American Institute of Mathematical Sciences

Advanced
December  2011, 4(4): 935-954. doi: 10.3934/krm.2011.4.935

Stagnation-point flow of a rarefied gas impinging obliquely on a plane wall

Kazuo Aoki 1, and  Yoshiaki Abe 2,

1. 

Department of Mechanical Engineering and Science, and Advanced Research Institute of Fluid Science and Engineering, Kyoto University, Kyoto 606-8501, Japan
2. 

School of Engineering Science, Faculty of Engineering, Kyoto University, Kyoto 606-8501, Japan

Received  June 2011 Published  November 2011

The steady two-dimensional stagnation-point flow of a rarefied gas impinging obliquely on an infinitely wide plane wall is investigated on the basis of kinetic theory. Assuming that the overall flow field has a length scale of variation much longer than the mean free path of the gas molecules and that the Mach number based on the characteristic flow speed is as small as the Knudsen number (the mean free path divided by the overall length scale of variation), one can exploit the result of the asymptotic theory (weakly nonlinear theory) for the Boltzmann equation, developed by Sone, that describes general steady behavior of a slightly rarefied gas over a smooth boundary [Y. Sone, in: D. Dini (ed.) Rarefied Gas Dynamics, Vol. 2, pp. 737--749. Editrice Tecnico Scientifica, Pisa (1971)]. By solving the fluid-dynamic system of equations given by the theory, the precise description of the velocity and temperature fields around the plane wall is obtained.
Keywords: kinetic theory of gases., slip flows, Stagnation-point flow, Boltzmann equation, rarefied gas flows.
Mathematics Subject Classification: Primary: 76P05, 82C40; Secondary: 76D0.
Citation: Kazuo Aoki, Yoshiaki Abe. Stagnation-point flow of a rarefied gas impinging obliquely on a plane wall. Kinetic & Related Models, 2011, 4 (4) : 935-954. doi: 10.3934/krm.2011.4.935
References:
[1]

R. Aris, "Vectors, Tensors, and the Basic Equations of Fluid Mechanics," Chap. 1, Dover, New York, 1989. Google Scholar

[2]

J. M. Dorrepaal, An exact solution of the Navier-Stokes equation which describes nonorthogonal stagnation-point flow in two dimensions, J. Fluid Mech., 163 (1986), 141-147. doi: 10.1017/S0022112086002240.  Google Scholar

[3]

K. Hiemenz, Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszlynder, Dinglers J., 326 (1911), 321-324, 344-348, 357-362, 372-376, 391-393, 407-410. Google Scholar

[4]

J. D. Hoffman, "Numerical Methods for Engineers and Scientists," McGraw-Hill, New York, 1993. Google Scholar

[5]

H. W. Liepmann and A. Roshko, "Elements of Gasdynamics. Galcit Aeronautical Series," John Wiley & Sons, Inc., New York, Chapman & Hall, Ltd., London, 1957.  Google Scholar

[6]

M. J. Martin and I. D. Boyd, Falkner-Skan flow over a wedge with slip boundary conditions, J. Thermophys. Heat Trans., 24 (2010), 263-270. Google Scholar

[7]

H. Okamoto, "Mathematical Analysis of the Navier-Stokes Equations," (Japanese) University of Tokyo Press, Tokyo, 2009. Google Scholar

[8]

L. Rosenhead, ed., "Laminar Boundary Layers," Clarendon, Oxford, 1966. Google Scholar

[9]

Y. Sone, Asymptotic theory of flow of rarefied gas over a smooth boundary I, in"Rarefied Gas Dynamics" (eds. L. Trilling and H. Y. Wachman), Vol. 1, Academic, New York, (1969), 243-253. Google Scholar

[10]

Y. Sone, Asymptotic theory of flow of rarefied gas over a smooth boundary II, in "Rarefied Gas Dynamics" (ed. D. Dini), Vol. 2, Editrice Tecnico Scientifica, Pisa, (1971), 737-749. Google Scholar

[11]

Y. Sone, Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers, in "Advances in Kinetic Theory and Continuum Mechanics" (eds. R. Gatignol and Soubbaramayer), Springer, Berlin, (1991), 19-31. Google Scholar

[12]

Y. Sone, "Kinetic Theory and Fluid Dynamics," Birkhäuser, Boston, 2002; Supplementary Notes and Errata: Kyoto University Research Information Repository. Available from: http://hdl.handle.net/2433/66099 . Google Scholar

[13]

Y. Sone, "Molecular Gas Dynamics: Theory, Techniques, and Applications," Birkhäuser, Boston, 2007; Supplementary Notes and Errata: Kyoto University Research Information Repository. Available from: http://hdl.handle.net/2433/66098. Google Scholar

[14]

Y. Sone and K. Aoki, Steady gas flows past bodies at small Knudsen numbers-Boltzmann and hydrodynamic systems, Transp. Theory Stat. Phys., 16 (1987), 189-199. Google Scholar

[15]

J. T. Stuart, The viscous flow near a stagnation point when the external flow has a uniform vorticity, J. Aerosp. Sci., 26 (1959), 124-125. Google Scholar

[16]

K. Tamada, Stagnation point flow of rarefied gas, J. Phys. Soc. Jpn., 22 (1967), 1284-1295. doi: 10.1143/JPSJ.22.1284.  Google Scholar

[17]

K. Tamada, Two-dimensional stagnation point flow impinging obliquely on a plane wall, J. Phys. Soc. Jpn., 46 (1979), 310-311. doi: 10.1143/JPSJ.46.310.  Google Scholar

[18]

C. Y. Wang, Exact solutions of the steady-state Navier-Stokes equations, in "Annu. Rev. Fluid Mech.," Vol. 23, 159-177, Annual Reviews, Palo Alto, CA, 1991.  Google Scholar

[19]

C. Y. Wang, Stagnation flows with slip: Exact solutions of the Navier-Stokes equations, Z. angew. Math. Phys., 54 (2003), 184-189. doi: 10.1007/PL00012632.  Google Scholar

[20]

C. Y. Wang, Similarity stagnation point solutions of the Navier-Stokes equations--review and extension, Eur. J. Mech. B Fluids, 27 (2008), 678-683. doi: 10.1016/j.euromechflu.2007.11.002.  Google Scholar

show all references

References:
[1]

R. Aris, "Vectors, Tensors, and the Basic Equations of Fluid Mechanics," Chap. 1, Dover, New York, 1989. Google Scholar

[2]

J. M. Dorrepaal, An exact solution of the Navier-Stokes equation which describes nonorthogonal stagnation-point flow in two dimensions, J. Fluid Mech., 163 (1986), 141-147. doi: 10.1017/S0022112086002240.  Google Scholar

[3]

K. Hiemenz, Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszlynder, Dinglers J., 326 (1911), 321-324, 344-348, 357-362, 372-376, 391-393, 407-410. Google Scholar

[4]

J. D. Hoffman, "Numerical Methods for Engineers and Scientists," McGraw-Hill, New York, 1993. Google Scholar

[5]

H. W. Liepmann and A. Roshko, "Elements of Gasdynamics. Galcit Aeronautical Series," John Wiley & Sons, Inc., New York, Chapman & Hall, Ltd., London, 1957.  Google Scholar

[6]

M. J. Martin and I. D. Boyd, Falkner-Skan flow over a wedge with slip boundary conditions, J. Thermophys. Heat Trans., 24 (2010), 263-270. Google Scholar

[7]

H. Okamoto, "Mathematical Analysis of the Navier-Stokes Equations," (Japanese) University of Tokyo Press, Tokyo, 2009. Google Scholar

[8]

L. Rosenhead, ed., "Laminar Boundary Layers," Clarendon, Oxford, 1966. Google Scholar

[9]

Y. Sone, Asymptotic theory of flow of rarefied gas over a smooth boundary I, in"Rarefied Gas Dynamics" (eds. L. Trilling and H. Y. Wachman), Vol. 1, Academic, New York, (1969), 243-253. Google Scholar

[10]

Y. Sone, Asymptotic theory of flow of rarefied gas over a smooth boundary II, in "Rarefied Gas Dynamics" (ed. D. Dini), Vol. 2, Editrice Tecnico Scientifica, Pisa, (1971), 737-749. Google Scholar

[11]

Y. Sone, Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers, in "Advances in Kinetic Theory and Continuum Mechanics" (eds. R. Gatignol and Soubbaramayer), Springer, Berlin, (1991), 19-31. Google Scholar

[12]

Y. Sone, "Kinetic Theory and Fluid Dynamics," Birkhäuser, Boston, 2002; Supplementary Notes and Errata: Kyoto University Research Information Repository. Available from: http://hdl.handle.net/2433/66099 . Google Scholar

[13]

Y. Sone, "Molecular Gas Dynamics: Theory, Techniques, and Applications," Birkhäuser, Boston, 2007; Supplementary Notes and Errata: Kyoto University Research Information Repository. Available from: http://hdl.handle.net/2433/66098. Google Scholar

[14]

Y. Sone and K. Aoki, Steady gas flows past bodies at small Knudsen numbers-Boltzmann and hydrodynamic systems, Transp. Theory Stat. Phys., 16 (1987), 189-199. Google Scholar

[15]

J. T. Stuart, The viscous flow near a stagnation point when the external flow has a uniform vorticity, J. Aerosp. Sci., 26 (1959), 124-125. Google Scholar

[16]

K. Tamada, Stagnation point flow of rarefied gas, J. Phys. Soc. Jpn., 22 (1967), 1284-1295. doi: 10.1143/JPSJ.22.1284.  Google Scholar

[17]

K. Tamada, Two-dimensional stagnation point flow impinging obliquely on a plane wall, J. Phys. Soc. Jpn., 46 (1979), 310-311. doi: 10.1143/JPSJ.46.310.  Google Scholar

[18]

C. Y. Wang, Exact solutions of the steady-state Navier-Stokes equations, in "Annu. Rev. Fluid Mech.," Vol. 23, 159-177, Annual Reviews, Palo Alto, CA, 1991.  Google Scholar

[19]

C. Y. Wang, Stagnation flows with slip: Exact solutions of the Navier-Stokes equations, Z. angew. Math. Phys., 54 (2003), 184-189. doi: 10.1007/PL00012632.  Google Scholar

[20]

C. Y. Wang, Similarity stagnation point solutions of the Navier-Stokes equations--review and extension, Eur. J. Mech. B Fluids, 27 (2008), 678-683. doi: 10.1016/j.euromechflu.2007.11.002.  Google Scholar

[1]

Raffaele Esposito, Mario Pulvirenti. Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas. Kinetic & Related Models, 2010, 3 (2) : 281-297. doi: 10.3934/krm.2010.3.281
[2]

Najwa Najib, Norfifah Bachok, Norihan Md Arifin, Fadzilah Md Ali. Stability analysis of stagnation point flow in nanofluid over stretching/shrinking sheet with slip effect using buongiorno's model. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 423-431. doi: 10.3934/naco.2019041
[3]

Etienne Bernard, Laurent Desvillettes, Franç cois Golse, Valeria Ricci. A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures. Kinetic & Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003
[4]

Evgenii S. Baranovskii. Steady flows of an Oldroyd fluid with threshold slip. Communications on Pure & Applied Analysis, 2019, 18 (2) : 735-750. doi: 10.3934/cpaa.2019036
[5]

Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345
[6]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic & Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025
[7]

Chun Liu, Jie Shen. On liquid crystal flows with free-slip boundary conditions. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 307-318. doi: 10.3934/dcds.2001.7.307
[8]

Marzia Bisi, Tommaso Ruggeri, Giampiero Spiga. Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics. Kinetic & Related Models, 2018, 11 (1) : 71-95. doi: 10.3934/krm.2018004
[9]

Michael Herty, Christian Ringhofer. Averaged kinetic models for flows on unstructured networks. Kinetic & Related Models, 2011, 4 (4) : 1081-1096. doi: 10.3934/krm.2011.4.1081
[10]

Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271
[11]

Gernot Greschonig. Real cocycles of point-distal minimal flows. Conference Publications, 2015, 2015 (special) : 540-548. doi: 10.3934/proc.2015.0540
[12]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483
[13]

Cédric Galusinski, Mazen Saad. A nonlinear degenerate system modelling water-gas flows in porous media. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 281-308. doi: 10.3934/dcdsb.2008.9.281
[14]

Alexander V. Bobylev, Sergey V. Meleshko. On group symmetries of the hydrodynamic equations for rarefied gas. Kinetic & Related Models, 2021, 14 (3) : 469-482. doi: 10.3934/krm.2021012
[15]

Céline Baranger, Marzia Bisi, Stéphane Brull, Laurent Desvillettes. On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases. Kinetic & Related Models, 2018, 11 (4) : 821-858. doi: 10.3934/krm.2018033
[16]

Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 219-251. doi: 10.3934/krm.2014.7.219
[17]

Karsten Matthies, George Stone, Florian Theil. The derivation of the linear Boltzmann equation from a Rayleigh gas particle model. Kinetic & Related Models, 2018, 11 (1) : 137-177. doi: 10.3934/krm.2018008
[18]

César Nieto, Mauricio Giraldo, Henry Power. Boundary integral equation approach for stokes slip flow in rotating mixers. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 1019-1044. doi: 10.3934/dcdsb.2011.15.1019
[19]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[20]

Gilberto M. Kremer, Wilson Marques Jr.. Fourteen moment theory for granular gases. Kinetic & Related Models, 2011, 4 (1) : 317-331. doi: 10.3934/krm.2011.4.317

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences