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Water entry of small hydrophobic spheres

Published online by Cambridge University Press:  25 January 2009

JEFFREY M. ARISTOFF
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
JOHN W. M. BUSH
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Corresponding
E-mail address:

Abstract

We present the results of a combined experimental and theoretical investigation of the normal impact of hydrophobic spheres on a water surface. Particular attention is given to characterizing the shape of the resulting air cavity in the low Bond number limit, where cavity collapse is driven principally by surface tension rather than gravity. A parameter study reveals the dependence of the cavity structure on the governing dimensionless groups. A theoretical description based on the solution to the Rayleigh–Besant problem is developed to describe the evolution of the cavity shape and yields an analytical solution for the pinch-off time in the zero Bond number limit. The sphere's depth at cavity pinch-off is also computed in the low Weber number, quasi-static limit. Theoretical predictions compare favourably with our experimental observations in the low Bond number regime, and also yield new insight into the high Bond number regime considered by previous investigators. Discrepancies are rationalized in terms of the assumed form of the velocity field and neglect of the longitudinal component of curvature, which together preclude an accurate description of the cavity for depths less than the capillary length. Finally, we present a theoretical model for the evolution of the splash curtain formed at high Weber number and couple it with the underlying cavity dynamics.

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Copyright © Cambridge University Press 2008

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References

Abelson, H. I. 1970 Pressure measurements in the water-entry cavity. J. Fluid Mech. 44, 129144.CrossRefGoogle Scholar
Ablett, R. 1923 An investigation of the angle of contact between paraffin wax and water. Philos. Mag. 46, 244256.CrossRefGoogle Scholar
Ashley, S. 2001 Warp drive underwater. Sci. Amer. 284, 7079.CrossRefGoogle Scholar
Bell, G. E. 1924 On the impact of a solid sphere with a fluid surface. Phil. Mag. J. Sci. 48, 753765.CrossRefGoogle Scholar
Bergmann, R., Meer, D., Stijnman, M., Sandtke, R., Prosperetti, A. & Lohse, D. 2006 Giant bubble pinch-off. Phys. Rev. Lett. 96 (154505), 14.CrossRefGoogle ScholarPubMed
Besant, W. H. 1859 Hydrostatics and Hydrodynamics. Cambridge University Press.Google Scholar
Birkhoff, G. & Caywood, T. E. 1949 Fluid flow patterns. J. Appl. Phys. 20, 646659.CrossRefGoogle Scholar
Birkhoff, G. & Isaacs, R. 1951 Transient cavities in air–water entry. Tech. Rep. 1490. Navord Rep.Google Scholar
Birkhoff, G. & Zarantonello, E. H. 1957 Jets, wakes and cavities. In Applied Mathematics and Mechanics (ed. Frenkiel, F. N.), pp. 1353. Academic.Google Scholar
Bush, J. W. M. & Hu, D. L. 2006 Walking on water: biolocomotion at the interface. Annu. Rev. Fluid Mech. 38, 339369.CrossRefGoogle Scholar
Bush, J. W. M., Prakash, M. & Hu, D. L. 2008 The surface structure of water-walking arthropods: form and function. Adv. Insect Physiol. 34, 117192.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Clanet, C. 2007 Waterbells and liquid sheets. Annu. Rev. Fluid Mech. 39, 469496.CrossRefGoogle Scholar
Cossali, G. E., Marengo, M., Coghe, A. & Zhdanov, S. 2004 The role of time in single drop splash on thin film. Exp. Fluids 36, 888900.CrossRefGoogle Scholar
Culick, F. E. C. 1960 Comments on a ruptured soap film. J. Appl. Phys. 31, 1128.CrossRefGoogle Scholar
De Gennes, P. G., Brochard-Wyart, F. & Quere, D. 2004 Capillarity and Wetting Phenomena. Springer.CrossRefGoogle Scholar
Duclaux, V., Caillé, F., Duez, C., Ybert, C., Bocquet, L. & Clanet, C. 2007 Dynamics of transient cavities. J. Fluid Mech. 591, 119.CrossRefGoogle Scholar
Duez, C., Ybert, C., Clanet, C. & Bocquet, L. 2007 Making a splash with water repellency. Nature Phys. 3, 180183.CrossRefGoogle Scholar
Gaudet, S. 1998 Numerical simulation of circular disks entering the free surface of a fluid. Phys. Fluids 10, 24892499.CrossRefGoogle Scholar
Gekle, S., van der Bos, A., Bergmann, R., van der Meer, D. & Lohse, D. 2008 Noncontinuous froude number scaling for the closure depth of a cylindrical cavity. Phys. Rev. Lett. 100, 084502.CrossRefGoogle ScholarPubMed
Gilbarg, D. & Anderson, R. 1948 Influence of atmospheric pressure on the phenomena accompanying the entry of spheres into water. J. Appl. Phys. 19, 127139.CrossRefGoogle Scholar
Glasheen, J. W. & McMahon, T. A. 1996 a A hydrodynamical model of locomotion in the basilisk lizard. Nature 380, 340342.CrossRefGoogle Scholar
Glasheen, J. W. & McMahon, T. A. 1996 b Vertical water entry of disks at low Froude numbers. Phys. Fluids 8, 20782083.CrossRefGoogle Scholar
Grumstrup, T., Keller, J. B. & Belmonte, A. 2007 Cavity ripples observed during the impact of solid objects into liquids. Phys. Rev. Lett. 99 (114502), 14.CrossRefGoogle ScholarPubMed
Hiemenz, P. C. & Rajagopalan, R. 1997 Principles of colloid and surface chemistry. CRC Press.CrossRefGoogle Scholar
Howison, S. D., Ockendon, J. R. & Wilson, S. K. 1991 Incompressible water-entry problems at small deadrise angles. J. Fluid Mech. 222, 215230.CrossRefGoogle Scholar
Korobkin, A. A. & Pukhnachov, V. V. 1988 Initial stage of water impact. Annu. Rev. Fluid Mech. 20, 159185.CrossRefGoogle Scholar
Kralchevsky, P. A. & Denkov, N. D. 2001 Capillary forces and structuring in layers of colloid particles. Curr. Opin. Coll. Interf. Sci. 6, 383401.CrossRefGoogle Scholar
Ku, T. C., Ramsey, J. H. & Clinton, W. C. 1968 Calculation of liquid droplet profiles from closed-form solution of young-laplace equation. IBM J. Res. Develop. 12, 441447.CrossRefGoogle Scholar
Lee, D. G. & Kim, H. Y. 2008 Impact of a superhydrophobic sphere onto water. Langmuir 24, 142145.CrossRefGoogle ScholarPubMed
Lee, M., Longoria, R. G. & Wilson, D. E. 1997 Cavity dynamics in high-speed water entry. Phys. Fluids 9, 540550.CrossRefGoogle Scholar
Lohse, D., Bergmann, R., Mikkelsen, R., Zeilstra, C., van der Meer, D. M., Versluis, M., van der Weele, K., van der Hoef, M. & Kuipers, H. 2004 a Impact on soft sand: void collapse and jet formation. Phys. Rev. Lett. 93, 198003, 14.CrossRefGoogle ScholarPubMed
Lohse, D., Rauhe, R., Bergmann, R. & van der Meer, D. 2004 b Granular physics: creating a dry variety of quicksand. Nature 432, 689690.CrossRefGoogle ScholarPubMed
Mallock, A. 1918 Sounds produced by drops falling on water. Proc. R. Soc. Lond. A 95, 138143.CrossRefGoogle Scholar
Mansfield, E. H., Sepangi, H. R. & Eastwood, E. A. 1997 Equilibrium and mutual attraction or repulsion of objects supported by surface tension. Phil. Trans. R. Soc. Lond. A 355, 869919.CrossRefGoogle Scholar
May, A. 1951 Effect of surface condition of a sphere on its water-entry cavity. J. Appl. Phys. 22, 12191222.CrossRefGoogle Scholar
May, A. 1952 Vertical entry of missiles into water. J. Appl. Phys. 23, 13621372.CrossRefGoogle Scholar
May, A. 1975 Water entry and the cavity-running behavior of missiles. Tech Rep. 20910. Naval Surface Weapons Center White Oak Laboratory.Google Scholar
Melosh, H. J. 1980 Cratering mechanics – observational, experimental, and theoretical. Annu. Rev. Earth Planet. Sci. 8, 6593.CrossRefGoogle Scholar
Melosh, H. J. & Ivanov, B. A. 1999 Impact crater collapse. Annu. Rev. Earth Planet. Sci. 27, 385415.CrossRefGoogle Scholar
Miloh, T. 1991 On the initial-stage slamming of a rigid sphere in a vertical water entry. Appl. Ocean Res. 13, 4348.CrossRefGoogle Scholar
Oliver, J. M. 2007 Second-order wagner theory for two-dimensional water-entry problems at small deadrise angles. J. Fluid Mech. 572, 5985.CrossRefGoogle Scholar
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145185.CrossRefGoogle Scholar
Rayleigh, LORD 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 9498.CrossRefGoogle Scholar
Richardson, E. G. 1948 The impact of a solid on a liquid surface. Proc. Phys. Soc. 61, 352367.CrossRefGoogle Scholar
Shi, H. H., Itoh, M. & Takami, T. 2000 Optical observation of the supercavitation induced by high-speed water entry. Trans. ASME 122, 806810.Google Scholar
Shi, H. H. & Kume, M. 2004 Underwater acoustics and cavitating flow of water entry. Acta Mechanica Sinica 20, 374382.Google Scholar
Taylor, G. I. 1950 The formation of a blast wave by a very intense explosion. Proc. R. Soc. Lond. A 201, 159186.CrossRefGoogle Scholar
Taylor, G. I. 1959 The dynamics of thin sheets of fluid: waves on fluid sheets. Proc. R. Soc. Lond. A 253, 296312.CrossRefGoogle Scholar
Thoroddsen, S. T., Etoh, T. G., Takehara, K. & Takano, Y. 2004 Impact jetting by a solid sphere. J. Fluid Mech. 499, 139148.CrossRefGoogle Scholar
Thoroddsen, S. T. & Shen, A. Q. 2001 Granular jets. Phys. Fluids 13, 46.CrossRefGoogle Scholar
Tomotika, S. 1935 On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. R. Soc. Lond. A 150, 322337.CrossRefGoogle Scholar
Vella, D., Lee, D. G. & Kim, H. Y. 2006 a The load supported by small floating objects. Langmuir 22, 59795981.CrossRefGoogle ScholarPubMed
Vella, D., Lee, D. G. & Kim, H. Y. 2006 b Sinking of a horizontal cylinder. Langmuir 22, 29722974.CrossRefGoogle ScholarPubMed
Vella, D. & Metcalfe, P. 2007 Surface tension dominated impact. Phys. Fluids 19, 072108.CrossRefGoogle Scholar
Von Karman, T. 1929 The impact on seaplane floats during landing. Tech Rep. 321. NACA.Google Scholar
Wagner, H. 1932 Phenomena associated with impacts and sliding on liquid surfaces. ZAMM 12, 193235.CrossRefGoogle Scholar
Ward, J. V. 1992 Aquatic Insect Ecology: Biology, and Habitat. John Wiley.Google Scholar
Whalley, I. A. 2002 Project upkeep – a review of the WWII dambuster weapon. In Proc. 38th AIAA/ASME/SAE/ASEE Joint Propulsion Conf. Exhibit. American Institute of Aeronautics and Astronautics Inc., Indianapolis, IN.Google Scholar
Worthington, A. M. & Cole, R. S. 1897 Impact with a liquid surface, studied by the aid of instantaneous photography. Phil. Trans. R. Soc. Lond. A 189, 137148.CrossRefGoogle Scholar
Worthington, A. M. & Cole, R. S. 1900 Impact with a liquid surface, studied by the aid of instantaneous photography: paper 2. Phil. Trans. R. Soc. Lond. A 194, 175199.CrossRefGoogle Scholar

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