Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-09T22:09:04.276Z Has data issue: false hasContentIssue false
  • English
  • Français

The dam-break problem for viscous fluids in the high-capillary-number limit

Published online by Cambridge University Press:  10 April 2009

C. ANCEY ,
S. COCHARD  and
N. ANDREINI
C. ANCEY*
Affiliation:
School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
S. COCHARD
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver V6K 2A5, Canada
N. ANDREINI
Affiliation:
School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Experiments were undertaken to investigate dam-break flows where a finite volume of highly viscous fluid (glucose with viscosity μ ≈ 350 Pa s) maintained behind a lock gate was released into a horizontal or inclined flume. The resulting sequence of flow-depth profiles was tracked using a three-dimensional visualization system. In the low-Reynolds-number and high-capillary-number limits, analytical solutions can be obtained from the Navier–Stokes equations using lubrication theory and matched asymptotic expansions. At shallow slopes, similarity solutions can also be worked out. While the variation in the front position scaled with time as predicted by theory for both horizontal and sloping flumes, there was a systematic delay in the front position observed. Moreover, taking a closer look at the experimental flow-depth profiles shows that they were similar, but they noticeably deviated from the theoretical similarity form for horizontal planes. For sloping beds, the flow-depth profile is correctly predicted provided that different scalings are used at shallow and large slopes.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 624 , 10 April 2009 , pp. 1 - 22
Copyright
Copyright © Cambridge University Press 2009

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

Barenblatt, G. I. 1996 Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press.CrossRefGoogle Scholar
Buckmaster, J. 1977 Viscous sheets advancing over dry beds. J. Fluid Mech. 81, 735–756.CrossRefGoogle Scholar
Cochard, S. 2007 Measurements of time-dependent free-surface viscoplastic flows down steep slopes. PhD thesis, Ecole Polytechnique Fédérale de Lausanne.Google Scholar
Cochard, S. & Ancey, C. 2008 Tracking the free surface of time-dependent flows: image processing for the dam-break problem. Exp. Fluids 44, 59–71.CrossRefGoogle Scholar
Courant, R. & Friedrich, K. O. 1948 Supersonic Flow and Shock Waves. Intersciences Publishers.Google Scholar
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169–194.CrossRefGoogle Scholar
Didden, N. & Maxworthy, T. 1982 The viscous spreading of plane and axisymmetric gravity currents. J. Fluid Mech. 121, 27–42.CrossRefGoogle Scholar
Diez, J. A., Gratton, R. & Gratton, J. 1992 Self-similar solution of the second kind for a convergent viscous gravity current. Phys. Fluids A 4, 1148–1155.CrossRefGoogle Scholar
Garabedian, P. R. 1966 Free boundary flows of a viscous liquid. Comm. Pure Appl. Math. 19, 421–434.CrossRefGoogle Scholar
Goodwin, R. & Homsy, G. M. 1991 Viscous flow down a slope in the vicinity of a contact line. Phys. Fluids A 3, 515–528.CrossRefGoogle Scholar
Gratton, J. & Minotti, F. 1990 Self-similar viscous gravity currents: phase-plane formalism. J. Fluid Mech. 210, 155–182.CrossRefGoogle Scholar
Grundy, R. E. 1983 a Asymptotic solution of a model non-linear diffusion equation. IMA J. Appl. Maths 31, 121–137.CrossRefGoogle Scholar
Grundy, R. E. 1983 b Local similarity solutions for the initial-value problem in non-linear diffusion. IMA J. Appl. Maths 31, 209–214.CrossRefGoogle Scholar
Grundy, R. E. & McLaughlin, R. 1982 Large-time solution of a nonlinear diffusion equation. Proc. R. Soc. London Ser. A 381, 395–406.Google Scholar
Hocking, L. M. 1990 Spreading and instability of a viscous fluid sheet. J. Fluid Mech. 211, 373–392.CrossRefGoogle Scholar
Hocking, L. M. & Rivers, A. D. 1982 The spreading of a drop by capillary action. J. Fluid Mech. 121, 425–442.Google Scholar
Hunt, B. 1994 Newtonian fluid mechanics treatment of debris flows and avalanches. J. Hydraul. Engng 120, 1350–1363.CrossRefGoogle Scholar
Huppert, H. E. 1982 a Flow and instability of a viscous current down a slope. Nature 300, 427–429.CrossRefGoogle Scholar
Huppert, H. E. 1982 b The propagation of two-dimensional and axisymetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.CrossRefGoogle Scholar
Huppert, H. E. 2006 Gravity currents: a personal perspective. J. Fluid Mech. 554, 299–322.CrossRefGoogle Scholar
Lister, J. R. 1992 Viscous flows down an inclined plane from point and line sources. J. Fluid Mech. 242, 631–653.CrossRefGoogle Scholar
Mathunjwa, J. S. & Hogg, A. J. 2006 Self-similar gravity currents in porous media: linear stability of the Barrenblatt–Pattle solution revisited. Eur. J. Mech. B 25, 360–378.CrossRefGoogle Scholar
Maxworthy, T. 1983 Gravity currents with variable inflow. J. Fluid Mech. 128, 247–257.CrossRefGoogle Scholar
Mei, C. C. 1966 Nonlinear gravity waves in a thin sheet of viscous fluid. J. Maths. Phys. 45, 266–288.CrossRefGoogle Scholar
Nakaya, C. 1974 Spread of fluid drops along a horizontal plane. J. Phys. Soc. Jpn. 37, 539–543.CrossRefGoogle Scholar
Nsom, B. 2002 Horizontal viscous dam-break flow: experiments and theory. J. Hydraul. Engng 128, 543–546.CrossRefGoogle Scholar
Nsom, B., Debiane, K. & Piau, J.-M. 2000 Bed slope effect on the dam break problem. J. Hydraul. Res. 38, 459–464.CrossRefGoogle Scholar
Ockendon, J. R. 1978 Numerical and analytical solutions of moving boundary problems. In Moving Boundary Problems (ed. Wilson, D.G., Solomon, A.D. & Boggs, P.T.). Academic Press.Google Scholar
Schenck, F. W. 2007 Glucose and glucose-containing syrups. In Ullmann's Encyclopedia of Industrial Chemistry (ed. Arpe, H.-J.), vol. A12, pp. 457–475. Wiley-VCH Verlag GmbH.Google Scholar
Simpson, J. E. 1997 Gravity Currents in the Environment and the Laboratory. Cambridge University Press.Google Scholar
Smith, H. S. 1969 On initial value problems for the flow in a thin sheet of viscous liquid. Z. Angew. Math. Mech. 20, 556–560.Google Scholar
Smith, P. C. 1973 A similarity solution for slow viscous flow down an inclined channel. J. Fluid Mech. 58, 275–288.CrossRefGoogle Scholar
Takagi, D. & Huppert, H. E. 2007 The effect of confining boundaries on viscous gravity currents. J. Fluid Mech. 577, 495–505.CrossRefGoogle Scholar
Veretennikov, I., Indeikina, A. & Chang, H.-C. 1998 Front dynamics and fingering of a driven contact line. J. Fluid Mech. 373, 81–110.CrossRefGoogle Scholar
van Vroonhoven, J. C. W. & Kuijpers, W. J. J. 1990 A free-boundary problem for viscous fluid flow in injection moulding. J. Eng. Math. 24, 151–165.CrossRefGoogle Scholar
Witelski, T. P. & Bernoff, A. J. 1998 Self-similar asymptotics for linear and nonlinear diffusion equations. Stud. Appl. Math. 100, 153–193.CrossRefGoogle Scholar