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Motion of a sphere normal to a wall in a second-order fluid

Published online by Cambridge University Press:  31 August 2007

A. M. ARDEKANI
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697-3975, USA
R. H. RANGEL
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697-3975, USA
D. D. JOSEPH
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697-3975, USA Department of Aerospace Engineering and Mechanics, University of Minnesota, MN 55455, USA

Abstract

The motion of a sphere normal to a wall is investigated. The normal stress at the surface of the sphere is calculated and the viscoelastic effects on the normal stress for different separation distances are analysed. For small separation distances, when the particle is moving away from the wall, a tensile normal stress exists at the trailing edge if the fluid is Newtonian, while for a second-order fluid a larger tensile stress is observed. When the particle is moving towards the wall, the stress is compressive at the leading edge for a Newtonian fluid whereas a large tensile stress is observed for a second-orderfluid. The contribution of the second-order fluid to the overall force applied to the particle is towards the wall in both situations. Results are obtained using Stokes equationswhen α12=0. In addition, a perturbation method has been utilized for a sphere very close to a wall and the effect of non-zero α12 is discussed. Finally, viscoelastic potential flow is used and the results are compared with the other methods.

Type
Papers
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Ardekani, A. M. & Rangel, R. H. 2006 Unsteady motion of two solid spheres in Stokes flow. Phys. Fluids. 18, 103306.Google Scholar
Ardekani, A. M. & Rangel, R. H. 2007 Numerical investigation of particle–particle and particle–wall collisions in a viscous fluid. J. Fluid Mech. (submitted).Google Scholar
Batchelor, G. K. & Green, J. T. 1972 The determination of the bulk stress in a suspension of spherical particles to order c2. J. Fluid Mech. 56, 401427.Google Scholar
Becker, D. L. E., McKinley, G. H. & Stone, H. A. 1996 Sedimentation of a sphere near a plane wall: Weak non-Newtonian and inertial effects. J. Non-Newtonian Fluid Mech. 63, 4586.CrossRefGoogle Scholar
Bird, R., Armstrong, R. & Hassager, O. 1987 Dynamics of Polymeric Liquids. John Wiley.Google Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Eng. Sci. 16, 242.CrossRefGoogle Scholar
Brindley, G., Davies, J. M. & Walters, K. 1976 Elastico-viscous squeeze films. Part I. J. Non-Newtonian Fluid Mech. 1, 1937.Google Scholar
Brunn, P. 1977 Interaction of spheres in a viscoelastic fluid. Rheologica Acta. 16, 461475.Google Scholar
Coleman, B. & Noll, W. 1960 An approximation theorem for functionals, with applications in continum mechanics. Arch. Rat. Mech. Anal. 6, 355370.Google Scholar
Davis, R. H. 1987 Elastohydrodynamic collisions of particles. PhysicoChem. Hydrodyn. 9, 4152.Google Scholar
Engmann, J., Servais, C. & Burbidge, A. S. 2005 Squeeze flow theory and applications to rheometry: A review. J. Non-Newtonian Fluid Mech. 132, 127.CrossRefGoogle Scholar
Fortes, A. F., Joseph, D. D. & Lundgren, T. S. 1987 Nonlinear mechanics of fluidization of beds of spherical particles. J. Fluid Mech. 177, 467483.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall. I. Motion through quiescent fluid. Chem. Engng. Sci. 22, 637651.Google Scholar
Ho, B. P. & Leal, L. G. 1976 Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-order fluid. J. Fluid Mech. 79, 783799.CrossRefGoogle Scholar
Jeffrey, D. J. 1973 Conduction through a random suspension of spheres. Proc. R. Soc. Lond. A 335, 355.Google Scholar
Jeffrey, D. J. & Corless, R. M. 1988 Forces and stresslets for the axisymmetric motion of nearly touching unequal spheres. PhysicoChemHydrodyn. 10, 461.Google Scholar
Joseph, D. D. 1990 Dynamics of Viscoelastic Liquids. Springer.Google Scholar
Joseph, D. D. 1992 Bernoulli equation and the competition of elastic and inertial pressure in the potential flow of a second-order fluid. J. Non-Newtonian Fluid Mech. 42, 358389.CrossRefGoogle Scholar
Joseph, D. D. & Feng, J. 1996 A note on the forces that move particles in a second-order fluid. J. Non-Newtonian Fluid Mech. 64, 299302.CrossRefGoogle Scholar
Joseph, D. D., Funada, T. & Wang, J. 2007 Potential Flows of Viscous and Viscoelastic Fluids. Cambridge University Press.CrossRefGoogle Scholar
Joseph, D. D., Liu, Y. J., Poletto, , & , M. Feng, J. 1994 Aggregation and dispersion of a spheres falling in viscoelastic liquids. J. Non-Newtonian Fluid Mech. 54, 4586.Google Scholar
Joseph, G. G., Zenit, R., Hunt, M. L. & Rosenwinkel, A. M. 2001 Particle-wall collisions in a viscous fluid. J. Fluid Mech. 433, 329346.Google Scholar
Koch, D. L. & Subramanian, 2006 The stress in a dilute suspension of spheres suspended in a second-order fluid subject to a linear velocity field. J. Non-Newtonian Fluid Mech. 138, 87.CrossRefGoogle Scholar
Lamb, H. 1945 Hydrodynamics. Dover.Google Scholar
Leal, L. G. 1975 The slow motion of slender rod-like particles in a second-order fluid. J. Fluid Mech. 69, 305337.CrossRefGoogle Scholar
Liu, Y. J. & Joseph, D. D. 1993 Sedimentation of particles in polymer-solutions. J. Fluid Mech. 255, 565595.CrossRefGoogle Scholar
Maude, A. D. 1961 End effects in a falling-sphere viscometer. Br. J. Appl. Phys. 12, 293.Google Scholar
Mifflin, R. T. 1985 Dissipation in a dilute suspension of spheres in a second-order fluid. J. Non-Newtonian Fluid Mech. 17, 267274.Google Scholar
Pasol, L., Chaoui, M., Yahiaoui, S. & Feuillebois, F. 2005 Analytic solution for a spherical particle near a wall in axisymmetrical polynomial creeping flows. Phys. Fluids. 17, 073602.Google Scholar
Riddle, M. J., Narvaez, C. & Bird, R. B. 1977 Interactions between two spheres falling along their line of centers in viscoelastic fluid. J. Non-Newtonian Fluid Mech. 2, 2325.Google Scholar
Rivlin, R. S. & Ericksen, J. L. 1955 Stress deformation relations for isotropic materials. J. Rat. Mech. Anal. 4, 323425.Google Scholar
Rodin, G. 1995 Squeeze film between two spheres in a power-law fluid. J. Non-Newtonian Fluid Mech. 63, 141152.CrossRefGoogle Scholar
Singh, P. & Joseph, D. D. 2000 Sedimentation of a sphere near a wall in Oldroyd-B fluid. J. Non-Newtonian Fluid Mech. 94, 179203.Google Scholar
Sun, K. & Jayaraman, K. 1984 Bulk rheology of dilute suspensions in viscoelastic liquids. Rheol. Acta. 23, 84.Google Scholar
Takagi, S., Oguz, H. N., Zhang, Z. & Prosperetti, A. 2003 A new method for particle simulation - part ii: Two-dimensional Navier-Stokes flow around cylinders. J. Comput. Phys. 187, 371390.Google Scholar
Tanner, R. I. 1985 Engineering Rheology. Clarendon.Google Scholar
Wang, J. & Joseph, D. D. 2004 Potential flow of a second-order fluid over a sphere or an eclipse. J. Fluid Mech. 511, 201215.CrossRefGoogle Scholar