Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-20T15:54:18.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Stokes wall effects for particles moving near cylindrical boundaries

Published online by Cambridge University Press:  20 April 2006

A. Falade  and
H. Brenner
A. Falade
Affiliation:
Department of Mechanical Engineering, University of Lagos, Lagos, Nigeria
H. Brenner
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
Get access Rights & Permissions [Opens in a new window]

Abstract

An asymptotic scheme is derived for calculating values of the ‘reflected’ Stokeslet-field velocity dyadic V and its gradient ${\boldmath \nabla V}$ back at the Stokeslet location for situations in which this singular point lies in close proximity to the wall of an infinitely long circular cylinder. The asymptotic formulas furnished by this scheme permit calculations of first- and second-order wall effects in the non-dimensional parameter κ = a/R0 (a ≡ characteristic particle radius, R0 ≡ cylinder radius) upon the Stokes resistance of a particle of arbitrary shape, location and orientation when translating and/or rotating near the wall of an otherwise quiescent fluid-filled or fluid-surrounded circular cylinder. This reflection-type calculation is applicable for circumstances in which the inequalities κ [Lt ] 1 and 1 − β [Lt ] 1 are each separately satisfied, while simultaneously K/(1 − β) [Lt ] 1. (Here β = b/R0 is the fractionally eccentric Stokeslet location, or equivalently the centre of reaction of the particle, with b its distance from the tube axis.) The main result of this paper is the development of the pair of asymptotic wall-correction formulas \[ W_{jk}\sim {}_0C_{jk}(1-\beta)^{-1}+{}_1C_{jk}+{}_2C_{jk}(1-\beta)+O(1-\beta)^2 \] and \[ W_{jk,\,l}\sim {}_0D_{jkl}(1-\beta)^{-2}+{}_1D_{jkl}(1-\beta)^{-1}+{}_2D_{jkl}+O(1-\beta) \] to particle resistance, with W ≡ Wjk and ${\boldmath \nabla W}\equiv W_{kl,j}$ respectively the non-dimensional normalized wall-effect dyadic and its gradient at the Stokeslet location β. The numerical, rational fraction, β-independent, nCjk and nDjkl coefficients (n = 0, 1, 2) appearing above are evaluated by solving a recursive sequence of Stokes-flow boundary-value problems in the semi-infinite fluid domain bounded by a plane wall. These simple asymptotic formulas are shown to agree excellently in the range near β = 1 with existing values derived from the exact solution of the original circular-cylinder boundary-value problem, involving tedious infinite-series summations of complicated Bessel-function integrands extended over infinite integration domains. Generalizations of the scheme to particle motion in the space external to a circular cylinder is briefly sketched, as too is the case of cylinders of non-circular cross-section.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 154 , May 1985 , pp. 145 - 162
Copyright
© 1985 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

Aderogba, K. 1977 On eigenstresses in dissimilar media. Phil. Mag. A 35, 281–292.Google Scholar
Aderogba, K. & Chou, T.-W. 1984 The generation of generalized prepotential functions and related problems. Proc. R. Soc. Lond. (in press).
Berker, R. 1963 Intégration des équations du mouvement d'un fluide visqueux incompressible. In Handbuch der Physik, vol. VIII/2: Fluid Dynamics II (ed. S. Flügge & C. Truesdell), pp. 242–244. Springer.
Brenner, H. 1984 Antisymmetric stresses induced by the rigid-body rotation of dipolar suspensions: vortex flows. Intl J. Engng Sci. 22, 645–682.CrossRefGoogle Scholar
Bungay, P. M. & Brenner, H. 1973a Pressure drop due to the motion of a sphere in proximity to a wall bounding a Poiseuille flow. J. Fluid Mech. 60, 81–96.Google Scholar
Bungay, P. M. & Brenner, H. 1973b The motion of a closely-fitting sphere in a fluid-filled tube. Intl J. Multiphase Flow 1, 25–56.Google Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press.
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
Falade, A. 1982 Arbitrary motion of an elliptic disc at a fluid interface. Int. J. Multiphase Flow 8, 543–551.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Hasimoto, H. & Sano, O. 1980 Stokeslets and eddies in creeping flow. Ann. Rev. Fluid Mech. 12, 335–363.Google Scholar
Hirschfeld, B. R. 1972 A theoretical study of the slow asymmetric settling of an arbitrarily positioned particle in a circular cylinder. Ph.D. dissertation, New York University.
Hirschfeld, B. R., Brenner, H. & Falade, A. 1984 First- and second-order wall effects upon the slow viscous asymmetric motion of an arbitrarily-shaped, -positioned and -oriented particle within a circular cylinder. PhysicoChem. Hydrodyn. 5, 99–133.Google Scholar
Lorentz, H. A. 1896 A general theorem concerning the motion of a viscous fluid with some applications. Zittingsverlag Akad. Wet. Amsterdam 5, 168–175; see also Abhandlungen über theoretische Physik Bd. 1, S. 23–42, Leipzig 1907.Google Scholar
Maude, A. D. 1963 The movement of a sphere in front of a plane at low Reynolds number. Brit. J. Appl. Phys. 14, 894–898.Google Scholar
Tözeren, H. 1982 Torque on eccentric spheres flowing in tubes. Trans. ASME E: J. Appl. Mech. 49, 279–283Google Scholar
Wakiya, S., Darabaner, C. L. & Mason, S. G. 1967 Particle motions in sheared suspensions. XXI: interactions of rigid spheres. Rheol. Acta 6, 264–273.Google Scholar