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

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 α1+α2=0. In addition, a perturbation method has been utilized for a sphere very close to a wall and the effect of non-zero α1+α2 is discussed. Finally, viscoelastic potential flow is used and the results are compared with the other methods.

[1]  A note on the forces that move particles in a second-order fluid , 1996 .

[2]  P. Brunn.  Interaction of spheres in a viscoelastic fluid , 1977 .

[3]  C. Servais,et al.  Squeeze flow theory and applications to rheometry: A review , 2005 .

[4]  R. Bird,et al.  Interactions between two spheres falling along their line of centers in a viscoelastic fluid , 1977 .

[5]  L. G. Leal,et al.  Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-order fluid , 1976, Journal of Fluid Mechanics.

[6]  Potential flow of a second-order fluid over a sphere or an ellipse , 2004, Journal of Fluid Mechanics.

[7]  B. D. Coleman,et al.  An approximation theorem for functionals, with applications in continuum mechanics , 1960 .

[8]  Gareth H. McKinley,et al.  SEDIMENTATION OF A SPHERE NEAR A PLANE WALL: WEAK NON-NEWTONIAN AND INERTIAL EFFECTS , 1996 .

[9]  L. G. Leal,et al.  The slow motion of slender rod-like particles in a second-order fluid , 1975, Journal of Fluid Mechanics.

[10]  Roger H. Rangel,et al.  Unsteady motion of two solid spheres in Stokes flow , 2006 .

[11]  Curtiss,et al.  Dynamics of Polymeric Liquids , .

[12]  Daniel D. Joseph,et al.  Sedimentation of particles in polymer solutions , 1993, Journal of Fluid Mechanics.

[13]  D. Koch,et al.  The stress in a dilute suspension of spheres suspended in a second-order fluid subject to a linear velocity field , 2006 .

[14]  Gregory J. Rodin,et al.  Squeeze film between two spheres in a power-law fluid , 1996 .

[15]  D. Joseph,et al.  Nonlinear mechanics of fluidization of beds of spherical particles , 1987, Journal of Fluid Mechanics.

[16]  R. G. Cox,et al.  Slow viscous motion of a sphere parallel to a plane wall—I Motion through a quiescent fluid , 1967 .

[17]  François Feuillebois,et al.  Analytical solutions for a spherical particle near a wall in axisymmetrical polynomial creeping flows , 2005 .

[18]  K. Jayaraman,et al.  Bulk rheology of dilute suspensions in viscoelastic liquids , 1984 .

[19]  Daniel D. Joseph,et al.  Potential flows of viscous and viscoelastic fluids , 1994, Journal of Fluid Mechanics.

[20]  Andrea Prosperetti,et al.  PHYSALIS: a new method for particle simulation part II: two-dimensional Navier--Stokes flow around cylinders , 2003 .

[21]  Daniel D. Joseph,et al.  Fluid Dynamics Of Viscoelastic Liquids , 1990 .

[22]  D. Jeffrey,et al.  Conduction through a random suspension of spheres , 1973, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[23]  H. Brenner.  The slow motion of a sphere through a viscous fluid towards a plane surface , 1961 .

[24]  K. Walters,et al.  Elastico-viscous squeeze films. part I , 1976 .

[25]  R. Rangel,et al.  Numerical investigation of particle–particle and particle–wall collisions in a viscous fluid , 2008, Journal of Fluid Mechanics.

[26]  D. Joseph.  Bernoulli equation and the competition of elastic and inertial pressures in the potential flow of a second-order fluid , 1992 .

[27]  G G J O S E P H,et al.  Particle – wall collisions in a viscous fluid , 2022 .

[28]  Daniel D. Joseph,et al.  Sedimentation of a sphere near a vertical wall in an Oldroyd-B fluid , 2000 .

[29]  Daniel D. Joseph,et al.  Aggregation and dispersion of spheres falling in viscoelastic liquids , 1994 .

[30]  B. Abu-Jdayil,et al.  Effects of electrode morphology on the slit flow of an electrorheological fluid , 1996 .

[31]  R. Rivlin,et al.  Stress-Deformation Relations for Isotropic Materials , 1955 .

[32]  G. Batchelor,et al.  The determination of the bulk stress in a suspension of spherical particles to order c2 , 1972, Journal of Fluid Mechanics.

[33]  A. Maude.  End effects in a falling-sphere viscometer , 1961 .