Abstract
The steady withdrawal of an inviscid fluid of finite depth into a line sink is considered for the case in which surface tension is acting on the free surface. The problem is solved numerically by use of a boundary-integral-equation method. It is shown that the flow depends on the Froude number, F B=m(gH 3 B)−1/2, and the nondimensional sink depth λ=H S/H B, where m is the sink strength, g the acceleration of gravity, H B is the total depth upstream, H S is the height of the sink, and on the surface tension, T. Solutions are obtained in which the free surface has a stagnation point above the sink, and it is found that these exist for almost all Froude numbers less than unity. A train of steady waves is found on the free surface for very small values of the surface tension, while for larger values of surface tension the waves disappear, leaving a waveless free surface. It the sink is a long way off the bottom, the solutions break down at a Froude number which appears to be bounded by a region containing solutions with a cusp in the surface. For certain values of the parameters, two solutions can be obtained.
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References
J. Imberger and P. F. Hamblin, Dynamics of lakes, reservoirs and cooling ponds. Ann. Rev. Fluid Mech. 14 (1982) 153-187.
A. Craya, Theoretical research on the flow of nonhomogeneous fluids. La Houille Blanche 4 (1949) 44-55.
P. Gariel, Experimental research on the flow of nonhomogeneous fluids. La Houille Blanche 4 (1949) 56-65.
D. G. Huber, Irrotational motion of two fluid strata towards a line sink. J. Engng. Mech. Div. ASCE 86 (1960) 71-86.
G. C. Hocking, Withdrawal from two-layer fluid through line sink. J. Hydraulic Engng, ASCE 117 (1991) 800-805.
G. H. Jirka, Supercritical withdrawal from two-layered fluid systems-Part 1: Two-dimensional skimmer wall. J. Hydraulic Res. 17 (1979) 43-51.
Q.-N. Zhou and W. P. Graebel, Axisymmetric draining of a cylindrical tank with a free surface. J. Fluid Mech. 221 (1990) 511-532.
G. C. Hocking, Cusp-like free-surface flows due to a submerged source or sink in the presence of a flat or sloping bottom, J. Austr. Math. Soc.-Series B 26 (1985) 470-486.
G. C. Hocking, Critical withdrawal from a two-layer fluid through a line sink. J. Engng Math. 25 (1991) 1-11.
C. Sautreaux, Mouvement d'un liquide parfait soumis à la pesanteur. Détermination des lignes de courant. J. Math. Pures Appl. 5 (1901) 125-159.
E. O. Tuck and J. M. Vanden-Broeck, A cusp-like free-surface flow due to a submerged source or sink. J. Austr. Math. Soc.-Series B 25 (1984) 443-450.
J. M. Vanden-Broeck and J. B. Keller, Free surface flow due to a sink. J. Fluid Mech. 175 (1987) 109-117.
G. C. Hocking, Supercritical withdrawal from a two-layer fluid through a line sink. J. Fluid Mech. 297 (1995) 37-45.
J. M. Vanden-Broeck, Cusp flow due to a submerged source with a free surface partially covered by a lid. Eur. J. Mech. B (Fluids) 16 (1997) 249-255.
G. C. Hocking and J. M. Vanden-Broeck, Withdrawal of a fluid of finite depth through a line sink with a cusp in the free surface. Comp. Fluids 27 (1998) 797-806.
D. H. Peregrine, A Line Beneath a Free Surface. University of Wisconsin: Mathematics Research Center Technical Summary Report 1248 (1972).
G. C. Hocking and L. K. Forbes, A note on the flow induced by a line sink beneath a free surface. J. Austr. Math. Soc.-Series B 32 (1991) 251-260.
L. K. Forbes and G. C. Hocking, Flow induced by a line sink in a quiescent fluid with surface-tension effects. J. Austr. Math. Soc.-Series B 34 (1993) 377-391.
H. Mekias and J. M. Vanden-Broeck, Supercritical free-surface flow with a stagnation point due to a submerged source. Phys. Fluids A 1 (1989) 1694-1697.
H. Mekias and J. M. Vanden-Broeck, Subcritical flow with a stagnation point due to a source beneath a free surface. Phys. Fluids A 3 (1991) 2652-2658.
J. M. Vanden-Broeck, Waves generated by a source below a free surface in water of finite depth. J. Engng Math. 30 (1996) 603-609.
G. C. Hocking and L. K. Forbes, Subcritical free-surface flow caused by a line source in a fluid of finite depth. J. Engng. Math. 26 (1992) 455-466.
T. Miloh and P. A. Tyvand, Nonlinear transient free-surface flow and dip formation due to a point sink. Phys. Fluids A 5 (1993) 1368-1375.
T. J. Singler and J. F. Geer, A hybrid perturbation-Galerkin solution to a problem in selective withdrawal. Phys. Fluids A 5 (1993) 1156-1166.
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Hocking, G., Forbes, L. Withdrawal from a fluid of finite depth through a line sink, including surface-tension effects. Journal of Engineering Mathematics 38, 91–100 (2000). https://doi.org/10.1023/A:1004612117673
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DOI: https://doi.org/10.1023/A:1004612117673