The Riemann Problem for Chaplygin Gas Flow in a Duct with Discontinuous Cross-Section

Abstract

The fluid flows in a variable cross-section duct are nonconservative because of the source term. Recently, the Riemann problem and the interactions of the elementary waves for the compressible isentropic gas in a variable cross-section duct were studied. In this paper, the Riemann problem for Chaplygin gas flow in a duct with discontinuous cross-section is studied. The elementary waves include rarefaction waves, shock waves, delta waves and stationary waves.

Keywords

Duct flow / Chaplygin gas equations / Nonconservative / Riemann problem / Delta wave / Stationary wave

References

[1] Andrianov N, Warnecke G. On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math., 2004, 64: 878-901

[2] Baer M R, Nunziato J W. A two-phase mixture theory for the deflagration-todetonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flows, 1986, 12: 861-889

[3] Brenier Y. Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas equations. J. Math. Fluid Mech., 2005, 7: 326-331

[4] Chaplygin S. On gas jets. Sci. Mem. Moscow Univ. Math. Phys., 1904, 21: 1-121

[5] Dal Maso G, LeFloch P G, Murat F. Definition and weak stability of nonconservative products. J. Math. Pures Appl., 1995, 74(9): 483-548

[6] Guo L H, Sheng W C, Zhang T. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system. Commun. Pure Appl. Anal., 2010, 9(2): 431-458

[7] Kong D X, Wang Y Z. Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases. Sci. China Math., 2010, 53(3): 719-738

[8] Kroner D, LeFloch P G, Thanh M D. The minimum entropy principle for fluid flows in a nozzle with discontinuous cross-section. M2AN Math. Model Numer. Anal., 2008, 42: 425-442

[9] Kroner D, Thanh M D. Numerical solutions to compressible flows in a nozzle with variable cross-section. SIAM J. Numer. Anal., 2005, 43: 796-824

[10] Lai G, Sheng W C, Zheng Y X. Simple waves and pressure delta waves for a Chaplygin gas in multi-dimensions. Discrete Contin. Dyn. Syst., 2011, 31(2): 489-523

[11] Lax, P. D., Shock waves and entropy, Contributions to Functional Analysis, Zarantonello, E. A. (ed.), Academic Press, New York, 1971, 603–634.

[12] LeFloch, P. G., Shock Waves for Nonlinear Hyperbolic Systems in Nonconservative Form, Institute for Mathematics and its Application, Univ. of Minnesota, 593, preprint, 1989.

[13] LeFloch P G, Thanh M D. The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Commun. Math. Sci., 2003, 1: 763-797

[14] Marchesin D, Paes-Leme P J. A Riemann problem in gas dynamics with bifurcation. Comput. Math. Appl. Part A, 1986, 12: 433-455

[15] Serre D. Multidimensional shock interaction for a Chaplygin gas. Arch. Ration. Mech. Anal., 2009, 191: 539-577

[16] Sheng W C, Zhang Q L. Interaction of the elementary waves of isentropic flow in a variable cross-section duct. Commun. Math. Sci., 2018, 16(6): 1659-1684

[17] Sheng, W. C. and Zhang, T., The Riemann problem for transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137 (654), 1999, viii+77pp.

[18] Thanh M D. The Riemann problem for a nonisentropic fluid in a nozzle with discontinuous cross-sectional area. SIAM J. Appl. Math., 2009, 69: 1501-1519

[19] Tsien H S. Two dimensional subsonic flow of compressible fluids. J. Aeronaut. Sci., 1939, 6: 399-407