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Axisymmetric unsteady stokes flow past an oscillating finite-length cylinder

Published online by Cambridge University Press:  26 April 2006

Michael Loewenberg
Michael Loewenberg
Affiliation:
Department of Physical Chemistry, University of Sydney, NSW, 2006, Australia
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Abstract

The flow field generated by axial oscillations of a finite-length cylinder in an incompressible viscous fluid is described by the unsteady Stokes equations and computed with a first-kind boundary-integral formulation. Numerical calculations were conducted for particle oscillation periods comparable with the viscous relaxation time and the results are contrasted to those for an oscillating sphere and spheroid. For high-frequency oscillations, a two-term boundary-layer solution is formulated that involves two, sequentially solved, second-kind integral equations. Good agreement is obtained between the boundary-layer solution and fully numerical calculations at moderate oscillation frequencies. The flow field and traction on the cylinder surface display several features that are qualitatively distinct from those found for smooth particles. At the edges, where the base joins the side of the cylinder, the traction on the cylinder surface exhibits a singular behaviour, characteristic of steady two-dimensional viscous flow. The singular traction is manifested by a sharply varying pressure profile in a near-field region. Instantaneous streamline patterns show the formation of three viscous eddies during the decelerating portion of the oscillation cycle that are attached to the side and bases of the cylinder. As deceleration proceeds, the eddies grow, coalesce at the edges of the particle, and thus form a single eddy that encloses the entire particle. Subsequent instantaneous streamline patterns for the remainder of the oscillation cycle are insensitive to particle geometry: the eddy diffuses outwards and vanishes upon particle reversal; a simple streaming flow pattern occurs during particle acceleration. The evolution of the viscous eddies is most apparent at moderate oscillation frequencies. Qualitative results are obtained for the oscillatory flow field past an arbitrary particle. For moderate oscillation frequencies, pathlines are elliptical orbits that are insensitive to particle geometry; pathlines reduce to streamline segments in constant-phase regions close to and far from the particle surface.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 265 , 25 April 1994 , pp. 265 - 288
Copyright
© 1994 Cambridge University Press

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References

Allergra, J. R. & Hawley, S. A. 1971 Attenuation of sound in suspensions and emulsions: theory and experiments. J. Acoust. Soc. Am. 51, 1545–1564.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Basset, A. B. 1888 A Treatise on Hydrodynamics, vol. 2. Cambridge: Deighton Bell.
Burgers, J. M. 1938 On the motion of small particles of elongated form suspended in a viscous liquid. Second report on viscosity and plasticity, Chap. III. Kon. Ned. Akad. Wet. 16, 113–184.Google Scholar
Chan, P. C., Leu, R. J. & Zargar, N. H. 1986 On the solution for the rotational motion of an axisymmetric rigid body at low Reynolds number with application to a finite length cylinder. Chem. Engng Commun. 49, 145–163.Google Scholar
Davis, R. H. 1991 Sedimentation of axisymmetric particles in shear flows. Phys. Fluids A 3, 2051–2060.Google Scholar
Dean, W. R. & Montagnon, P. E. 1949 On the steady motion of viscous liquid in a corner. Proc. Camb. Phil. Soc. 45, 389–394.Google Scholar
Edwardes, D. 1892 Steady motion of a viscous liquid in which an ellipsoid is constrained to rotate about a principal axis. Q. J. Maths 26, 70–78.Google Scholar
Gavze, E. 1990 The accelerated motion of rigid bodies in non-steady Stokes flow. J. Multiphase Flow 16, 153–166.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., Weinbaum, S. & Pfeffer, R. 1972 Axisymmetric slow viscous flow past arbitrary convex body of revolution. J. Fluid Mech. 55, 677–709.Google Scholar
Heiss, J. F. Coull, J. 1952 The effect of orientation and shape on the settling velocity of non-isometric particles in a viscous medium. Chem. Engng Prog. 48, 133–140.Google Scholar
Hess, J. L. 1962 Calculation of potential flow about bodies of revolution having axes perpendicular to the free-stream direction. J. Aerospace Sci. 29, 726–742.Google Scholar
Hocquart, R. & Hinch, E. J. 1983 The long-time tail of the angular-velocity autocorrelation function for a rigid Brownian particle of arbitrary centrally symmetric shape. J. Fluid Mech. 137, 217–220.Google Scholar
Hurd, A. J., Clark, N. A., Mockler, R. C. & O'Sullivan, W. J. 1985 Friction factors for a lattice of Brownian particles. J. Fluid Mech. 153, 401–416.Google Scholar
Jackson, J. D. 1962 Classical Electrodynamics. John Wiley & Sons.
Jeffery, G. B. 1922 The motion of ellipsoid particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
Karrila, S. J. & Kim, S. 1989 Integral equations of the second kind for the Stokes flow: direct solution for physical variables and removal of inherent accuracy limitations. Chem. Engng Commun. 82, 123–161.Google Scholar
Kasper, G., Niida, T. & Yang, M. 1985 Measurements of viscous drag on cylinders and chains of spheres with aspect ratios between 2 and 50. Aerosol. Sci. 16, 515–556.Google Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach.
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
Loewenberg, M. 1993 The unsteady Stokes resistance of arbitrarily oriented, finite-length cylinders. Phys. Fluids A 5, 3004–3006.Google Scholar
Loewenberg, M. 1994 Asymmetric, oscillatory motion of a finite-length cylinder: the macroscopic effect of particle edges. Phys. Fluids 6, 1095–1107.Google Scholar
Loewenberg, M. & O'Brien, R. W. 1992 The dynamic mobility of nonspherical particles. J. Colloid Interface Sci. 150, 158–168.Google Scholar
Oberbeck, A. 1876 Über stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inneren Reibung (On steady-state flow under consideration of inner friction). J. Reine. Angew. Math. 81, 62–80.Google Scholar
O'Brien, R. W. 1990 The electroacoustic equations for a colloidal suspension. J. Fluid Mech. 212, 81–93.Google Scholar
Pozrikidis, C. 1989a A singularity method for unsteady linearized flow. Phys. Fluids A 1, 1508–1520.Google Scholar
Pozrikidis, C. 1989b A study of linearized oscillatory flow past particles by the boundary-integral method. J. Fluid Mech. 202, 17–41.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods. Cambridge University Press.
Smith, A. M. O. & Pierce, J. 1958 Exact solution of the Neumann problem. Calculation of plane and axially symmetric flow about or within arbitrary boundaries. Proc. 3rd US Nat Congr. of Appl. Mech., pp. 807–815.
Stokes, G. G. 1851 On the effect of internal friction of fluids on the motion of pendulum. Trans. Camb. Phil. Soc. 9, 8.Google Scholar
Ui, T. J., Hussey, R. G. & Roger, R. P. 1984 Stokes drag on a cylinder in axial motion. Phys. Fluids 27, 787–795.Google Scholar
Williams, W. E. 1966 A note of slow vibrations in a viscous fluid. J. Fluid Mech. 25, 589–590.Google Scholar
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