Vortices and boundaries

Chapman, S. J.; Hunton, B. J.; Ockendon, J. R.

doi:10.1090/qam/1637052

Authors: S. J. Chapman, B. J. Hunton and J. R. Ockendon
Journal: Quart. Appl. Math. 56 (1998), 507-519
MSC: Primary 76B47; Secondary 76A25, 76D17, 82D55
DOI: https://doi.org/10.1090/qam/1637052
MathSciNet review: MR1637052
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper develops a framework in which vortex/boundary interactions can be analysed for the cases of inviscid fluid dynamics and type II superconductors. Comments are made concerning the possibility of reconnection, defined as an interaction in which an initially single-component vortex eventually emerges with two components.

W. T. Ashurst and D. I. Meiron, Numerical study of vortex reconnection, Phys. Rev. Lett. 58 (16), 1632–1635 (1987)

G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638

L. P. Bernal and J. T. Kwon, Vortex ring dynamics beneath a free surface, Phys. Fluids A 1 (3), 449–451 (1989)

A. J. Callegari and L. Ting, Motion of a curved vortex filament with decaying vortical core and axial velocity, SIAM J. Appl. Math. 35 (1978), no. 1, 148–175. MR 495588, DOI https://doi.org/10.1137/0135013
S. Jonathan Chapman and G. Richardson, Motion of vortices in type II superconductors, SIAM J. Appl. Math. 55 (1995), no. 5, 1275–1296. MR 1349310, DOI https://doi.org/10.1137/S0036139994263872
W. Craig and B. J. Hunton, Modulated waves on a vortex filament beneath a fluid surface, Appl. Math. Lett. 8 (1995), no. 3, 7–11. MR 1356799, DOI https://doi.org/10.1016/0893-9659%2895%2900021-H
W. Craig, C. Sulem, and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5 (1992), no. 2, 497–522. MR 1158383

S. C. Crow, Stability theory for a pair of trailing vortices, A.I.A.A.A. Journal 8 (12), 2172–2179 (1970)

Hidenori Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1972), no. 3, 477–485. MR 3363420, DOI https://doi.org/10.1017/S0022112072002307

J. P. Hirth and J. Lothe, Theory of Dislocations, second edition, John Wiley and Sons, New York, 1982

B. J. Hunton, Vortex Dynamics, D. Phil. thesis, Oxford University, 1994

A. S. Joseph and W. T. Tomasch, Experimental evidence for delayed entry of flux into a type II superconductor, Phys. Rev. Lett. 12 (9), 219–222 (1964)

Robert M. Kerr, Evidence for a singularity of the three-dimensional, incompressible Euler equations, Phys. Fluids A 5 (1993), no. 7, 1725–1746. MR 1223050, DOI https://doi.org/10.1063/1.858849

S. Kida and M. Takaoka, Bridging in vortex reconnection, Phys. Fluids 30 (10), 2911–2914 (1987)

Rupert Klein, Andrew J. Majda, and Kumaran Damodaran, Simplified equations for the interaction of nearly parallel vortex filaments, J. Fluid Mech. 288 (1995), 201–248. MR 1325356, DOI https://doi.org/10.1017/S0022112095001121
R. Klein and L. Ting, Theoretical and experimental studies of slender vortex filaments, Appl. Math. Lett. 8 (1995), no. 2, 45–50. MR 1357250, DOI https://doi.org/10.1016/0893-9659%2895%2900009-F

J. Koeplik and H. Levine, Vortex reconnection in superfluid helium, Phys. Rev. Lett. 71 (9), 1375–1378 (1993)

Martin D. Kruskal and Harvey Segur, Asymptotics beyond all orders in a model of crystal growth, Stud. Appl. Math. 85 (1991), no. 2, 129–181. MR 1115186, DOI https://doi.org/10.1002/sapm1991852129

J. B. McLeod, private communication, 1995

D. I. Meiron, M. Shelley, W. T. Ashurst, and S. Orzsag, Numerical studies of vortex reconnection, in Mathematical Aspects of Vortex Dynamics, 1989, pp. 183–194

M. V. Melander and F. Hussain, Cross-linking of two antiparallel vortex tubes, Phys. Fluids A1 (4), 633–636 (1989)

D. W. Moore and P. G. Saffman, The motion of a vortex filament with axial flow, Phil. Trans. 272, 403–429 (1972)

P. G. Saffman, A model of vortex reconnection, J. Fluid Mech. 212 (1990), 395–402. MR 1051337, DOI https://doi.org/10.1017/S0022112090002026
P. G. Saffman, Vortex dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992. MR 1217252
M. J. Shelley, D. I. Meiron, and S. A. Orszag, Dynamical aspects of vortex reconnection of perturbed anti-parallel vortex tubes, J. Fluid Mech. 246 (1993), 613–652. MR 1200632, DOI https://doi.org/10.1017/S0022112093000291

E. D. Siggia, Collapse and amplification of a vortex filament, Phys. Fluids 28 (3), 794–805 (1985)

E. D. Siggia and A. Pumir, Vortex dynamics and the existence of solutions to the Navier-Stokes equations, Phys. Fluids 30 (6), 1606–1626 (1987)

J. G. Skifstead, Numerical treatment of line singularities for modelling a jet in a low speed cross flow. In: Analysis of a jet in a subsonic crosswind, NASA SP-218

L. Ting and R. Klein, Viscous vortical flows, Lecture Notes in Physics, vol. 374, Springer-Verlag, Berlin, 1991. MR 1146212
N. J. Zabusky and M. V. Melander, Three-dimensional vortex tube reconnection: morphology for orthogonally-offset tubes, Phys. D 37 (1989), no. 1-3, 555–562. Advances in fluid turbulence (Los Alamos, NM, 1988). MR 1024405, DOI https://doi.org/10.1016/0167-2789%2889%2990160-7

V. E. Zakharov, Wave collapse, Usp. Fiz. Nauk. 155, 529–533 (1988)