Theory of Images in Spherical-Layered Axisymmetric Viscous Hydrodynamics

K. Aderogba

doi:10.1115/1.3130450

Journal of Applied Mechanics | Volume 76 | Issue 6 | Research Paper

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Theory of Images in Spherical-Layered Axisymmetric Viscous Hydrodynamics

K. Aderogba

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K. Aderogba

Department of Mathematics, University of Lagos, Lagos 23401, Nigeriakanmiaderogba@yahoo.com

J. Appl. Mech 76(6), 061007 (Jul 23, 2009) (9 pages) doi:10.1115/1.3130450 History: Received November 19, 2007; Revised February 09, 2009; Published July 23, 2009

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Abstract

A spherical drop of viscosity $μ^{(1)}$ and radius $a_{1}$ is surrounded by a spherical shell of viscosity $μ^{(2)}$ , internal radius $a_{1}$ , and external radius $a_{2}$ , beyond which there is an unbounded matrix fluid of viscosity $μ^{(3)}$ . An arbitrary axisymmetric singularity acts inside or outside the viscous spherical shell. It is established that the Stokes’ stream function induced in this heterogeneous medium is explicitly expressible solely in terms of the corresponding stream function for the unperturbed homogeneous unbounded medium. As an application of the general solution, it is shown that there exists a homogeneous spherical drop of viscosity $μ$ and radius $a_{2}$ , which is equivalent to the spherical drop of viscosity $μ^{(1)}$ and radius $a_{1}$ , surrounded by the spherical shell of viscosity $μ^{(2)}$ and outer radius $a_{2}$ . A new formula is also established for the effective viscosity of a multiphase medium comprising an incompressible fluid of viscosity $μ^{(3)}$ in which are embedded $N$ small identical spherical drops of viscosity $μ^{(1)}$ , surrounded by spherical shells of viscosity $μ^{(2)}$ , assuming that the interference effects of these coated spherical drops are negligible, and that their arrangement is axisymmetrical. The corresponding two-dimensional results are also presented, by slightly modifying the three-dimensional results.

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References

Milne-Thomson, L. M., 1940, “On Hydrodynamical Images,” Proc. Cambridge Philos. Soc., 36 , pp. 246–248. [CrossRef]

Weiss, P., 1944, “Arbitrary Irrotational Flow Disturbed by a Sphere,” Proc. Cambridge Philos. Soc., 40 , pp. 259–263. [CrossRef]

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Collins, W. D., 1958, “Viscous Stokes Flow Past a Rigid Sphere,” Mathematika, 5 , pp. 118–124.

Stokes, G. G., 1851, “On the Effect of the Internal Friction of Fluids on the Motion of Pendulums,” Trans. Cambridge Philos. Soc., 9 , pp. 8–96.

Maxwell, J. C., 1873, "A Treatise on Electricity and Magnetism", Vol. 1 , Dover, New York, pp. 244–450.

Aderogba, K., 2003, “An Image Treatment of Elastostatic Transmission From an Interface Layer,” J. Mech. Phys. Solids, 51 , pp. 267–279. [CrossRef]

Aderogba, K., 2007, “Theory of Images in Plane-Layered Axisymmetric Elastostatics,” Int. J. Solids Struct., 44 , pp. 3920–3927. [CrossRef]

Aderogba, K., 1972, “Some Results for Two-Phase Elastic Plates,” Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci., 71 , pp. 21–35.

Eshelby, J. D., 1957, “The Determination of the Elastic Field of an Ellipsoidal Inhomogeneity and Related Problems,” Proc. R. Soc. London, Ser. A, 241 , pp. 376–396. [CrossRef]

Eshelby, J. D., 1961, “Inclusions and Inhomogeneities,” Progress in Solid Mechanics, 2 , pp. 89–140.

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Spherical polar coordinates

Geometry of problem with singularity in third medium

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Geometry of problem with singularity in third medium

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Aderogba KK. Theory of Images in Spherical-Layered Axisymmetric Viscous Hydrodynamics. ASME. J. Appl. Mech. 2009;76(6):061007-061007-9. doi:10.1115/1.3130450.

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Theory of Images in Spherical-Layered Axisymmetric Viscous Hydrodynamics

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