Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T04:03:36.135Z Has data issue: false hasContentIssue false
  • English
  • Français

A study of linearized oscillatory flow past particles by the boundary-integral method

Published online by Cambridge University Press:  26 April 2006

C. Pozrikidis
C. Pozrikidis
Affiliation:
Department of Applied Mechanics and Engineering Sciences, R-011, University of California at San Diego, La Jolla, CA 92093, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Viscous oscillatory flow past particles, governed by the unsteady Stokes equation, is considered. The problem is addressed in its general form for arbitrary flows and particle shapes using the boundary-integral method. It is shown that the leading-order correction to the force exerted on a particle in unsteady flow may be inferred directly from the drag in steady translational motion. For axisymmetric flow, a numerical procedure for solving the boundary-integral equation is developed, and is applied to study streaming oscillatory flow past spheroids, dumbbells, and biconcave disks. The effect of the particle geometry on the structure of the flow is studied by comparing the streamline pattern associated with these particles to that for the sphere. The results reveal the existence of travelling stagnation points on the surface of non-spherical particles, and the formation of unsteady viscous eddies in the interior of the flow. These eddies grow during the decelerating flow period, and shrink during the accelerating flow period. For particles with concave boundaries, unsteady free eddies may originate from an expansion of wall eddies that reside within the concave regions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 202 , May 1989 , pp. 17 - 41
Copyright
© 1989 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

Basset, A. B.: 1888 A Treatise on Hydrodynamics, vol. 2. Cambridge: Deighton Bell.
Batchelor, G. K.: 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bentwich, M. & Miloh, T., 1978 The unsteady matched Stokes-Oseen solution for the flow past sphere. J. Fluid Mech. 88, 17–32.Google Scholar
Buchanan, J.: 1891 The oscillations of a spheroid in a viscous fluid. Proc. Lond. Math. Soc. 22, 181–214.Google Scholar
Chwang, A. T. & Wu, T. Y., 1975 Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows. J. Fluid Mech. 67, 787–815.Google Scholar
Clift, R., Grace, J. R. & Weber, M. E., 1978 Bubbles, Drops and Particles. Academic.
Dabros, T.: 1985 A singularity method for calculating hydrodynamic forces and particle velocities in low-Reynolds-number flows. J. Fluid Mech. 156, 1–21.Google Scholar
Davis, A. M. J., O'Neill, M. E., Dorrepaal, J. M. & Ranger, K. B., 1976 Separation from the surface of two equal spheres in Stokes flow. J. Fluid Mech. 77, 625–644.Google Scholar
Dorrepaal, J. M., Majumdar, S. R., O'Neill, M. E. & Ranger, K. B. 1976a A closed turns in Stokes flow. Q. J. Mech. Appl. Maths 24, 381–396.Google Scholar
Dorrepaal, J. M., O'Neill, M. E. & Ranger, K. B. 1976b Axisymmetric Stokes flow past a spherical cap. J. Fluid Mech. 75, 273–286.Google Scholar
Ghaddar, N. K., Magen, M., Mikic, B. B. & Patera, A. T., 1986 Numerical investigation of incompressible flow in grooved channels. Part 2. Resonance and oscillatory heat-transfer enhancement. J. Fluid Mech. 168, 541–567.Google Scholar
Gluckman, M. J., Pfeffer, R. & Weinbaum, S., 1971 A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids. J. Fluid Mech. 50, 705–740.Google Scholar
Hall, P.: 1974 Unsteady viscous flow in a pipe of slowly varying cross-section. Fluid Mech. 64, 209–226.Google Scholar
Happel, J. & Brenner, H., 1983 Low Reynolds Number Hydrodynamics. Martinus Nijhoff.
Hasimoto, H. & Sano, O., 1980 Stokeslets and eddies in creeping flow. Ann. Rev. Fluid Mech. 12, 335–363.Google Scholar
Higdon, J. J. L.: 1985 Stokes flow in arbitrary two-dimensional domains: shear flow over ridges and cavities. Fluid Mech. 159, 195–226.Google Scholar
Howells, I. D.: 1974 Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects. J. Fluid Mech. 64, 449–475.Google Scholar
Kanwal, R. P.: 1955 Rotatory and longitudinal oscillations of axi-symmetric bodies in a viscous fluid. Q. J. Mech. Appl. Maths 8, 147–163.Google Scholar
Kanwal, R. P.: 1964 Drag on an axially symmetric body vibrating slowly along its axis in a viscous fluid. J. Fluid Mech. 19, 631–636.Google Scholar
Kim, S.: 1986 Singularity solutions for ellipdoids in low-Reynolds-number flows: with applications to the calculation of hydrodynamic interactions in suspensions of ellipsoids Intl J. Multiphase Flow 12, 469–491.
Kim, S. & Russel, W. B., 1985 The hydrodynamic interactions between two spheres in a Brinkman medium. J. Fluid Mech. 154, 253–268.Google Scholar
Ladyzhenskaya, O. A.: 1969 The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach.
Lai, R. Y. S. & Mockros, L. F. 1972 The Stokes-flow drag on prolate and oblate spheroids during axial translatory accelerations. J. Fluid Mech. 52, 1–15.Google Scholar
Lamb, H.: 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Lawrence, C. J. & Weinbaum, S., 1986 The force on an axisymmetric body in linearized, time-dependent motion: 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 axisymmetric motion of a spheroid. J. Fluid Mech. 189, 463–489.Google Scholar
Oseen, C. W.: 1927 Hydrodynamic. Akademische verlagsgesellschaft M. B. H., Leipzig.
Padmanabhan, N. & Pedley, T. J., 1987 Three-dimensional steady streaming in a uniform tube with an oscillating elliptical cross-section. J. Fluid Mech. 178, 325–343.Google Scholar
Pedley, T. J. & Stephanoff, K. D., 1985 Flow along a channel with a time-dependent indentation in one wall: the generation of vorticity waves. J. Fluid Mech. 160, 337–367.Google Scholar
Pienkowska, I.: 1984 Unsteady friction and mobility relations for Stokes flows. Arch. Mech. 36 (5–6), 749–769.Google Scholar
Pozrikidis, C.: 1988a Flow of a liquid layer along a wavy wall. J. Fluid Mech. 188, 275–300.Google Scholar
Pozrikidis, C.: 1988b Linear oscillatory flow past particles and drops by the singularity method. UCSD manuscript, submitted for publication.Google Scholar
Ralph, M. E.: 1986 Oscillatory flows in wavy-walled tubes. J. Fluid Mech. 168, 515–540.Google Scholar
Riley, N.: 1967 Oscillatory viscous flows. Review and Extension. J. Inst. Math. Applic. 3, 419–434.Google Scholar
Smith, S. H.: 1987 Unsteady Stokes' flow in two dimensions. J. Engng Maths 21, 271–285.Google Scholar
Sobey, I. J.: 1980 On flow through furrowed channels. Part 1. Calculated flow patterns. J. Fluid Mech. 96, 1–26.Google Scholar
Sobey, I. J.: 1982 Oscillatory flow at intermediate Reynolds number in asymmetric channels. J. Fluid Mech. 125, 359–373.Google Scholar
Sobey, I. J.: 1983 The occurrence of separation in oscillatory flow. J. Fluid Mech. 134, 247–257.Google Scholar
Sobey, I. J.: 1985a Observation of waves during oscillatory channel flow. J. Fluid Mech. 134, 247–257.Google Scholar
Sobey, I. J.: 1985b Dispersion caused by separation during oscillatory flow through a furrowed channel. Chem. Engng Sci. 40, 2129–2134.Google Scholar
Stokes, G. C.: 1851 On the effect of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8.Google Scholar
Wakiya, S.: 1976 Axisymmetric flow of a viscous fluid near the vertex of a body. J. Fluid Mech. 78, 737–747.Google Scholar
Williams, W. E.: 1966 A note on slow vibrations in a viscous fluid. J. Fluid Mech. 25, 589–590.Google Scholar
Yih, C. C.: 1979 Fluid Mechanics. West River Press.
Youngren, G. K. & Acrivos, A., 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69, 377–403.Google Scholar