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A Green's Function for the Domain Bounded by Nonconcentric Spheres

[+] Author and Article Information
Jeng-Tzong Chen

Department of Harbor and River Engineering,
National Taiwan Ocean University,
20224 Keelung, Taiwan;
Department of Mechanical and Mechatronic Engineering,
National Taiwan Ocean University,
Keelung 20224, Taiwan
e-mail: jtchen@mail.ntou.edu.tw

Hung-Chih Shieh

Department of Harbor and River Engineering,
National Taiwan Ocean University,
20224 Keelung, Taiwan

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received February 14, 2011; final manuscript received February 20, 2012; accepted manuscript posted July 6, 2012; published online October 31, 2012. Assoc. Editor: Pradeep Sharma.

J. Appl. Mech 80(1), 014503 (Oct 31, 2012) (6 pages) Paper No: JAM-11-1048; doi: 10.1115/1.4007071 History: Received February 14, 2011; Revised February 20, 2012; Accepted July 06, 2012

The main result is the analytical derivation of Green's function for the domain bounded by nonconcentric spheres in terms of bispherical coordinates. Both surfaces, inner and outer boundaries, are specified by the Dirichlet boundary conditions. This work can be seen as an extension study for the Green's function of eccentric annulus derived by Heyda (1959, “A Green's Function Solution for the Case of Laminar Incompressible Flow Between Non-Concentric Circular Cylinders,” J. Franklin Inst., 267, pp. 25–34). To verify the solution, a semianalytical solution using the image method and a numerical solution using the method of fundamental solutions (MFS) are utilized for comparisons. Good agreement is made.

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References

Courant, R., and Hilbert, D., 1953, Method of Mathematical Physics, Interscience, New York.
Lebedev, N. N., Skalskaya, I. P., and Uflyand, Y. S., 1979, Worked Problems in Applied Mathematics, Dover, New York.
Morse, P., and Feshbach, H., 1953, Method of Theoretical Physics, McGraw-Hill, New York.
Heyda, J. F., 1959, “A Green's Function Solution for the Case of Laminar Incompressible Flow Between Non-Concentric Circular Cylinders,” J. Franklin Inst., 267, pp. 25–34. [CrossRef]
Chen, J. T., Lee, Y. T., Yu, S. R., and Shieh, S. C., 2009, “Equivalence Between Trefftz Method and Method of Fundamental Solution for the Annular Green's Function Using the Addition Theorem and Image Concept,” Eng. Anal. Boundary Elem., 33, pp. 678–688. [CrossRef]
Chen, J. T., Shieh, H. C., Tsai, J. J., and Lee, J. W., 2010, “A Study on the Method of Fundamental Solutions Using an Image Concept,” Appl. Math. Model., 34, pp. 4253–4266. [CrossRef]
Chen, J. T., Shieh, H. C., Lee, Y. T., and Lee, J. W., 2011, “Bipolar Coordinates, Image Method and the Method of Fundamental Solutions for Green's Functions of Laplace Problems With Circular Boundaries,” Eng. Anal. Boundary Elem., 35, pp. 236–243. [CrossRef]
Chen, J. T., and Wu, C. S., 2006, “Alternative Derivations for the Poisson Integral Formula,” Int. J. Math. Educ. Sci. Technol., 37, pp. 165–185. [CrossRef]
Shieh, H. C., 2009, “A Study on the Green's Functions for Laplace Problems With Circular and Spherical Boundaries by Using the Image Method,” Master's thesis, Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan.
Karageorghis, A., and Fairweather, G., 1999, “The Method of Fundamental Solutions for Axisymmetric Potential Problems,” Int. J. Numer. Methods Eng., 44, pp. 1653–1669. [CrossRef]

Figures

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Fig. 1

An eccentric annulus in the bipolar coordinate system (2D case)

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Fig. 2

A nonconcentric problem in the bispherical coordinate system (3D case)

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Fig. 3

Sketch of the solution by using the superposition technique

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Fig. 4

Successive image points of the nonconcentric spheres and two frozen images at c1 and c2

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Fig. 5

Distribution of fictitious sources of the MFS

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Fig. 6

Potential contour (x=0 plane) for a concentrated source at the z axis

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Fig. 7

Potential contour (x=0 plane) for a concentrated source at s2

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