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

< Previous Article Next Article >

Research Papers

Theory of Images in Spherical-Layered Axisymmetric Viscous Hydrodynamics

K. Aderogba

[+] Author and Article Information

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

Article

References

Figures

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.

FIGURES IN THIS ARTICLE

Purchase this Content

$25.00

Purchase

Learn about subscription and purchase options

Your Session has timed out. Please sign back in to continue.

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]

Butler, S. E. J., 1953, “A Note for Stokes’ Stream Function for Motion With a Spherical Boundary,” Proc. Cambridge Philos. Soc., 49 , pp. 169–171. [CrossRef]

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.

Figures

View Large | Save Figure | Download Slide (.ppt)

Figure 1

Spherical polar coordinates

Geometry of problem with singularity in third medium

View Large | Save Figure | Download Slide (.ppt)

Figure 2

Geometry of problem with singularity in third medium

Tables

Errata

Discussions

Web of Science® Times Cited: 0

Some tools below are only available to our subscribers or users with an online account.

PDF
Email

You must be logged in as an individual user to share content.

Return to: Theory of Images in Spherical-Layered Axisymmetric Viscous Hydrodynamics

Copyright in the material you requested is held by the American Society of Mechanical Engineers (unless otherwise noted). This email ability is provided as a courtesy, and by using it you agree that you are requesting the material solely for personal, non-commercial use, and that it is subject to the American Society of Mechanical Engineers' Terms of Use. The information provided in order to email this topic will not be used to send unsolicited email, nor will it be furnished to third parties. Please refer to the American Society of Mechanical Engineers' Privacy Policy for further information.
Share
Get Citation

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.

Download citation file:

RIS (Zotero)

EndNote

BibTex

Medlars

ProCite

RefWorks

Reference Manager

© Copyright
Get Alerts

Processing your request... Please Wait.

Theory of Images in Spherical-Layered Axisymmetric Viscous Hydrodynamics

Abstract

FIGURES IN THIS ARTICLE

References

Figures

Tables

Errata

Discussions

Related Content

Journals

Conference Proceedings

eBooks