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Creeping motion of a sphere along the axis of a closed axisymmetric container

Published online by Cambridge University Press:  07 August 2007

N. LECOQ ,
K. MASMOUDI ,
R. ANTHORE  and
F. FEUILLEBOIS
N. LECOQ
Affiliation:
UMR 6634 CNRS Université de Rouen, 76801 Mont Saint Aignan, France
K. MASMOUDI
Affiliation:
UMR 6634 CNRS Université de Rouen, 76801 Mont Saint Aignan, France
R. ANTHORE
Affiliation:
UMR 6634 CNRS Université de Rouen, 76801 Mont Saint Aignan, France
F. FEUILLEBOIS*
Affiliation:
PMMH, ESPCI, 10, rue Vauquelin, 75005 Paris, France
*
† Author to whom correspondence should be addressed: [email protected].
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Abstract

The creeping flow around a sphere settling along the axis of a closed axisymmetric container is obtained both theoretically and experimentally. The numerical technique for solving the Stokes equations uses the classical Sampson expansion; the boundary conditions on the sphere are satisfied exactly and those on the container walls are applied in the sense of least squares. This is an extension to the axisymmetric case of the technique for solving various two-dimensional flow problems. Two types of axisymmetric container are considered here as examples: circular cylinders closed by planes at both ends, and cones closed by a base plane. Calculated streamlines patterns show various sets of eddies, depending upon the geometry and the sphere position. Results are in agreement with earlier Stokes flow calculations of eddies in corners and in closed containers. Experiments use laser interferometry to measure the vertical displacement of a steel bead a few millimetres in diameter settling in a container filled with a very viscous silicone oil. The Reynolds number based on the sphere radius is typically of the order of 10−5. The accuracy on the vertical displacement is 50nm. Experiments show that the motion towards the apex of a cone is much slower than that towards a plane; the bead takes hours to reach the micrometre size roughness asperities on a conical wall, as compared with minutes to reach those on a plane wall. The numerical results for the drag force are in excellent agreement with experiments both for the cylindrical and the conical containers. With standard computer accuracy, the present numerical technique applies when the gap between the sphere and the nearby wall is larger than about one radius. For a sphere in the vicinity of any plane horizontal wall, these results also match with a previous analytical solution. That solution is in excellent agreement with our experimental results at small distances from the wall (typically less than a diameter, depending on the sphere size).

Type
Papers
Information
Journal of Fluid Mechanics , Volume 585 , 25 August 2007 , pp. 127 - 152
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Adamczyk, Z., Adamczyk, M., & Van de Ven, T. G. M. 1983 Resistance coefficient of a solid sphere approaching plane and curved boundaries. J. Colloid Interface Sci 96, 204–213.CrossRefGoogle Scholar
Ambari, A., Gauthier-Manuel, B. & Guyon, E. 1984 Wall effects on a sphere translating at constant velocity. J. Fluid Mech. 149, 235–253.CrossRefGoogle Scholar
Ambari, A., Gauthier-Manuel, B. & Guyon, E. 1985 Direct measurement of tube wall effect on the Stokes force. Phys. Fluids 28 (5), 156–159.CrossRefGoogle Scholar
Axelsson, A. 1985 A survey of preconditioned iterative methods for linear systems of equations. BIT 25, 166–187.CrossRefGoogle Scholar
Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C. & Van der Vorst, H. 1994 Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd edn. SIAM, Philadelphia.CrossRefGoogle Scholar
Blake, J. R. 1979 On the generation of viscous toroidal eddies in a cylinder. J. Fluid Mech. 95, 209–222.CrossRefGoogle Scholar
Bourot, J. M. 1969 Sur l'application d'une méthode de moindres carrés à la résolution approchée du problème aux limites, pour certaines catégories d'écoulements. J. Méc. Théor. Appl. 8, 301–322.Google Scholar
Bourot, J. M. & Moreau, F. 1987 Sur l'utilisation de la série cellulaire pour le calcul d'écoulements plans de Stokes en canal indéfini: application au cas d'un cylindre circulaire en translation. Mech. Res. Commun. 14 (3), 187–197.CrossRefGoogle Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242–251.CrossRefGoogle Scholar
Coutanceau, M. 1971 Contribution à l'étude théorique et expérimentale de l'écoulement autour d'une sphère qui se déplace dans l'axe d'un cylindre à faible nombre de Reynolds ou en régime irrotationnel. Thèse de Doctorat d'État, Université de Poitiers.Google Scholar
Coutanceau, M. 1987 Confined creeping flow around an axisymmetric body: increase of the shape effect by a tube wall. Fluid Dyn. Res. 2, 153–174.CrossRefGoogle Scholar
Davey, K. & Ward, M. J. 2000 A successive preconditioned conjugate gradient method for the minimization of quadratic and nonlinear functions. Appl. Numer. Maths 35, 129–156.CrossRefGoogle Scholar
Davis, A. M. J., O'Neill, M. E., Dorrepaal, J. M. & Ranger, K. B. 1976 Separation from the surface of two equal spheres in Stokes flow. J. Fluid Mech. 77, 625–644.CrossRefGoogle Scholar
Dean, W. R. & Montagnon, P. E. 1949 On the steady motion of a viscous liquid in a corner. Proc. Camb. Phil. Soc. 45, 389–394.CrossRefGoogle Scholar
Ekiel-Jeżewska, M. L., Feuillebois, F., Lecoq, N., Masmoudi, K., Anthore, R., Bostel, F. & Wajnryb, E. 1999 Hydrodynamic interactions between two spheres at contact. Phys. Rev. E 59, 3182–3191. Erratum: 1999, 60, 4994.Google Scholar
Ekiel-Jeżewska, M. L., Lecoq, N., Anthore, R., Bostel, F. & Feuillebois, F. 2002 a Rotation due to hydrodynamic interactions between two spheres at contact. Phys. Rev. E 66 (051504), 1–14.Google ScholarPubMed
Ekiel-Jeżewska, M. L., Lecoq, N., Anthore, R., Bostel, F. & Feuillebois, F. 2002 b Interactions between two close spheres in Stokes flow. In Tubes, Sheets and Singularities in Fluid Dynamics, Proc. IUTAM Symp. Zakopane, Poland, 2–7 September vol. 9 ofFluid Mechanics and its Applications (ed. Bajer, K. & Moffatt, H. K.), pp. 343–348. Kluwer.Google Scholar
Elasmi, L., Berzig, M. & Feuillebois, F. 2003 Stokes flow for the axisymmetric motion of several spherical particles perpendicular to a plane wall. Z. Angew. Math. Phys. 54, 304–327.CrossRefGoogle Scholar
Ganatos, P., Pfeffer, R. & Weinbaum, S. 1980 A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 1. Perpendicular motion. J. Fluid Mech. 99, 739–754.CrossRefGoogle Scholar
Graham, A. L., Mondy, L. A., Miller, J. D. Wagner, N. J. & Cook, W. A. 1989 Numerical simulations of eccentricity and end effects in falling ball rheometry. J. Rheol. 33, 1107–1128.CrossRefGoogle Scholar
Habermann, W. L. & Sayre, R. M. 1958 David Taylor model basin. Rep. 1143, Washington DC, US Navy Dept.Google Scholar
Happel, J. & Brenner, H. 1991 Low Reynolds Number Hydrodynamics, 5th edn. Kluwer.Google Scholar
Hellou, M. 1988 Etude numérique et expérimentale de l'écoulement à structure cellulaire, engendré par la rotation d'un cylindre dans un canal. Thèse de Doctorat, Université de Poitiers.Google Scholar
Hellou, M. & Coutanceau, M. 1992 Cellular Stokes flow induced by rotation of a cylinder in a closed channel. J. Fluid Mech. 236, 557–577.CrossRefGoogle Scholar
Kim, M. U. 1979 Slow viscous flow due to the motion of a sphere on the axis of a circular cone. J. Phys. Soc. Japan 47, 1670–1675.CrossRefGoogle Scholar
Kim, S. & Karrila, S. J. 1993 Microhydrodynamics: Principles and Selected Applications. Butterworth–Heinemann.Google Scholar
Lecoq, N. & Feuillebois, F. 2007 Lubrication in a flat cone: the transition from a cone to a plane wall. Phys. Fluids 19, 038103.CrossRefGoogle Scholar
Lecoq, N., Feuillebois, F., Anthore, N., Anthore, R. Bostel, F. & Petipas, C. 1993 Precise measurement of particle–wall hydrodynamic interactions at low Reynolds number using laser interferometry. Phys. Fluids A 5 (1), 3–12.CrossRefGoogle Scholar
Lecoq, N., Feuillebois, F., Anthore, R., Petipas, C. & Bostel, F. 1995 Experimental investigation of the hydrodynamic interactions between a sphere and a large spherical obstacle. J. Phys. Paris II 5, 323–334.Google Scholar
Lecoq, N., Anthore, R., Cichocki, B., Szymczak, P. & Feuillebois, F. 2004 Drag force on a sphere moving towards a corrugated wall. J. Fluid Mech. 513, 247–264.CrossRefGoogle Scholar
Lobry, L. & Ostrowsky, N. 1996 Diffusion of Brownian particles trapped between two walls: theory and dynamic-light-scattering measurements. Phys. Rev. B 53, 12050–12056.CrossRefGoogle ScholarPubMed
Masmoudi, K., Lecoq, N., Anthore, R., May, S. & Feuillebois, F. 1998 Lubricating motion of a sphere in a conical vessel. Phys. Fluids 10 (5), 1231–1233.CrossRefGoogle Scholar
Masmoudi, K., Lecoq, N., Anthore, R., Bostel, F. & Feuillebois, F. 2002 Accurate measurement of hydrodynamic interactions between a particle and walls. Exps. Fluids 32 (1), 55–65.CrossRefGoogle Scholar
Maude, A. D. 1961 End effects in falling-sphere viscometer. Br. J. Appl. Phys. 12, 293–295.CrossRefGoogle Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–17.CrossRefGoogle Scholar
Moreau, F. 1988 Etude de la série cellulaire, bidimensionnelle et complexe, et de ses applications aux écoulements de Stokes en canal plan. Thèse de Doctorat, Université de Poitiers.Google Scholar
Moreau, F. & Bourot, J. M. 1993 Ecoulements cellulaires de Stokes produits en canal plan illimité par la rotation de deux cylindres Z. Angew. Math. Phys. 44, 777–798.CrossRefGoogle Scholar
O'Neill, M. E. 1983 On angles of separation in Stokes flow. J. Fluid Mech. 133, 427–442.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Method for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1990 Numerical Recipes in C, the Art of Scientific Computing. Cambridge University Press.Google Scholar
Sampson, R. A. 1891 On Stokes's current function, Phil. Trans. R. Soc. A 182, 449.Google Scholar
Sano, O. 1987 Mobility of a small sphere in a viscous fluid confined in a rigid circular cylinder of finite length. J. Phys. Soc. Japan 56, 2713–2720.CrossRefGoogle Scholar
Shankar, P. N. 1993 The eddy structure in Stokes flow in a cavity. J. Fluid Mech. 250, 371–383.CrossRefGoogle Scholar
Sigli, D. 1970 Contribution à la mise au point d'une technique de résolution approchée du problème aux limites, pour un écoulement de révolution en régime de Stokes. Thèse de 3ème Cycle, Université de Poitiers.Google Scholar
Tullock, D. L., Phan-Thien, N. & Graham, A. L. 1992 Boundary element simulations of spheres settling in circular, square and triangular conduits. Rheol Acta 31, 139–150.CrossRefGoogle Scholar
Wakiya, S. 1976 Axisymmetric flow of a viscous fluid near the vertex of a body. J. Fluid Mech. 78, 737–747.CrossRefGoogle Scholar
Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: numerical method of solution. J. Fluid Mech. 69, 377–403.CrossRefGoogle Scholar
Zimmerman, W. B. 2004 On the resistance of a spherical particle settling in a tube of viscous fluid. Intl J. Engng Sci. 42, 1753–1778.CrossRefGoogle Scholar
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