Analysis of propagation of weakly nonlinear waves in a two-layer fluid with free surface

Authors

  • Ольга Валентинівна Авраменко Volodymyr Vynnychenko Kirovohrad State Pedagogical University 1 Shevchenko str., Kirovohrad, Ukraine, 25002, Ukraine https://orcid.org/0000-0002-7960-1436
  • Володимир Володимирович Нарадовий Volodymyr Vynnychenko Kirovohrad State Pedagogical University 1 Shevchenko str., Kirovohrad, Ukraine, 25002, Ukraine https://orcid.org/0000-0001-5187-8831

DOI:

https://doi.org/10.15587/1729-4061.2015.48282

Keywords:

nonlinear waves, two-layer fluid, wave packet form, free surface

Abstract

The study of wave motions in stratified fluids is one of the main tasks of hydrodynamics, which is caused by both theoretical and practical needs.

Analytical analysis of propagation of weakly nonlinear wave packets in a two-layer fluid of finite depth in the presence of a free surface is performed. As a result, evolution equations of wave packets on the interface and the free surface in the form of the second-order nonlinear differential Schrödinger-type equations were derived. The form of internal and surface waves depending on the ratio of layer densities and the wave number considering the surface tension was analyzed. As a result, the effects of taking into account the second approximation in modeling wave motions in the two-layer system, which leads to blunting or sharpening of the wave crests and troughs were revealed. The analytical results are confirmed by field observations.

Author Biographies

Ольга Валентинівна Авраменко, Volodymyr Vynnychenko Kirovohrad State Pedagogical University 1 Shevchenko str., Kirovohrad, Ukraine, 25002

Doctor of physical and mathematical sciences, Professor, head of the department

Department of applied mathematics, statistics and economics

Володимир Володимирович Нарадовий, Volodymyr Vynnychenko Kirovohrad State Pedagogical University 1 Shevchenko str., Kirovohrad, Ukraine, 25002

Teacher

Department of applied mathematics, statistics and economics

References

  1. Docenko, S. F., Ivanov, V. A., Poberegnuy, Y. A. (2010). Svyaz obrazovaniya voln-ybiyc I meterologicheskih ysloviy v severo-zapadnoy chasti Chernogo moray. Dopovidi NAN Ykrainu, 12, 105–109.
  2. Docenko, S. F., Sannikova, N. K. V. (2011). Analiz dvymernogo rasprostraneniya voln cynami iz elipticheskogo ochaga v pryamolineynuy kanal. MGI NAN Ykrainu, 419–428.
  3. Kilinichenko, V. A., Sekerg-Zenkovich, S. Ya. (2007). Eksperimentalnoe issledovanie voln Faradeya maksimalnoy vusotu Izv. RAN MGG, 6, 103–110.
  4. Kilinichenko, V. A., Sekerg-Zenkovich, S. Ya. (2008). Eksperimentalnoe issledovanie vtorichnuh stacionarnuh techeniy v poverhnostnuh volnah Faradeya. Izv. RAN MGG, 1, 141–148.
  5. Nayfeh, A. H. (1976). Nonlinear Propagation of Wave-Packets on Fluid Interfaces. Journal of Applied Mechanics, 43 (4), 584–588. doi: 10.1115/1.3423936
  6. Selezov, I. T., Avramenko, O. V. (2002). Stryktyra nelineynuh volnovuh paketov na poverhnosti kontakta gidkih sred. Prukladna gidromehanika, 4 (76), 3–13.
  7. Selezov, I. T., Avramenko, O. V. (2001). Ystoychivost volnovuh paketov v sloistuh gidrodinamicheskih sistemah s ychetom poverhnostnogo natyageniya. Prukladna gidromehanika, 3 (75), 38–46.
  8. Selezov, I. T., , O. V. (2001). Evolycionnoe yravnenie tretego poryadka dlya nelineynuh volnovuh paketov pri okolokriticheskih volnovuh chislah. Dinamicheskie sistemu, 19, 58–67.
  9. Selezov, I. T., Avramenko, O. V. (2001). Evolyciya nelineynuh volnovuh paketov v gidrodinamicheskoy sisteme “sloy – polyprostranstvo” s ychetom poverhnostnogo natyageniya. Mat. metodu i fizuko-mehanichni polya, 44, 113–122.
  10. Vincze, M., Kozma, P., Gyüre, B., Jánosi, I. M., Gábor Szabó, K., Tél, T. (2007). Amplified internal pulsations on a stratified exchange flow excited by interaction between a thin sill and external seiche. Journal of Applied Mechanics, 19 (10), 108108-1–108108-4. doi: 10.1063/1.2796182
  11. Makarenko, N. N., Malceva, J. L. (2008). Asimptoticheskie modeli vnytrennih stacionarnuh voln. Prikl. mehanika i teh. fizika, 49 (4), 151–161.
  12. Debsarma, S., Das, K. P. (2007). Fourth-order nonlinear evolution equations for a capillary-gravity wave packet in the presence of another wave packet in deep water. Physics of Fluids, 19 (9), 097101–097101. doi: 10.1063/1.2772252
  13. Carr, M., Davies, P. A. (2006). The motion of an internal solitary wave of depression over a fixed bottom boundary in a shallow, two-layer fluid. Phys. Fluids, 18 (1), 016601–016601. doi: 10.1063/1.2162033
  14. Sutherland, B. R., Nault, J. T. (2007). Intrusive gravity currents propagating along thin and thick interfaces. Journal of Fluid Mechanics, 586, 109–118. doi: 10.1017/s0022112007007288
  15. Haciyev, B. I. (2006). Unstationary waves in two-layered fluid caused by normal loading at the interface. Proc. IMM (Inst. Mathematics and Mechanics) NAS of Azerbaijan, 119–126.
  16. Selezov, I. T., Avramenko, O. V.,Gyrtovyy, Y. V., Naradovyy, V. V. (2009). Nelineynoe vzaimodeystvie vnytrennih i poverhnostnuh gravitacionnuh voln v dvyhsloynoy gidkosti so svobodnoy poverhnosty. Mat. metodu i fizuko-mehanichni polya, 52, 72–83.
  17. Selezov, I. T., Krivonos, Y. G. (2012). Matimaticheskie metodu v zadachah rasprostraneniya I difrakcii voln. Kiev: Naykova dymka, 232.
  18. Naradovyy, V. (2013) Interaction of internal and surface waves in two-layer fluid with free surface/Challenges of Modern Tehnology, 3 (4), 103–106.

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Published

2015-08-19

How to Cite

Авраменко, О. В., & Нарадовий, В. В. (2015). Analysis of propagation of weakly nonlinear waves in a two-layer fluid with free surface. Eastern-European Journal of Enterprise Technologies, 4(7(76), 39–44. https://doi.org/10.15587/1729-4061.2015.48282

Issue

Vol. 4 No. 7(76) (2015): Applied mechanics

Section

Applied mechanics

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Copyright (c) 2015 Ольга Валентинівна Авраменко, Володимир Володимирович Нарадовий

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