The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics

Fluid Dynamics Research

Volumn 18, Issue 5, 1996, Pages 245-268

  Ricca, Renzo L ^a  

^a UNIVERSITY COLLEGE LONDON   (United Kingdom)

Author keywords

Hasimoto soliton; Intrinsic kinematics; Localized induction approximation (LIA); Newtonian potential; Vortex filament

Indexed keywords

APPROXIMATION THEORY; EQUATIONS OF MOTION; KINEMATICS; MATHEMATICAL MODELS; SOLITONS; TUBES (COMPONENTS); VORTEX FLOW;

ASYMPTOTIC POTENTIAL THEORY; HASIMOTO SOLITON; LOCALIZED INDUCTION APPROXIMATION; NEWTONIAN POTENTIAL; VORTEX FILAMENT DYNAMICS; VORTEX TUBES;

FLUID DYNAMICS;

REVIEWS; VORTICES;

EID: 0030268006     PISSN: 01695983     EISSN: None     Source Type: Journal    
DOI: 10.1016/0169-5983(96)82495-6     Document Type: Article

References (53)

1. Abramowitz, M. and I.A. Stegun (1972) Handbook of Mathematical Functions (Dover Publications, New York).
2. Arms, R.J. and F.R. Hama (1965) Localized induction concept on a curved vortex and motion of an elliptic vortex ring, Phys. Fluids 8, 553-559.
3. Batchelor, G.K. (1967) An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge).
4. Betchov, R. (1965) On the curvature and torsion of an isolated vortex filament, J. Fluid Mech. 22, 471-479.
5. Cisotti, U. (1913) Su alcune recenti ricerche in idrodinamica (On some recent researches in hydrodynamics), Atti Soc. Ital. Progresso Scienze 1, 491-511.
6. Cisotti, U. (1924) Über den Anteil Italiens an dem Fortschritt der Klassischen Hydrodynamik in den Letzen fünfzehn Jahren (On the Italian contributions to the progress of classical hydrodynamics in the last fifteen years), in: von Karman, T. and Levi-Civita, T., eds., Vortrage aus dem Gebiete der Hydro- und Aerodynamik (Innsbruck, 1922) (Springer, Berlin) pp. 1-17.
7. Da Rios, L.S. (1906) Sul moto d'un liquido indefinito con un filetto vorticoso di forma qualunque (On the motion of an unbounded liquid with a vortex filament of any shape), Rend. Circ. Mat. Palermo 22, 117-135.
8. Da Rios, L.S. (1909) Sul moto dei filetti vorticosi di forma qualunque (On the motion of vortex filaments of any shape), Rend. R. Acc. Lincei 18, 75-79.
9. Da Rios, L.S. (1910) Sul moto dei filetti vorticosi di forma qualunque (On the motion of vortex filaments of any shape), Rend. Circ. Mat. Palermo 29, 354-368.
10. Da Rios, L.S. (1911) Sul moto intestino dei filetti vorticosi (On the interior motion of vortex filaments), Giornale di Matematiche del Battaglini 49, 300-308.
11. Da Rios, L.S. (1916a) Sezioni trasversali stabili dei filetti vorticosi (Stable cross-sections of vortex filaments), Atti R. 1st. Veneto Sci. Lett. Arti 75, 299-308.
12. Da Rios, L.S. (1916b) Sui tubi vorticosi rettilinei posti a raffronto (Comparative study of rectilinear vortex tubes), Atti R. Acc. Sci. Lett. Arti Padova 32, 343-350.
13. Da Rios, L.S. (1916c) Vortici ad elica (Helical vortices), Il Nuovo Cimento 11, 419-431.
14. Da Rios, L.S. (1930) Sur la théorie des tourbillons, Comptes Rendus Acad. Sci. Paris 191, 399-401.
15. Da Rios, L.S. (1931a) Sui vortici piani indeformabili (On planar vortices of invariant shape), Atti Pontificia Acc. Sci. Nuovi Lincei 84, 720-731.
16. Da Rios, L.S. (1931b) Sui vortici piani indeformabili (On planar vortices of invariant shape), Atti Soc. Ital. Progresso Scienze 2, 7-8.
17. Da Rios, L.S. (1931c) Anelli vorticosi ruotanti (Rotating vortex rings), Rend. Sem. Matern. Univ. Padova 9, 142-151.
18. Da Rios, L.S. (1933a) Ancora sugli anelli vorticosi ruotanti (Again on rotating vortex rings), Rend. R. Acc. Lincei 17, 924-926.
19. Da Rios, L.S. (1933b) Sui vortici gobbi indeformabili (On hunched vortices of invariant shape), Atti Pontificia Acc. Sci. Nuovi Lincei 86, 162-168.
20. Germano, M. (1983) Sulla riscoperta delle equazioni del Da Rios concernenti il moto di un filetto vorticoso (On the re-discovery of the Da Rios equations concerning the motion of a vortex filament), in: Proc. VII Natl. Cong. Assoc. Ital. Aeronaut. Astronaut., 1, 163-170.
21. Hasimoto, H. (1971) Motion of vortex filament and its relation to elastica, J. Phys. Soc. Jpn. 31, 293-294.
22. Hasimoto, H. (1972) A soliton on a vortex filament, J. Fluid Mech. 51, 477-485.
23. Helmholtz, H. (1858) Über Integrale der hydrodynamischen Gleighungen, welche den Wirbelbewegungen entsprechen, J. Reine Angew. Math. 55, 25-55.
24. Kambe, T. and T. Takao (1971) Motion of distorted vortex rings, J. Phys. Soc. Jpn. 31, 591-599.
25. Keener, J.P. (1990) Knotted vortex filaments in an ideal fluid, J. Fluid Mech. 211, 629-651.
26. Kelvin, Lord (Thomson, W.) (1867) The translatory velocity of a circular vortex ring, Phil. Mag. 33, 511-512.
27. Kelvin, Lord (Thomson, W.) (1869) On vortex motion, Trans. Roy. Soc. Edinb. 25, 217-260.
28. Kelvin, Lord (Thomson, W.) (1880) Vibrations of a columnar vortex, Phil. Mag. 10, 155-168.
29. Kida, S. (1981) A vortex filament moving without change of form, J. Fluid Mech. 112, 397-409.
30. Landkof, N.S. (1972) Foundations of Modern Potential Theory (Springer, Berlin).
31. Langer, J. and R. Perline (1991) Poisson geometry of the filament equation, J. Nonlin. Sci. 1, 71-93.
32. Levi-Civita, T. (1908a) Sull'attrazione esercitata da una linea materiale in punti prossimi alia linea stessa (On the attraction of a material line at points placed near to the line), Rend. R. Acc. Lincei 17, 3-15.
33. Levi-Civita, T. (1908b) Sull'attrazione Newtoniana di un tubo sottile (On the Newtonian attraction of a slender tube), Rend. R. Acc. Lincei 17, 413-426; 535-551.
34. Levi-Civita, T. (1909) Sulle azioni meccaniche dovute ad un flusso filiforme di elettricità (On the mechanical effects due to a filiform flow of electric current), Rend. R. Acc. Lincei 18, 41-50.
35. Levi-Civita, T. (1912) Sulla gravitazione di un tubo sottile con applicazione all'anello di Saturno (On the gravitation of a slender tube with application to the Saturnian ring), Rend. Circ. Mat. Palermo 33, 354-374.
36. Levi-Civita, T. (1932a) Attrazione Newtoniana dei tubi sottili e vortici filiformi (Newtonian attraction of slender tubes and filiform vortices), Annali R. Scuola Norm. Sup. Pisa 1, 1-33; 229-250.
37. Levi-Civita, T. (1932b) Teoremi di unicità e di esistenza per le piccole oscillazioni di un filetto vorticoso prossimo alla forma circolare (Uniqueness and existence theorems for small oscillations of a vortex filament near the circular shape), Rend. R. Acc. Lincei 15, 409-416.
38. Love, A.E.H. (1912) Dèveloppments concernant l'Hydrodynamique (Irans, by P. Appell and H. Villat), in: Encyclopédie des Sciences Mathématiques Pures et Appliquées, Tome IV, 5 (Gauthier-Villars, Paris) pp. 143-144.
39. Love, A.E.H. (1944) A Treatise on the Mathematical Theory of Elasticity (Dover, New York).
40. Nakayama, K., H. Segur and M. Wadati (1992) Integrability and the motion of curves, Phys. Rev. Lett. 69, 2603-2606.
41. Nastasi, P., ed. (1988) Documenti della vita del Circolo Matematico di Palermo (Documents on the life of Circolo Matematico di Palermo), Rend. Circ. Mat. Palermo 20, Supplement.
42. Ricca, R.L. (1991a) Rediscovery of Da Rios equations, Nature 352, 561-562.
43. Ricca, R.L. (1991b) Intrinsic equations for the kinematics of a classical vortex string in higher dimensions, Phys. Rev. A 43, 4281-4288.
44. Ricca, R.L. (1992) Physical interpretation of certain invariants for vortex filament motion under LIA, Phys. Fluids A 4, 938-944.
45. Ricca, R.L. (1993) Torus knots and polynomial invariants for a class of soliton equations, Chaos 3, 83-91 [1995 Erratum. Chaos 5, 346].
46. Ricca, R.L. (1994) The effect of torsion on the motion of a helical vortex filament, J. Fluid Mech. 273, 241-259.
47. Ricca, R.L. (1995) Geometric and topological aspects of vortex filament dynamics under LIA, in: Meneguzzi, M. et al., eds., Small-Scale Structures in Three-Dimensional Hydro and Magnetohydrodynamics Turbulence, Lecture Notes in Physics, Vol. 462 (Springer, Berlin) pp. 99-104.
48. Sym, A. (1984) Soliton surfaces. VI - Gauge invariance and final formulation of the approach, Lett. Nuovo Cimento 41, 353-360.
49. Truesdell, C. (1954) The Kinematics of Vorticity (Indiana University Press, Indiana).
50. Truesdell, C. (1983) The influence of elasticity on analysis: the classic heritage, Bull. Amer. Math. Soc. 9, 293-310.
51. Vasil'ev, N.S. (1913) Motion of infinitely thin vortex lines (in Russian), Thesis (submitted to Moscow University in 1916) (Komercheskaia tipografiia Sapozhnikova, Odessa).
52. Villat, H. (1930) Leçons sur la Théorie des Tourbillons (Gauthier-Villars, Paris).
53. Wehner, R. (1939) Die italienische Literatur über die Bewegung isolierten Wirbel und isolierter Quellen in der Ebene, Dissertation, Technischen Hochshule Dresden.