Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-27T18:18:50.849Z Has data issue: false hasContentIssue false
  • English
  • Français

Pressure drop due to the motion of a sphere near the wall bounding a Poiseuille flow

Published online by Cambridge University Press:  29 March 2006

Peter M. Bungay  and
Howard Brenner
Peter M. Bungay
Affiliation:
Department of Chemical Engineering and Biotechnology Program, Carnegie–Mellon University, Pittsburgh, Pennsylvania 15213 Present address: University Medical Clinic, Montreal General Hospital, 1650 Cedar Avenue, Montreal 109.
Howard Brenner
Affiliation:
Department of Chemical Engineering and Biotechnology Program, Carnegie–Mellon University, Pittsburgh, Pennsylvania 15213
Get access Rights & Permissions [Opens in a new window]

Abstract

An expression is derived for the (low Reynolds number) additional pressure drop created by a relatively small sphere moving near the wall of a circular tube through which there is a Poiseuille flow. Two specific applications are examined: (i) the sedimentation of a homogeneous non-neutrally buoyant sphere in a quiescent fluid; and (ii) the motion of a neutrally buoyant sphere. In the latter case a pronounced increase in the additional pressure drop is predicted when the separation between the sphere and the tube wall is reduced to zero.

This analysis, which includes the behaviour for a sphere in contact with the tube wall, supplements previous ‘method of reflexions’ treatments valid only when the distance from the sphere centre to the wall is large compared with the sphere radius.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 60 , Issue 1 , 21 August 1973 , pp. 81 - 96
Copyright
© 1973 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

Brenner, H. 1964 The Stokes resistance of an arbitrary particle. II. An extension. Chem. Engng Sci. 19, 599–629.Google Scholar
Brenner, H. 1966 Hydrodynamic resistance of particles at small Reynolds numbers. In Advances in Chemical Engineering, vol. VI (ed. T. B. Drew, J. W. Hoopes & T. Vermuelen), pp. 287–438. Academic.
Brenner, H. 1970 Pressure drop due to the motion of neutrally buoyant particles in duct flows. J. Fluid Mech. 43, 641–660.Google Scholar
Brenner, H. & Happel, J. 1958 Slow viscous flow past a sphere in a cylindrical tube. J. Fluid Mech. 4, 195–213.Google Scholar
Bungay, P. M. 1970 Ph.D. thesis, Carnegie—Mellon University, Pittsburgh.
Bungay, P. M. & Brenner, H. 1973a Pressure drop due to the motion of neutrally buoyant particles in duct flows. III. Non-neutrally buoyant spherical droplets and bubbles. Z. angew. Math. Mech. 53, 187–192.Google Scholar
Bungay, P. M. & Brenner, H. 1973b The motion of a closely-fitting sphere through a fluid-filled tube. Int. J. Multiphase Flow, 1 (in press).Google Scholar
Cox, R. G. & Brenner, H. 1967 Effect of finite boundaries on the Stokes resistance of an arbitrary particle. Part 3. Translation and rotation. J. Fluid Mech. 28, 391–411.Google Scholar
Cox, R. G. & Mason, S. G. 1971 Suspended particles in fluid flow through tubes. Ann. Rev. Fluid Mech. 3, 291–316.Google Scholar
Feldman, G. A. & Brenner, H. 1968 Experiments on the pressure drop created by a sphere settling in a viscous liquid. Part 2. Reynolds numbers from 0·2 to 21000. J. Fluid Mech. 32, 705–720.Google Scholar
Goldman, A. J. 1966 Investigations in low Reynolds number fluid-particle dynamics. Ph.D. thesis, New York University.
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, 637–651Google 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, 653–660.Google Scholar
Goldsmith, H. L. & Mason, S. G. 1962 The flow of suspensions through tubes. I. Single spheres, rods, and discs. J. Colloid Sci. 17, 448–476.Google Scholar
Goren, S. L. & O'Neill, M. E. 1971 On the hydrodynamic resistance to a particle of a dilute suspension when in the neighbourhood of a large obstacle. Chem. Engng Sci. 26, 325–338.Google Scholar
Greenstein, T. & Happel, J. 1968 Theoretical study of the slow motion of a sphere and a fluid in a cylindrical tube. J. Fluid Mech. 34, 705–710.Google Scholar
Greenstein, T. & Happel, J. 1970 Viscosity of dilute uniform suspensions of uniform spheres. Phys. Fluids, 13, 18–21.Google Scholar
Haberman, W. L. & Sayre, R. M. 1958 Motion of rigid and fluid spheres in stationary and moving liquids inside cylindrical tubes. David Taylor Model Basin Rep. no. 1143.Google Scholar
Hinch, E. J. 1972 Note on the symmetries of certain material tensors for a particle in Stokes flow. J. Fluid Mech. 54, 423–425.Google Scholar
Hirschfeld, B. R. 1972 A theoretical study of the slow, asymmetric settling motion of an arbitrarily positioned particle in a circular cylinder. Ph.D. thesis, New York University.
Hochmuth, R. M. & Sutera, S. P. 1970 Spherical caps in low Reynolds number tube flow. Chem. Engng Sci. 25, 593–604.Google Scholar
O'Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika, 11, 67–74.Google Scholar
O'Neill, M. E. 1968 A sphere in contact with a plane wall in a slow linear shear flow. Chem. Engng Sci. 23, 1293–1298.Google Scholar
O'Neill, M. E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27, 705–724.Google Scholar
Sonshine, R. M., Cox, R. G. & Brenner, H. 1966 The Stokes translation of a particle of arbitrary shape along the axis of a circular cylinder filled to a finite depth with viscous liquid. Appl. Sci. Res. 16, 273–300.Google Scholar
Wakiya, S., Darabaner, C. L. & Mason, S. G. 1967 Particle motions in sheared suspensions. XXI. Interactions of rigid spheres (theoretical). Rheol. Acta, 6, 264–273.Google Scholar
Wang, H. & Skalar, R. 1969 Viscous flow in a cylindrical tube containing a line of spherical particles. J. Fluid Mech. 38, 75–96.Google Scholar