Interaction of elementary waves with a weak discontinuity in an isothermal drift-flux model of compressible two-phase flows
Authors:
Minhajul and T. Raja Sekhar
Journal:
Quart. Appl. Math. 77 (2019), 671-688
MSC (2010):
Primary 35L45, 35L65, 58J45; Secondary 35Q35, 35L67, 76T10
DOI:
https://doi.org/10.1090/qam/1539
Published electronically:
April 16, 2019
MathSciNet review:
3962588
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Abstract: In this paper, we study the interaction of elementary waves of the Riemann problem with a weak discontinuity for an isothermal no-slip compressible gas-liquid drift flux equation of two-phase flows. We construct the solution of the Riemann problem in terms of a one parameter family of curves. Using the properties of elementary waves, we prove a necessary and sufficient condition on initial data for which the solution of the Riemann problem consists of a left shock, contact discontinuity, and a right shock. Moreover, we derive the amplitudes of weak discontinuity and discuss the interactions of weak discontinuity with shocks and contact discontinuity. Finally, we carry out some tests to investigate the effect of shock strength and initial data on the jump in shock acceleration and the amplitudes of reflected and transmitted waves.
References
- B. Bira, T. Raja Sekhar, and G. P. Raja Sekhar, Collision of characteristic shock with weak discontinuity in non-ideal magnetogasdynamics, Comput. Math. Appl. 75 (2018), no. 11, 3873–3883. MR 3797029, DOI https://doi.org/10.1016/j.camwa.2018.02.034
- Guy Boillat and Tommaso Ruggeri, On the evolution law of weak discontinuities for hyperbolic quasilinear systems, Wave Motion 1 (1979), no. 2, 149–152. MR 533471, DOI https://doi.org/10.1016/0165-2125%2879%2990017-9
- Guy Boillat and Tommaso Ruggeri, Reflection and transmission of discontinuity waves through a shock wave. General theory including also the case of characteristic shocks, Proc. Roy. Soc. Edinburgh Sect. A 83 (1979), no. 1-2, 17–24. MR 538581, DOI https://doi.org/10.1017/S0308210500011331
- Maren Hantke, Wolfgang Dreyer, and Gerald Warnecke, Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition, Quart. Appl. Math. 71 (2013), no. 3, 509–540. MR 3112826, DOI https://doi.org/10.1090/S0033-569X-2013-01290-X
- Mamoru Ishii and Takashi Hibiki, Thermo-fluid dynamics of two-phase flow, 2nd ed., Springer, New York, 2011. With a foreword by Lefteri H. Tsoukalas. MR 3203021
- Alan Jeffrey, The propagation of weak discontinuities in quasi-linear hyperbolic systems with discontinuous coefficients. I. Fundamental theory, Applicable Anal. 3 (1973), 79–100. MR 393868, DOI https://doi.org/10.1080/00036817308839058
- Alan Jeffrey, The propagation of weak discontinuities in quasi-linear hyperbolic systems with discontinuous coefficients part II-special cases and application, Applicable Analysis 3 (1974), no. 4, 359–375.
- Alan Jeffrey, Quasilinear hyperbolic systems and waves, Pitman Publishing, London (1976).
- J. Jena and V. D. Sharma, Interaction of a characteristic shock with a weak discontinuity in a relaxing gas, J. Engrg. Math. 60 (2008), no. 1, 43–53. MR 2374229, DOI https://doi.org/10.1007/s10665-007-9182-2
- Sahadeb Kuila, T. Raja Sekhar, and D. Zeidan, A robust and accurate Riemann solver for a compressible two-phase flow model, Appl. Math. Comput. 265 (2015), 681–695. MR 3373515, DOI https://doi.org/10.1016/j.amc.2015.05.086
- Sahadeb Kuila, T. Raja Sekhar, and D. Zeidan, On the Riemann problem simulation for the drift-flux equations of two-phase flows, International Journal of Computational Methods 13 (2016), no. 01, 1650009.
- S. Kuila and T. Raja Sekhar, Interaction of weak shocks in drift-flux model of compressible two-phase flows, Chaos Solitons Fractals 107 (2018), 222–227. MR 3759506, DOI https://doi.org/10.1016/j.chaos.2017.12.030
- L. D. Landau and E. M. Lifshitz, Fluid mechanics, Course of Theoretical Physics, Vol. 6, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959. Translated from the Russian by J. B. Sykes and W. H. Reid. MR 0108121
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI https://doi.org/10.1002/cpa.3160100406
- Yujin Liu and Wenhua Sun, Elementary wave interactions in magnetogasdynamics, Indian J. Pure Appl. Math. 47 (2016), no. 1, 33–57. MR 3477809, DOI https://doi.org/10.1007/s13226-016-0172-9
- Vítor Matos and Dan Marchesin, Compositional flow in porous media: Riemann problem for three alkanes, Quart. Appl. Math. 75 (2017), no. 4, 737–767. MR 3686519, DOI https://doi.org/10.1090/qam/1477
- V. V. Menon, V. D. Sharma, and A. Jeffrey, On the general behavior of acceleration waves, Applicable Anal. 16 (1983), no. 2, 101–120. MR 709814, DOI https://doi.org/10.1080/00036818308839462
- Andrea Mentrelli, Tommaso Ruggeri, Masaru Sugiyama, and Nanrong Zhao, Interaction between a shock and an acceleration wave in a perfect gas for increasing shock strength, Wave Motion 45 (2008), no. 4, 498–517. MR 2406845, DOI https://doi.org/10.1016/j.wavemoti.2007.09.005
- Minhajul, D. Zeidan, and T. Raja Sekhar, On the wave interactions in the drift-flux equations of two-phase flows, Appl. Math. Comput. 327 (2018), 117–131. MR 3761030, DOI https://doi.org/10.1016/j.amc.2018.01.021
- Marko Nedeljkov, Singular shock waves in interactions, Quart. Appl. Math. 66 (2008), no. 2, 281–302. MR 2416774, DOI https://doi.org/10.1090/S0033-569X-08-01109-5
- L. Pan, S. W. Webb, and C. M. Oldenburg, Analytical solution for two-phase flow in a wellbore using the drift-flux model, Advances in Water Resources 34 (2011), no. 12, 1656–1665.
- Manoj Pandey and V. D. Sharma, Interaction of a characteristic shock with a weak discontinuity in a non-ideal gas, Wave Motion 44 (2007), no. 5, 346–354. MR 2311431, DOI https://doi.org/10.1016/j.wavemoti.2006.12.002
- Ch. Radha, V. D. Sharma, and A. Jeffrey, On interaction of shock waves with weak discontinuities, Appl. Anal. 50 (1993), no. 3-4, 145–166. MR 1278323, DOI https://doi.org/10.1080/00036819308840191
- R. Radha and V. D. Sharma, Interaction of a weak discontinuity with elementary waves of Riemann problem, J. Math. Phys. 53 (2012), no. 1, 013506, 12. MR 2919545, DOI https://doi.org/10.1063/1.3671383
- T. Raja Sekhar and Minhajul, Elementary wave interactions in blood flow through artery, J. Math. Phys. 58 (2017), no. 10, 101502, 20. MR 3707594, DOI https://doi.org/10.1063/1.5004666
- T. Raja Sekhar and V. D. Sharma, Riemann problem and elementary wave interactions in isentropic magnetogasdynamics, Nonlinear Anal. Real World Appl. 11 (2010), no. 2, 619–636. MR 2571237, DOI https://doi.org/10.1016/j.nonrwa.2008.10.036
- B. Riemann, Über die fortpflanzung ebener luftwellen von endlicher schwingungsweite, Gott. Abh. Math. Cl. 8 (1860), 43–65.
- Tommaso Ruggeri, Interaction between a discontinuity wave and a shock wave: critical time for the fastest transmitted wave, example of the polytropic fluid, Applicable Anal. 11 (1980), no. 2, 103–112. MR 599258, DOI https://doi.org/10.1080/00036818008839323
- T. Raja Sekhar and V. D. Sharma, Interaction of shallow water waves, Stud. Appl. Math. 121 (2008), no. 1, 1–25. MR 2428542, DOI https://doi.org/10.1111/j.1467-9590.2008.00402.x
- Anupam Sen, T. Raja Sekhar, and V. D. Sharma, Wave interactions and stability of the Riemann solution for a strictly hyperbolic system of conservation laws, Quart. Appl. Math. 75 (2017), no. 3, 539–554. MR 3636168, DOI https://doi.org/10.1090/qam/1466
- Vishnu D. Sharma, Quasilinear hyperbolic systems, compressible flows, and waves, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 142, CRC Press, Boca Raton, FL, 2010. MR 2668539
- V. D. Sharma and V. V. Menon, Further comments on the behavior of acceleration waves of arbitrary shape, Journal of Mathematical Physics 22 (1981), no. 4, 683–684.
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
- Meina Sun, Interactions of elementary waves for the Aw-Rascle model, SIAM J. Appl. Math. 69 (2009), no. 6, 1542–1558. MR 2487160, DOI https://doi.org/10.1137/080731402
- N. Virgopia and F. Ferraioli, Evolution of weak discontinuity waves in self-similar flows and formation of secondary shocks. The “point explosion model”, Z. Angew. Math. Phys. 33 (1982), no. 1, 63–80 (English, with Italian summary). MR 652923, DOI https://doi.org/10.1007/BF00948313
- N. Zuber and J. Findlay, Average volumetric concentration in two-phase flow systems, Journal of heat transfer 87 (1965), no. 4, 453–468.
References
- B. Bira, T. Raja Sekhar, and G. P. Raja Sekhar, Collision of characteristic shock with weak discontinuity in non-ideal magnetogasdynamics, Comput. Math. Appl. 75 (2018), no. 11, 3873–3883. MR 3797029, DOI https://doi.org/10.1016/j.camwa.2018.02.034
- Guy Boillat and Tommaso Ruggeri, On the evolution law of weak discontinuities for hyperbolic quasilinear systems, Wave Motion 1 (1979), no. 2, 149–152. MR 533471, DOI https://doi.org/10.1016/0165-2125%2879%2990017-9
- Guy Boillat and Tommaso Ruggeri, Reflection and transmission of discontinuity waves through a shock wave. General theory including also the case of characteristic shocks, Proc. Roy. Soc. Edinburgh Sect. A 83 (1979), no. 1-2, 17–24. MR 538581, DOI https://doi.org/10.1017/S0308210500011331
- Maren Hantke, Wolfgang Dreyer, and Gerald Warnecke, Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition, Quart. Appl. Math. 71 (2013), no. 3, 509–540. MR 3112826, DOI https://doi.org/10.1090/S0033-569X-2013-01290-X
- Mamoru Ishii and Takashi Hibiki, Thermo-fluid dynamics of two-phase flow, 2nd ed., Springer, New York, 2011. With a foreword by Lefteri H. Tsoukalas. MR 3203021
- Alan Jeffrey, The propagation of weak discontinuities in quasi-linear hyperbolic systems with discontinuous coefficients. I. Fundamental theory, with Collection of articles dedicated to Eberhard Hopf on the occasion of his 70th birthday, Applicable Anal. 3 (1973), 79–100. MR 0393868, DOI https://doi.org/10.1080/00036817308839058
- Alan Jeffrey, The propagation of weak discontinuities in quasi-linear hyperbolic systems with discontinuous coefficients part II-special cases and application, Applicable Analysis 3 (1974), no. 4, 359–375.
- Alan Jeffrey, Quasilinear hyperbolic systems and waves, Pitman Publishing, London (1976).
- J. Jena and V. D. Sharma, Interaction of a characteristic shock with a weak discontinuity in a relaxing gas, J. Engrg. Math. 60 (2008), no. 1, 43–53. MR 2374229, DOI https://doi.org/10.1007/s10665-007-9182-2
- Sahadeb Kuila, T. Raja Sekhar, and D. Zeidan, A robust and accurate Riemann solver for a compressible two-phase flow model, Appl. Math. Comput. 265 (2015), 681–695. MR 3373515, DOI https://doi.org/10.1016/j.amc.2015.05.086
- Sahadeb Kuila, T. Raja Sekhar, and D. Zeidan, On the Riemann problem simulation for the drift-flux equations of two-phase flows, International Journal of Computational Methods 13 (2016), no. 01, 1650009.
- S. Kuila and T. Raja Sekhar, Interaction of weak shocks in drift-flux model of compressible two-phase flows, Chaos Solitons Fractals 107 (2018), 222–227. MR 3759506, DOI https://doi.org/10.1016/j.chaos.2017.12.030
- L. D. Landau and E. M. Lifshitz, Fluid mechanics, Translated from the Russian by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, Vol. 6, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1959. MR 0108121
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 0093653, DOI https://doi.org/10.1002/cpa.3160100406
- Yujin Liu and Wenhua Sun, Elementary wave interactions in magnetogasdynamics, Indian J. Pure Appl. Math. 47 (2016), no. 1, 33–57. MR 3477809, DOI https://doi.org/10.1007/s13226-016-0172-9
- Vítor Matos and Dan Marchesin, Compositional flow in porous media: Riemann problem for three alkanes, Quart. Appl. Math. 75 (2017), no. 4, 737–767. MR 3686519, DOI https://doi.org/10.1090/qam/1477
- V. V. Menon, V. D. Sharma, A. Jeffrey, , and On the general behavior of acceleration waves, Applicable Anal. 16 (1983), no. 2, 101–120. MR 709814, DOI https://doi.org/10.1080/00036818308839462
- Andrea Mentrelli, Tommaso Ruggeri, Masaru Sugiyama, and Nanrong Zhao, Interaction between a shock and an acceleration wave in a perfect gas for increasing shock strength, Wave Motion 45 (2008), no. 4, 498–517. MR 2406845, DOI https://doi.org/10.1016/j.wavemoti.2007.09.005
- Minhajul, D. Zeidan, and T. Raja Sekhar, On the wave interactions in the drift-flux equations of two-phase flows, Appl. Math. Comput. 327 (2018), 117–131. MR 3761030, DOI https://doi.org/10.1016/j.amc.2018.01.021
- Marko Nedeljkov, Singular shock waves in interactions, Quart. Appl. Math. 66 (2008), no. 2, 281–302. MR 2416774, DOI https://doi.org/10.1007/s00205-009-0281-2
- L. Pan, S. W. Webb, and C. M. Oldenburg, Analytical solution for two-phase flow in a wellbore using the drift-flux model, Advances in Water Resources 34 (2011), no. 12, 1656–1665.
- Manoj Pandey and V. D. Sharma, Interaction of a characteristic shock with a weak discontinuity in a non-ideal gas, Wave Motion 44 (2007), no. 5, 346–354. MR 2311431, DOI https://doi.org/10.1016/j.wavemoti.2006.12.002
- Ch. Radha, V. D. Sharma, and A. Jeffrey, On interaction of shock waves with weak discontinuities, Appl. Anal. 50 (1993), no. 3-4, 145–166. MR 1278323, DOI https://doi.org/10.1080/00036819308840191
- R. Radha and V. D. Sharma, Interaction of a weak discontinuity with elementary waves of Riemann problem, J. Math. Phys. 53 (2012), no. 1, 013506, 12. MR 2919545, DOI https://doi.org/10.1063/1.3671383
- T. Raja Sekhar and Minhajul, Elementary wave interactions in blood flow through artery, J. Math. Phys. 58 (2017), no. 10, 101502, 20. MR 3707594, DOI https://doi.org/10.1063/1.5004666
- T. Raja Sekhar and V. D. Sharma, Riemann problem and elementary wave interactions in isentropic magnetogasdynamics, Nonlinear Anal. Real World Appl. 11 (2010), no. 2, 619–636. MR 2571237, DOI https://doi.org/10.1016/j.nonrwa.2008.10.036
- B. Riemann, Über die fortpflanzung ebener luftwellen von endlicher schwingungsweite, Gott. Abh. Math. Cl. 8 (1860), 43–65.
- Tommaso Ruggeri, Interaction between a discontinuity wave and a shock wave: critical time for the fastest transmitted wave, example of the polytropic fluid, Applicable Anal. 11 (1980), no. 2, 103–112. MR 599258, DOI https://doi.org/10.1080/00036818008839323
- T. Raja Sekhar and V. D. Sharma, Interaction of shallow water waves, Stud. Appl. Math. 121 (2008), no. 1, 1–25. MR 2428542, DOI https://doi.org/10.1111/j.1467-9590.2008.00402.x
- Anupam Sen, T. Raja Sekhar, and V. D. Sharma, Wave interactions and stability of the Riemann solution for a strictly hyperbolic system of conservation laws, Quart. Appl. Math. 75 (2017), no. 3, 539–554. MR 3636168, DOI https://doi.org/10.1090/qam/1466
- Vishnu D. Sharma, Quasilinear hyperbolic systems, compressible flows, and waves, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 142, CRC Press, Boca Raton, FL, 2010. MR 2668539
- V. D. Sharma and V. V. Menon, Further comments on the behavior of acceleration waves of arbitrary shape, Journal of Mathematical Physics 22 (1981), no. 4, 683–684.
- Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
- Meina Sun, Interactions of elementary waves for the Aw-Rascle model, SIAM J. Appl. Math. 69 (2009), no. 6, 1542–1558. MR 2487160, DOI https://doi.org/10.1137/080731402
- N. Virgopia and F. Ferraioli, Evolution of weak discontinuity waves in self-similar flows and formation of secondary shocks. The “point explosion model”, Z. Angew. Math. Phys. 33 (1982), no. 1, 63–80 (English, with Italian summary). MR 652923, DOI https://doi.org/10.1007/BF00948313
- N. Zuber and J. Findlay, Average volumetric concentration in two-phase flow systems, Journal of heat transfer 87 (1965), no. 4, 453–468.
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Additional Information
Minhajul
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
MR Author ID:
1232942
Email:
minhaz@maths.iitkgp.ac.in
T. Raja Sekhar
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
MR Author ID:
831418
Email:
trajasekhar@maths.iitkgp.ac.in
Received by editor(s):
April 14, 2018
Received by editor(s) in revised form:
January 30, 2019
Published electronically:
April 16, 2019
Additional Notes:
The first author is highly thankful to the Ministry of Human Resource Development, Government of India, for the institute fellowship (grant no. IIT/ACAD/PGS& R/F.II/2/14MA90J08).
Research support from the Science and Engineering Research Board, Department of Science and Technology, Government of India (Ref. No. SB/FTP/MS-047/2013) is gratefully acknowledged by the second author.
Dedicated:
Dedicated to Professor V. D. Sharma on the occasion of his $70^{th}$ birthday
Article copyright:
© Copyright 2019
Brown University
