Abstract
The uniform ideal flow past an impermeable sphere in contact with an impermeable plane is calculated, and the potential Φ is expressed as an integral over solutions in tangent sphere coordinates, up to an unknown function α. An ordinary differential equation satisfied by α is solved numerically to a high degree of accuracy and then a detailed presentation of the potential near the sphere is given. The potential near to the point of contact is analysed separately using a theory appropriate to crevice regions, and the two solutions are matched. The potential Φ is interpreted as the pressure impulse, P, for which the uniform gradient is specified far from the sphere. This provides information about the impulsive velocity and the net fluid impulse on the sphere or other blunt body near a region of wave impact.
Similar content being viewed by others
References
V.P. Zenkovich, Processes of Coastal Development. London: Oliver & Boyd (1967) 738 pp.
S.J. Cox and M.J. Cooker, The Motion of a Rigid Body Impelled by Sea-Wave Impact. Appl. Ocean Res. 21 (1999) 113-125.
R.A. Bagnold, Beach Formation by Waves; Some Model-Experiments in a Wave Tank. J. Inst. Civil Eng. 12 (1939) 201-226.
P.A. Blackmore and P.J. Hewson, Experiments on full-scale wave impact pressures. Coastal Eng. 8 (1984) 331-346.
E.S. Chan and W.K. Melville, Deep-water plunging wave pressures on a vertical plane wall. Proc. R. Soc. London A 417 (1988) 95-131.
M.J. Cooker and D.H. Peregrine, Pressure-impulse theory for liquid impact problems. J. Fluid Mech. 297 (1995) 193-214.
M.J. Cooker and D.H. Peregrine, Wave impact pressure and its effect upon bodies lying on the sea bed. Coastal Eng. 18 (1992) 205-229.
S.T. Grilli, M.A. Losada and F. Martin, Wave Impact Forces on Mixed Breakwaters Proc. 23rd Intl. Conf. Coastal Eng., ASCE, Venice, Italy (1993) 1161-1174.
W.D. Collins, On the solution of some axisymmetric boundary value problems by means of integral equations III. Some electrostatic and hydrodynamic problems for two spherical caps. Proc. London Math. Soc. (3) 10 (1960) 428-460.
H. Lamb, Hydrodynamics. 6th edition. Cambridge: Cambridge University Press (1932) 738pp.
I. Eames, J.C.R. Hunt and S.E. Belcher, Displacement of inviscid fluid by a sphere moving away from a wall. J. Fluid Mech. 324 (1996) 333-353.
L. Li, W.W. Schultz and H. Merte, The velocity potential and the interacting force for two spheres moving perpendicularly to the line joining their centers. J. Eng. Maths. 27 (1993) 147-160.
Y. Solomentsev, D. Velegol and J.L. Anderson, Conduction in the small gap between two spheres. Phys. Fluids 9 (1997) 1209-1217.
G.E. Latta and G.B. Hess, Potential flow past a sphere tangent to a plane. Phys. Fluids 16 (1973) 974-976.
A.M.J. Davis, High frequency limiting virtual-mass coefficients of heaving half-immersed spheres. J. Fluid Mech. 80 (1977) 305-319.
J.J. Stoker, Water Waves. New York: Wiley (1992) 567 pp.
P. Moon and D.E. Spencer, Field Theory Handbook. Berlin: Springer (1961) 236 pp.
S.J. Cox, Pressure Impulses caused by Wave Impact. PhD Thesis: School of Mathematics, University of East Anglia (1998) 199 pp.
I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products. New York: Academic Press (1965) 1160 pp.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cox, S.J., Cooker, M.J. Potential flow past a sphere touching a tangent plane. Journal of Engineering Mathematics 38, 355–370 (2000). https://doi.org/10.1023/A:1004738528787
Issue Date:
DOI: https://doi.org/10.1023/A:1004738528787