On Green’s function for Laplace’s equation in a rigid tube
Author:
P. A. Martin
Journal:
Quart. Appl. Math. 80 (2022), 87-98
MSC (2020):
Primary 31B20
DOI:
https://doi.org/10.1090/qam/1603
Published electronically:
November 16, 2021
MathSciNet review:
4360551
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Abstract: A classical problem from potential theory (a point source inside a long rigid tube) is revisited. It has an extensive literature but its resolution is not straightforward: standard approaches lead to divergent integrals or require the discarding of infinite constants. We show that the problem can be solved rigorously using classical methods.
References
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- H. S. Carslaw, Integral equations and the determination of Green’s functions in the theory of potential, Proc. Edinburgh Math. Soc. 31 (1912), 71–89.
- J. Dougall, The determination of Green’s function by means of cylindrical or spherical harmonics, Proc. Edinburgh Math. Soc. 18 (1900), 33–83.
- R. S. Eisenberg and E. A. Johnson, Three-dimensional electrical field problems in physiology, Prog. Biophys. & Molecular Biology 20 (1970), 1–65.
- T. R. Goodman, Aerodynamic characteristics of a slender body traveling in a tube, AIAA J. 9 (1971), 712–717.
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 4th ed., Academic Press, New York, 1980.
- Silvana Ilie and David J. Jeffrey, A note on Laplace’s equation inside a cylinder, Appl. Math. Lett. 18 (2005), no. 1, 55–59. MR 2121554, DOI 10.1016/j.aml.2003.05.015
- P. Kalinay and J. K. Percus, Mapping of diffusion in a channel with abrupt change of diameter, Phys. Rev. E 82 (2010), 031143.
- R. C. Knight, The potential of a sphere inside an infinite circular cylinder, Quart. J. Math. 7 (1936), 124–133.
- H. Lamb, On the effect of the walls of an experimental tank on the resistance of a model, Aeronautical Research Committee Reports & Memoranda No. 1010, 1926.
- L. Landweber, Axisymmetric potential flow in a circular tube, J. Hydronautics 8 (1974), 137–145.
- T. Miloh, Irrotational axisymmetric flow about a prolate spheroid in cylindrical duct, J. Eng. Math. 8 (1974), 315–327.
- Philip M. Morse and Herman Feshbach, Methods of theoretical physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0059774
- NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.1.2 of 2021-06-15.
- A. Peskoff, Green’s function for Laplace’s equation in an infinite cylindrical cell, J. Mathematical Phys. 15 (1974), 2112–2120. MR 367246, DOI 10.1063/1.1666591
- A. Peskoff, R. S. Eisenberg, and J. D. Cole, Matched asymptotic expansions of the Green’s function for the electric potential in an infinite cylindrical cell, SIAM J. Appl. Math. 30 (1976), no. 2, 222–239. MR 391734, DOI 10.1137/0130024
- W. R. Smythe, Static and dynamic electricity, 2nd edition, McGraw-Hill, New York, 1950.
- G. N. Watson, The use of series of Bessel functions in problems connected with cylindrical wind-tunnels, Proc. Roy. Soc. London, Ser. A 130 (1930), 29–37.
- G. N. Watson, The diffraction of electric waves by the Earth, Proc. Roy. Soc. London, Ser. A 95 (1918), 83–99.
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
References
- C. J. Bouwkamp and N. G. de Bruijn, The electrostatic field of a point charge inside a cylinder, in connection with wave guide theory, J. Appl. Phys. 18 (1947), 562–577. MR 20933
- H. S. Carslaw, Integral equations and the determination of Green’s functions in the theory of potential, Proc. Edinburgh Math. Soc. 31 (1912), 71–89.
- J. Dougall, The determination of Green’s function by means of cylindrical or spherical harmonics, Proc. Edinburgh Math. Soc. 18 (1900), 33–83.
- R. S. Eisenberg and E. A. Johnson, Three-dimensional electrical field problems in physiology, Prog. Biophys. & Molecular Biology 20 (1970), 1–65.
- T. R. Goodman, Aerodynamic characteristics of a slender body traveling in a tube, AIAA J. 9 (1971), 712–717.
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 4th ed., Academic Press, New York, 1980.
- Silvana Ilie and David J. Jeffrey, A note on Laplace’s equation inside a cylinder, Appl. Math. Lett. 18 (2005), no. 1, 55–59. MR 2121554, DOI 10.1016/j.aml.2003.05.015
- P. Kalinay and J. K. Percus, Mapping of diffusion in a channel with abrupt change of diameter, Phys. Rev. E 82 (2010), 031143.
- R. C. Knight, The potential of a sphere inside an infinite circular cylinder, Quart. J. Math. 7 (1936), 124–133.
- H. Lamb, On the effect of the walls of an experimental tank on the resistance of a model, Aeronautical Research Committee Reports & Memoranda No. 1010, 1926.
- L. Landweber, Axisymmetric potential flow in a circular tube, J. Hydronautics 8 (1974), 137–145.
- T. Miloh, Irrotational axisymmetric flow about a prolate spheroid in cylindrical duct, J. Eng. Math. 8 (1974), 315–327.
- Philip M. Morse and Herman Feshbach, Methods of theoretical physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0059774
- NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.1.2 of 2021-06-15.
- A. Peskoff, Green’s function for Laplace’s equation in an infinite cylindrical cell, J. Mathematical Phys. 15 (1974), 2112–2120. MR 367246, DOI 10.1063/1.1666591
- A. Peskoff, R. S. Eisenberg, and J. D. Cole, Matched asymptotic expansions of the Green’s function for the electric potential in an infinite cylindrical cell, SIAM J. Appl. Math. 30 (1976), no. 2, 222–239. MR 391734, DOI 10.1137/0130024
- W. R. Smythe, Static and dynamic electricity, 2nd edition, McGraw-Hill, New York, 1950.
- G. N. Watson, The use of series of Bessel functions in problems connected with cylindrical wind-tunnels, Proc. Roy. Soc. London, Ser. A 130 (1930), 29–37.
- G. N. Watson, The diffraction of electric waves by the Earth, Proc. Roy. Soc. London, Ser. A 95 (1918), 83–99.
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
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Additional Information
P. A. Martin
Affiliation:
Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, Colorado 80401
Email:
pamartin@mines.edu
Received by editor(s):
September 13, 2021
Received by editor(s) in revised form:
October 5, 2021
Published electronically:
November 16, 2021
Article copyright:
© Copyright 2021
Brown University
