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Far-field disturbance flow induced by a small non-neutrally buoyant sphere in a linear shear flow

Published online by Cambridge University Press:  15 January 2010

EVGENY S. ASMOLOV  and
FRANÇOIS FEUILLEBOIS
EVGENY S. ASMOLOV*
Affiliation:
Central Aero-Hydrodynamics Institute, 1 Zhukovsky str., Zhukovsky, Moscow region, 140180, Russia Institute of Mechanics, Lomonosov Moscow State University, 1 Michurinsky prosp., Moscow, 119192, Russia
FRANÇOIS FEUILLEBOIS
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), ESPCI, 10 rue Vauquelin, 75231 Paris cedex 05, France
*
†Email address for correspondence: aes@an.aerocentr.msk.su
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Abstract

The disturbance flow due to the motion of a small sphere parallel to the streamlines of an unbounded linear shear flow is evaluated at small Reynolds number using the method of matched expansions. Decaying laws are obtained for all velocity components in a far inviscid region and viscous wakes. The z component (in the direction of the shear-rate gradient) of the disturbance velocity is cylindrically symmetric in the inviscid region. It decays with the distance r from the sphere like r−5/3, while the y component (in the direction of vorticity) decays like r−4/3. The widths of two viscous wakes, located upstream and downstream of the sphere, grow with the longitudinal coordinate x as yw ~ zw ~ |x|1/3. The maximum x and z components of the velocity are located in the wake cores; they scale like |x|−2/3 and |x|−1 respectively. For two particles interacting through their outer regions, the migration velocity of each particle is the sum of the velocity of an isolated particle and of a disturbance velocity induced by the other one. Particles placed in the normal or transversal directions repel each other. When each particle is located in a wake of the other one, they may either attract or repel each other.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 643 , 25 January 2010 , pp. 449 - 470
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Abramowitz, M. & Stegun, I. 1964 Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. US Government Printing Office: Nat. Bur. Standards.Google Scholar
Asmolov, E. S. 1990 Dynamics of a spherical particle in a laminar boundary layer. Fluid Dyn. 25, 886–890.CrossRefGoogle Scholar
Asmolov, E. S. 1999 The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number. J. Fluid Mech. 381, 63–87.CrossRefGoogle Scholar
Conte, S. D. 1966 The numerical solution of linear boundary value problems. SIAM Rev. 8, 309.CrossRefGoogle Scholar
Elrick, D. E. 1962 Source functions for diffusion in uniform shear flow. Austral. J. Phys. 15, 283.CrossRefGoogle Scholar
Feuillebois, F. 2004 Perturbation Problems at Low Reynolds Number. Lecture Notes of Center of Excellence of Advanced Materials and Structures, vol. 15. Polish Academy of Sciences.Google Scholar
Godunov, S. K. 1961 Numerical solution of boundary-value problems for systems of linear ordinary differential equations. Usp. Mat. Nauk 16 (3), 171–174 (in Russian).Google Scholar
Harper, E. Y. & Chang, I.-D. 1968 Maximum dissipation resulting from lift in a slow viscous shear flow. J. Fluid Mech. 33, 209–225.CrossRefGoogle Scholar
Kaneda, Y. & Ishii, K. 1982 The hydrodynamic interaction of two spheres moving in an unbounded fluid at small but finite Reynolds number. J. Fluid Mech. 124, 209–217.CrossRefGoogle Scholar
Kulkarni, P. M. & Morris, J. F. 2008 Pair-sphere trajectories in finite-Reynolds-number shear flow. J. Fluid Mech. 596, 413–435.CrossRefGoogle Scholar
Lin, C. J., Peery, J. H. & Schowalter, W. R. 1970 Simple shear flow round a rigid sphere: inertial effects and suspension rheology. J. Fluid Mech. 44, 1–17.CrossRefGoogle Scholar
Matas, J.-P., Glezer, V., Guazzelli, E. & Morris, J. F. 2004 Trains of particles in finite-Reynolds-number pipe flow. Phys. Fluids 11, 4192–4195.CrossRefGoogle Scholar
McLaughlin, J. B. 1991 Inertial migration of a small sphere in linear shear flows. J. Fluid Mech. 224, 261–274.CrossRefGoogle Scholar
McLaughlin, J. B. 1993 The lift on a small sphere in wall-bounded linear shear flows. J. Fluid Mech. 246, 249–265.CrossRefGoogle Scholar
Mikulencak, D. R. & Morris, J. F. 2004 Stationary shear flow around fixed and free bodies at finite Reynolds number. J. Fluid Mech. 520, 215–242.CrossRefGoogle Scholar
Miyazaki, K., Bedeaux, D. & Avalos, J. B. 1995 Drag on a sphere in slow shear flow. J. Fluid Mech. 296, 373–390.CrossRefGoogle Scholar
Poe, G. G. & Acrivos, A. 1975 Closed-streamline flows past rotating single cylinders and spheres: inertia effects. J. Fluid Mech. 72, 605–623.CrossRefGoogle Scholar
Proudman, I. & Pearson, J. R. A. 1957 Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, 237–262.CrossRefGoogle Scholar
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22, 385–400.CrossRefGoogle Scholar
Saffman, P. G. 1968 Corrigendum. J. Fluid Mech. 31, 624.Google Scholar
Schonberg, J. A. & Hinch, E. J. 1989 Inertial migration of a sphere in Poiseuille flow. J. Fluid Mech. 203, 517–524.CrossRefGoogle Scholar
Sidorov, Y. V., Fedoryuk, M. V. & Shabunin, M. I. 1976 Lectures on the Theory of the Functions of Complex Variable. In Russian. Nauka.Google Scholar
Subramanian, G. & Koch, D. L. 2006 Inertial effects on the transfer of heat or mass from neutrally buoyant spheres in a steady linear velocity field. Phys. Fluids 18, 073302.CrossRefGoogle Scholar
Vasseur, P. & Cox, R. G. 1977 The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. J. Fluid Mech. 80, 561–591.CrossRefGoogle Scholar