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

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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

Abstract
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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 and 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.
[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.
[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.
[4]	J. D. Hoffman, "Numerical Methods for Engineers and Scientists," McGraw-Hill, New York, 1993.
[5]	H. W. Liepmann and A. Roshko, "Elements of Gasdynamics. Galcit Aeronautical Series," John Wiley & Sons, Inc., New York, Chapman & Hall, Ltd., London, 1957.
[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.
[7]	H. Okamoto, "Mathematical Analysis of the Navier-Stokes Equations," (Japanese) University of Tokyo Press, Tokyo, 2009.
[8]	L. Rosenhead, ed., "Laminar Boundary Layers," Clarendon, Oxford, 1966.
[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.
[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.
[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.
[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 .
[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.
[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.
[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.
[16]	K. Tamada, Stagnation point flow of rarefied gas, J. Phys. Soc. Jpn., 22 (1967), 1284-1295. doi: 10.1143/JPSJ.22.1284.
[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.
[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.
[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.
[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.

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References:

[1]	R. Aris, "Vectors, Tensors, and the Basic Equations of Fluid Mechanics," Chap. 1, Dover, New York, 1989.
[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.
[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.
[4]	J. D. Hoffman, "Numerical Methods for Engineers and Scientists," McGraw-Hill, New York, 1993.
[5]	H. W. Liepmann and A. Roshko, "Elements of Gasdynamics. Galcit Aeronautical Series," John Wiley & Sons, Inc., New York, Chapman & Hall, Ltd., London, 1957.
[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.
[7]	H. Okamoto, "Mathematical Analysis of the Navier-Stokes Equations," (Japanese) University of Tokyo Press, Tokyo, 2009.
[8]	L. Rosenhead, ed., "Laminar Boundary Layers," Clarendon, Oxford, 1966.
[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.
[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.
[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.
[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 .
[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.
[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.
[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.
[16]	K. Tamada, Stagnation point flow of rarefied gas, J. Phys. Soc. Jpn., 22 (1967), 1284-1295. doi: 10.1143/JPSJ.22.1284.
[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.
[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.
[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.
[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.

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