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Journal of Engineering Mathematics

, Volume 61, Issue 1, pp 55–68 | Cite as

On starting conditions for a submerged sink in a fluid

  • Lawrence K. ForbesEmail author
  • Graeme C. Hocking
  • Tim E. Stokes
Article

Abstract

Withdrawal of a fluid through an isolated line sink (for planar flow) or a point sink (for axi-symmetric flow in three dimensions) is considered. A linearized solution is presented in both cases, under the assumption that the sink strength is small and the sink is turned on gradually. The results show that the behaviour for small times is as if an image source were present above the surface. Asymptotic results indicate that, in both cases, the flow rapidly develops an effective image sink above the free surface.

Keywords

Asymptotic behaviour Heuristic derivation Linearized solution Method of stationary phase Withdrawal flows 

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References

  1. 1.
    Tuck EO and Vanden-Broeck J-M (1984). A cusp-like free-surface flow due to a submerged source or sink. J Austral Math Soc, Ser B 25: 443–450 zbMATHMathSciNetGoogle Scholar
  2. 2.
    Hocking GC (1995). Supercritical withdrawal from a two-layer fluid through a line sink. J Fluid Mech 297: 37–47 zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Forbes LK and Hocking GC (1993). Flow induced by a line sink in a quiescent fluid with surface tension effects. J Austral Math Soc, Ser B 34: 377–391 zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Forbes LK and Hocking GC (1990). Flow caused by a point sink in a fluid having a free surface. J Austral Math Soc, Ser B 32: 231–249 zbMATHMathSciNetGoogle Scholar
  5. 5.
    Vanden-Broeck J-M and Keller JB (1997). An axisymmetric free surface with a 120 degree angle along a circle. J Fluid Mech 342: 403–409 zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Forbes LK and Hocking GC (2005). Flow due to a sink near a vertical wall, in infinitely deep fluid. Comput Fluids 34: 684–704 zbMATHCrossRefGoogle Scholar
  7. 7.
    Lucas SK and Kucera A (1996). A boundary integral method applied to the 3D water coning problem. Phys Fluids 8: 3008–3022 zbMATHCrossRefADSGoogle Scholar
  8. 8.
    Tyvand PA (1992). Unsteady free-surface flow due to a line source. Phys Fluids A 4: 671–676 zbMATHCrossRefADSGoogle Scholar
  9. 9.
    Sozer EM and Greenberg MD (1995). The time-dependent free surface flow induced by a submerged line source or sink. J Fluid Mech 284: 225–237 zbMATHCrossRefADSGoogle Scholar
  10. 10.
    Zhou Q-N and Graebel WP (1990). Axisymmetric draining of a cylindrical tank with a free surface. J Fluid Mech 221: 511–532 zbMATHCrossRefADSGoogle Scholar
  11. 11.
    Baek JH and Chung HY (1998). Numerical analysis on axisymmetric draining from a cylindrical tank with a free surface. Comput Fluid Dyn J 6: 413–425 Google Scholar
  12. 12.
    Xue M and Yue DKP (1998). Nonlinear free-surface flow due to an impulsively started submerged point sink. J Fluid Mech 364: 325–347 zbMATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Miloh T and Tyvand PA (1993). Nonlinear transient free-surface flow and dip formation due to a point sink. Phys Fluids A 5: 1368–1375 zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Stokes TE, Hocking GC and Forbes LK (2003). Unsteady free-surface flow induced by a line sink. J Eng Math 47: 137–160 zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Tyvand PA and Haugen KB (2005). An impulsive bathtub vortex. Phys Fluids 17: 062105 CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Lighthill MJ (1980). An introduction to Fourier analysis and generalised functions. Cambridge University Press, Cambridge Google Scholar
  17. 17.
    Wehausen JV, Laitone EV (1960) Surface waves. In: Encyclopaedia of physics, vol 9. Springer Verlag, BerlinGoogle Scholar
  18. 18.
    Marsden JE (1973). Basic complex analysis. Freeman, San Francisco zbMATHGoogle Scholar
  19. 19.
    Andrews GE, Askey R and Roy R (2000). Special functions. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  20. 20.
    Abramowitz M and Stegun IA (1972). Handbook of mathematical functions. Dover, New York zbMATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  • Lawrence K. Forbes
    • 1
    Email author
  • Graeme C. Hocking
    • 2
  • Tim E. Stokes
    • 3
  1. 1.School of Mathematics and PhysicsUniversity of TasmaniaHobartAustralia
  2. 2.Division of Science, School of Mathematics and StatisticsMurdoch UniversityMurdochAustralia
  3. 3.University of WaikatoHamiltonNew Zealand

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