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Supersonic flow of a Chaplygin gas past a delta wing

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Abstract

We consider the problem of supersonic flow of a Chaplygin gas past a delta wing with a shock or a rarefaction wave attached to the leading edges. The flow under study is described by the three-dimensional steady Euler system. In conical coordinates, this problem can be reformulated as a boundary value problem for a nonlinear equation of mixed type. The type of this equation depends fully on the solutions of the problem itself, and thus it cannot be determined in advance. We overcome the difficulty by establishing a crucial Lipschitz estimate, and finally prove the unique existence of the solution via the method of continuity.

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References

  1. Ashley H, Landahl M. Aerodynamics of Wings and Bodies. New York: Dover, 1985

    MATH  Google Scholar 

  2. Babaev D A. Numerical solution of the problem of supersonic flow past the lower surface of a delta wing. AIAA J, 1963, 1: 2224–2231

    Article  Google Scholar 

  3. Beeman E R, Powers S A. A method for determining the complete flow field around conical wings at supersonic/hypersonic speeds. https://doi.org/10.2514/6.1969-646, 1969

  4. Brenier Y. Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations. J Math Fluid Mech, 2005, 7: S326–S331

    Article  MathSciNet  Google Scholar 

  5. Chaplygin S. On gas jets. Sci Mem Moscow Univ Math Phys, 1904, 21: 1–121

    Google Scholar 

  6. Chen G-Q, Fang B. Stability of transonic shocks in steady supersonic flow past multidimensional wedges. Adv Math, 2017, 314: 493–539

    Article  MathSciNet  Google Scholar 

  7. Chen G-Q, Feldman M. Global solutions of shock reflection by large-angle wedges for potential flow. Ann of Math (2), 2010, 171: 1067–1182

    Article  MathSciNet  Google Scholar 

  8. Chen S. Existence of local solution to supersonic flow past a three-dimensional wing. Adv Appl Math, 1992, 13: 273–304

    Article  MathSciNet  Google Scholar 

  9. Chen S. Linear approximation for supersonic flow past a delta wing. Arch Ration Mech Anal, 1997, 140: 319–333

    Article  MathSciNet  Google Scholar 

  10. Chen S, Qu A. Riemann boundary value problems and reflection of shock for the Chaplygin gas. Sci China Math, 2012, 55: 671–685

    Article  MathSciNet  Google Scholar 

  11. Chen S, Qu A. Two-dimensional Riemann problems for Chaplygin gas. SIAM J Math Anal, 2012, 44: 2146–2178

    Article  MathSciNet  Google Scholar 

  12. Chen S, Yi C. Global solutions for supersonic flow past a delta wing. SIAM J Math Anal, 2015, 47: 80–126

    Article  MathSciNet  Google Scholar 

  13. Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. New York: Springer-Verlag, 1984

    MATH  Google Scholar 

  14. Elling V, Liu T-P. The ellipticity principle for self-similar potential flows. J Hyperbolic Differ Equ, 2005, 2: 909–917

    Article  MathSciNet  Google Scholar 

  15. Fowell L R. Exact and approximate solutions for the supersonic delta wing. J Aeronaut Sci, 1956, 23: 709–720

    Article  MathSciNet  Google Scholar 

  16. Gad-el-Hak M, Ho C-M. The pitching delta wing. AIAA J, 1985, 23: 1660–1665

    Article  Google Scholar 

  17. Gilbarg D, Hörmander L. Intermediate Schauder estimates. Arch Ration Mech Anal, 1980, 74: 297–314

    Article  MathSciNet  Google Scholar 

  18. Gilbarg D, Trudinger N. Elliptic Partial Differential Equations of Second Order. New York: Springer-Verlag, 2003

    Google Scholar 

  19. Hui W H. Supersonic and hyperbolic flow with attached shock waves over delta wings. Proc R Soc A, 1971, 325: 251–268

    MATH  Google Scholar 

  20. Kamenshchik A, Moschella U, Pasquier V. An alternative to quintessence. Phys Lett B, 2001, 511: 265–268

    Article  Google Scholar 

  21. Lai G, Sheng W. Elementary wave interactions to the compressible Euler equations for Chaplygin gas in two dimensions. SIAM J Appl Math, 2016, 76: 2218–2242

    Article  MathSciNet  Google Scholar 

  22. Li L, Xu G, Yin H. On the instability problem of a 3-D transonic oblique shock wave. Adv Math, 2015, 282: 443–515

    Article  MathSciNet  Google Scholar 

  23. Lieberman G M. Oblique derivative problems in Lipschitz domains: II. Discontinuous boundary data. J Reine Angew Math, 1988, 389: 1–21

    MathSciNet  MATH  Google Scholar 

  24. Miller D S, Wood R M. Leeside flows over delta wings at supersonic speeds. J Aircraft, 1984, 21: 680–686

    Article  Google Scholar 

  25. Popov V A. Dark energy and dark matter unification via superfluid Chaplygin gas. Phys Lett B, 2010, 686: 211–215

    Article  Google Scholar 

  26. Serre D. Multidimensional shock interaction for a Chaplygin gas. Arch Ration Mech Anal, 2009, 191: 539–577

    Article  MathSciNet  Google Scholar 

  27. Serre D. Three-dimensional interaction of shocks in irrotational flows. Conflu Math, 2011, 3: 543–576

    Article  MathSciNet  Google Scholar 

  28. Voskresenskii G P. Numerical solution of the problem of supersonic gas flow past an arbitrary delta wing surface in the compression region. Fluid Dyn, 1968, 3: 90–95

    Article  Google Scholar 

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Acknowledgements

The authors are indebted to Professor Shuxing Chen for many helpful comments. The first author also acknowledges the Center for Mathematical Sciences of Huazhong University of Science and Technology for the invitation and hospitality.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huanggang Normal University, Huanggang, 438000, China

    Bingsong Long

  2. Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, 430074, China

    Chao Yi

Authors
  1. Bingsong Long
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  2. Chao Yi
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Correspondence to Bingsong Long.

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Long, B., Yi, C. Supersonic flow of a Chaplygin gas past a delta wing. Sci. China Math. 65, 2271–2296 (2022). https://doi.org/10.1007/s11425-021-1881-7

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  • DOI: https://doi.org/10.1007/s11425-021-1881-7

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