Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T07:16:58.886Z Has data issue: false hasContentIssue false
  • English
  • Français

The unsteady motion of solid bodies in creeping flows

Published online by Cambridge University Press:  26 April 2006

J. Feng  and
D. D. Joseph
J. Feng
Affiliation:
Department of Aerospace Engineering and Mechanics and The Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN 55455, USA
D. D. Joseph
Affiliation:
Department of Aerospace Engineering and Mechanics and The Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN 55455, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In treating unsteady particle motions in creeping flows, a quasi-steady approximation is often used, which assumes that the particle's motion is so slow that it is composed of a series of steady states. In each of these states, the fluid is in a steady Stokes flow and the total force and torque on the particle are zero. This paper examines the validity of the quasi-steady method. For simple cases of sedimenting spheres, previous work has shown that neglecting the unsteady forces causes a cumulative error in the trajectory of the spheres. Here we will study the unsteady motion of solid bodies in several more complex flows: the rotation of an ellipsoid in a simple shear flow, the sedimentation of two elliptic cylinders and four circular cylinders in a quiescent fluid and the motion of an elliptic cylinder in a Poiseuille flow in a two-dimensional channel. The motion of the fluid is obtained by direct numerical simulation and the motion of the particles is determined by solving their equations of motion with solid inertia taken into account. Solutions with the unsteady inertia of the fluid included or neglected are compared with the quasi-steady solutions. For some flows, the effects of the solid inertia and the unsteady inertia of the fluid are importanty quantitatively but not qualitatively. In other cases, the character of the particles' motion is changed. In particular, the unsteady effects tend to suppress the periodic oscillations generated by the quasi-steady approximation. Thus, the results of quasi-steady calculatioins are never uniformly valid and can be completely misleading. The conditions under which the unsteady effects at small Reynolds numbers are important are explored and the implications for modelling of suspension flows are addressed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 303 , 25 November 1995 , pp. 83 - 102
Copyright
© 1995 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

Arminski, L. & Weinbaum, S. 1979 Effect of waveform and duration of impulse on the solution to the Basset-Langevin equation. Phys. Fluids 22, 404–411.Google Scholar
Baggio, T. 1907 Integrazione dell'equazione funzionale che regge la caduta di una sfera in un liquido viscose. Atti. Accad. naz. Lincei 16, 730–737.Google Scholar
Basset, A. B. 1888 On the motion of a sphere in a viscous liquid. Phil. Trans. R. Soc. Lond. A 179, 43–63.(Also in A Treatise on Hydrodynamics, Chap. 22. Dover, 1961.)Google Scholar
Bossis, G. & Brady, J. F. 1984 Dynamic simulation of sheared suspensions. J. Chem. Phys. 80, 5141–5154.Google Scholar
Brady, J. F. & Bossis, G. 1985 The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation. J. Fluid Mech. 155, 105–129.Google Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Ann. Rev. Fluid Mech. 20, 111–157.Google Scholar
Brady, J. F., Phillips, R. J., Lester, J. C. & Bossis, G. 1988 Dynamic simulation of hydrodynamically interacting suspensions. J. Fluid Mech. 195, 257–280.Google Scholar
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14, 284–304.Google Scholar
Chang, C. & Powell, R. L. 1993 Dynamic simulation of bimodal suspensions of hydrodynamically interacting spherical particles. J. Fluid Mech. 253, 1–25.Google Scholar
Chwang, A. T. 1975 Hydromechanics of low-Reynolds-number flow. Part 3. Motion of a spheroidal particle in quadratic flows. J. Fluid Mech. 72, 17–34.Google Scholar
Claeys, I. L. & Brady, J. F. 1993a Suspensions of prolate spheroids in Stokes flow. Part 1. Dynamics of a finite number of particles in an unbounded fluid. J. Fluid Mech. 251, 411–442.Google Scholar
Claeys, I. L. & Brady, J. F. 1993b Suspensions of prolate spheroids in Stokes flow. Part 2. Statistically homogeneous dispersions. J. Fluid Mech. 251, 443–477.Google Scholar
Claeys, I. L. & Brady, J. F. 1993c Suspensions of prolate spheroids in Stokes flow. Part 3. Hydrodynamic transport properties of crystalline dispersions. J. Fluid Mech. 251, 479–500.Google Scholar
Cox, R. G., Zia, I. Y. & Mason, S. G. 1968 Particle motions in sheared suspensions. XXV. Streamlines around cylinders and spheres. J. Colloid Interface Sci. 27, 7–18.Google Scholar
Durlofsky, L., Brady, J. F. & Bossis, G. 1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 21–49.Google Scholar
Feng, J., Hu, H. H. & Joseph, D. D. 1994a Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation. J. Fluid Mech. 261, 95–134.Google Scholar
Feng, J., Hu, H. H. & Joseph, D. D. 1994b Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2. Couette and Poiseuille flows. J. Fluid Mech. 277, 271–301.Google Scholar
Feng, J., Huang, P. Y. & Joseph, D. D. 1995 Dynamic simulation of the motion of capsules in pipelines. J. Fluid Mech. 286, 201–227.Google Scholar
Ganatos, P., Pfeffer, R. & Weinbaum, S. 1978 A numerical-solution technique for three-dimensional Stokes flows, with application to the motion of strongly interacting spheres in a plane. J. Fluid Mech. 84, 79–111.Google Scholar
Gavze, E. 1990 The accelerated motion of rigid bodies in non-steady Strokes flow. Intl. J. Multiphase Flow 16, 153–166.Google Scholar
Hassonijee, Q., Pfeffer, R. & Ganatos, P. 1992 Behavior of multiple spheres in shear and Poiseuille flow field at low Reynolds number. Intl. J. Multiphase Flow 18, 353–370.Google Scholar
Hinch, E. J. 1975 Application of the langevin equation to fluid suspensions. J. Fluid Mech. 72, 499–511.Google Scholar
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65, 365–400.Google Scholar
Hocking, L. M. 1964 The behavior of clusters of spheres falling in a viscous fluid. J. Fluid Mech. 20, 129–139.Google Scholar
Hsu, R. 1985 Hydrodynamic interaction of an arbitrary particle with a planar-wall at low Reynolds numbers. PhD Thesis, City College of New York.
Hu, H. H., Joseph, D. D. & Crochet, M. J. 1992 Direct simulation of fluid particle motions. Theoret. Comput. Fluid Dyn 3, 285–306.Google Scholar
Jayaweera, K. O. L. F. & Manson B. J. 1965 The behavior of freely falling cylinders and cones in a viscous fluid. J. Fluid Mech. 22, 709–720.Google Scholar
Jefery, G. B. 1922 The motion of ellipsoidal particles immersed ina viscous fluid. Proc. R. Soc. Land. A 102, 161–179.Google Scholar
Karnis A., Goldsmith, H. L. & Mason, S. G. 1963 Axial migration of particles in Poiseuille flow. Nature 200, 159–160.Google Scholar
Karnis, A., Goldsmith, H. L. & Mason, S. G. 1966 The flow of suspensions through tubes. V. Interal effects. Can. J. Chem. Engng 44, 181–193.Google Scholar
Kim. S. 1985 Sedimentation of two arbitrary oriented spheroids in a viscous fluid. Intl. J. Multiphase Flow 11, 699–712.Google Scholar
Kim. S. & Karrila, S. J. 1991 Micro hydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.
Lawrence, C. J. & Weinbaum, S. 1986 The force on an axisymmertric body in linerized, time dependent motion of a new memory term. J. Fluid Mech. 171, 209–218.Google Scholar
Lawrence, C. J. & Weinbaum, S.. 1988 The unsteady force on a body at low Reynolds number, the axisymmertric motion of a spheroid. J. Fluid Mech. 189, 463–489.Google Scholar
Leal, L. G. 1980 Particle motion in a viscous fluid. Ann. Rev Fluid Mech. 12, 435–476.Google Scholar
Leichtberg, S., Weinbaum, S.Pfeefer, R. & Gluckman, M. J. 1976 A steady of unsteady forces at low Reynolds number: a strong interaction theory for the coaxial selting of three or more sphere. Phil. Trans. R. Soc. Land. A 282, 585–613.Google Scholar
Lovalenti, P. M. & Brady, J. F. 1993 The hydrodynamic force of a rigid particle undergoing arbitiory time-dependent motion at small Reynolds number. J. Fluid. Mech 256, 561–605.Google Scholar
Lovalenti, P. M. & Brady, J. F. 1995 The temporal behavior of the hydrodynamic force on a body in response to an abrupt charge in velocity at small but finite Renolds number. J. Fluid. Mech 293, 35–46.Google Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883–889.Google Scholar
Mazur, P. & Bedeaux, D. 1974 A generalization of Faxens theorem to nonsteady motion of a sphere through an incompressible fluid in arbitary flow. Phyisca 76, 235–246.Google Scholar
Mei, R., Adrin, R. J. & Hanratty, T. J. 1991 Particle dispersion in isotropic turbulence under strokes drag and Basset force with gravitational setting. J. Fluid Mech. 225, 481–495.Google Scholar
Phillips, R. J., Brady, J. F. & Bossis, G. 1988a Hydrodynamic transport properties of hard-sphere desperations, I. Suspensions of freely mobile particles. Phys. Fluids 31, 3462–3472.Google Scholar
Phillips, R. J., Brady, J. F. & Bossis, G. 1988b Hydrodynamic transport properties of hard-sphere desperations, II Porous media. Phys. Fluids 31, 3473–3479.Google Scholar
Pittman, J. F. T. & Kasiri, N. 1992 Motion of rigid rod-like particles suspended in non-homogeneous flow fields. Intl. J. Multiphase Flow 18, 1077–1091.Google Scholar
Strokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9 (11), 8–106.Google Scholar
Sugihara-Seki, M. 1993 The motion of an elliptical cylinder in channel flow at low Reynolds numbers. J. Fluid Mech. 257, 575–596.Google Scholar
Yasg, S.-M. & Leal, L. G. 1984 Particle motion in Strokes flow near a plane fluid-fluid interface. Part-2.Lineae shear and axisymmertric straining. J. Fluid Mech. 149, 275–304.Google Scholar