Self-Similar Solutions to the Lin-Reissner-Tsien Equation

John Haussermann; K. Vajravelu; Robert A. Van Gorder

doi:10.3879/j.issn.1000-0887.2011.11.009

Volume 32 Issue 11

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John Haussermann, K. Vajravelu, Robert A. Van Gorder. Self-Similar Solutions to the Lin-Reissner-Tsien Equation[J]. Applied Mathematics and Mechanics, 2011, 32(11): 1352-1360. doi: 10.3879/j.issn.1000-0887.2011.11.009

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Self-Similar Solutions to the Lin-Reissner-Tsien Equation

doi: 10.3879/j.issn.1000-0887.2011.11.009

Department of Mathematics, University of Central Florida, P. O. Box 161364, Orlando, FL 32816-1364, USA

Received Date: 2011-01-24
Rev Recd Date: 2011-07-08
Publish Date: 2011-11-15

Abstract

Abstract

The Lin-Reissner-Tsien equation describes unsteady transonic flows under the transonic approximation.The equation was reduced to an ordinary differential equation via a similarity transformation. The resulting equation was then solved analytically,and in some cases even exactly.Numerical simulations were provided for the cases in which there were no exact solutions.Traveling wave solutions were also obtained.
- Lin-Reissner-Tsien equation,
- self-similar solutions,
- transonic approximation,
- nonlinear partial differential equation

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References(12)

References

[1]	Lin C C, Reissner E, Tsien H S. On two-dimensional non-steady motion of a slender body in a compressible fluid[J]. J Math Phys, 1948, 27: 220-231.
[2]	Ames W F, Nucci W N. Analysis of fluid equations by group methods[J]. J Eng Mech, 1985, 20(2): 181-187.
[3]	Kuz'min A G, Ivanova A V. The structural instability of transonic flow associated with amalgamation/splitting of supersonic regions[J]. Fluid Dynamics, 2004, 18(5): 335-344.
[4]	Bogdanov A N, Duesperov V N. The properties of asymptotic expansions for the parameters of unsteady transonic flow with axial symmetry[J]. Journal of Applied Mathematics and Mechanics, 2008, 72(1): 33-35. doi: 10.1016/j.jappmathmech.2008.03.009
[5]	Bibik Yu V, Duesperov V N, Popov S P. Structure of time dependent transonic flows in pane channels[J]. Fluid Dynamics, 2005, 40(2): 315-325. doi: 10.1007/s10697-005-0071-y
[6]	Glazatov S N. On solvability of a spatial periodic problem for the Lin-Reissner-Tsien equation of transonic gas dynamics[J]. Mathematical Notes, 2010, 87(1/2): 130-134. doi: 10.1134/S0001434610010177
[7]	Bogdanov A N, Duesperov V N. Higher approximations of the transonic expansion in problems of unsteady transonic flow[J]. Journal of Applied Mathematics and Mechanics, 1997, 61(5): 775-786. doi: 10.1016/S0021-8928(97)00101-9
[8]	Diesperov V N, Korolev G L. Triple-deck analysis of formation and evolution of supersonic zones and local separation zones in unsteady transonic flow over a surface roughness element[J]. Computational Mathematics and Mathematical Physics, 2005, 45(3): 516-523.
[9]	Bogdanov A N, Duesperov V N. Tollmien-Schlichting waves in a transonic boundary layer: excitation from the outer flow and from the surface[J]. Journal of Applied Mathematics and Mechanics, 2007, 71(2): 258-268. doi: 10.1016/j.jappmathmech.2007.06.009
[10]	Titov S S. A method finite-dimensional rings for solving nonlinear equations of mathematical physics[C] Ivanova T P. Aerodynamics. Saratov: Saratov University, 1988: 104-110. (in Russian)
[11]	Svirshchevskii S R. Lie-Bcklund symmetries of linear ODEs and generalized separation of variables in nonlinear equations[J]. Phys Lett A, 1995, 199(5/6): 344-348. doi: 10.1016/0375-9601(95)00136-Q
[12]	Polyanin A D, Zaitsev V F. Handbook of Nonlinear Partial Differential Equations[M]. Boca Raton: Chapman & Hall/CRC, 2004.