Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-20T00:14:37.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Particle motion in Stokes flow near a plane fluid–fluid interface. Part 2. Linear shear and axisymmetric straining flows

Published online by Cambridge University Press:  20 April 2006

Seung-Man Yang  and
L. Gary Leal
Seung-Man Yang
Affiliation:
Department of Chemical Engineering, California Institute of Technology, Pasadena, California 91125
L. Gary Leal
Affiliation:
Department of Chemical Engineering, California Institute of Technology, Pasadena, California 91125
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the motion of a sphere or a slender body in the presence of a plane fluid–fluid interface with an arbitrary viscosity ratio, when the fluids undergo a linear undisturbed flow. First, the hydrodynamic relationships for the force and torque on the particle at rest in the undisturbed flow field are determined, using the method of reflections, from the spatial distribution of Stokeslets, rotlets and higher-order singularities in Stokes flow. These fundamental relationships are then applied, in combination with the corresponding solutions obtained in earlier publications for the translation and rotation through a quiescent fluid, to determine the motion of a neutrally buoyant particle freely suspended in the flow. The theory yields general trajectory equations for an arbitrary viscosity ratio which are in good agreement with both exact-solution results and experimental data for sphere motions near a rigid plane wall. Among the most interesting results for motion of slender bodies is the generalization of the Jeffrey orbit equations for linear simple shear flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 149 , December 1984 , pp. 275 - 304
Copyright
© 1984 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44, 419.Google Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242.Google Scholar
Chwang, A. T. & Wu, T. Y-T. 1975 Hydrodynamics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows. J. Fluid Mech. 67, 787.Google Scholar
Cox, R. G. 1971 The motion of long slender bodies in a viscous fluid. Part 2. Shear flow. J. Fluid Mech. 45, 625.Google Scholar
Darabaner, C. L. & Mason, S. G. 1967 Particle motions in sheared suspensions. XXII: interactions of rigid sphere. Rheol. Acta 6, 273.Google Scholar
Dukhin, S. S. & Rulev, N. N. 1977 Hydrodynamic interaction between a solid spherical particle and a bubble in the elementary act of flotation. Colloid J. USSR 39, 270.Google Scholar
Faxén, H. 1921 Dissertation, Uppsala University.
Fulford, G. R. & Blake, J. R. 1983 On the motion of a slender body near an interface between two immiscible liquids at very low Reynolds number. J. Fluid Mech. 127, 203.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967a Slow viscous motion of a sphere parallel to a plane wall. I. Motion through a quiescent fluid. Chem. Engng Sci. 22, 637.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967b Slow viscous motion of a sphere parallel to a plane wall. II. Couette flow. Chem. Engng Sci. 22, 653.Google Scholar
Goren, S. L. & O'Neill, M. E. 1971 On the hydrodynamic resistance to a particle of a dilute suspension when in the neighborhood of a large obstacle. Chem. Engng Sci. 26, 325.Google Scholar
Jeffery, G. B. 1912 On a form of the solution of Laplace's equation suitable for problems relating to two spheres. Proc. R. Soc. Lond. A 87, 109.Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161.Google Scholar
Leal, L. G. 1975 The slow motion of slender rod-like particles in a second-order fluid. J. Fluid Mech. 69, 305.Google Scholar
Lee, S. H., Chadwick, R. S. & Leal, L. G. 1979 Motion of a sphere in the presence of a plane interface. Part 1. An approximate solution by generalization of the method of Lorentz. J. Fluid Mech. 93, 705.Google Scholar
Lee, S. H. & Leal, L. G. 1980 Motion of a sphere in the presence of a plane interface. Part 2. An exact solution in bipolar coordinates. J. Fluid Mech. 98, 193.Google Scholar
Lorentz, H. A. 1907 A general theory concerning the motion of a viscous fluid. Abhandl. Theor. Phys. 1, 23.Google Scholar
Spielman, L. A. 1977 Particle capture from low-speed laminar flows. Ann. Rev. Fluid Mech. 9, 297.Google Scholar
Wakiya, S. 1957 Viscous flows past a spheroid. J. Phys. Soc. Japan 12, 1130.Google Scholar
Yang, S.-M. & Leal, L. G. 1983 Particle motion in Stokes flow near a plane fluid—fluid interface. Part 1. Slender body in a quiescent fluid. J. Fluid Mech. 136, 393.Google Scholar
Supplementary material: PDF

Yang and Leal supplementary material

Supplementary Appendix

Download Yang and Leal supplementary material(PDF)
PDF 387.9 KB