Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Three-phase eccentric annulus subjected to a potential field induced by arbitrary singularities

Author: Yu. V. Obnosov
Journal: Quart. Appl. Math. 69 (2011), 771-786
MSC (2010): Primary 30E25, 76T30
DOI: https://doi.org/10.1090/S0033-569X-2011-01242-8
Published electronically: July 12, 2011
MathSciNet review: 2894000
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Abstract | References | Similar Articles | Additional Information

Abstract: An infinite planar, three-component heterogeneous medium with a pair of circles as interfaces between homogeneous zones forming an eccentric annulus is considered for refraction of a potential field on the two interfaces. The velocity field is generated by an arbitrary system of singularities of arbitrary order, in congruity with the Milne-Thomson case of a two-component medium and a single circular interface. An exact analytical solution of the corresponding $ {\mathbb{R}}$-linear conjugation problem of two Laplacian fields in the eccentrical annulus structure is derived in the class of piecewise meromorphic functions with fixed principal part. Three general cases of loci of the singularities with respect to the interfaces are investigated. Flow nets (isobars and streamlines) are presented.

References [Enhancements On Off] (What's this?)

  • 1. Crowdy D.G., Analytical solutions for uniform potential flow past multiple cylinders. European Journal of Mechanics, B/Fluids 25 (4), 459-470 (2006) MR 2241382 (2007a:76025)
  • 2. Dagan G., Flow and Transport in Porous Formations, Springer-Verlag, Heidelberg Berlin New York, 465 pp. (1989).
  • 3. De Zwart B.-R., Investigation of clogging processes in unconsolidated aquifers near water supply wells. Ph.D. thesis, Delft Univ. of Technology. 200 pp. (2007))
  • 4. De Zwart A.H., Currie P.K., De Boer J., Naeini A.F., Schotting R.J., Experimental and theoretical investigation of clogging processes near production wells using X-ray tomography. SPE paper 116411 (2008).
  • 5. Drygaś P., Mityushev V., Effective conductivity of unidirectional cylinders with interfacial resistance, Quarterly Journal of Mechanics and Applied Mathematics 62 (3), 235-262 (2009). MR 2524804
  • 6. Emets Y.P., Boundary-value problems of electrodynamics of anisotropically conducting media, Naukova dumka, Kiev (1987). (In Russian)
  • 7. Emets Yu.P., Obnosov Yu.V., An accuratly solvable problem of the mutual effect of inclusions in the theory of heterogeneous media. Journal of Appl. Mech. and Tekhn. Physics, v.31, No 1, 21-29, (1990).(DOI: 10.1007/BF00852740 ) MR 1055573 (92a:78003)
  • 8. Golubeva O.V., Generalization of the theorem about a circle on seepage flow. Izv. AN USSR, MGhG, No 1, 113 -116 (1966). (In Russian)
  • 9. Honein E., Honein T., Herrmann G., On two circular inclusions in harmonic problems, Quart. Appl. Math., 50, No 3, 479-499 (1992). MR 1178429 (93f:73031)
  • 10. Honein E., Honein T., Herrmann G., Energetics of two circular inclusions in anti-plane electrostatics, Int. J. of Solids and Structures., 37, 3667-3679 (2000).
  • 11. Lee D.K., Image singularity system to represent two circular cylinders of different diameter. Journal of Fluids Engineering, Transactions of the ASME, Volume 122, Issue 4, December 2000, pages 715-719.
  • 12. Maltseva A.M., Obnosov Yu.V., Rogozin S.V., A generalization of Milne-Thomson theorem on the case of concentric annulus, Uchen. zap. Kazan University, Ser. phys.-math. nauk, 148, book.4, 35-50 (2006). (In Russian)
  • 13. Maxwell J.C., A Treatise on Electricity and Magnetism, 3rd edn. Oxford University Press, 1, 440 pp. (1904)
  • 14. Milne-Thomson L.M., Theoretical hydrodynamics. 5th ed., Macmillan (1968).
  • 15. Milton G.W., The Theory of Composites, Cambridge University Press (2002). MR 1899805 (2003d:74077)
  • 16. Obnosov Yu.V., A generalized Milne-Thomson theorem. Applied Mathematics Letters, 19, 581-586 (2006). MR 2221517 (2007a:30018)
  • 17. Obnosov Yu.V., Solution of a problem of a seepage fields distribution into infinite porous massif with two circular inclusions. Uch. Zap. Kazan. Gos. Univ., Ser. Fiz.-Mat. Nauki, 148, No. 2, 109-123 (2006). (In Russian)
  • 18. Obnosov Yu.V., A generalized Milne-Thomson theorem for the case of parabolic inclusion, Applied Mathematical Modelling, 33, 1970-1981 (2009). MR 2488259 (2010b:76123)
  • 19. Obnosov Yu.V., Boundary-value problems of heterogeneous medium theory, Kazan University Press, Kazan (2009). (In Russian)
  • 20. Obnosov Yu.V., Kasimova R. G., Al-Maktoumi A., Kacimov A.R., Can heterogeneity of the near-wellbore rock cause extrema of the Darcian fluid inflow rate from the formation (the Polubarinova-Kochina problem revisited)?, Computers & Geosciences, Elsevier. Doi:10.1016/j.cageo.2010.01.014 (2010).
  • 21. Palaniappan D., Electrostatics of two intersecting conducting cylinders. Mathematical and Computer Modelling 36 (7-8), pp. 821-830 (2002). MR 1950735 (2003m:78009)
  • 22. Polubarinova-Kochina P.Ya., Theory of Ground-water Movement, Nauka, Moscow, (1977). (in Russian) MR 0670100 (58:32303)
  • 23. Lord Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of medium. Phil. Mag. 34, 481-502 (1892).
  • 24. Strack O.D.L., Groundwater Mechanics. Prentice Hall, Englewood Cliffs (1989).
  • 25. Wu L., Interaction of two circular cylindrical inhomogeneities under anti-plane shear. Composites Science and Technology 60 (12-13), 2609-2615 (2000)

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Additional Information

Yu. V. Obnosov
Affiliation: Institute of Mathematics and Mechanics, Kazan State University, Prof. Nughin Str.,1/37, Kazan, 420008, Russia
Email: Yurii.Obnosov@ksu.ru

DOI: https://doi.org/10.1090/S0033-569X-2011-01242-8
Keywords: Refraction, heterogeneous media, $\mathbb R$-linear conjugation problem, analytic functions
Received by editor(s): May 3, 2010
Published electronically: July 12, 2011
Article copyright: © Copyright 2011 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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