layer. Physica A 387, 1750–1766 (2008)
Gu X.J., Emerson D.R.: A high-order moment approach for capturing non-equilibrium phenomena in the transition regime. J. Fluid Mech. 636, 177–216 (2009)
Cercignani C.: Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations. Cambridge University Press, Cambridge (2000)
Ohwada T., Sone Y., Aoki K.: Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1, 2042–2049 (1989)
Cercignani C., Lampis M., Lorenzani S.: Variational approach to gas flows in microchannels. Phys. Fluids 16, 3426–3437 (2004)
Mansour M.M., Baras F., Garcia A.L.: On the validity of hydrodynamics in plane Poiseuille flow. Physica A 240, 255–267 (1997)
Albertoni S., Cercignani C., Gotusso L.: Numerical evaluation of the slip coefficient. Phys. Fluids 6, 993–996 (1963)
Loyalka S.K.: Velocity profile in the Knudsen layer for the Kramer’s problem. Phys. Fluids 18, 1666–1669 (1975)
Loyalka S.K., Petrellis N., Storvick T.S.: Some numerical results for BGK model: thermal creep and viscous slip problems with arbitrary accomodation at the surface. Phys. Fluids 18, 1094–1099 (1975)
Loyalka S.K., Hickey K.A.: Velocity slip and defect: hard sphere gas. Phys. Fluids A 1, 612–614 (1989)
Siewert C.E.: Kramers’ problem for a variable collision frequency model. Eur. J. Appl. Math. 12, 179–191 (2001)
Loyalka S.K., Tompson R.V.: The velocity slip problem: accurate solutions of the BGK model integral equation. Eur. J. Mech. B Fluids 28, 211–213 (2009)
Reynolds, M.A., Smolderen, J.J., Wendt, J.F.: Velocity profile measurements in the Knudsen layer for the Kramers problem. In: Becker, M., Fiebig, M. (eds.) Rarefied Gas Dynamics, vol. I, A.21-1-14. DFVLR-Press, Porz-Wahn (1974)
Lockerby D.A., Reese J.M.: On the modelling of isothermal gas flows at the microscale. J. Fluid Mech. 604, 235–261 (2008)
Tang G.H., Zhang Y.H., Gu X.J., Emerson D.R.: Lattice Boltzmann modelling Knudsen layer effect in non-equilibrium flows. EPL 83, 40008 (2008)
Guo Z.L., Shi B.C., Zheng C.G.: An extended Navier–Stokes formulation for gas flows in the Knudsen layer near a wall. EPL 80, 24001 (2007)
Xu K., Liu H.: A multiple-temperature kinetic model and its application to near continuum flows. Commun. Comput. Phys. 4, 1069–1085 (2008)
Truesdell C., Muncaster R.G.: Fundamentals of Maxwell’s Kinetic Theory of a Simple Monotomic Gas. Academic Press, New York (1980)
Sone Y.: Kinetic Theory and Fluid Dynamics. Birkhäuser, Boston (2002)
Stops D.W.: The mean free path of gas molecules in the transition regime. J. Phys. D 3, 685–696 (1970)
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Gu, XJ., Emerson, D.R. & Tang, GH. Kramers’ problem and the Knudsen minimum: a theoretical analysis using a linearized 26-moment approach. Continuum Mech. Thermodyn. 21, 345–360 (2009). https://doi.org/10.1007/s00161-009-0121-5
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DOI: https://doi.org/10.1007/s00161-009-0121-5