Abstract
The accurate Riemann solutions in fully explicit forms are obtained for the one-dimensional inviscid, compressible, isothermal liquid–gas two-phase flow model of drift-flux type under the action of the unique external force of gravity for a pipeline placed on an inclined plane. It is also shown that the Riemann solution involves either a curved delta shock wave or the vacuum state for the pressureless case. Moreover, the asymptotic limits of Riemann solutions are investigated in detail by letting the pressure drop to zero, in which the formation of curved delta shock wave is obtained from the curved Riemann solution made up of backward shock wave, middle contact discontinuity and forward shock as well as the formation of vacuum state is also achieved from the curved Riemann solution consisting of backward rarefaction wave, middle contact discontinuity and forward rarefaction wave. In addition, some representative numerical results are offered to confirm the formation of delta shock wave and vacuum state.
Similar content being viewed by others
Availability of data and materials
Not applicable.
References
Andrianov, N., Warnecke, G.: The Riemann problem for the Baer–Nunziato two-phase flow model. J. Comput. Phys. 195, 434–464 (2004)
Pudasaini, S.P.: A general two-phase debris flow model. J. Geophys. Res. 117, F03010 (2012)
Baer, M.R., Nunziato, J.W.: A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flow 12, 861–889 (1986)
Evje, S., Flatten, T.: On the wave structure of two-phase flow models. SIAM J. Appl. Math. 67, 487–511 (2007)
Flatten, T., Munkejord, S.T.: The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model. ESAIM Math. Model. Numer. Anal. 40, 735–764 (2006)
Friis, H.A., Evje, S.: Global weak solutions for a gas-liquid model with external forces and general pressure law. SIAM J. Appl. Math. 71, 409–442 (2011)
Huang, F., Wang, D., Yuan, D.: Nonlinear stability and existence of vortex sheets for inviscid liquid–gas two-phase flow. Discrete Contin. Dyn. Syst. 39, 3535–3575 (2019)
Ruan, L., Trakhinin, Y.: Elementary symmetrization of inviscid two-fluid flow equations giving a number of instant results. Physica D 391, 66–71 (2019)
Zeidan, D., Romenski, E., Slaouti, A., Toro, E.F.: Numerical study of wave propagation in compressible two-phase flow. Int. J. Numer. Methods Fluids 54, 393–417 (2007)
Zeidan, D.: The Riemann problem for a hyperbolic model of two-phase flow in conservative form. Int. J. Comput. Fluid Dyn. 25, 299–318 (2011)
Kuila, S., Sekhar, T.R., Zeidan, D.: A robust and accurate Riemann solver for a compressible two-phase flow model. Appl. Math. Comput. 265, 681–695 (2015)
Kuila, S., Sekhar, T.R., Zeidan, D.: On the Riemann problem simulation for the drift-flux equations of two-phase flows. Int. J. Comput. Methods 13, 1650009 (2016)
Minhajul, D., Zeidan, T.R.: Sekhar, On the wave interactions in the drift-flux equations of two-phase flows. Appl. Math. Comput. 327, 117–131 (2018)
Rohde, C., Zeiler, C.: On Riemann solvers and kinetic relations for isothermal two-phase flows with surface tension. Z. Angew. Math. Phys. 69, 76 (2018)
Hantke, M., Thein, F.: A general existence result for isothermal two-phase flows with phase transition. J. Hyperbolic Differ. Equ. 16, 595–637 (2019)
Shen, C., Sun, M.: Exact Riemann solutions for the drift-flux equations of two-phase flow under gravity. J. Differ. Equ. 314, 1–55 (2022)
Saurel, R., Chinnayya, A., Carmouze, Q.: Modelling compressible dense and dilute two-phase flows. Phys. Fluids 29, 063301 (2017)
Shen, C.: The asymptotic limits of Riemann solutions for the isentropic drift-flux model of compressible two-phase flows. Math. Methods Appl. Sci. 43, 3673–3688 (2020)
Shen, C.: The singular limits of solutions to the Riemann problem for the liquid–gas two-phase isentropic flow model. J. Math. Phys. 61, 081502 (2020)
Sun, M.: The intrinsic phenomena of cavitation and concentration in Riemann solutions for the isentropic two-phase model with the logarithmic equation of state. J. Math. Phys. 62, 101502 (2021)
Zhang, Q.: Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model, Commun. Pure. Appl. Anal. 20, 3235–3258 (2021)
Hantke, M., Matern, C., Ssemaganda, V., Warnecke, G.: The Riemann problem for a weakly hyperbolic two-phase flow model of a dispersed phase in a carrier fluid. Q. Appl. Math. 78, 431–467 (2020)
Shen, C.: The transition of Riemann solutions for the drift-flux model with the pressure law for the extended Chaplygin gas. Phys. Fluids 35, 046105 (2023)
Wei, Z., Sun, M.: The Riemann problem for a simplified two-phase flow model with the Chaplygin pressure law under the external force. Int. J. Non-Linear Mech. 144, 104082 (2022)
Chen, G.Q., Liu, H.: 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, 925–938 (2003)
Mitrovic, D., Nedeljkov, M.: Delta-shock waves as a limit of shock waves. J. Hyperbolic Differ. Equ. 4, 629–653 (2007)
Sahoo, M.R., Sen, A.: Limiting behavior of scaled general Euler equations of compressible fluid flow. Z. Angew. Math. Phys. 71, 51 (2020)
Sheng, W., Wang, G., Yin, G.: Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes. Nonlinear Anal. RWA 22, 115–128 (2015)
Sun, M.: Concentration and cavitation phenomena of Riemann solutions for the isentropic Euler system with the logarithmic equation of state. Nonlinear Anal. RWA 53, 103068 (2020)
Yang, H., Liu, J.: Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation. SCIENCE CHINA Math. 58, 2329–2346 (2015)
Guo, L., Li, T., Yin, G.: The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. Commun. Pure Appl. Anal. 16, 295–309 (2017)
Guo, L., Li, T., Yin, G.: The limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term. J. Math. Anal. Appl. 455, 127–140 (2017)
Sen, A., Raja Sekhar, T.: The limiting behavior of the Riemann solution to the isentropic Euler system for the logarithmic equation of state with a source term. Math. Methods Appl. Sci. 44, 7207–7227 (2021)
Sheng, S., Shao, Z.: The vanishing adiabatic exponent limits of Riemann solutions to the isentropic Euler equations for power law with a Coulomb-like friction term. J. Math. Phys. 60, 101504 (2019)
Sheng, S., Shao, Z.: Concentration of mass in the pressureless limit of the Euler equations of one-dimensional compressible fluid flow. Nonlinear Anal. RWA 52, 103039 (2020)
Shen, C., Sun, M.: The Riemann problem for the one-dimensional isentropic Euler system under the body force with varying gamma law. Physica D 448, 133731 (2023)
Zhang, Q.: Concentration in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equations with friction. J. Math. Phys. 60, 101508 (2019)
Sheng, W., Zhang, T.: The Riemann problem for the transportation equations in gas dynamics. Mem. Amer. Math. Soc. 137(N654) (1999) AMS: Providence
Danilov, V.G., Shelkovich, V.M.: Dynamics of propagation and interaction of \(\delta \)-shock waves in conservation law systems. J. Differ. Equ. 211, 333–381 (2005)
Danilov, V.G., Mitrovic, D.: Delta shock wave formation in the case of triangular hyperbolic system of conservation laws. J. Differ. Equ. 245, 3704–3734 (2008)
Nilsson, B., Shelkovich, V.M.: Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks. Appl. Anal. 90, 1677–1689 (2011)
Nilsson, B., Rozanova, O.S., Shelkovich, V.M.: Mass, momentum and energy conservation laws in zero-pressure gas dynamics and \(\delta \)-shocks:II. Appl. Anal. 90, 831–842 (2011)
Kalisch, H., Mitrovic, D.: Singular solutions of a fully nonlinear \(2\times 2\) system of conservation laws. Proc. Edinb. Math. Soc. 55, 711–729 (2012)
Kalisch, H., Mitrovic, D.: Singular solutions for the shallow-water equations. IMA J. Appl. Math. 77, 340–350 (2012)
Kalisch, H., Mitrovic, D., Teyekpiti, V.: Existence and uniqueness of singular solutions for a conservation law arising in magnetohydrodynamics. Nonlinearity 31, 5463–5483 (2018)
Zeidan, D., Jana, S., Kuila, S., Raja Sekhar, T.: Solution to the Riemann problem for the drift-flux model with modified Chaplygin two-phase flows. Int. J. Numer. Methods Fluids 95, 242–261 (2023)
Acknowledgements
The authors are very grateful to the anonymous reviewer and editor for their very helpful comments and suggestions, which improve the original manuscript greatly.
Funding
This work is partially supported by Natural Science Foundation of Shandong Province (ZR2023MA067).
Author information
Authors and Affiliations
Contributions
The authors wrote the main manuscript text. Both authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Ethics approval and consent to participate
Not applicable.
Consent for publication
All authors read and approved the final manuscript.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is partially supported by Natural Science Foundation of Shandong Province (ZR2023MA067).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sun, M., Wei, Z. The vanishing pressure limits of Riemann solutions to the isothermal two-phase flow model under the external force. Z. Angew. Math. Phys. 74, 222 (2023). https://doi.org/10.1007/s00033-023-02115-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-023-02115-5