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Fundamental Singularities in a Two-Fluid Stokes Flow with a Plane Interface

Published online by Cambridge University Press:  05 May 2011

H. Y. Yu
H. Y. Yu*
Affiliation:
U.S. Army Research Laboratory—Far East Research Office, 7–23–17 Roppongi, Minato-Ku, Tokyo 106–0032, Japan
*
*Scientist and Acting Director
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Abstract

A modified image method is presented for obtaining the solutions of the fundamental singularities in the neighborhood of a plane interface between two semi-infinite, immiscible, and incompressible viscous fluids. The fundamental singularities considered are the stokeslet, rotlet, stresslet, stokes-doublet, source, and source-doublet. The Galerkin vector function representation introduced reduces the complexity of the expressions for the solutions. Moreover, the physical meaning of each solution is clearly identified by these new expressions.

Keywords

StokesletsSingularitiesTwo-fluid flowStokes flow
Type
Articles
Information
Journal of Mechanics , Volume 19 , Issue 1 , March 2003 , pp. 263 - 270
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2003

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References

REFERENCES

1Lorentz, H. A., “A General Theorem Concerning the Motion of a Viscous Fluid and a Few Consequences Derived from it,” Versl. Kon. Akad. Wet. Amst., 5, pp. 168–182 (1879).Google Scholar
2Oseen, C. W., “Naure Methoden und Ergebnisse in der Hydrodynamik,” Akademishe Verlag (1927).Google Scholar
3Burgers, J. M., “On the Motion of Small Particles of Elongated form Suspended in a Viscous Liquid,” Chap. III, Second Report on Viscosity and Plasticity. Kon. Ned. Akad. Wet., Verhand, 16, pp. 113–145 (1938).Google Scholar
4Hancock, G. J., “The Self-Propulsion of Microscopic Organisms Through Liquids,” Proc. Roy. Soc. Lond., A217, pp. 96–121 (1953).Google Scholar
5Brenner, H., “The Stokes Resistance of an Arbitrary Particle,” Chem. Engng. Sci., 18, pp. 1–15 (1963).CrossRefGoogle Scholar
6Batchelor, G. K., “The Stress System in a Suspension of Force-Free Particles,” J. Fluid Mech., 41, pp. 545–570 (1970).CrossRefGoogle Scholar
7Batchelor, G. K., “Slender-Body Theory for Particles of Arbitrary Cross-Section in Stokes Flow,” J. Fluid Mech., 44, pp. 419–440 (1970).CrossRefGoogle Scholar
8Cox, R. G., “The Motion of Long Slender Bodies in a Viscous Fluid, Part 1, General Theory,” J. Fluid Mech., 44, pp. 791–810 (1970).CrossRefGoogle Scholar
9Cox, R. G., “The Motion of Long Slender Bodies in a Viscous Fluid, Part 2, Shear Flow,” J. Fluid Mech., 45, pp. 625–657 (1971).CrossRefGoogle Scholar
10Blake, J. R., “A Note on the Image System for a Stokeslet in a No-Slip Boundary,” Proc. Camb. Phil. Soc., 70, pp. 303–310 (1971).CrossRefGoogle Scholar
11Blake, J. R., “Singularities of Viscous Flow, Part II, Applications to Slender Body Theory,” J. Engng Maths., 8, pp. 113–124 (1974).CrossRefGoogle Scholar
12Blake, J. R. and Chwang, A. T., “Fundamental Singularities of Viscous Flow, Part I, The Image System in the Vicinity of a Stationary No-Slip Boundary,” J. Engng Maths., 8, pp. 23–29 (1974).CrossRefGoogle Scholar
13Chwang, A. T. and Wu, T. Y., “Hydrodynamic of Low-Reynolds-Number Flow, Part 2, Singularity Method for Stokes Flows,” J. Fluid Mech., 67, pp. 787–815 (1975).CrossRefGoogle Scholar
14Aderogba, K., “On Stokeslets in a Two-Fluid Space,” J. Engng Maths., 10, pp. 143–151 (1976).CrossRefGoogle Scholar
15Lee, S. H., Chadwick, R. S. and Leal, L. G., “Motion of a Sphere in the Presence of a Plane Interface, Part 1, An Approximate Solution by Generalization of the Method of Lorentz,” J. Fluid Mech., 93 pp. 705–726 (1979).CrossRefGoogle Scholar
16Yang, S. M. and Leal, L. G., “Particle Motion in Stokes Flow Near a Plane Fluid-Fluid Interface, Part 1, Slender Body in a Quiescent Fluid,” J. Fluid Mech., 136, pp. 393–421 (1983).CrossRefGoogle Scholar
17Yang, S. M. and Leal, L. G., “Particle Motion in Stokes Flow near a Plane Fluid-Fluid Interface. Part 2, Linear Shear and Axisymmetric Straining Flows,” J. Fluid Mech., 149, pp. 275–304 (1984).CrossRefGoogle Scholar
18Dorrepaal, J. M., O'Neill, M. E. and Ranger, K. B., “Two-Dimensional Stokes Flows with Cylinders and Line Singularities,” Mathematika, 31, pp. 65–75 (1984).CrossRefGoogle Scholar
19Dabros, T., “A Singularity Method for Calculating Hydrodynamics Forces and Particle Velocities in Low-Reynolds-Number Flow,” J. Fluid Mech., 156, pp. 1–21 (1985).CrossRefGoogle Scholar
20Kim, S., and Arunachalam, P. V., “The General Solution for an Ellipsoid in Low-Reynolds-Number Flow,” J. Fluid Mech., 178, pp. 535–546 (1987).CrossRefGoogle Scholar
21Avudainayagam, A. and Jothiram, B., “Non-Slip Images of Certain Line Singularities in a Circular Cylinder,” Int. J. Engng. Sci., 25, pp. 1193–1205 (1987).CrossRefGoogle Scholar
22Pozikidis, C., “A Singularity Method for Unsteady Linearized Flow,” Phys. Fluids., A1, pp. 1508–1520 (1989).CrossRefGoogle Scholar
23Nitsche, L. C. and Brenner, H., “Hydrodynamics of a Particulate Motion in Sinusoidal Pores via a Singularity Method,” AIChE J., 36, pp. 1403–1419 (1990).CrossRefGoogle Scholar
24Jeffery, D. J., “The Lubrication Analysis for Two Spheres in a Two Dimensional Pure-Straining Motion,” Phys. Fluids, A3, pp. 1819–1821 (1991).CrossRefGoogle Scholar
25Love, A. E. H., Mathematical Theory of Elasticity, fourth edition, New York, Dover Publications (1927).Google Scholar
26Jeffery, G. B., “On a Form of the Solution of Laplace's Equation Suitable for Problems Relating to Two Spheres,” Proc. Roy. Soc. Lond., A87, pp. 109–120 (1912).Google Scholar
27Jeffery, G. B., “On the Steady Rotation of a Solid of Revolution in a Viscous Fluid,” Proc. Lond. Math. Soc., 14, pp. 327–338 (1915).CrossRefGoogle Scholar
28Bart, E., “The Slow Unsteady Settling of a Fluid Sphere toward a Flat Fluid Interface,” Chem. Engng Sci., 23, pp. 193–210 (1968).CrossRefGoogle Scholar
29Faxen, H., Dissertation, Uppsala University (1921).Google Scholar
30Happel, J. and Brenner, H., Low Reynolds Number Hydrodynamic, Noordhoff International Publishers (1983).CrossRefGoogle Scholar
31Lorentz, H. A., “A General Theorem Concerning the Motion of a Viscous Fluid and a Few Consequences Derived from It,” Zittingsverlag. Akad. v. Wet., 5, pp. 168–182 (1896).Google Scholar
32Yu, H. Y. and Sanday, S. C., “Elastic Fields in Joined Half-Spaces due to Nuclei of Strain,” Proc. Roy. Soc. Lond., A434, pp. 503–519 (1991).Google Scholar
33Galerkin, B., “Contribution " la solution gėnėrale du problėme de la th'orie de l'ėlasticitė dans le cas de trois dimensions,” Comptes Rendus, 190, pp. 1047–1049 (1930).Google Scholar
34Mindlin, R. D., “Force at a Point in the Interior of a Semi-Infinite Solid,” Phys., 7, pp. 195–202 (1936).CrossRefGoogle Scholar