Applied Mathematics and Mechanics

, Volume 36, Issue 8, pp 1105–1112 | Cite as

Exact solutions to drift-flux multiphase flow models through Lie group symmetry analysis

  • B. Bira
  • T. R. SekharEmail author


In the present paper, Lie group symmetry method is used to obtain some exact solutions for a hyperbolic system of partial differential equations (PDEs), which governs an isothermal no-slip drift-flux model for multiphase flow problem. Those symmetries are used for the governing system of equations to obtain infinitesimal transformations, which consequently reduces the governing system of PDEs to a system of ODEs. Further, the solutions of the system of ODEs which in turn produces some exact solutions for the PDEs are presented. Finally, the evolutionary behavior of weak discontinuity is discussed.


multiphase flow drift-flux models Lie group analysis exact solution weak scontinuity 

Chinese Library Classification

TV 314 

2010 Mathematics Subject Classification

22E05 35L02 76T99 37D99 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bluman, G. W. and Cole, J. D. Similarity Methods for Differential Equations, Springer, New York (1974)CrossRefzbMATHGoogle Scholar
  2. [2]
    Bluman, G. W. and Kumei, S. Symmetries and Differential Equations, Springer, New York (1989)CrossRefzbMATHGoogle Scholar
  3. [3]
    Abd-el-Malek, M. B. and Amin, A. M. Lie group analysis of nonlinear inviscid flows with a free surface under gravity. Journal of Computational and Applied Mathematics, 258, 17–29 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Abd-el-Malek, M. B. and Amin, A. M. Lie group method for solving generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equations. Applied Mathematics and Computation, 224, 501–516 (2013)MathSciNetCrossRefGoogle Scholar
  5. [5]
    Bira, B. and Sekhar, T. R. Symmetry group analysis and exact solutions of isentropic magnetogasdynamics. Indian Journal of Pure and Applied Mathematics, 44(2), 153–165 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Bira, B. and Sekhar, T. R. Lie group analysis and propagation of weak discontinuity in onedimensional ideal isentropic magnetogasdynamics. Applicable Analysis, 93(12), 2598–2607 (2014)MathSciNetCrossRefGoogle Scholar
  7. [7]
    Zhang, Y., Liu, X., and Wang, G. Symmetry reductions and exact solutions of the (2+1)- dimensional Jaulent-Miodek equation. Applied Mathematics and Computation, 219(3), 911–916 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Sharma, V. D. and Radha, R. Exact solutions of Euler equations of ideal gasdynamics via Lie group analysis. Zeitschrift für angewandte Mathematik und Physik ZAMP, 59(6), 1029–1038 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [10]
    [9] Jena, J. Lie group transformations for self-similar shocks in a gas with dust particles. Mathematical Methods in the Applied Sciences, 32(16), 2035–2049 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Donato, A. and Oliveri, F. Reduction to autonomous form by group analysis and exact solutions of axisymmetric MHD equations. Mathematical and Computer Modelling, 18(10), 83–90 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Afify, A. A. Some new exact solutions for MHD aligned creeping flow and heat transfer in second grade fluids by using Lie group analysis. Nonlinear Analysis, 70(9), 3298–3306 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Sharma, V. D. and Radha, R. Similarity solutions for converging shocks in a relaxing gas. International Journal of Engineering Science, 33(4), 535–553 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Pandey, M. and Sharma, V. D. Interaction of a characteristic shock with a weak discontinuity in a non-ideal gas. Wave Motion, 44(5), 346–354 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Sekhar, T. R. and Sharma, V. D. Similarity solutions for three dimensional Euler equations using Lie group analysis. Applied Mathematics and Computation, 196(1), 147–157 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Sekhar, T. R. and Sharma, V. D. Evolution of weak discontinuities in shallow water equations. Applied Mathematics Letters, 23(3), 327–330 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Sharma, V. D. Quasilinear Hperbolic Systems, Compressible Flows and Waves, CRC Press, Boca Raton (2010)CrossRefGoogle Scholar
  17. [17]
    Banda, M. K., Herty, M., and Ngotchouye, J. M. T. Toward a mathematical analysis for drift-flux multi-phase models in networks. SIAM Journal on Scientific Computing, 31(6), 4633–4653 (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Science and TechnologyBerhampurIndia
  2. 2.Department of MathematicsIndian Institute of Technology Kharagpurnew delhiIndia

Personalised recommendations