Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-30T23:14:48.506Z Has data issue: false hasContentIssue false
  • English
  • Français

SUPERCRITICAL TWO-FLUID INTERACTIONS WITH SURFACE TENSION AND GRAVITY

Part of: Incompressible inviscid fluids

Published online by Cambridge University Press:  10 December 2009

Jean-Marc Vanden-Broeck  and
Frank T. Smith
Jean-Marc Vanden-Broeck
Affiliation:
Department of Mathematics, University College London, Gower St, London WC1E 6BT, U.K. (email: [email protected])
Frank T. Smith
Affiliation:
Department of Mathematics, University College London, Gower St, London WC1E 6BT, U.K. (email: [email protected])
Get access

Abstract

Gravity and surface-tension effects are examined for inviscid–inviscid interactions between two fluids close to a wall. The ratios of density and viscosity of the two fluids are taken to be small. A nonlinear integro–differential equation is found to govern the near-wall flow velocity, interface shape and pressure; analysis, computation and comparisons are then applied. Travelling-state solutions are of particular interest.

MSC classification

Secondary: 76B07: Free-surface potential flows 76B45: Capillarity (surface tension)
Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010 , pp. 93 - 106
Copyright
Copyright © University College London 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

[1]Chen, B. and Saffman, P. G., Steady gravity-capillary waves on deep water, Part II: numerical results for finite amplitude. Stud. Appl. Math. 62 (1980), 95–111.Google Scholar
[2]Gent, R. W., Dart, N. P. and Cansdale, G. T., Aircraft icing. Philos. Trans. R. Soc. A 358 (2000), 2873–2991.Google Scholar
[3]Gueyffier, D., Li, J., Nadim, A., Scardovelli, R. and Zaleski, S., Volume of fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J. Comput. Phys. 152 (1999), 423–456.Google Scholar
[4]Hogan, S. J., Some effects of surface tension on steep water waves, Part 2. J. Fluid Mech. 96 (1980), 417–445.Google Scholar
[5]Howison, S. D., Ockendon, J. R. and Oliver, J. M., Deep- and shallow-water slamming at small and zero deadrise angles. J. Engrg. Math. 42 (2002), 373–388.CrossRefGoogle Scholar
[6]Howison, S. D., Ockendon, J. R. and Wilson, S. K., Incompressible water entry problems at small deadrise angles. J. Fluid Mech. 222 (1991), 215–230.Google Scholar
[7]Josserand, C. and Zaleski, S., Droplet splashing on a thin liquid film. Phys. Fluids 15 (2003), 1650–1657.Google Scholar
[8]King, A. and Tuck, E. O., Thin liquid layers supported by steady air-flow surface traction. J. Fluid Mech. 251 (1993), 709–718.CrossRefGoogle Scholar
[9]King, A., Tuck, E. O. and Vanden-Broeck, J.-M., Air-blown waves on thin viscous sheets. Phys. Fluids A5 (1993), 973–978.CrossRefGoogle Scholar
[10]Korobkin, A. A., Shallow water impact problems. J. Engrg. Math. 35 (1999), 233–250.Google Scholar
[11]Korobkin, A. A., Asymptotic theory of liquid–solid impact. Philos. Trans. R. Soc. A 355 (1997), 507–522.CrossRefGoogle Scholar
[12]Lesser, M. B. and Field, J. E., The impact of compressible fluids. Annu. Rev. Fluid Mech. 15 (1983), 97–122.Google Scholar
[13]Liow, J. L., Splash formation by spherical drops. J. Fluid Mech. 427 (2001), 73–105.Google Scholar
[14]Purvis, R. and Smith, F. T., Large droplet impact on water layers. AIAA Paper 2004-0414, American Institute of Aeronautics and Astronautics 42nd Aerospace Sciences Meeting, 2004.CrossRefGoogle Scholar
[15]Purvis, R. and Smith, F. T., Air–water interactions near droplet impact. European J. Appl. Math. 15 (2004), 853–871.Google Scholar
[16]Purvis, R. and Smith, F. T., Droplet impact on water layers: post-impact analysis and computations. Philos. Trans. R. Soc. A 363 (2005), 1209–1221.Google Scholar
[17]Quero, M., Hammond, D. W., Purvis, R. and Smith, F. T., Analysis of super-cooled water droplet impact on a thin layer and ice growth. AIAA Paper 2006-466, American Institute of Aeronautics and Astronautics 44th Aerospace Sciences Meeting, 2006.Google Scholar
[18]Schwartz, L. W. and Vanden-Broeck, J.-M., Numerical solutions of the exact equations for capillary-gravity waves. J. Fluid Mech. 95 (1979), 119–139.Google Scholar
[19]Smith, F. T., Li, L. and Wu, G.-X., Air cushioning with a lubrication/inviscid balance. J. Fluid Mech. 482 (2003), 291–318.Google Scholar
[20]Smith, F. T. and Purvis, R., Air effects on droplet impact. AIAA Paper 2005-5184, American Institute of Aeronautics and Astronautics 4th Theoretical Fluid Mechanics Meeting, 2005.Google Scholar
[21]Vanden-Broeck, J.-M., Damped waves generated by a moving pressure distribution. European J. Appl. Math. 12 (2001), 357–366.Google Scholar
[22]Vanden-Broeck, J.-M. and Miloh, T., The influence of a layer of mud on the train of waves generated by a moving distribution of pressure. J. Engrg. Math. 30 (1996), 378–400.Google Scholar
[23]Vanden-Broeck, J.-M. and Smith, F. T., Surface tension effects on interaction between two fluids near a wall. Quart. J. Mech. Appl. Math. 61 (2008), 117–128.Google Scholar
[24]Wilson, S. K., A mathematical model for the initial stages of fluid impact in the presence of a cushioning fluid layer. J. Engrg. Math. 25 (1991), 265–285.Google Scholar
[25]Wilton, J. R., On ripples. Phil. Mag. 29 (1915), 688–700.Google Scholar