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In a reference system moving with the bubble vertex we investigate the effects of fluid viscosity and surface tension on the bubble velocity in the nonlinear Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities, by extending the ideal fluid model [Goncharov V N, Phys. Rev. Lett. 88 134502 (2002)] to the non-ideal fluid case. First of all, the governing equation (i.e. self-consistent differential equations) describing the dynamic of the bubble front in RT and RM instabilities is obtained. Then, the numerical and asymptotic solutions of the bubble velocity in two-dimensional planar geometry and three-dimensional cylindrical geometry are obtained. Moreover, we quantitatively study the effects of fluid viscosity and surface tension on the RT and RM bubble velocities. It is found that in the fully nonlinear evolutions of RT and RM instabilities, the bubble velocity and amplitude in the non-ideal fluid are both less than those in its ideal fluid counterpart. That is to say, the effects of fluid viscosity and surface tension tend to stabilize the RT and RM instabilities.
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Keywords:
- Rayleigh-Taylor instability /
- Richtmyer-Meshkov instability /
- bubble velocity /
- non-ideal fluids
[1] Taylor G I 1950 Proc. R. Soc. London A 201 192
[2] Ramaprabhu P, Andrews M J 2004 J. Fluid Mech. 502 233
[3] Richtmyer R D 1960 Commun. Pure Appl. Math. 13 297
[4] Meshkov E E 1969 Sov.Fluid Dyn. 4 101
[5] Ye W H, Wang L F, He X T 2010 Phys. Plasmas 17 122704
[6] Wang L F, Ye W H, He X T, Zhang W Y, Sheng Z M, Yu M Y 2012 Phys. Plasmas 19 100701
[7] Layzer D 1955 Astrophys. J. 122 1
[8] Zufiria J 1988 Phys. Fluids 31 440
[9] Wang L F, Ye W H, Sheng Z M, Don W S, Li Y J, He X T 2010 Phys. Plasmas 17 122706
[10] Wang L F, Ye W H, Li Y J 2010 Phys. Plasmas 17 052305
[11] Wang L F, Ye W H, Fan Z F, Li Y J 2010 EPL. 90 15001
[12] Liu W H, Wang L F, Ye W H, He X T 2012 Phys. Plasmas 19 042705
[13] Wang L F, Ye W H, Li Y J 2010 Chin. Phys. Lett. 27 025203
[14] Zhang Q 1998 Phys. Rev. Lett. 81 3391
[15] Goncharov V N 2002 Phys. Rev. Lett. 88 134502
[16] Karnig O Mikaelian 2010 Phys. Rev. E 81 016325
[17] Young Y N, Ham F E 2006 J. Turbul. 7 1
[18] Niebling M J, Flekkoy E G, Måloy K J, Toussaint R 2010 Phys. Rev. E 82 051302
[19] Garnier J 2003 Phys. Rev. E 68 036401
[20] Sohn S I 2004 Phys. Rev. E 70 045301
[21] Sohn S I 2009 Phys. Rev. E 80 055302(R)
[22] Oron D, Arazi L, Kartoon D, Rikanati A, Alon U, Shrarts D 2001 Phys. Plasmas 8 2883
[23] Tao Y S, Wang L F, Ye W H, Zhang G C, Zhang J C, Li Y J 2012 Acta Phys. Sin. 61 075207 (in Chinese) [陶烨晟, 王立锋, 叶文华, 张广财, 张建成, 李英骏 2012 61 075207]
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[1] Taylor G I 1950 Proc. R. Soc. London A 201 192
[2] Ramaprabhu P, Andrews M J 2004 J. Fluid Mech. 502 233
[3] Richtmyer R D 1960 Commun. Pure Appl. Math. 13 297
[4] Meshkov E E 1969 Sov.Fluid Dyn. 4 101
[5] Ye W H, Wang L F, He X T 2010 Phys. Plasmas 17 122704
[6] Wang L F, Ye W H, He X T, Zhang W Y, Sheng Z M, Yu M Y 2012 Phys. Plasmas 19 100701
[7] Layzer D 1955 Astrophys. J. 122 1
[8] Zufiria J 1988 Phys. Fluids 31 440
[9] Wang L F, Ye W H, Sheng Z M, Don W S, Li Y J, He X T 2010 Phys. Plasmas 17 122706
[10] Wang L F, Ye W H, Li Y J 2010 Phys. Plasmas 17 052305
[11] Wang L F, Ye W H, Fan Z F, Li Y J 2010 EPL. 90 15001
[12] Liu W H, Wang L F, Ye W H, He X T 2012 Phys. Plasmas 19 042705
[13] Wang L F, Ye W H, Li Y J 2010 Chin. Phys. Lett. 27 025203
[14] Zhang Q 1998 Phys. Rev. Lett. 81 3391
[15] Goncharov V N 2002 Phys. Rev. Lett. 88 134502
[16] Karnig O Mikaelian 2010 Phys. Rev. E 81 016325
[17] Young Y N, Ham F E 2006 J. Turbul. 7 1
[18] Niebling M J, Flekkoy E G, Måloy K J, Toussaint R 2010 Phys. Rev. E 82 051302
[19] Garnier J 2003 Phys. Rev. E 68 036401
[20] Sohn S I 2004 Phys. Rev. E 70 045301
[21] Sohn S I 2009 Phys. Rev. E 80 055302(R)
[22] Oron D, Arazi L, Kartoon D, Rikanati A, Alon U, Shrarts D 2001 Phys. Plasmas 8 2883
[23] Tao Y S, Wang L F, Ye W H, Zhang G C, Zhang J C, Li Y J 2012 Acta Phys. Sin. 61 075207 (in Chinese) [陶烨晟, 王立锋, 叶文华, 张广财, 张建成, 李英骏 2012 61 075207]
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