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A line sink in a flowing stream with surface tension effects

Published online by Cambridge University Press:  30 October 2015

R. J. HOLMES  and
G. C. HOCKING
R. J. HOLMES
Affiliation:
Mathematics and Statistics, Murdoch University, Perth, Western Australia email: [email protected], [email protected]
G. C. HOCKING
Affiliation:
Mathematics and Statistics, Murdoch University, Perth, Western Australia email: [email protected], [email protected]
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Abstract

We examine a problem in which a line sink causes a disturbance to an otherwise uniform flowing stream of infinite depth. We consider the fully non-linear problem with the inclusion of surface tension and find the maximum sink strength at which steady solutions exist for a given stream flow, before examining non-unique solutions. The addition of surface tension allows for a more thorough investigation into the characteristics of the solutions. The breakdown of steady solutions with surface tension appears to be caused by a curvature singularity as the flow rate approaches the maximum. The non-uniqueness in solutions is shown to occur for a range of parameter values in all cases with non-zero surface tension.

Keywords

free surface flowsubmerged sinksurface tensionselective withdrawal
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 2 , April 2016 , pp. 248 - 263
Copyright
Copyright © Cambridge University Press 2015 

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References

[1] Craya, A. (1949) Theoretical research on the flow of nonhomogeneous fluids. La Houille Blanche 4, 44–55.Google Scholar
[2] Forbes, L. K. (1985) On the effects of non-linearity in free-surface flow about a submerged point vortex. J. Eng. Math. 19, 139–155.Google Scholar
[3] Forbes, L. K. & Hocking, G. C. (1993) Flow induced by a line sink in a quiescent fluid with surface tension effects. J. Austral. Math. Soc. Ser. B 34, 377–391.Google Scholar
[4] Gariel, P. (1949) Experimental research on the flow of nonhomogeneous fluids. La Houille Blanche 4, 56–65.Google Scholar
[5] Harleman, D. R. F. & Elder, R. E. (1965) Withdrawal from two-layer stratified flow. J. Hydraul. Div. ASCE 91, HY4, 43–58.Google Scholar
[6] Hocking, G. C. (1991) Withdrawal from two-layer fluid through line sink. J. Hydraul. Engng ASCE 117, 800–805.Google Scholar
[7] Hocking, G. C. & Forbes, L. K. (1991) A note on the flow induced by a line sink beneath a free surface. J. Austral. Math. Soc. Ser. B 32, 251–260.Google Scholar
[8] Hocking, G. C. & Forbes, L. K. (1992) Subcritical free-surface flow caused by a line source in a fluid of finite depth. J. Eng. Math. 26, 455–466.Google Scholar
[9] Hocking, G. C. & Forbes, L. K. (2000) Withdrawal from a fluid of finite depth through a line sink, including surface tension effects. J. Eng. Math. 38, 91–100.Google Scholar
[10] Hocking, G. C., Forbes, L. K. & Stokes, T. E. (2014) A note on steady flow into a submerged point sink. ANZIAM J. 56, 150–159.Google Scholar
[11] Hocking, G. C., Stokes, T. E. & Forbes, L. K. (2010) A rational approximation to the evolution of a free surface during fluid withdrawal through a point sink. ANZIAM J. 51, E31–E36.Google Scholar
[12] Hocking, G. C. & Vanden-Broeck, J.-M. (1998) Withdrawal of a fluid of finite depth through a line sink with a cusp in the free surface. Comp. Fluids 27, 797–806.Google Scholar
[13] Imberger, J. & Hamblin, P. F. (1982) Dynamics of lakes, reservoirs and cooling ponds. Ann. Rev. Fluid Mech. 14, 153–187.CrossRefGoogle Scholar
[14] Jirka, G. H. (1979) Supercritical withdrawal from two-layered fluid systems, Part 1. Two-dimensional skimmer wall. J. Hydraul. Res. 17, 53–62.Google Scholar
[15] Lustri, C. J., McCue, S. W. & Chapman, S. J. (2013) Exponential asymptotics of free surface flow due to a line source. IMA J. Appl. Math. 78, 697–713.Google Scholar
[16] Mekias, H. & Vanden-Broeck, J.-M. (1991) Subcritical flow with a stagnation point due to a source beneath a free surface. Phys. Fluids Ser. A 3, 2652–2658.Google Scholar
[17] Peregrine, D. H. (1972) A line source beneath a free surface. Mathematics Research Center Technical Summary Report. University of Wisconsin Report, 1248.Google Scholar
[18] Sautreaux, C. (1901) Mouvement d'un liquide parfait soumis à la pesanteur. Dé termination des lignes de courant. J. Math. Pures. Appl. 7, 125–159.Google Scholar
[19] Stokes, T. E., Hocking, G. C. & Forbes, L. K. (2003) Unsteady free surface flow induced by a line sink. J. Eng. Math. 47, 137–160.Google Scholar
[20] Stokes, T. E., Hocking, G. C. & Forbes, L. K. (2005) Unsteady flow induced by a withdrawal point beneath a free surface. ANZIAM J. 47, 185–202.Google Scholar
[21] Stokes, T. E., Hocking, G. C. & Forbes, L. K. (2008) Unsteady free surface flow induced by a line sink in a fluid of finite depth. Comput. Fluids 37, 236–249.Google Scholar
[22] Trinh, P. H. & Chapman, S. J. (2013) New gravity-capillary waves at low speeds. Part 1. Linear geometries. J. Fluid Mech. 724, 367–391.Google Scholar
[23] Trinh, P. H. & Chapman, S. J. (2013) New gravity-capillary waves at low speeds. Part 2. Nonlinear theory. J. J. Fluid Mech. 724, 392–424.Google Scholar
[24] Tuck, E. O. & 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.Google Scholar
[25] Vanden-Broeck, J. -M. (1996) Waves generated by a source below a free surface in water of finite depth. J. Eng. Math. 30, 603–609.Google Scholar
[26] Vanden-Broeck, J. -M. (1998) A model for the free surface flow due to a submerged source in water of infinite depth. J. Austral. Math. Soc. Ser. B 39, 528–538.Google Scholar
[27] Vanden-Broeck, J.-M., Schwartz, L. W. & Tuck, E. O. (1978) Divergent low-Froude-number series expansion of nonlinear free-surface flow problems. Proc. R. Soc. Lond. A 361, 207–224.Google Scholar
[28] Wehausen, J. V. & Laitone, E. V. (1960) Surface waves. In: Flügge, S. (editor), Handbuch der Physik, vol. 9, Springer, Berlin, pp. 446–778.Google Scholar
[29] Wood, I. R. & Lai, K. K. (1972) Selective withdrawal from a two-layered fluid. J. Hydraul. Res. 10, 475–496.Google Scholar
[30] Xue, X. & Yue, D. K. P. (1998) Nonlinear free-surface flow due to an impulsively started submerged point sink. J. Fluid Mech. 364, 325–347.Google Scholar