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Lie symmetry analysis for two-phase flow with mass transfer

  • Andronikos Paliathanasis ORCID logo EMAIL logo
Published/Copyright: October 6, 2022
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 5

Abstract

We perform a complete symmetry classification for the hyperbolic system of partial differential equations, which describes a drift-flux two-phase flow in a one-dimensional pipe, with a mass-transfer term between the two different phases of the fluid. In addition, we consider the polytropic equation of states parameter and gravitational forces. For general values of the polytropic indices, we find that the fluid equations are invariant under the elements of a three-dimensional Lie algebra. However, additional Lie point symmetries follow for specific values of the polytropic indices. The one-dimensional systems are investigated in each case of the classification scheme, and the similarity transformations are calculated in order to reduce the fluid equations into a system of ordinary differential equations. Exact solutions are derived, while the reduced systems are studied numerically.

Keywords: exact solutions; invariants; Lie symmetries; optimal system; two-phase flow
Corresponding author: Andronikos Paliathanasis, Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, Republic of South Africa; and Departamento de Matemáticas, Universidad Católica del Norte, Avda. Angamos 0610, 1280 Casilla, Antofagasta, Chile, E-mail:

Funding source: National Research Foundation of South Africa

Award Identifier / Grant number: 131604

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work was supported by the National Research Foundation of South Africa (grant number 131604).

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

  4. Data availability statement: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

[1] G. F. Hewitt, “Two-phase flow and its applications: past, present, and future,” Heat Tran. Eng., vol. 4, p. 67, 1983. https://doi.org/10.1080/01457638108939596.Search in Google Scholar

[2] A. Poullikkas, “Two phase flow performance of nuclear reactor cooling pumps,” Prog. Nucl. Energy, vol. 36, p. 123, 2000. https://doi.org/10.1016/s0149-1970(00)00007-x.Search in Google Scholar

[3] I. L. Ryhming, “On plane, laminar two-phase jet flow,” Acta Mech., vol. 11, p. 117, 1971. https://doi.org/10.1007/bf01368122.Search in Google Scholar

[4] N. F. Okechi and S. Asghar, “Two-phase flow in a groovy curved channel,” Eur. J. Mech. B, vol. 88, p. 191, 2021. https://doi.org/10.1016/j.euromechflu.2021.03.004.Search in Google Scholar

[5] C. Dong and T. Hibiki, “Drift-flux parameter modeling of vertical downward gas-liquid two-phase flows for interfacial drag force formulation,” Nucl. Eng. Des., vol. 378, p. 111185, 2021. https://doi.org/10.1016/j.nucengdes.2021.111185.Search in Google Scholar

[6] B. M. Dia, B. Saad, and M. Saad, “Modeling and simulation of partially miscible two-phase flow with kinetics mass transfer,” Math. Comput. Simulat., vol. 186, p. 29, 2021. https://doi.org/10.1016/j.matcom.2020.04.023.Search in Google Scholar

[7] M. F. Trujillo, “Reexamining the one-fluid formulation for two-phase flows,” Int. J. Multiphas. Flow, vol. 141, p. 103672, 2021. https://doi.org/10.1016/j.ijmultiphaseflow.2021.103672.Search in Google Scholar

[8] C. Shao, G. Zhong, and J. Zhou, “Study on gas-liquid two-phase flow in the suction chamber of a centrifugal pump and its dimensionless characteristics,” Nucl. Eng. Des., vol. 380, p. 111298, 2021. https://doi.org/10.1016/j.nucengdes.2021.111298.Search in Google Scholar

[9] K. Gjerstad and O. M. Aamo, “A dynamic model of two-phase flow for simulating gas kick and facilitating model-based control in drilling operations,” J. Petoleum Sci. Eng., vol. 205, p. 108902, 2021. https://doi.org/10.1016/j.petrol.2021.108902.Search in Google Scholar

[10] K. Fukagata, N. Kasagi, P. Ua-arayaporn, and T. Himeno, “Numerical simulation of gas-liquid two-phase flow and convective heat transfer in a micro tube,” Int. J. Heat Fluid Flow, vol. 28, p. 72, 2007.10.1016/j.ijheatfluidflow.2006.04.010Search in Google Scholar

[11] S. Schmelter, M. Olbrich, E. Schmeyer, and M. Bar, “Numerical simulation, validation, and analysis of two-phase slug flow in large horizontal pipes,” Flow Meas. Instrum., vol. 73, p. 101722, 2020. https://doi.org/10.1016/j.flowmeasinst.2020.101722.Search in Google Scholar

[12] D. Maochang, Y. Xijun, C. Dawei, Q. Fang, and Z. Shijun, “Numerical simulation of gas-particle two-phase flow in a nozzle with DG method,” Discrete Dynam Nat. Soc., vol. 2019, p. 706041, 2019. https://doi.org/10.1155/2019/7060481.Search in Google Scholar

[13] S. G. Hajra, S. Kandel, and S. P. Pudasaini, “On analytical solutions of a two-phase mass flow model,” Nonlinear Anal. R. World Appl., vol. 41, p. 412, 2017.10.1016/j.nonrwa.2017.09.009Search in Google Scholar

[14] S. G. Hajra, S. Kandel, and S. P. Pudasaini, “Lie symmetry solutions for two-phase mass flows,” Int. J. Non Lin. Mech., vol. 77, p. 325, 2015.10.1016/j.ijnonlinmec.2015.09.010Search in Google Scholar

[15] L. V. Ovsiannikov, Group Analysis of Differential Equations, New York, Academic Press, 1982.10.1016/B978-0-12-531680-4.50012-5Search in Google Scholar

[16] N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws, Florida, CRS Press LLC, 2000.Search in Google Scholar

[17] G. W. Bluman and S. Kumei, Symmetries of Differential Equations, New York, Springer-Verlag, 1989.10.1007/978-1-4757-4307-4Search in Google Scholar

[18] H. Stephani, Differential Equations: Their Solutions Using Symmetry, New York, Cambridge University Press, 1989.Search in Google Scholar

[19] P. J. Olver, Applications of Lie Groups to Differential Equations, New York, Springer-Verlag, 1993.10.1007/978-1-4612-4350-2Search in Google Scholar

[20] P. G. L. Leach and V. M. Gorringe, “A conserved Laplace–Runge–Lenz-like vector for a class of three-dimensional motions,” Phys. Lett. A, vol. 133, p. 289, 1988. https://doi.org/10.1016/0375-9601(88)90446-x.Search in Google Scholar

[21] R. O. Popovych, O. O. Vaneeva, and N. M. Ivanova, “Potential nonclassical symmetries and solutions of fast diffusion equation,” Phys. Lett. A, vol. 362, p. 166, 2007. https://doi.org/10.1016/j.physleta.2006.10.015.Search in Google Scholar

[22] A. R. Chowdhury and P. K. Chanda, “On the Lie symmetry approach to Small’s equation of nonlinear optics,” J. Phys. A: Math. Gen., vol. 18, p. L117, 1985. https://doi.org/10.1088/0305-4470/18/3/004.Search in Google Scholar

[23] S. V. Meleshko and N. F. Samatova, “Invariant solutions of the two-dimensional shallow water equations with a particular class of bottoms,” AIP Conf. Proc., vol. 2164, p. 050003, 2019.10.1063/1.5130801Search in Google Scholar

[24] A. Bihlo, N. Poltavets, and R. O. Popovych, “Lie symmetries of two-dimensional shallow water equations with variable bottom topography,” Chaos, vol. 30, p. 073132, 2020. https://doi.org/10.1063/5.0007274.Search in Google Scholar PubMed

[25] A. V. Aksenov and K. P. Druzhkov, “Conservation laws and symmetries of the shallow water system above rough bottom,” J. Phys. Conf. Ser., vol. 722, p. 012001, 2016. https://doi.org/10.1088/1742-6596/722/1/012001.Search in Google Scholar

[26] V. A. Dorodnitsym and E. I. Kaptsov, “Shallow water equations in Lagrangian coordinates: symmetries, conservation laws and its preservation in difference models,” Commun. Nonlinear Sci. Numer. Simul., vol. 89, p. 105343, 2020.10.1016/j.cnsns.2020.105343Search in Google Scholar

[27] J. G. Liu, Z. F. Zeng, Y. He, and G. P. Ai, “A class of exact solution of (3+1)-dimensional generalized shallow water equation system,” Int. J. Nonlinear Sci. Numer Simul., vol. 16, p. 114, 2013.10.1515/ijnsns-2013-0114Search in Google Scholar

[28] G. M. Webb, G. P. Zank, E. K. Kaghashvili, and R. E. Ratkiewicz, “Magnetohydrodynamic waves in non-uniform flows I: a variational approach,” J. Plasma Phys., vol. 71, p. 785, 2005. https://doi.org/10.1017/s0022377805003739.Search in Google Scholar

[29] G. M. Webb, G. P. Zank, E. K. Kaghashvili, and R. E. Ratkiewicz, “Magnetohydrodynamic waves in non-uniform flows II: stress-energy tensors, conservation laws and Lie symmetries,” J. Plasma Phys., vol. 71, p. 811, 2005. https://doi.org/10.1017/s0022377805003740.Search in Google Scholar

[30] G. M. Webb and G. P. Zank, “Fluid relabelling symmetries, Lie point symmetries and the Lagrangian map in magnetohydrodynamics and gas dynamics,” J. Phys. A: Math. Theor., vol. 40, p. 545, 2006. https://doi.org/10.1088/1751-8113/40/3/013.Search in Google Scholar

[31] G. M. Webb and S. C. Anco, “Conservation laws in magnetohydrodynamics and fluid dynamics: Lagrangian approach,” AIP Conf. Proc., vol. 2153, p. 020024, 2019.10.1063/1.5125089Search in Google Scholar

[32] A. Paliathanasis, “Similarity inner solutions for the pulsar equation,” Math. Methods Appl. Sci., vol. 43, p. 716, 2020. https://doi.org/10.1002/mma.5951.Search in Google Scholar

[33] A. Paliathanasis, “One-dimensional optimal system for 2D rotating ideal gas,” Symmetry, vol. 11, p. 1115, 2019. https://doi.org/10.3390/sym11091115.Search in Google Scholar

[34] B. Bira and T. R. Sekhar, “Exact solutions to drift-flux multiphase flow models through Lie group symmetry analysis,” Appl. Math. Mech. Engl. Ed., vol. 36, p. 1105, 2015. https://doi.org/10.1007/s10483-015-1968-7.Search in Google Scholar

[35] B. Bira, T. R. Sekhar, and D. Zeidan, “Application of Lie groups to compressible model of two-phase flows,” Comput. Math. Appl., vol. 71, p. 46, 2016.10.1016/j.camwa.2015.10.016Search in Google Scholar

[36] A. Paliathanasis, “Shallow-water equations with complete Coriolis force: group properties and similarity solutions,” Math. Methods Appl. Sci., vol. 44, p. 11631, 2021. https://doi.org/10.1002/mma.7168.Search in Google Scholar

[37] S. Kumar, D. Kumar, and A. Kumar, “Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons for a higher-dimensional Fokas equation,” Chaos, Solit. Fractals, vol. 142, p. 110507, 2021. https://doi.org/10.1016/j.chaos.2020.110507.Search in Google Scholar

[38] T. Raja Sekhar and P. Satapathy, “Group classification for isothermal drift flux model of two phase flows,” Compt. Math. Appl., vol. 72, p. 1436, 2016. https://doi.org/10.1016/j.camwa.2016.07.017.Search in Google Scholar

[39] P. Satapathy and T. R. Sekhar, “Optimal system, invariant solutions and evolution of weak discontinuity for isentropic drift flux model,” Appl. Math. Compt., vol. 334, p. 107, 2018. https://doi.org/10.1016/j.amc.2018.03.114.Search in Google Scholar

[40] M. Kumar and A. K. Tiwari, “On group-invariant solutions of Konopelchenko–Dubrovsky equation by using Lie symmetry approach,” Nonlinear Dynam., vol. 94, p. 475, 2018. https://doi.org/10.1007/s11071-018-4372-1.Search in Google Scholar

[41] Y. Taitel and D. Barena, Encyclopedia of Two-Phase Heat Transfer and Flow I, Fundamentals and Methods Volume 1: Modeling of Gas Liquid Flow in Pipes, J. R. Thome, Ed., Singapore, World Scientific Publishing Co. Pte. Ltd., 2015.Search in Google Scholar

[42] V. V. Morozov, “Classification of six-dimensional nilpotent Lie algebras,” Izv. Vysshikh Uchebnykh Zavend. Mat., vol. 5, p. 161, 1958.Search in Google Scholar

[43] G. M. Mubarakzyanov, “On solvable Lie algebras,” Izv. Vysshikh Uchebnykh Zavend. Mat., vol. 32, p. 114, 1963.Search in Google Scholar

[44] G. M. Mubarakzyanov, “Classification of real structures of five-dimensional Lie algebras,” Izv. Vysshikh Uchebnykh Zavend. Mat., vol. 34, p. 99, 1963.Search in Google Scholar

[45] G. M. Mubarakzyanov, “Classification of solvable six-dimensional Lie algebras with one nilpotent base element,” Izv. Vysshikh Uchebnykh Zavend. Mat., vol. 35, p. 104, 1963.Search in Google Scholar

[46] R. Cherniha and S. Kovalenko, “Lie symmetries of nonlinear boundary value problems,” Commun. Nonlinear Sci. Numer. Simul., vol. 17, p. 71, 2012. https://doi.org/10.1016/j.cnsns.2011.04.028.Search in Google Scholar
Received: 2022-03-27
Accepted: 2022-09-18
Published Online: 2022-10-06

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijnsns-2022-0126
Keywords for this article
exact solutions; invariants; Lie symmetries; optimal system; two-phase flow

Articles in the same Issue

  1. Frontmatter
  2. Original Research Article
  3. Hybrid solitary wave solutions of the Camassa–Holm equation
  4. Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions
  5. A Chebyshev collocation method for solving the non-linear variable-order fractional Bagley–Torvik differential equation
  6. Higher order codimension bifurcations in a discrete-time toxic-phytoplankton–zooplankton model with Allee effect
  7. Numerical modeling of the dam-break flood over natural rivers on movable beds
  8. A class of piecewise fractional functional differential equations with impulsive
  9. Lie symmetry analysis for two-phase flow with mass transfer
  10. Asymptotic behavior for stochastic plate equations with memory in unbounded domains
  11. Discussion on controllability of non-densely defined Hilfer fractional neutral differential equations with finite delay
  12. A linearized finite difference scheme for time–space fractional nonlinear diffusion-wave equations with initial singularity
  13. Ground state solutions of Schrödinger system with fractional p-Laplacian
  14. Bifurcation analysis of a new stochastic traffic flow model
  15. An uncertainty measure based on Pearson correlation as well as a multiscale generalized Shannon-based entropy with financial market applications
  16. Hilfer fractional stochastic evolution equations on infinite interval
  17. Iterative learning control for conformable stochastic impulsive differential systems with randomly varying trial lengths
  18. Chebyshev wavelet-Picard technique for solving fractional nonlinear differential equations
  19. Theoretical assessment of the impact of awareness programs on cholera transmission dynamic
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  23. A (2 + 1)-dimensional variable-coefficients extension of the Date–Jimbo–Kashiwara–Miwa equation: Lie symmetry analysis, optimal system and exact solutions
  24. Mathematical model of fluid flow in a double constricted tapered tube with permeable boundary
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