# American Institute of Mathematical Sciences

September  2021, 20(9): 3235-3258. doi: 10.3934/cpaa.2021104

## Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model

 School of Mathematics and Statistics, Ningbo University, Ningbo, Zhejiang Province, 315211, China

Received  September 2020 Revised  May 2021 Published  September 2021 Early access  June 2021

Fund Project: The author is supported by K. C. Wong Magna Fund in Ningbo University

The phenomena of concentration and cavitation for the Riemann problem of the Baer-Nunziato (BN) two-phase flow model has been investigated in this paper. By using the characteristic analysis method, the formation of $\delta-$waves and vacuum states are obtained as the pressure for both phases vanish in the BN model. The solid contact wave is carefully dealt. The comparison with the solutions of pressureless two-phase model shows that, two shock waves tend to a $\delta-$shock solution, and two rarefaction waves tend to a two contact discontinuity solution when the solid contact discontinuity is involved. Moreover, the detailed Riemann solutions for two-phase flow model are given as the double pressure parameters vanish. This may contribute to the design of numerical schemes in the future research.

Citation: Qinglong Zhang. Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3235-3258. doi: 10.3934/cpaa.2021104
##### References:
 [1] N. Andrianov and G. Warnecke, On the solution to the Riemann problem for the compressible duct flow, SIAM J. Appl. Math., 64 (2004), 878-901.  doi: 10.1137/S0036139903424230.  Google Scholar [2] N. Andrianov and G. Warnecke, The Riemann problem for the Baer-Nunziato two-phase flow model, J. Comput. Phys., 195 (2004), 434-464.  doi: 10.1016/j.jcp.2003.10.006.  Google Scholar [3] M. R. Baer and J. W. Nunziato, A two-phase mixture theory for the deflagration-todetonation transition (DDT) in reactive granular materials, Int. J. Multiphase Flows, 12 (1986), 861-889.   Google Scholar [4] T. Chang and L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, Pitman Monographs, Longman Scientific and technica, 1989.  Google Scholar [5] G. Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34 (2003), 925-938.  doi: 10.1137/S0036141001399350.  Google Scholar [6] G. Q. Chen and H. Liu, Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Phys. D, 189 (2004), 141-165.  doi: 10.1016/j.physd.2003.09.039.  Google Scholar [7] P. Embid and M. Baer, Mathematical analysis of a two-phase continuum mixture theory, Continuum Mech. Thermodyn., 4 (1992), 279-312.  doi: 10.1007/BF01129333.  Google Scholar [8] L. H. Guo, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal., 9 (2010), 431-458.  doi: 10.3934/cpaa.2010.9.431.  Google Scholar [9] P. G. LeFloch and M. D. Thanh, The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Commun. Math. Sci., 1 (2003), 763-797.   Google Scholar [10] J. Q. Li, Note on the compressible Euler equations with zero temperature, Appl. Math. Lett., 14 (2001), 519-523.  doi: 10.1016/S0893-9659(00)00187-7.  Google Scholar [11] R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys., 150 (1999), 425-467.  doi: 10.1006/jcph.1999.6187.  Google Scholar [12] R. Saurel and O. Lemetayer, A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation, J. Fluid Mech., 431 (2001), 239-271.   Google Scholar [13] C. Shen and M. Sun, Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model, J. Diff. Equ., 249 (2010), 3024-3051.  doi: 10.1016/j.jde.2010.09.004.  Google Scholar [14] W. Sheng, G. Wang and G. Yin, Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes, Nonlinear Anal., 22 (2015), 115-128.  doi: 10.1016/j.nonrwa.2014.08.007.  Google Scholar [15] W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, in: Mem. Amer. Math. Soc., AMS, Providence, 1999. doi: 10.1090/memo/0654.  Google Scholar [16] W. Sheng and Q. Zhang, Interaction of the elementary waves of isentropic flow in a variable cross-section duct, Commun. Math. Sci., 16 (2008), 1659-1684.  doi: 10.4310/CMS.2018.v16.n6.a8.  Google Scholar [17] J. Smoller, Shock waves and Reaction-Diffusion Equations, 2nd ed., Sringer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [18] D. Tan, T. Zhang and Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differ. Equ., 112 (1994), 1-32.  doi: 10.1006/jdeq.1994.1093.  Google Scholar [19] M. D. Thanh, The Riemann problem for a nonisentropic fluid in a nozzle with discontinuous cross-sectional area, SIAM J. Appl. Math., 69 (2009), 1501-1519.  doi: 10.1137/080724095.  Google Scholar [20] G. D. Wang, The Riemann problem for one dimensional generalized Chaplygin gas dynamics, J. Math. Anal. Appl., 403 (2013), 434-450.  doi: 10.1016/j.jmaa.2013.02.026.  Google Scholar [21] H. Yang and J. Wang, Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas., J. Math. Anal. Appl., 413 (2014), 800-820.  doi: 10.1016/j.jmaa.2013.12.025.  Google Scholar [22] G. Yin and W. Sheng, Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations, Chin. Ann. Math. Ser. B, 29 (2008), 611-622.  doi: 10.1007/s11401-008-0009-x.  Google Scholar

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##### References:
 [1] N. Andrianov and G. Warnecke, On the solution to the Riemann problem for the compressible duct flow, SIAM J. Appl. Math., 64 (2004), 878-901.  doi: 10.1137/S0036139903424230.  Google Scholar [2] N. Andrianov and G. Warnecke, The Riemann problem for the Baer-Nunziato two-phase flow model, J. Comput. Phys., 195 (2004), 434-464.  doi: 10.1016/j.jcp.2003.10.006.  Google Scholar [3] M. R. Baer and J. W. Nunziato, A two-phase mixture theory for the deflagration-todetonation transition (DDT) in reactive granular materials, Int. J. Multiphase Flows, 12 (1986), 861-889.   Google Scholar [4] T. Chang and L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, Pitman Monographs, Longman Scientific and technica, 1989.  Google Scholar [5] G. Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34 (2003), 925-938.  doi: 10.1137/S0036141001399350.  Google Scholar [6] G. Q. Chen and H. Liu, Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids, Phys. D, 189 (2004), 141-165.  doi: 10.1016/j.physd.2003.09.039.  Google Scholar [7] P. Embid and M. Baer, Mathematical analysis of a two-phase continuum mixture theory, Continuum Mech. Thermodyn., 4 (1992), 279-312.  doi: 10.1007/BF01129333.  Google Scholar [8] L. H. Guo, The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system, Commun. Pure Appl. Anal., 9 (2010), 431-458.  doi: 10.3934/cpaa.2010.9.431.  Google Scholar [9] P. G. LeFloch and M. D. Thanh, The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Commun. Math. Sci., 1 (2003), 763-797.   Google Scholar [10] J. Q. Li, Note on the compressible Euler equations with zero temperature, Appl. Math. Lett., 14 (2001), 519-523.  doi: 10.1016/S0893-9659(00)00187-7.  Google Scholar [11] R. Saurel and R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Phys., 150 (1999), 425-467.  doi: 10.1006/jcph.1999.6187.  Google Scholar [12] R. Saurel and O. Lemetayer, A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation, J. Fluid Mech., 431 (2001), 239-271.   Google Scholar [13] C. Shen and M. Sun, Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model, J. Diff. Equ., 249 (2010), 3024-3051.  doi: 10.1016/j.jde.2010.09.004.  Google Scholar [14] W. Sheng, G. Wang and G. Yin, Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes, Nonlinear Anal., 22 (2015), 115-128.  doi: 10.1016/j.nonrwa.2014.08.007.  Google Scholar [15] W. Sheng and T. Zhang, The Riemann problem for the transportation equations in gas dynamics, in: Mem. Amer. Math. Soc., AMS, Providence, 1999. doi: 10.1090/memo/0654.  Google Scholar [16] W. Sheng and Q. Zhang, Interaction of the elementary waves of isentropic flow in a variable cross-section duct, Commun. Math. Sci., 16 (2008), 1659-1684.  doi: 10.4310/CMS.2018.v16.n6.a8.  Google Scholar [17] J. Smoller, Shock waves and Reaction-Diffusion Equations, 2nd ed., Sringer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [18] D. Tan, T. Zhang and Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differ. Equ., 112 (1994), 1-32.  doi: 10.1006/jdeq.1994.1093.  Google Scholar [19] M. D. Thanh, The Riemann problem for a nonisentropic fluid in a nozzle with discontinuous cross-sectional area, SIAM J. Appl. Math., 69 (2009), 1501-1519.  doi: 10.1137/080724095.  Google Scholar [20] G. D. Wang, The Riemann problem for one dimensional generalized Chaplygin gas dynamics, J. Math. Anal. Appl., 403 (2013), 434-450.  doi: 10.1016/j.jmaa.2013.02.026.  Google Scholar [21] H. Yang and J. Wang, Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas., J. Math. Anal. Appl., 413 (2014), 800-820.  doi: 10.1016/j.jmaa.2013.12.025.  Google Scholar [22] G. Yin and W. Sheng, Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations, Chin. Ann. Math. Ser. B, 29 (2008), 611-622.  doi: 10.1007/s11401-008-0009-x.  Google Scholar
The case $u_{s_-}<u_{s_+}$ and $u_{g_-}<u_{g_+}$
The limit Riemann solution as $u_{g_-}<u_{g_+}$ and $u_{s_-}<u_{s_+}$
The case $u_{g_-} = u_{g_+}$ and $U_{g_+}^{*}\in {\rm I\!I\!I}$
The formation of $\delta$ shock wave of phase $g$
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