Abstract
We investigate the regularization of Moore’s singularities by surface tension in the evolution of vortex sheets and its dependence on the Weber number (which is inversely proportional to surface tension coefficient). The curvature of the vortex sheet, instead of blowing up at finite time t 0, grows exponentially fast up to a O(We) limiting value close to t 0. We describe the analytic structure of the solutions and their self-similar features and characteristic scales in terms of the Weber number in a O(We−1) neighborhood of the time at which, in absence of surface tension effects, Moore’s singularity would appear. Our arguments rely on asymptotic techniques and are supported by full numerical simulations of the PDEs describing the evolution of vortex sheets.
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References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Applied Mathematics Series, vol. 55, National Bureau of Standards (1964)
Ambrose, D.M.: Well-posedness of vortex sheets with surface tension. SIAM J. Math. Anal. 35(1), 211–244 (2003)
Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32-3, 797–823 (1995)
Caflisch, R.E., Orellana, O.F.: Singular solutions and ill-posedness for the evolution of vortex sheets. SIAM J. Math. Anal. 20(2), 293–307 (1989)
Ceniceros, H.D., Hou, T.Y.: Dynamic generation of capillary waves. Phys. Fluids 11, 1042–1050 (1999)
Ceniceros, H.D., Roma, A.M.: Study of long-time dynamics of a viscous vortex sheet with a fully adaptive non-stiff method. Phys. Fluids 16(12), 4285–4318 (2004)
Cowley, S.J., Baker, G.R., Tanveer, S.: On the formation of Moore curvature singularities in vortex sheets. J. Fluid Mech. 378, 233–267 (1999)
Hou, T.Y., Lowengrub, J.S., Shelley, M.J.: Removing the stiffness from interfacial flow with surface-tension. J. Comput. Phys. 114(2), 312–338 (1994)
Hou, T.Y., Lowengrub, J.S., Shelley, M.J.: The long-time motion of vortex sheets with surface tension. Phys. Fluids 9(7), 1933–1954 (1997)
Hou, T.Y., Lowengrub, J., Shelley, M.: Boundary integral methods for multi-phase problems in fluid dynamics and materials science. J. Comput. Phys. 169(2), 302–362 (2001)
Krasny, R.: Desingularization of periodic vortex sheet foll-up. J. Comput. Phys. 65, 292–313 (1986)
Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)
Moore, D.W.: Spontaneous appearance of a singularity in the shape of an evolving vortex sheet. In: Proceedings of the Royal Society of London, Series A—Mathematical, Physical and Engineering Sciences, 365–1720, pp. 105–119 (1979)
Siegel, M.: A study of singularity formation in the Kelvin–Helmholtz instability with surface-tension. SIAM J. Appl. Math. 55(4), 865–891 (1995)
Velazquez, J.J.L.: Point dynamics in a singular limit of the Keller–Segel model 1: Motion of the concentration regions. SIAM J. Appl. Math. 64(4), 1198–1223 (2004)
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de la Hoz, F., Fontelos, M.A. & Vega, L. The Effect of Surface Tension on the Moore Singularity of Vortex Sheet Dynamics. J Nonlinear Sci 18, 463–484 (2008). https://doi.org/10.1007/s00332-008-9020-3
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DOI: https://doi.org/10.1007/s00332-008-9020-3