Abstract
The problem of withdrawing water through a line sink from a region containing two homogenous layers of different density is considered. Assuming steady, irrotational flow of an ideal fluid, a nonlinear integral equation is derived and solved numerically. Confirmation of earlier research is given, and some new results obtained in which the interface between the two layers rises up and then enters the sink vertically from above, even when the sink is located above the undisturbed level of the interface. A diagram is presented which summarises the work on this problem to this time.
Similar content being viewed by others
References
Abramowitz M. and Stegun I.A. eds., Handbook of Mathematical Functions, Dover, New York (1970).
“CE-QUAL-R1: A numerical one-dimensional model of reservoir water quality: User's manual.” Instruction Rept. E-82-1, Envir. Lab., U.S. Army Engr. Wtrwy. Expt. Sta., CE, Vicksburg, Miss. (1982).
Craya A., Theoretical research on the flow of nonhomogeneous fluids, La Houille Blanche 4 (1949) 44–55.
Collings I.L., Two infinite Froude number cusped free surface flows due to a submerged line source or sink. J. Aust. Math Soc. Ser. B 28 (1986) 260–270.
Forbes L.K. and Hocking G.C., Flow caused by a point sink in a fluid having a free surface, J. Aust. Math Soc. Ser. B 32 (1990) 233–252.
Hocking G.C., Cusp-like free-surface flows due to a submerged source or sink in the presence of a flat or sloping bottom, J. Aust. Math Soc. Ser. B 26 (1985) 470–486.
Hocking G.C., Infinite Froude number solutions to the problem of a submerged source or sink, J. Aust. Math. Soc. Ser. B 29 (1988) 401–409.
Hocking G.C., Sherman B.S., and Patterson J.C. (1988), Algorithm for selective withdrawal from stratified reservoir, J. Hydr. Engng., A.S.C.E. 114 (1988) 707–719.
Hocking, G.C. and Forbes, L.K., A note on the flow induced by a line sink beneath a free surface. J. Aust. Math Soc. Ser. B (1991) (in press).
Hocking, G.C., Withdrawal from a two layer fluid through a line sink, submitted to J. Hydr. Engng., A.S.C.E. (1989).
Hocking, G.C., Flow induced by a line sink in a fluid of finite depth. University of Western Australia, Dept. of Math. Res. Rept. 13, April, 1990.
Imberger J. and Hamblin P.F., Dynamics of lakes, reservoirs and cooling ponds, Ann. Rev. Fluid Mech. 14 (1982) 153–187.
Imberger J., Reservoir Dynamics Modelling, in O'Loughlin E.M. and Cullen P. (eds), Prediction in Water Quality, Australian Acad. of Sci., Canberra, Australia (1982) 223–248.
Imberger J., Patterson J., Hebbert R., and Loh I., Dynamics of reservoir of medium size, J. Hydr. Div., ASCE, 104 (1978) 725–743.
Imberger J. and Patterson J.C., Physical limnology, Adv. Appl. Mech. 27 (1989) 303–475.
Jirka G.H. and Katavola D.S., Supercritical withdrawal from two-layered fluid systems, Part 2-Three dimensional flow into a round intake, J. Hyd. Res. 17 (1979) 53–62.
King A.C. and Bloor M.I.G., A note on the free surface induced by a submerged source at infinite Froude number, J. Aust. Math Soc. Ser. B 30 (1988) 147–156.
Peregrine, D.H., A line source beneath a free surface, Mathematics Research Center, Univ. Wisconsin Rept. 1248 (1972).
Ryan P.J. and Harleman D.R.F., Prediction of the annual cycle of temperature changes in a stratified lake or reservoir: Mathematical model and user's manual, M.I.T. Rept. No. 137, M.I.T., Cambridge, Mass. (1971).
Tuck, E.O. and Vanden-Broeck, J.M., A cusp-like free-surface flow due to a submerged source or sink, J. Aust. Math Soc. Ser. B 25 (18984) 443–450.
Vanden-Broeck J.M., Schwartz L.W. and Tuck E.O., Divergent low-Froude number series expansion of non-linear free-surface flow problems, Proc. Roy. Soc. London. Ser. A 361 (1978) 207–224.
Wood I.R. and Lai K.K., Selective withdrawal from a two-layered fluid, J. Hyd. Res. 10 No. 4 (1972) 475–496.
Yih, C.S., Stratified Flows, Academic Press (1980).
Author information
Authors and Affiliations
Additional information
Part of this work was carried out while the author was at the Centre for Water Research, University of Western Australia.
Rights and permissions
About this article
Cite this article
Hocking, G.C. Critical withdrawal from a two-layer fluid through a line sink. J Eng Math 25, 1–11 (1991). https://doi.org/10.1007/BF00036598
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00036598