Skip to main content

Solutions in spherical coordinates

  • Chapter
Electrically Induced Vortical Flows

Part of the book series: Mechanics of Fluids and Transport Processes ((MFTP,volume 9))

Abstract

With this chapter we begin a systematic study of axisymmetric electrically induced vortex flows, the governing equations of which have been presented in the previous chapter. The study will begin with so-called exact solutions, by which we mean those solutions obtained from the full equations of motion without a priori estimates and omission of any terms, say, after the order-of-magnitude analysis (this does not apply to the electrodynamic quantities, which will conform to assumptions simplifying the electrodynamic part of problem).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ajayi O. O.: The transient Stokes flow induced by a point source of electric current. Meccanica (1985), 20, pp. 12–17.

    Article  MATH  Google Scholar 

  2. Ajayi O. O., Sozou C, and Pickering W. M.: Nonlinear fluid motions in a container due to the discharge of an electric current. J. Fluid Mech. (1984), 148, pp. 285–300.

    Article  ADS  MATH  Google Scholar 

  3. Andrews J. G. and Craine R. E.: Fluid flow in a hemisphere induced by a distributed source of current. J. Fluid Mech. (1978), 84(2), pp. 281–290.

    Article  ADS  MATH  Google Scholar 

  4. Atthey D. R.: A mathematical model for fluid flow in a weld pool at high currents. J. Fluid Mech. (1980), 98(4), pp. 787–801.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Batchelor G. K. and Gill A. E.: Analysis of the stability of axisymmetric jets. J. Fluid Mech. (1962), 14(4), pp. 529–551.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Batchelor G. K.: An Introduction to Fluid Dynamics. Cambridge at the University Press, 1970.

    Google Scholar 

  7. Bojarevičs V. V.: On the application limits of an exact solution of MHD equations. Magnitnaya Gidrodinamika (1976), No. 4, pp. 140–141.

    Google Scholar 

  8. Bojarevičs V. V.: Electrovortex flow at a hemispherical electrode. Magnitnaya Gidrodinamika (1978), No. 4, pp. 77–81.

    Google Scholar 

  9. Bojarevičs V. V. and Millere R.: Amplification of rotation of the meridional electrovortex flow in a hemisphere. Magnitnaya Gidrodinamika (1982), No. 4, pp. 51–56.

    Google Scholar 

  10. Bojarevičs V. V. and Shilova E. I.: Landau-Squire jet in radially diverging electric current. Magnitnaya Gidrodinamika (1977), No. 3, pp. 89–94.

    Google Scholar 

  11. Cantwell B. J.: Transition in the axisymmetric jet. J. Fluid Mech. (1981), 104, pp. 369–386.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Clayton B. R. and Massey B. S.: Exact similar solution for an axisymmetric laminar boundary layer on a circular cone. AIAA J. (1979), 17(7), pp. 785–786.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Cooke J. C.: On Pholhausen’s method with application to a swirl problem of Taylor. J. Aeronautical Sci. (1952), 19(7), pp. 486–490.

    MathSciNet  MATH  Google Scholar 

  14. Cowley M. D.: On electrode jets. In: Proc. 11th Intern, conf. phenomena ionized gases. Prague, 1973, p. 249.

    Google Scholar 

  15. Dangey J. W.: Cosmic Electrodynamics. Cambridge at the University Press, 1958.

    Google Scholar 

  16. Germain P.: Cours de Mechanique des Milieux Continus. Paris: Massan et Cie, 1973.

    Google Scholar 

  17. Goldshtik M. A.: A paradoxical solution of Navier-Stokes equation. Prikladnaya Mekhanika i Matematika (1960), 24(4), pp. 610–621.

    Google Scholar 

  18. Goldshtik M. A.: Vortex Flows. Novosibirsk: Nauka, 1981 (In Russian).

    Google Scholar 

  19. Goldshtik M. A. and Silant’ev B. A.: To the theory of laminar submerged jets. Prikladnaya Matematika i Teoreticheskaya Fizika (1965), No. 5, pp. 149–152.

    Google Scholar 

  20. Golubinskij A. A. and Sichev V. V.: About a similarity solution of Navier-Stokes equations. Uchenye Zapiski TSAGI (1976), 7(6), pp. 11–17.

    ADS  Google Scholar 

  21. Guilloud J. C. and Arnault J.: Sur une nouvelle classe de solutions exactes des equations de Navier-Stokes. C. R. Acad. Sci., ser. A (1971), 273, pp. 517–520.

    MATH  Google Scholar 

  22. Hamel G.: Spiralformige Bewegung zäher Flüssigkeiten. Iber. Dt. Math.-Ver. (1916), 25, S. 34–60.

    MATH  Google Scholar 

  23. Jeffery G. B.: Steady motion of a viscous fluid. Phil. Mag., ser. 6 (1915), 29, p. 455.

    Article  Google Scholar 

  24. Joukov M. F., Koroteev A. S., and Uryukov B. A.: Applied Dynamics of Thermal Plasma. Novosibirsk: Nauka, 1975 (In Russian).

    Google Scholar 

  25. Kashkarov V. P.: Some exact solutions in the theory of incompressible fluid jets. Investigations in physical basis of working process in stoves and furnaces. Ed. by Vulis L. A. Alma-Ata: Kazakhian Academy of Sciences (1957), pp. 54–63 (In Russian).

    Google Scholar 

  26. Kidd G. J. and Farris G. J.: Potential Vortex Flow Adjacent to a Stationary Surface. Trans, of the ASME. Ser. E. J. Appl. Mech. (1968), 35, No. 2.

    Google Scholar 

    Google Scholar 

  27. Landau L. D.: New exact solution of Navier-Stokes equations. Doklady Akademii Nauk SSSR (1944), 43(7), pp. 299–301.

    Google Scholar 

  28. Landau L. D. and Lifshits E. M.: Hydrodynamics. Moscow: Nauka, 1986 (In Russian).

    Google Scholar 

  29. Leibowich S.: The structure of vortex breakdown. Ann. Rev. Fluid Mech. (1978), 10, pp. 221–246.

    Article  ADS  Google Scholar 

  30. Levakov V. S. and Lyubavskij K. V.: The effect of longitudinal magnetic field on electric arc with nonmelting tungsten electrode. Svarochnoe Proizvodstvo (1965), No. 10, pp. 9–12.

    Google Scholar 

  31. Loitsyanskij L. G.: Propagation of swirling jet in unbounded space filled with the same liquid. Prikladnaya Matematika i Mekhanika (1953), 17(1), pp. 3–16.

    Google Scholar 

  32. Loitsyanskij L. G.: Radial jet in space filled by the same liquid. Trudy Leningradskogo Politekhnicheskogo Instituta (1953), No. 5, pp. 5–14.

    Google Scholar 

  33. Loitsyanskij L. G.: Fluid and Gas Mechanics. Moscow: Nauka, 1973 (In Russian).

    Google Scholar 

  34. Lundquist S.: On the hydromagnetic viscous flow generated by a diverging electric current. Ark. Fys. (1969), 40(5), pp. 89–95.

    Google Scholar 

  35. Moffatt H. K.: Some problems in the magnetohydrodynamics of liquid metals. Ztschr. Angew. Math. Mech. (1978), 58, S. T65–T71.

    Google Scholar 

  36. Morgan A. J. A.: On a class of laminar viscous flows within one or two bounding cones. Aeronautical Quart. (1956), 7, pp. 225–239.

    Google Scholar 

  37. Narain J. P. and Uberoi M. S.: Magnetohydrodynamics of conical flows. Phys. Fluids (1971), 14(12), pp. 2687–2692.

    Article  ADS  MATH  Google Scholar 

  38. Narain J. P. and Uberoi M. S.: Fluid motion caused by conical currents. Phys. Fluids (1973), 16(6), pp. 940–942.

    Article  ADS  Google Scholar 

  39. Pao H. P. and Long R. R.: Magnetohydrodynamic jet-vortex in a viscous conducting fluid. Quart. J. Mech. Appl. Math. (1966), 19(1), pp. 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  40. Potsch K.: Schwache Auftriebseffekte in laminaren, vertikalen, runden Freistrahlen. Acta Mech. 36(1/2), pp. 1–14.

    Google Scholar 

  41. Ranger K. B.: Hydromagnetic momentum source. Phys. Fluids (1965), 8(9), pp. 1747–1748.

    Article  ADS  Google Scholar 

  42. Rumer Yu. B.: Submerged jet problem. Prikladnaya Matematika i Mekhanika (1952), 16(2), pp. 255–256.

    MathSciNet  MATH  Google Scholar 

  43. Rumer Yu. B.: Convective diffusion in submerged jet. Prikladnaya Matematika i Mekhanika (1953), 17(6), pp. 743–744.

    MathSciNet  MATH  Google Scholar 

  44. Samerville G. M.: Electric Arc. London: Methuen and Co., 1959.

    Google Scholar 

  45. Slezkin N. A.: Viscous fluid motion within two cones. Uchenye Zapiski MGU (1934), No. 2, pp. 83–87.

    Google Scholar 

  46. Schlichting H.: Boundary Layer Theory. New York: McGraw-Hill, 1960.

    MATH  Google Scholar 

  47. Schneider W.: Flow induced by jets and plumes. J. Fluid Mech. (1981), 108, pp. 55–65.

    Article  ADS  MATH  Google Scholar 

  48. Schwiderski E. W.: On the axisymmetric vortex flow over a flat surface. Trans, of the ASME. Ser. E, J. Appl. Mech. (1969), 36(3), pp. 614–619.

    MATH  Google Scholar 

  49. Shcherbinin E. V.: On a kind of exact solutions in MHD. Magnitnaya Gidrodinamika (1969), No. 4, pp. 46–58.

    Google Scholar 

  50. Shcherbinin E. V.: Viscous Fluid Jet Flows in Magnetic Field. Riga: Zinatne, 1973 (In Russian).

    Google Scholar 

  51. Shcherbinin E. V.: Jet flows in an electric arc. Magnitnaya Gidrodinamika (1973), No. 4, pp. 66–72.

    Google Scholar 

  52. Shcherbinin E. V. and Yakovleva E. E.: An electrovortex flow in a spheroidal container. Magnitnaya Gidrodinamika (1986), No. 4, pp. 64–69.

    Google Scholar 

  53. Shercliff J. A.: Fluid motions due to an electric current source. J. Fluid Mech. (1970), 40(2), pp. 241–250.

    Article  ADS  MATH  Google Scholar 

  54. Shilova E. I. and Shcherbinin E. V.: Some exact solutions of the equations of motion in MHD. Magnitnaya Gidrodinamika (1969), No. 4, pp. 59–64.

    Google Scholar 

  55. Shilova E. I. and Shcherbinin E. V.: Some aspects of theoretical study of a three-dimensional MHD flow in a diffuser. Magnitnaya Gidrodinamika (1971), No. 1, pp. 11–17.

    Google Scholar 

  56. Shilova E. I. and Shcherbinin E. V.: MHD vortex flow in a cone. Magnitnaya Gidrodinamika (1971), No. 2, pp. 33–38.

    Google Scholar 

  57. Sozou C.: On fluid motions induced by an electric current source. J. Fluid Mech. (1971), 46(1), pp. 25–32.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. Sozou C.: On some similarity solutions in magnetohydrodynamics. J. Plasma Phys. (1971), 6(2), pp. 331–341.

    Article  ADS  Google Scholar 

  59. Sozou C.: On magnetohydrodynamic flows generated by an electric current discharge. J. Fluid Mech. (1974), 63(4), pp. 665–671.

    Article  ADS  MATH  Google Scholar 

  60. Sozou C.: Development of the flow field of a point force in an infinite fluid. J. Fluid Mech. (1979), 91(3), pp. 541–546.

    Article  MathSciNet  ADS  Google Scholar 

  61. Sozou C. and English H.: Fluid motions induced by an electric current discharge. Proc. Roy. Soc. London (1972), A329, pp. 71–81.

    ADS  Google Scholar 

  62. Sozou C. and Pickering W. M.: The development of magnetohydrodynamic flow due to an electric current discharge. J. Fluid Mech. (1975), 70(3), pp. 509–517.

    Article  ADS  MATH  Google Scholar 

  63. Sozou C. and Pickering W. M.: Magnetohydrodynamic flow due to the discharge of an electric current in a hemispherical container. J. Fluid Mech. (1976), 73(4), pp. 641–650.

    Article  ADS  MATH  Google Scholar 

  64. Sozou C. and Pickering W. M.: The round laminar jet: the development of the flow field. J. Fluid Mech. (1977), 80(4), pp. 673–683.

    Article  ADS  MATH  Google Scholar 

  65. Sozou C. and Pickering W. M.: Magnetohydrodynamic flows in a container due to the discharge of an electric current from a finite size electrode. Proc. Roy. Soc. London (1978), A362, pp. 509–523.

    ADS  Google Scholar 

  66. Squire H. B.: The round laminar jet. Quart. J. Mech. Appl. Math. (1951), 4(3), pp. 321–329.

    MathSciNet  MATH  Google Scholar 

  67. Squire H. B.: Jet emerging from a hole in a plane wall. Phil. Mag., ser. 7 (1952), 43(343), pp. 942–945.

    MathSciNet  MATH  Google Scholar 

  68. Squire H. B.: Radial jets. In: 50 Jahre Grenaschichforschung. Braunschweig, 1954, S. 47–54.

    Google Scholar 

  69. Tsuker M. S.: Swirling jet propagating in space filled by the same liquid. Prikladnaya Matematika i Mekhanika (1955), 19(4), pp. 500–503.

    Google Scholar 

  70. Vulis L. A. and Kashkarov V. P.: Theory of Viscous Fluid Jets. Moscow: Nauka, 1965 (In Russian).

    Google Scholar 

  71. Weber H. E.: The boundary layer inside a conical surface due to swirl. J. Fluid Mech. (1956), 23(4), pp. 587–592.

    MATH  Google Scholar 

  72. Wu Ch.-Sh.: A class of exact solutions of the magnetohydrodynamic Navier-Stokes equations. Quart. J. Mech. Appl. Math. (1961), 14(1), pp. 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  73. Yatseyev V. I.: About a class of exact solutions of viscous fluid equations of motion. Zhurnal Tekhnicheskoj Fiziki (1950), 20(11), pp. 1031–1034.

    Google Scholar 

  74. Yih C.-S., Wu F., Garg A. K., and Leibovich S.: Conical vortices: a class of exact solutions of the Navier-Stokes equations. Phys. Fluids (1982), 25(12), pp. 2147–2158.

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Kluwer Academic Publishers

About this chapter

Cite this chapter

Bojarevičs, V., Freibergs, J.A., Shilova, E.I., Shcherbinin, E.V. (1989). Solutions in spherical coordinates. In: Electrically Induced Vortical Flows. Mechanics of Fluids and Transport Processes, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1163-5_3

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-1163-5_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7017-1

  • Online ISBN: 978-94-009-1163-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics