2017/11/13 by Adriana I. Pesci, Pesci, Adriana I., Raymond E. Goldstein +3
Earth and Planetary Sciences · Engineering · #Aeolian processes and effects #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fluid Dynamics and Turbulent Flows #Geological formations and processes
paper · pdf · doi:10.48550/arxiv.1711.04549
openalex publication_date 2017/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A classical problem in fluid mechanics is the motion of an axisymmetric\nvortex sheet evolving under the action of surface tension, surrounded by an\ninviscid fluid. Lagrangian descriptions of these dynamics are well-known,\ninvolving complex nonlocal expressions for the radial and longitudinal\nvelocities in terms of elliptic integrals. Here we use these prior results to\narrive at a remarkably compact and exact Eulerian evolution equation for the\nsheet radius r(z,t) in an explicit flux form associated with the conservation\nof enclosed volume. The flux appears as an integral involving the pairwise\nmutual induction formula for vortex loop pairs first derived by Helmholtz and\nMaxwell. We show how the well-known linear stability results for cylindrical\nvortex sheets in the presence of surface tension and streaming flows [A.M.\nSterling and C.A. Sleicher, J.~Fluid~Mech. bf 68, 477 (1975)] can be\nobtained directly from this formulation. Furthermore, the inviscid limit of the\nempirical model of Eggers and Dupont [J.~Fluid~Mech. \262 205\n(1994); SIAM~J.~Appl.~Math. bf 60, 1997 (2000)], which has served as the\nbasis for understanding singularity formation in droplet pinchoff, is derived\nwithin the present formalism as the leading order term in an asymptotic\nanalysis for long slender axisymmetric vortex sheets, and should provide the\nstarting point for a rigorous analysis of singularity formation.\n