Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-09T22:18:34.847Z Has data issue: false hasContentIssue false
  • English
  • Français

Free streamline flows with singularities

Published online by Cambridge University Press:  18 March 2010

J. EGGERS  and
A. F. SMITH
J. EGGERS*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
A. F. SMITH
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We rederive and expand upon a method for finding solutions to the two-dimensional irrotational (inviscid) flow equations in the presence of a free surface, found by Hopkinson. This method allows the flow to be driven by placing singularities, like sources or vortices, in the interior of the flow domain. We then apply the method to find a number of novel solutions: separated flow driven by a source, vortices behind a plate and free-surface flow stirred by a double vortex. Free surfaces generically exhibit cusp singularities with a 2/3 power index, similar to those found in very viscous flow.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 647: Dedicated to Professor S. H. Davis on his 70th birthday , 25 March 2010 , pp. 187 - 200
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Birkhoff, G. & Zarantonello, E. H. 1957 Jets, Wakes and Cavities. Academic.Google Scholar
Brocchini, M. & Peregrine, D. H. 2001 The dynamics of strong turbulence at free surfaces. Part 1. Description. J. Fluid Mech. 449, 225–254.CrossRefGoogle Scholar
Carrier, G. F., Krook, M. & Pearson, C. E. 1966 Functions of a Complex Variable. McGraw-Hill.Google Scholar
Chapman, S. J. & Vanden-Broeck, J.-M. 2002 Exponential asymptotics and capillary waves. SIAM J. Appl. Math. 62, 1872–1898.Google Scholar
Craya, A. 1949 Theoretical research on the flow of nonhomogeneous fluids. La Houille Blanche 4, 44–55.Google Scholar
Eggers, J. 2001 Air entrainment through free-surface cusps. Phys. Rev. Lett. 86, 4290.CrossRefGoogle ScholarPubMed
Eggers, J. & Fontelos, M. 2009 Cusps in interfacial problems. http://arxiv:math-ph/0910.3499.Google Scholar
Elcrat, A. R., Fornberg, B. & Miller, K. G. 2001 Some steady axisymmetric vortex flows past a sphere. J. Fluid Mech. 433, 315–328.CrossRefGoogle Scholar
Forbes, L. K. & Hocking, G. C. 1990 Flow caused by a point sink in a fluid having a free surface. J. Aust. Math. Soc. B 32, 231–249.CrossRefGoogle Scholar
Gurevich, M. I. 1966 Theory of Jets in an Ideal Fluid. Pergamon.Google Scholar
Hopkinson, B. 1898 Discontinuous fluid motions involving sources and vortices. Proc. Lond. Math. Soc. 29, 142–164.Google Scholar
Hudson, J. D. & Dennis, S. C. R. 1985 The flow of a viscous incompressible fluid past a normal flat plate at low and intermediate Reynolds numbers: the wake. J. Fluid Mech. 160, 369–383.Google Scholar
Jeong, J.-T. & Moffatt, H. K. 1992 Free-surface cusps associated with a flow at low Reynolds numbers. J. Fluid Mech. 241, 1–22.CrossRefGoogle Scholar
Kirchhoff, G. 1869 Zur Theorie freier Flüssigkeitsstrahlen. J. reine angew. Math. 70, 289.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1984 Fluid Mechanics. Pergamon.Google Scholar
Lorenceau, E., Quéré, D. & Eggers, J. 2004 Air entrainment by a viscous jet plunging into a bath. Phys. Rev. Lett. 93, 254501.CrossRefGoogle ScholarPubMed
Love, A. E. H. 1891 On the theory of discontinuous fluid motion in two dimensions. Proc. Camb. Phil. Soc. 7, 175–201.Google Scholar
Mekias, H. & Vanden-Broeck, J.-M. 1991 Subcritical flow with a stagnation point due to a source beneath a free surface. Phys. Fluids A 3, 2652–2658.CrossRefGoogle Scholar
Milne-Thompson, L. M. 1962 Theoretical Hydrodynamics, 4th edn. Macmillan & Co.Google Scholar
Planck, M. 1884 Zur Theorie der Flüssigkeitsstrahlen. Wied. Ann. 21, 499–509.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Sautreaux, C. 1901 Mouvement d'un liquide parfait soumis à la pesanteur. Détermination des lignes de courant. J. Math. Pures Appl. 7, 125–159.Google Scholar
Sébilleau, J., Limat, L. & Eggers, J. 2009 Flow separation from a stationary meniscus. J. Fluid Mech. 633, 137.CrossRefGoogle Scholar
Smith, F. T. 1986 Steady and unsteady boundary layer separation. Annu. Rev. Fluid Mech. 18, 197–220.CrossRefGoogle Scholar
Taneda, S. 1987 Irregular flows. Sadhana 10, 349–375.Google Scholar
Vanden-Broeck, J.-M. & Keller, J. B. 1987 Free surface flow due to a sink. J. Fluid Mech. 175, 109–117.CrossRefGoogle Scholar
Verga, A. 2004 Singularity formation in vortex sheets and interfaces. Nonlinear Phenom. Complex Syst. 9, 361–388.CrossRefGoogle Scholar