Skip to main content

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

  • A. M. ARDEKANI (a1), R. H. RANGEL (a1) and D. D. JOSEPH (a1) (a2)

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.

Hide All
Ardekani A. M. & Rangel R. H. 2006 Unsteady motion of two solid spheres in Stokes flow. Phys. Fluids. 18, 103306.
Ardekani A. M. & Rangel R. H. 2007 Numerical investigation of particle–particle and particle–wall collisions in a viscous fluid. J. Fluid Mech. (submitted).
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.
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.
Bird R., Armstrong R. & Hassager O. 1987 Dynamics of Polymeric Liquids. John Wiley.
Brenner H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Eng. Sci. 16, 242.
Brindley G., Davies J. M. & Walters K. 1976 Elastico-viscous squeeze films. Part I. J. Non-Newtonian Fluid Mech. 1, 1937.
Brunn P. 1977 Interaction of spheres in a viscoelastic fluid. Rheologica Acta. 16, 461475.
Coleman B. & Noll W. 1960 An approximation theorem for functionals, with applications in continum mechanics. Arch. Rat. Mech. Anal. 6, 355370.
Davis R. H. 1987 Elastohydrodynamic collisions of particles. PhysicoChem. Hydrodyn. 9, 4152.
Engmann J., Servais C. & Burbidge A. S. 2005 Squeeze flow theory and applications to rheometry: A review. J. Non-Newtonian Fluid Mech. 132, 127.
Fortes A. F., Joseph D. D. & Lundgren T. S. 1987 Nonlinear mechanics of fluidization of beds of spherical particles. J. Fluid Mech. 177, 467483.
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.
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.
Jeffrey D. J. 1973 Conduction through a random suspension of spheres. Proc. R. Soc. Lond. A 335, 355.
Jeffrey D. J. & Corless R. M. 1988 Forces and stresslets for the axisymmetric motion of nearly touching unequal spheres. PhysicoChemHydrodyn. 10, 461.
Joseph D. D. 1990 Dynamics of Viscoelastic Liquids. Springer.
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.
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.
Joseph D. D., Funada T. & Wang J. 2007 Potential Flows of Viscous and Viscoelastic Fluids. Cambridge University Press.
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.
Joseph G. G., Zenit R., Hunt M. L. & Rosenwinkel A. M. 2001 Particle-wall collisions in a viscous fluid. J. Fluid Mech. 433, 329346.
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.
Lamb H. 1945 Hydrodynamics. Dover.
Leal L. G. 1975 The slow motion of slender rod-like particles in a second-order fluid. J. Fluid Mech. 69, 305337.
Liu Y. J. & Joseph D. D. 1993 Sedimentation of particles in polymer-solutions. J. Fluid Mech. 255, 565595.
Maude A. D. 1961 End effects in a falling-sphere viscometer. Br. J. Appl. Phys. 12, 293.
Mifflin R. T. 1985 Dissipation in a dilute suspension of spheres in a second-order fluid. J. Non-Newtonian Fluid Mech. 17, 267274.
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.
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.
Rivlin R. S. & Ericksen J. L. 1955 Stress deformation relations for isotropic materials. J. Rat. Mech. Anal. 4, 323425.
Rodin G. 1995 Squeeze film between two spheres in a power-law fluid. J. Non-Newtonian Fluid Mech. 63, 141152.
Singh P. & Joseph D. D. 2000 Sedimentation of a sphere near a wall in Oldroyd-B fluid. J. Non-Newtonian Fluid Mech. 94, 179203.
Sun K. & Jayaraman K. 1984 Bulk rheology of dilute suspensions in viscoelastic liquids. Rheol. Acta. 23, 84.
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.
Tanner R. I. 1985 Engineering Rheology. Clarendon.
Wang J. & Joseph D. D. 2004 Potential flow of a second-order fluid over a sphere or an eclipse. J. Fluid Mech. 511, 201215.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 28 *
Loading metrics...

Abstract views

Total abstract views: 129 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd November 2017. This data will be updated every 24 hours.