Abstract
Singularities are found in solutions of three systems that model the equations for axi-symmetric swirling flow. The first example is a simple first order system in “Jordan form.” The second is a one dimensional analogue of the 2D Boussinesq system. In the third example, a complex solution of the axi-symmetric swirling flow equations is numerically constructed. This solution is a traveling wave with a complex wave speed that brings a singularity from the complex plane to a real position at a finite real time. The complex-valued flow can be understood as coming from a solution Moore’s approximation for axi-symmetric swirling flow.
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References
C.R. Anderson and C. Greengard. The vortex ring merger problem at infinite Reynolds number. Communications on Pure and Applied Mathematics, 42:1123–1139, 1989.
David H. Bailey. Mpfun: A portable high performance multiprecision package. RNR Technical Report RNR-90-022, 1991a.
David H. Bailey. Automatic translation of fortran programs to multiprecision. RNR Technical Report RNR-91-025, 1991b.
G.R. Baker, Russel E. Caflisch, and Michael Siegel. Singularity formation during Rayleigh-Taylor instability. 1992.
Claude Bardos and S. Benachour. Domaine d’analycite des solutions de l’equation d’Euler dans un ouvert de R n. Annali della Scuola Normale Superiore di Pisa, IV,4:647–687, 1977.
J.T. Beale, T. Kato, and A. Majda. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Communications on Mathematical Physics, 94:61–66, 1984.
John Bell and Daniel Marcus. Communications on Mathematical Physics.
A. Bhattacharjee and Xiaogang Wang. Finite-time vortex singularity in a model of three-dimensional Euler flows. 1992.
M.E. Brachet, D. Meiron, S. Orszag, B. Nickel, R. Morf, and U. Frisch. Small-scale structure of the Taylor-Green vortex. Journal of Fluid Mechanics, 130:411–452, 1983.
R.E. Caflisch. Singularity formation for complex solutions of the 3D incompressible Euler equations. preprint, 1992.
R.E. Caflisch, N. Ercolani, T.Y. Hou, and Yelena Landis. Multi-valued solutions and branch point singularities for nonlinear hyperbolic and elliptic systems. Communica tions on Pure and Applied Mathematics, 1992.
R.E. Caflisch, Xiaofan Li, and M.J. Shelley. The collapse of an axi-symmetric swirling vortex sheet. 1992.
R.E. Caflisch, O.F. Orellana, and M. Siegel. Singularity formulation and ill-posedness for vortex sheets. SIAM J. Math. Anal., 20:293–307, 1989.
S. Childress, G.R. Ierley, E.A. Spiegel, and W.R. Young. Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation point form. Journal of Fluid Mechanics, 203:1–22, 1989.
A. Chorin. Estimates of intermittency, spectra and blowup in developing turbulence. Communications on Pure and Applied Mathematics, 24:853–856, 1981.
A. Chorin. The evolution of a turbulent vortex. Communications on Mathematical Physics, 83:517–535, 1982.
A. Forestier and P. LeFloch. Multivalued solutions to some nonlinear and nonstrictly hyperbolic systems. Japan J. Applied Math., 1992.
R. Grauer and T. Sideris. Numerical computation of 3d incompressible ideal fluids with swirl. Physical Review Letters, 25:3511–3514, 1991.
Robert M. Kerr. Evidence for a singularity of the three-dimensional, incompressible Euler equations. 1992.
Robert Krasny. A study of singularity formation in a vortex sheet by the point-vortex approximation. Journal of Fluid Mechanics, 167:65–93, 1986a.
Robert Krasny. Desingularization of periodic vortex sheet roll-up. Journal of Computational Physics, 65:292–313, 1986b.
D.I. Meiron and M.J. Shelley. personal communication. 1992.
D.W. Moore. The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc Roy. Soc. London A, 365:105–119, 1979.
D.W. Moore. Numerical and analytical aspects of Helmholtz instability. In Niordson and Olhoff, editors, Theoretical and Applied Mechanics, pages 629–633. Proc. XVIICTAM, North-Holland, 1985.
David A. Pugh. Development of vortex sheets in Boussinesq flows - Formation of singularities. PhD thesis, Imperial College, 1989.
David A. Pugh and Stephen J. Cowley. On the formation of an interface singularity in the rising 2-d boussinesq bubble. 1992.
Alain Pumir and Eric D. Siggia. Collapsing solutions to the 3-d Euler equations. Physics of Fluids, A2:220, 1990.
Alain Pumir and Eric D. Siggia. Development of singular solutions to the axisymmetric Euler equations. 1992.
M.J. Shelley, D.I. Meiron, and S.A. Orszag. Dynamical aspects of vortex reconnection.1992.
Michael Siegel. An analytical and numerical study of singularity formation in the Rayleigh-Taylor problem. PhD thesis, NYU, 1989.
Eric D. Siggia. Collapse and amplification of a vortex filament. Physics of Fluids, 28:794–805, 1985.
J.T. Stuart. Nonlinear Euler partial differential equations: Singularities in their solution.Symposium in honor of C. C. Lin, 1989.
C. Sulem, P.L. Sulem, and H. Frisch. Tracing complex singularities with spectral methods.Journal of Computational Physics, 50:138–161, 1983.
Xiaogang Wang and A. Bhattacharjee. Is there a finite-time singularity in axisymmetric Euler flows? 1991.
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Caflisch, R.E., Smereka, P. (1993). Singularity Formation for Models of Axi-Symmetric Swirling Flow. In: Caflisch, R.E., Papanicolaou, G.C. (eds) Singularities in Fluids, Plasmas and Optics. NATO ASI Series, vol 404. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2022-7_4
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DOI: https://doi.org/10.1007/978-94-011-2022-7_4
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