Abstract
A 1919 contribution of Viktor Trkal on Beltrami fields is contextualized and shown to be of significance in fluid mechanics, time-harmonic electromagnetism and astrophysics.
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Reed D.: Archetypal vortex topology in nature. Speculat. Sci. Technol. (accepted for publication 1993).
Trkal V.: Poznámka k hydrodynamice vazkých tekutin. Časopis pro pěstování mathematiky a fysiky48 (1919) 302–311.
Lugt H. J.: Vortex Flow in Nature and Technology. Wiley, New York, 1983, Chap. 1.
Siegel D. M.: Innovation in Maxwell's Electromagnetic Theory: Molecular Vortices, Displacement Current, and Light. Cambridge University Press, New York, 1991.
Reynolds O.: On an Inversion of Ideas as to the Structure of the Universe. Cambridge University Press, Cambridge, 1903.
Beltrami E.: Considerazioni idrodinamiche. Rend. Inst. Lombardo Acad. Sci. Lett.22 (1889) 122–131. [This paper was reprinted as Paper No. 87in: Beltrami E.: Opere Mathematiche di Eugenio Beltrami, Tomo Quarto. Ulrico Hoepli, Milan, 1920. An English translation of the paper by G. Filipponi is available: Beltrami E.: Considerations on hydrodynamics. Int. J. Fusion Energy3 (1985), No. 3, 53–57.]
Strang J. A.: Self-superposable motion of viscous incompressible fluid referred to rotating axes. Proc. Benares Math. Soc. (N.S.)4 (1942) 9–18.
Wang C. Y.: Exact solutions of the Navier-Stokes equations — the generalized Beltrami flows, review and extension. Acta Mechanica81 (1990) 69–74.
Ballabh R.: Self superposable motions of the typeξ=λu etc. Proc. Benares Math. Soc. (N.S.)2 (1940) 85–89.
Ballabh R.: On coincidence of vortex and stream lines in ideal liquids. Ganita1 (1950) 1–4.
Dombre T., Frisch U., Greene J. M., Hénon M., Mehr A., and Soward A. M.: Chaotic streamlines in the ABC flows. J. Fluid Mech.167 (1986) 353–391.
Galloway D. and Frisch U.: A note on the stability of a family of space-periodic flows. J. Fluid Mech.180 (1987) 557–564.
Fresnel A.-J.: Extrait d'un mémoire sur la double réfraction particulière que présente le cristal de roche dans la direction de son axe. Bull. Soc. Philomat. (1822) 191–198.
MacCullagh J.: On the laws of the double refraction in quartz. Trans. R. Irish Acad.17 (1837) 461–469.
Silberstein L.: Elektromagnetische Grundgleichungen in Bivektorieller Behandlung. Ann. Phys. Leipzig22 (1907) 579–587.
Rumsey V. H.: A new way of solving Maxwell's equations. IEEE Trans. Antennas Propagat.9 (1961) 461–465.
Chambers Ll. G.: The generalized Beltrami equation. J. Math. Anal. Applics.36 (1971) 241–250.
Baum C. E.: The PARTES concept in EMP simulation. Electromagnetics3 (1983) 1–19.
Lakhtakia A. (ed.): Selected Papers on Natural Optical Activity. SPIE Opt. Engg. Press, Bellingham, WA, USA, 1990.
Lakhtakia A.: Beltrami Fields in Chiral Media. World Scientific, Singapore, 1994 (in press).
Lakhtakia A. and Ma Y.: Scattering by an isotropic noncentrosymmetric solid sphere immersed in a fluid. J. Acoust. Soc. Am.92 (1992) 3000–3002.
Chandrasekhar S.: On cosmic magnetic fields. Proc. Nat. Acad. Sci. USA43 (1957) 24–27.
Lüst R. and Schlüter A.: Kraftfreie Magnetfelder. Z. Astrophys.34 (1954) 263–282.
Chandrasekhar S.: On force-free magnetic fields. Proc. Nat. Acad. Sci. USA42 (1956) 1–5.
Woltjer L.: A theorem on force-free magnetic fields. Proc. Nat. Acad. Sci. USA44 (1958) 489–491.
Zaghloul H. and Barajas O.: Force-free magnetic fields.In: Essays on the Formal Aspects of Electromagnetic Theory (ed. A. Lakhtakia). World Scientific, Singapore, 1993.
DeBroglie L.: The Revolution in Physics. Noonday Press, New York, 1953, p. 250.
Hillion P.: Spinor form of electromagnetism in nonhomogeneous media. Int. J. Engng. Sci.29 (1991) 1157–1165.
Lakhtakia A.: The Maxwell postulates and chiral worlds.In: Essays on the Formal Aspects of Electromagnetic Theory (ed. A. Lakhtakia). World Scientific, Singapore, 1993.
Lakhtakia A.: Being a message from J. Chiral-Maxwell. Speculat. Sci. Technol.16 (1993) 145–153.
Lakhtakia A.: Time-dependent Beltrami fields in material continua: The Beltrami-Maxwell postulates. Int. J. Infrared Millim. Waves15 (1994) (in press).
Trkal V.: A note on the hydrodynamics of viscous fluids. Czech. J. Phys.44 (1994) 97–106 (this issue).
Gould S. J.: The Razumovsky Duet. Natural History102 (1993), No. 10, 10–19.
Bjørgum O.: On Beltrami vector fields and flows\((\nabla \times \vec \upsilon = \Omega \vec \upsilon )\). Part I. A comparative study of some basic types of vector fields. Universitetet i Bergen Årbok 1951. Naturvitenskapelig rekke Nr. 1. (The monograph is written in English.)
Bjørgum O. and Godal T.: On Beltrami vector fields and flows\((\nabla \times \vec \upsilon = \Omega \vec \upsilon )\). Part II. The case whenΩ is constant in space. Universitetet i Bergen Årbok 1952, Naturvitenskapelig rekke Nr. 13. (The monograph is written in English.)
Dryden H. L., Murnaghan F. D., and Bateman H.: Hydrodynamics. Dover, New York, 1956. (This book was originally published as a committee report in 1932.)
Aris R.: Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Prentice-Hall, Englewood Cliffs, NJ, 1962, p. 65.
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This paper introduces the English translation of an article by V. Trkal from 1919 — see next paper in this issue, p. 97.
I gratefully acknowledge the assistance of Drs. Per Lindstrom (Bergen, Norway), Avadh B. Saxena (Los Alamos, New Mexico), Chandra S. Vikrarn (Huntsville, Alabama) and Miloslav Znojil (Prague) in locating old publications. I also thank the editors of this journal for asking me to write this introductory piece as well as for readily agreeing to publish an English translation of Trkal's paper.
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Lakhtakia, A. Viktor Trkal, Beltrami fields, and Trkalian flows. Czech J Phys 44, 89–96 (1994). https://doi.org/10.1007/BF01701185
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DOI: https://doi.org/10.1007/BF01701185