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Contact in a viscous fluid. Part 1. A falling wedge

Published online by Cambridge University Press:  08 March 2010

C. J. CAWTHORN  and
N. J. BALMFORTH
C. J. CAWTHORN
Affiliation:
DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
N. J. BALMFORTH*
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada Department of Earth and Ocean Science, University of British Columbia, 6339 Stores Road, Vancouver, BC, V6T 1Z4, Canada
*
†Email address for correspondence: njb@math.ubc.ca
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Abstract

Computations are presented of the upward force on a two-dimensional wedge descending towards a plane surface due to the Stokes flow of an intervening viscous fluid. The predictions are compared with those of lubrication theory and an approximate analytical solution; all three predict a logarithmic divergence of the force with the minimum separation. An object falling vertically under gravity will therefore make contact with an underlying plane surface in finite time if roughened by asperities with sharp corners (with smooth surfaces, contact is made only after infinite time). Contact is still made in finite time if the roughened object also moves horizontally or rotates as it falls.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 646 , 10 March 2010 , pp. 327 - 338
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Balmforth, N. J., Bush, J. W. M., Vener, D. & Young, W. R. 2007 Dissipative descent: rocking and rolling down an incline. J. Fluid Mech. 590, 295–318.Google Scholar
Balmforth, N. J., Cawthorn, C. J. & Craster, R. V. 2010 Contact in a viscous fluid. Part 2. A compressible fluid and an elastic solid. J. Fluid Mech. 646, 339–361.Google Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242.CrossRefGoogle Scholar
Davis, R. H., Zhao, Y., Galvin, K. P. & Wilson, H. J. 2003 Solid–solid contacts due to surface roughness and their effects on suspension behaviour. Phil. Trans. Roy. Soc.: Math. Phys. Engng Sci. 361, 871–894.Google Scholar
Gérard-Varet, D. & Hillairet, M. 2008 Regularity issues in the problem of fluid–structure interaction. Arch. Rat. Mech. Anal., doi:10.1007/s00205-008-0202-9.Google Scholar
Liu, C. H. & Joseph, D. D. 1977 Stokes flow in wedge-shaped trenches. J. Fluid Mech. 80, 443–463.CrossRefGoogle Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.Google Scholar
Reynolds, O. 1886 On the theory of lubrication and its application to Mr. Beauchamp Tower's experiments, including an experimental determination of the viscosity of olive oil. Phil. Trans. Roy. Soc. Lond. 177, 157–234.Google Scholar
Smart, J. R., Beimfohr, S. & Leighton, D. T. 1993 Measurement of the translational and rotational velocities of a noncolloidal sphere rolling down a smooth inclined plane at low Reynolds number. Phys. Fluids A 5, 13–24.CrossRefGoogle Scholar